ANALYSIS OF TESTS FOR TWO FORMS OF SPECIFICATION ERROR IN LINEAR REGRESSION ANALYSIS Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY RONALD L TRACY ‘ 1975 4 LIBRARY Michigan State University This is to certify that the thesis entitled ANALYSIS OF TESTS FOR TM) FORMS OF SPECIFICATION ERROR IN LINEAR REGRESSION ANALYSIS presented by Ronald L. Tracy has been accepted towards fulfillment of the requirements for Ph.D. . Economics degree in (f \ I RYE—(fl " (7 "‘2 I) :3 Major professor Z- 3'57 ”I” Date qu It: I97s’ ,,,,,- 0-7639 ABSTRACT ANALYSIS OF TESTS FOR TWO FORMS OF SPECIFICATION ERROR IN LINEAR REGRESSION ANALYSIS By Ronald L. Tracy In this study two new specification error tests based on a Power Series Expansion Medel (POSEX) are develOped. The first test is designed to detect a misspecified conditional mean of the dependent variable and the second to detect heteroskedastic disturbance terms. TWO versions of the test fer a misspecified conditional mean are presented. One of these versions is shown to yield the same results as the procedures currently in use yet offers the advantage of being easier to implement. The two versions of the test are then compared on six misspecified models using a sample experiment. It was found that both tests have an extremely high probability of correctly rejecting the null hypothesis if the misspecified conditional mean is caused by using the wrong functional form of either the regressand or regressors. In contrast, when the specification error is caused by omitting a variable, the power of the test is a fUnction of the relation between the omitted variable and those included in the model. Four versions of the test for heteroskedastic disturbance terms are presented. These four tests are then compared with various Ronald L. Tracy versions of Goldfeld G Quant's parametric and non-parametric test, Glejser's test, Park's test, and Ramsey's test (BAMSET) by using a sample experiment on ten heteroskedastic models. It was discovered that when no information about the form of the heteroskedasticity is available, the most powerful test is BAMSET with the observations reordered by ranking the dependent variable. However, since this is a non-constructive test, if heteroskedasticity is found, no corrective procedure is suggested. Of the constructive tests, two versions of the test formulated in this study were found to be the most powerful. ANALYSIS OF TESTS FOR TWO FORMS OF SPECIFICATION ERROR IN LINEAR REGRESSION ANALYSIS By r" I)“ Ronald L5 Tracy A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1975 ACKNOWLEDGEMENTS From the inception of this study to its conclusion I have received keen advice, criticism, and encouragement from James Ramsey, my thesis chairman. I would also like to thank Robert Rasche and William.Ha1ey for their valuable comments on an earlier draft of this study. The extremely costly computer work on which this study is based was funded by a National Science Foundation grant which had been awarded to James Ramsey. For the use of these fUnds I am very grateful. I have appreciated the assistance of my wife Karen in editing and prOof-reading. Any errors remaining in the study are, of course, my own responsibility. ii TABLE OF CONTENTS Page LIST OF TABLES .......................... v LIST OF FIGURES ......................... viii Chapter I. INTRODUCTION AND REVIEW OF THE LITERATURE ........ 1 1.1 Introduction .................... 1 1.2 Effects and Causes of a Misspecified Mean Vector . . 6 I 3 Effects and Causes of a Misspecified Variance Vector ...................... 10 1.4 Review of Literature ................ 12 4.1 Different Residuals Being Used in Specification Error Test ............ 12 4.2 Present Procedures to Test for the Disturbance Terms Having a Non-Zero Mean . . . . 15 4.3 Heteroskedasticity ................ 19 4.4 Studies Comparing Tests for Heteroskedasticity . . 40 1.5 Summary ....................... 52 II. A NEW APPROACH ...................... 54 11.1 Estimation Using a Power Series Expansion Model . . 54 1.1. Development of Power Series Eicpansion Model . . . 55 1.2 Similarities to Ramsey's RESET Model ....... 59 1.3 A Suggested Instrument ..... . ........ 61 11.2 POSEX Test for a Non-Zero Mean ........... 65 2.1 Formulating the POSEX Model and Testing Procedure .............. . . . 66 2.2 Comparison with Previous Testing Procedures . . . 68 2.3 Examination of the New Testing Procedure Under H1 .................... 70 2.4 Summary. . ................... 81 11.3 POSEX Test to Determine if the Disturbance Terms are Heteroskedastic ............ 81 3.1 Estimators of the Variance of ui ......... 83 iii n! \n "I “\ NV TABLE C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Test Results Page Model 1, Sample Size 30 ........... 201 Medel 1, Sample Size 60 ........... 202 Model 1, Sample Size 90 ........... 203 Mbdels 2, 3, and 4 .............. 204 Medels S, 6, and 7. . . . .......... 205 Mbdel 8, Sample Size 30 ........... 206 Medel 8, Sample Size 60 ........... 207 Medel 8, Sample Size 90 ........... 208 Model 9. Sample Size 30 ........... 209 MOdel 9, Sample Size 60 ........... 210 Medel 9, Sample Size 90 ........... 211 MOdel 10, Sample Size 30 ........... 212 Mbdel 10, Sample Size 60 ........... 213 Medel 10, Sample Size 90 ........... 214 Model 11, Sample Size 30 ........... 215 Medel 11, Sample Size 60 ........... 216 Medel 11, Sample Size 90 ........... 217 Model 12, Sample Size 30 ........... 218 Model 12, Sample, Size 60. . _ ......... 219 IModel 12, Sample Size 90 ........... 220 Medel 13, Sample Size 30 ........... 221 'Model 13, Sample Size 60 ........... 222 Medel 13, Sample Size 90 ........... 223 Mbdel 14, Sample Size 30 ........... 224 Model 14, Sample Size 60 ........... 225 Medel 14, Sample Size 90 ........... 226 vi TABLE C28 C29 C30 C31 C32 C33 C34 C35 C36 Test Test Test Test Test Test Test Test Test Results Results Results Results Results Results Results Results Results Model Model Medel Model Model Model Model Model Model 15, Sample 15, Sample 15 , Sample 16, Sample l6,Swmfle 16, Sample 17,8wmfle 17, Sample 17, Sample vii Size Size Size Size Size Size Size Size Size FIGURE 1 A Schematic 2 A Schematic 3 A Schematic 4 A Schematic A Schematic 6 A Schematic 7 A Schematic 8 A Schematic 9 A Schematic 10 A Schematic 11 A Schematic Diagram Diagram Diagram Diagram Diagram Diagram Diagram Diagram Diagram Diagram Diagram LIST OF FIGURES of Test Results of Test Results of Test Results of Test Results of Test Results of Test Results of Test Results of Test Results of Test Results of Test Results of Test Results viii for Model 1 ..... for Model 8 ..... for Model 9 ..... for Model for Model for Model for Model for Model for Model for Model for Model 10.... 11.... 12.... 13.. 14.... 15.... 16.... 17.... PAGE 135 155 156 158 160 161 163 167 169 173 175 CHAPTER I INTRODUCTION AND REVIEW OF THE LITERATURE I . 1 Introduction Linear regression analysis is one of many statistical procedures that can be used to indicate a relationship among different variables. This method requires specification of the variable whose conditional mean is to be estimated (the dependent variable), of the variables that affect the mean of the dependent variable (the independent variables), and of the distribution of the unexplained variation (the disturbance term). One such regression model is y = X_B_ + 11. (1.1) where y_ is the n x 1 vector of observed dependent variables, X is the n x k matrix of nonstochastic independent variables of rank k, g is the k x 1 vector of parameters to be estimated, and u is the n x 1 vector of disturbance terms . 7 If the method of least squares is employed to estimate the regres- A sion model in (1.1), the estimator for the parameter _8_, g = (X'X)_ X'y, and the model's variance oz, 82 = (y - Xé)‘(y - XE/(n-k) can be obtained. If, however, the statistical properties of these estimators are to be ascertained and tests of significance carried out, the distribution of the disturbance terms must also be lmown. If, for 2 example, the vector of disturbance terms has a normal distribution with a mean of zero and a covariance matrix of 021 (hereafter denoted N(fl, 021)), the resulting estimators are unbiased, efficient, and consistent. Difficulties arise when the disturbance term has a different distribution than that which has been hypothesized. When an incorrect assumption is made about the distribution of the disturbance term, a Specification error has been committed. It must be emphaSized that a specification error arises only because the exact distribution of the disturbance term.is incorrectly assumed, not because it is distributed differently than required by the classical assumption (that u .. N(¢, 021)). Typically there are two major types of specification error. The first type concerns the distributional form of the disturbances and the second deals with the parameters of that distribution. In the first case, a specification error of incorrect distributional form.is made when the vector of disturbance terms 2.15 actually distributed differently than has been hypothesized. An example of this is if the disturbance terms are assumed to be distributed normally whereas they are actually distributed as log normal. The second type of specification error is committed if an incorrect assumption is made about the parameters that define the exact distribu- tion of the disturbance terms. In the context of the classical assumptions that 9;” N(0, 021), where only two parameters are needed to define the distribution completely, this second type of specification error can be divided into three types. 3 The first error arises when an incorrect assumption is made about the papulation.mean. IMOst commonly, this type of error occurs when the expected value of the vector u_is assumed to be zero instead of some non-zero vector 5, The second error occurs when one makes an incorrect assumption about the population variance. The most common form of this error arises when it is incorrectly assumed that the variance of each disturb- ance term is identical (homoskedastic) whereas the true variances would compose a non-constant vector y_(heteroskedastic). The third and last error incurred involves the correlation between the disturbance terms u1,...,un. In its most common form, this error occurs when it is assumed that the disturbance terms are independent of one another whereas elements of the disturbance vector that are adjacent are actually correlated (first order autocorrelation). The purpose of this study is to examine, compare and prepare statistical tests designed to help the researcher determine if a given regression.model is misspecified because the vector of disturbance terms has an incorrectly specified mean.or variance vector. The remaining two forms of specification errors involving the disturbance tenms, incorrect distributional form, and autocorrelation have been studied in great detail by other authors. The reader is referred to Shapiro, Wilk, 8 Chen [1968] and to Huang 8 Bolch for more information on.distributiona1 form errors and to Kramer [1969], Berenblut & webb [1973], and Abrahamse G Louter [1971] for more information on autoregressive errors. Netation When tests are examined to determine if'a model has been.misspec- ified, the null hypothesis (hereafter H6) is that no specification error exists. This null hypothesis will be tested against two alterna- tive hypotheses. The first alternative (hereafter H1) is that the disturbance terms have an incorrectly specified mean vector; the second alternative (hereafter H2) is that the disturbance terms have an incorrectly specified variance vector. To simplify the complexity of the statistical discussion, certain notational conventions are used throughout this study. First, matrices are always denoted by either Upper case Greek or Latin letters. Second, any Greek or Latin letter that is underscored denotes a column vector, (e.g., y;or g). Third, any lower case Greek or Latin letter not underscored represents a scalar. Fourth, parameters are denoted by Greek letters, whereas random variables are represented by Latin letters. An estimator of a parameter is signified by that parameter topped by a.symbol (for example, 9, S, g, 8, g are all estimators for B). In a like manner, the predictor of a random variable is denoted by a symbol over that random variable. When the inverse of a matrix is required, the symbol -'immediately to the right of the matrix is used (for example, the inverse of the matrix A is A7). The Operator DIAG denotes that the diagonal elements of the specified matrix are formed into a column vector. The operator E denotes the expected value operator. A prime ' to the right of a vector or a.matrix denotes the transpose of that vector or matrix. The capital letter I denotes the identity matrix while the vector i_denotes a column of ones. 5 Sane standard notation on tests will be reviewed as this notation will be used extensively throughout this study. The probability of incorrectly rejecting the null hypothesis (H6) (type I error) is denoted as alpha (a) or is referred to as the alpha level of the test. The prObability of incorrectly accepting the alternative hypothesis (type 11 error) is denoted as beta (8). The probability of correctly accepting the alternative hypothesis then becomes 1-8 and is referred to as the power of the test. Outline Before the various testing procedures designed to detect an incorrectly specified mean or variance vector are compared, a detailed discussion of eaCh specification error is given. This discussion is fellowed by a review of the pertinent literature on different predictors of the true disturbance terms, on.various tests for detecting an incorrectly specified mean vector, and on various tests for detecting an incorrectly specified variance vector. In the second Chapter, a new test fer each of the two ferms of specification error under discussion is described. Following a detailed explanation of the new testing procedure, the test is applied to the case of H6 vs. H1, with careful attention paid to developing the exact distribution theory. The new procedure is applied to testing H6 vs. Hz with special attention focused on certain aspects of the distribution theory. The third chapter begins with a restatement of the hypotheses posed in Chapters 1 and II. A sampling experiment is presented that compares the two new tests with the previously discussed tests for 6 H6 vs. H1 and H6 vs. H2. Since all of the tests presented were designed for particular situations, special attention is given to the experimental design so that all tests can.be compared fairly. In Chapter IV, the experimental results are presented. Compari- sons and contrasts between the various tests as well as between the various models tested are made. The hypothesis presented in the previous chapter are examined. Finally, in Chapter V, a summary of the entire study is given. This is followed by a discussion of the major conclusions of this study and the inferences which can be drawn from them. Some suggestions for further research are given. 1.2 Effects and Causes of a Misspecified Mean Vector Assume that one hypothesizes the regression model y1 = 81 + X12 82 + ..... + xik Bk + ui , i = 1,...n, 2 (1.2) 1_.1_~ NM, 0 I). If these assumptions are correct, model (1.2) would be the 'true' model; that is, the model which generated each element of the vector of dependent variables yi. The regression model would thus be correctly specified and the resultant least squares estimators, §_and 32, would be unbiased, consistent, and efficient. It is evident that if the hypothesized model had had a disturbance term with a constant mean.vector r = r}, it could be transformed into an hypothesized.model with a zero mean vector by subtracting the vector 5 from the dependent variable y_or by incorporating r into the constant coefficient. Hence, it will be assumed from this point on, and without loss of generality, that the disturbance term in the hypothesized.model 7 has a zero mean vector and that the alternate hypothesis (H1) is that of a non-zero, non-constant mean vector. Therefore, if E;15 actually distributed as N(g,ozl), §_being a.non-constant mean vector, then the model hypothesized in (1.2) is misspecified because of an incorrectly specified.mean vector. The effect of this specification error on the least squares estimators §_and 82 can be demonstrated by examining the effect of regressing the vector §_on the matrix X. The resulting regression model is 3 = X1 + 1, 1 ~ N(¢, 021). (1.3) Thus, the bias in the least squares estimators caused by the misspeci- fication is seen to be 13(5) - g = l and so?) - o?- = E(y - xfiy (x - x_é_)/(n-k) - oz = 5' _z_/(n-k). Hence, as a result of an incorrectly specified mean vector, 3 has an upward bias and hence always causes a loss in efficiency, which in turn causes tests of significance to be unduly conservative. In addition, the extent to which any parameter Bi is biased by the mdsspecification is directly related to the correlation between the corresponding independent variablezxi and the vector 5, Further, the constant vector will always be biased unless all the variation in 5_ can be explained by the other independent variables. If the vector 5- is a constant vector, that is, §_= zi_where i_is a column vector of ones, only the constant term is biased and by the amount z. Similarly, if 5_is a non-constant vector and uncorrelated with all the independent variables, 51”"’§k’ only the intercept term will be biased and by the amount 2 = z zi/n. l 8 Since the estinators of g and 02 are affected by the error of a non- zero mean vector , it must be determined under what circunstances such an error can occur. One such circumstance is when the original data is collected or transcribed incorrectly. Typically, it is assured that these errors are distributed normally and have an expected value of zero. If this is not true, however, and, in fact, the data contains an upward (downward) bias , only the intercept term and the variance are affected since the bias will presumably be uncorrelated with the independent variables in the model . Another situation in which a non-zero mean occurs is when a variable is omitted from the hypothesized model. This may occur if the hypothesized model is given by (1.2), x= X13) 2» 9. ~ NW» 021), whereas the true model (the model that actually generated the dependent variable y) is x = 113+ ng- g, Q ~ N(¢, 021), (1.4) where X, g and y are as previously defined, W is an n x m natrix of 111 additional independent variables, and _<_S_ is a colum vector (m elements long) of additional parameters. The non-zero mean of u in this case is equal to W3. Such an error can be comnitted if there is no data available on the variable(s) 111, . . .,v_vm or if the variable(s) are erroneously excluded from the hypothesized model because the researcher was not aware of their occurence in the true model. Note that the omitted variables cannot be included in the model and have their significance tested because the researcher is either unaware of their occurence in the true model or cannot obtain the necessary data. 9 One final way that a specification error due to a non-zero mean vector can occur is when the incorrect functional form of the regressors or regressand is used. Given the true model y1 = 71 + yz ln(xiz) + ... + yk1n(xik) + Vi’ i = 1,...,n, !’~ N(¢, 021) (1.5) and the hypothesized model, given in (1.2), x= Xa+ 2.11 ~ 1103.021), it is obvious that the hypothesized model has been misspecified. The independent variables have taken on the wrong functional form. .As a result, the vector u will have a mean given by E(ui) = Y1 + y21n(xiz) + ...... + ykln(xik) - (81 + 82 xik + ..... Bk Xik) # 0, i = 1,...,n. .Although the mean is non-zero, it may result in a relatively small bias in eaCh of the estimated parameters because of the high correlation between the hypothesized independent variable and the true independent variables. It is interesting to note that a similar violation is caused.when the incorrect form of the regressand is used. (This error can also cause the additional specification error of incorrect distributional fbrm.) If, fer example, the true model is eXp(y_)=Xy_ + 1. 1 ~ No. 021), (1.6) whereas the hypothesized model is given by (1.2), then the hypothesized model has been misspecified because the wrong functional form for the dependent variables has been assumed. The mean of u would, in this case, be 10 = Ecl°ge(Yl 1 X12 Y2 + "° + Xik Yk + V1) ' (81 + 82x12 + ... + Skxik)), i = 1,...,n, which in general is non-zero for any set of x's. .Although this non- zero value is different from that which occurs when the misspecification is due to the incorrect functional form of the regressand, the relation- ship is strikingly similar. One final point is that though incorrectly including an independent variable in a model is committing a specification error, this error does not affect the mean of the disturbance term; hence, the model is not misspecified because of a non-zero mean vector. This can be demonstrated by hypothesizing the model X.= x§.+ np_+ 9, 9_~ th, 021), (1.7) where y, X, g, W, and g are as previously defined, whereas the true model is y_= x§_+ u, u_~ N(¢, 021). The expected value of the hypothesized model would be B(y)=X§_+w-0=Xg which is exactly the true model; thus, the only cost of this specifica- tion error is a loss of efficiency in estimating the vector of parameters _E and the variance oz. 1.3 Effects and Causes of a Misspecified variance Vector Given the model X: XRT 9.1 9. " NW» 021), it should be noted that a constant variance vector ozi_(=DIAG (021)) is assumed. This does not, however, imply that a specification error 11 is made when a non-constant variance vector (v) is correctly hypothesized. Rather, just as a method exists of transforming any hypothesized model with a non- zero mean vector into a model with a zero mean vector, a transformation exists that will change any model with a hypothesized non-constant variance vector into a model with a constant variance vector. One simply divides each observed dependent and independent variable by the square root of the corresponding hypothesized variance (Vi)' This transforms the model x= X83 2, 2" NW: 02V), where DIAG (V) = L into the model y1 = 81 1 + 82 X12 + ... + Bk xik + wi, i=1,...,n, w;~ N(¢, 021). 717-? r? r; We Hence it can now be assumed without loss of generality that the hypothe- sized model will always have a constant variance vector. 2, i=1,...,n) A regression model with constant variance (var(ui) = o is Said to be homoskedastic. Since estimation using classical least S(Wares requires the assumption that the E(ui) = 0, i = 1,. . .,n, a homOSkedastic model conforming to the classical assumptions has BLUE) = oz, 1 = 1,...,n. If the model violates this assumption, it is said to be heteroskedastic (non—constant variance vector). If a model suffers from heteroskedasticity, it is known that the leaSt squares estimators of g, E, are unbiased and consistent but are inefficient and asymptotically inefficient. Further, the least squares esumaltor of the variance of the model is inappropriate since Eco ) = 35; ECX‘Xél' (x - XS) = if}? (z 0% - 13(3' (X(X'X)—X')_g)) 7‘ Z Gig/n, where CI is the variance of the i'th disturbance term. 12 Heteroskedasticity is generally believed to be a more serious problem when cross-sectional data is used than when time-series data is used. This belief is held because the magnitude of the dependent variable over each observation differs, in general, mulch more in cross- sectional data than in time-series data. This belief, however, is not always justified. The dependent variable in time-series data can be heteroskedastic if it covers a large number of years or if major changes have occurred since its conception. 1.4 Review of Literature The following section is divided into four parts. First, the different types of residuals that are currently being used in testing bOth alternative hypothesis Ho vs. H1 and HD vs. H2 are discussed. SeCond, the testing procedures now being utilized for testing HD vs. H1 are discussed. Third, the different testing procedures now being used to test HD vs. H2 are reviewed. Finally, previous sampling experiments that have compared various tests for HD vs. H2 are discussed. IAxllDifferent Residuals Being Used in Specification Error Test If one could observe the vector of disturbances 3, either hypothesis could be easily tested. For example, to test for a non-zero mean vector, only a simple t test of E(=§ui) about zero is required. In a similar vein, testing for a non-conStant variance can be done by stratifying the ui's and using an F test for equal variances. Unfortunately, how- ever, since the disturbance terms are not observable, another testing Pmcedure must be devised. The procedure that most often suggests itSelf is to use some predictor of the vector 1_1_ as a proxy for the unObserved disturbance term. 13 So far, three residuals have been used in the literature. The first of these, which is both the easiest to compute and most frequently used, is the residual obtained from ordinary least squares (hereafter OLS). It is defined as =y_-X_B_. IC>|c§> is normally distributed with HR.) = y - Xg = O and ac); - x5) (x - xiv/(n - k) E(y_ X' - xcx'xflc' XX' - X X'xot'xfx' + X(X'X)_ x'y_ yX(X'X)—X') Under Ho’ Var (é) 13(1 - X(X'X)—X') yy' (I - X(X'X)—X') M E(y y')M = M oz I M = 02M, that is, é~ NM, 02M). The second technique utilized was developed by Theil [1965, 1968] and Koerts [1967] . These residuals, denoted u}, are called the Best Linear Unbiased Scalar-covariance (BLUS) predictors of the true disturb- ance terms 2. They are defined as 2* = A'x. Where A is an n x (n - k) dimensional matrix satisfying the conditions a) A'X = 0 b) A'A = In _ k, and c) AA' = M. UHider HO the E*'S are normally distributed with 501*) = A'(E()L)) = New = o mug) = E(A'yy' A) = A' B(yy')A _ 2 _ 2 -A'OIA-UIn-k, 14 that is 2* ~ N(¢, Ozln-k)° It is important to note that although this orthonormalization process ensures that the 3* 's are independent of one another, it also limits the number of residuals to only (n-k) instead of n. The third technique was developed by Hedayat 81 Robson [1970] and is called stepwise or recursive residuals, denoted by 1:1. The basic idea of this method is to "obtain (residuals) by a stepwise fitting of the linear model to successively more observations" [Hedayat and Robson, 1970, p. 1574]. The first step of the procedure is to estimate the model using OLS and only k+l observations. The least Squares residual that corresponds to the (k+l) 'th observation becomes the first stepwise residual, {11. The next step is to reestimate the "Ddel using k+2 observations. As before, the stepwise residual is the one that corresponds to the last observation,((k+2) in this case), and is denoted by {‘2' As this process is continued, n-k independent step- wise residuals are generated, 131,” . ’fin-k' These same n-k residuals can be obtained with only a single matrix inVersion by using a recursive teclmique develOped by Harvey 5 Phillips [1973]. The first step of the procedure is to estimate the model using OLS and k+1 observations, just as before, denoting the estimate of the vector g as 9(1). The least squares residual that Corresponds to the (k+1) 'th observation becomes the first recursive residual, denoted by 131. The second step is to calculate a new estimate of the vector .3; This is done by using the recursive formula (with i=2) . o - ' 0 ~ ~ X'. X. _x. -x B. 3(1) = E(j-l) “ ( 3‘1 3‘1) ’3 (y) ‘3‘“) 1+!xtx.-. §-J ( J-l 1-1) *1 15 where (X'. _1)_ denotes the inverse matrix used to calculate 8(j- _1), i- -X1 J' and x_! is the row vector that corresponds to the (k+j)' th observation. To obtain the next inverse matrix (Xij)—, the recursive formula _ (X X-' _) xj x5 (X3, _) -J' j-lx 1-1 -J' is used. These n-k residuals are distributed under HO as Nm, TZIn-k) , 2 where r is the associated variance. As in the case of the BLUS residuals, the stepwise (recursive) residuals are independent and k observations have been lost. 141.2 Present Procedures to Test for the Disturbance Terms Having a Non—Zero Mean The first test for Ho vs. H1 was developed by Ramsey [1969] using BLUS residuals, 3*. Recall that 3* = A'y where A'X = 0, A'A = In-k and AA' = M. Ramsey hypothesized that if the disturbance terms had an incorrectly specified non-zero mean vector, 3, "then the mean of the i'th disturbance terms 2 i can be expressed as a linear function of the Merits of y i’ the least squares estimator of the conditional mean of Y1." [Ramsey, 1968, p. 66]. Stated formally, + E(“1) = zi = °‘o + 0‘1 m110 I °‘2 mizo o‘3 1m130 + (1 3) i = 1,... ,n, I Whfi‘re mijO is the j'th moment about the origin of yi. Given that BLUS residuals have the property that if 13(3) = 5 7‘ 0, then 12th = E(A'x) = A'th) = A'E(g) = A'_Z_. he suggested pre-multiplying equation (1.8) by the matrix A'. This Yiekled the equation my) = =A'o0 + A o1m_10 + A'o2 1320 + A'o3 1330 + A'o4 1340 + (1.8') l6 Removing the expected value operator from equation (1.8') and noting that A'i = A'X_§_ = 0, Ramsey formulated the errors in variable model (2) + “3 1*(3) + a4 y1(4) + w (1.9) 2* = 0‘2 Z".I where 91 ~ NM, EZIn-k) under HO. In this formulation [*(i) = A'yfi) = A'{y§,...,yril}. Given that under H0, E(u*) 7‘ 0, it follows that under HO, the E(o2) = E(o3) = E(o.4) = 0. Hence, an F-test was proposed by Ramsey to test for the joint significance of oz, as, and o4. This procedure he named RESET (Regression Specification Error Test). The RESET procedure has been examined by Ramsey and Gilbert [1972] 11$ng a monte carlo sampling procedure. Their results (as ammended by unPublished results of this author) have indicated that just as exPeCted, under the null hypothesis, the test was not biased (the Percent rejection corresponded to the a level); second that for the alternative model examined, the power was close to 100 percent under the alternative hypothesis of incorrect functional form; third that the test had virtually no power under the alternative hypothesis of an 0"fitted variable for the model examined (the reason for this result W111 be explained later). Because BLUS residuals are utilized in this procedure, three difficulties associated with those residuals are inherent in RESET. First, since the A matrix is difficult to calculate, the 1_._1_*'S are not easily computed. Second, the BLUS procedure can be used to generate only n-k residuals from the original observations. Third, because there are only n-k residuals, in order to find a one to one correspond- emce between the residuals and the n observations, the k observations that are discarded in calculating the matrix A must be carefully noted. 17 Since all of the problems just outlined are caused by the use of BLUS residuals, Ramsey and Gilbert [1972] suggested substituting the standard least squares residuals, §= (I - x (x'xf x') X= My, for the BLUS residuals in the RESET technique. With this substitution, equation (1 . 9) becomes A A2 A3 A 9=M9=al (“azzmwgz where w ~ N(0,oZM). (4) + E = Q E + E, (1.10) This procedure, however, creates another problem. The standard F-test used to test the hypothesis that ol = 012 = as = 0 breaks down because of nonindependence between the numerator and denominator. To ShOW this nonindependence one can eXpress the F statistic as a ratio of quadratics in the disturbance terms 3. The F-statistic in this Particular case is g‘CM'QcQ'QfQ'M) 9/3 2' W-M'QCQ'Q)_ Q'M) g/(n-k-Sl Where Q is defined implicitly as in equation (1.10). F: Since (M'QCQ'Q)- Q'M) 04-M' QCQ'Ql— Q'M) 7‘ 0, it follows that the numerator and denominator are not independent as a necessary and sufficient condition for their independence is that the PTOduct of the two quadratics be identically zero. To correct for this non-independence, Ramsey E Schmidt [1974] haVe suggested pro-multiplying equation (1.10) by the matrix M. This reslllts in: A 63 .4 ME=E=91MXUI “ZMX(293MX(1ME=MQE+ME where M w ~ N00, 02M) . It is easily seen from the quadratic form that 18 the F-statistic has a numerator and denominator which are independent. writing F once again as a ratio of quadratic forms one gets F = u' GW' ( ' )_"M_ u 3 . 2' W-MIQIQIMQI“ Q9/5117 (11-15-35 Since independence of two quadratic forms is proven if their product is identically zero and given that M is idempotent (that is, ARFiD one obtains (M'QLQ'MQ)_Q'M)(I‘d-M'Q(Q'MQ)—Q'M) = M'QLQ'MQ)_Q'M-I‘I'QLQ'MQl-Q'I‘I E 0, thus proving that the numerator and denominator are independent. .All the initial problems associated with the original formulation of the RESET technique are thus rectified in this newly defined RESET test. However, this procedure still requires the calculation of the well- defined.matrix M = (I - X (X'X)_ X'). Although this is not a difficult process, it is a time-consuming and cumbersome one. Mbreover, it is important to note that although the BLUS and OLS residuals are unbiased predictors of the error vector u_under H0, they are biased under H1; that is, though the expected value of the residual vector is equal to the expected value of the disturbance term.under the null hypothesis, the two sets of expected values are unequal under H1. This can be clearly shown by defining a general set of residuals E.= B u, where B is a matrix with n columns. Under the alternative hypothesis of non- zero mean (E(u) = g_# 0), the expected value of the general set of residuals is ECQ)=BE(2)=B3_#3. It can thus be inferred that with any test in which a predictor (such as OLS or BLUS residuals) of the true disturbance term u_is used, an incorrect measure of the non-zero vector §_is being employed. Hence, 19 a procedure that is unbiased under both H0 and H1 and where the calculation of the matrix M is not required would be preferred. 1.4.3 Heteroskedasticity There are two different types of tests for heteroskedasticity; constructive tests and non-constructive tests. Simply stated, a non- constructive test for heteroskedasticity enables one to test the null hypothesis of homoskedasticity but does not help one to estimate the individual variances if H0 is rejected. In contrast, a constructive test not only enables one to test for H0 vs. H2, but also provides an estimate of 01’ i=1,...,n, (the variance of the i'th disturbance term); if the null hypothesis is rejected. These estimates of the variance can then be used to reestimate the model using Aiken's Generalized Least Squares (hereafter GLS) technique. However, it should be noted that since fewer assumptions about the form of the heteroskedasticity are usually necessary to use non-constructive than constructive tests, the former tend to be more widely applicable. Non-Constructive Tests There are three different types of non-constructive tests employed to test H0 vs. H2; they are an F-test, a likelihood ratio test, and a nonrparametric peak test. 99__- The first test utilizing the F-test was designed by Goldfeld G Quant [1965]. It can be used by a researcher who knows, or hypothe- sizes, that the individual variances oi,...,ofi are monotonically related to one of the variables, say 55, and that the error term.is normally distributed. The procedure is first to order the observations of variable xi in increasing magnitude (decreasing magnitude if it is 20 hypothesized that xj is inversely related to the variance) so that xij < xkj where i < k. The remaining variables are reordered to confonm to this ordering. Second, the observations are separated into two groups (denoted as group I and II, respectively) omitting the central p,(%-< p < g), observations. Each group will have m = (n-p)/2 > k observations. Third, using OLS, the model is estimated using each subset of the data. Fourth, the OLS estimate of the variance of the disturbance term from the first group of data is calculated and denoted as 51 while the variance from the second group is calculated and denoted as 52. The ratio of these two independent, scaled, chi squared variables, denoted by R1 = 52/51, defines a statistic that has an F distribution with mrk and m-k degrees of freedom. Under H0 of homoskedasticity, s1 and 52 have the same scaled chi squared distribution, whereas under H2 of heteroskedasticity of the form hypothesized, s1 and 52 will have different scaled chi squared distributions. There are, however, two difficulties with this procedure. First, the technique requires knowledge (or at least an hypothesis) about which single independent variable is causing the heteroskedasticity. .Although this knowledge is sometimes available, it usually is not. Second, though it has been found that omitting the central p observa- tions increased the power of this test, the technique should prove less powerful (in correctly rejecting H0) than tests that do not discard information. Finally, in the test's favor, it should be mentioned that the distribution of R1 is independent of the values of the regression coefficient and, under the null hypothesis, is independent of the value of the variance of the disturbance term. Zl IEEELL’ A similar test has been suggested by Theil [1965] using BLUS residuals. He suggested that the (n-k) BLUS residuals be divided into two equal groups of m observations after the central p,(%-< p < g), observations have been omitted. Denoting t1 as the sum of squared residuals from the first group and 1:2 as the sum of squared residuals from the second, the statistic R2 = :3 is calculated. It is distributed as F with m and m degrees of freedom under the null hypothesis. Under the alternative hypothesis that the heteroskedast- icity is a fUnction of the order of the observations (for example, a function of time in time series data), R2 is distributed as scaled F with m and m degrees of freedom. The problem of the loss of information associated with the GQP procedure is thus partially solved by using this procedure. If one does not omit the central p observations in both tests, the F-statistic using the GQP procedure has (n-2k)/2 and (n-2k)/2 degrees of freedom, whereas with the Theil procedure, the F-statistic has (n-k)/2 and (n-k)/2 degrees of freedom. The reason for this is that in order to use the GQP procedure, one must calculate the residuals after the observations have been divided into groups. By contrast, since the BLUS residuals are independent of one another, they can be calculated before the data is divided into groups. However, to use this procedure effectively, one must still discard p observations. Finally, it must be recalled that two problems are added because BLUS residuals are used. First, it is difficult to calculate the vector 9?. Second, it is difficult to reorder the n-k residuals when some variable, say xi, is related to the heteroskedastic disturbance terms u1,...,un. It can, however, be accomplished by carefully noting the k Observations 22 that are discarded in calculating the matrix A. Since the remaining n-k observations correspond to the n-k BLUS residuals, reordering can be done. RECURSIVE-P - The final technique utilizing an F-statistic was developed by Harvey G Phillips [1973]. In this technique the F- statistic is defined in terms of recursive residuals. The prerequisite for using this procedure, just as for the previous two procedures, is that one have knowledge as to which variable, say xj, is monotonically related to the heteroskedastic variances 01’ and that the disturbance terms be normally distributed. If these prerequisites are met, the test can be carried out. First the n-k recursive residuals are calculated. Second, the first k observations of the vector xi are discarded and the remaining n-k observations are reordered in increas- ing magnitude (decreasing magnitude if xj is inversely related to the variances oi,...,ofi). Third, the n-k residuals are reordered to confbrm.to this new ordering. Fourth, the residuals are divided into two equal groups of’m observations, after the central p observations (§-> p > 2) have been omitted. Finally, denoting t1 and t2 as the sum of the squared residuals from group one and two respectively, the ratio R3 = ;%_is defined. This ratio has an F distribution with m and m degrees of freedom under H0, whereas under H2, R3 is distributed as scaled F with.m and m degrees of freedom. To use this test, like Theil's, it is not required that the residuals be recalculated. Hence, k degrees of freedom are saved. Also, even though the recursive residuals are easier to calculate and reorder than the BLUS residuals, they are still not as easily manipulated as the OLS residuals. Finally, since the BLUS residuals 23 have the prOperty of having the minimum variance for the class of residuals which have a scalar covariance matrix, the BLUS procedure will probably have more power against H2 than will the recursive residual technique. BAMSET - In the next procedure, Bartlet's M statistic is used. Developed by Ramsey [1969], the test, which he named BAMSET (Bartlet's M Specification Error Test), requires use of BLUS residuals as did the Theil procedure. This procedure involves first calculating the n-k BLUS residuals and then separating the residuals into three mutually exclusive and exhaustive groups of approximately equal size (sample size n1, n2 and n3 respectively). Denoting $1, 52 and 53 as the sum of squared residuals from groups one, two, and three respectively, one can form a likelihood ratio test. The ratio used in the test is 9*, defined as (131” (3.32” (a (s. + s. + 5.)? n1 n2 “3 nl + n2 + “3 Since 2* is a likelihood ratio, it is well known that -2 loge 9* is asymptotically distributed as x2 with, in this case, 2 degrees of freedom. Under HO, the values of 51, $2 and 53 are found to be statistically equal, whereas under H2, they are found to be statistically different from one another. As an alternative form of this same procedure, Ramsey 5 Gilbert [1972] have suggested that OLS residuals instead of BLUS residuals be used. They have, however, pointed out that since under H0 the OLS residuals are heteroskedastic and not independent (recall that E(§§f) = 02M), the asymptotic distribution of the resulting ratio cannot be determined. I... M\ . h: .0; Hi 'I "A .c" 24 At this point, some remarks about this test must be made. First, since the observations are not reordered, the three groups are a function of the index 1. Hence, the test should prove most powerful against the alternative hypothesis when the heteroskedasticity is a function of the observation number 1. Nevertheless, this form of heteroskedasticity was not what the test was specifically designed for. Rather, it was designed as a general test to detect any form of heteroskedasticity. Because in using this procedure, one makes no assumption as to the form of the heteroskedasticity, it should be expected that BAMSET will prove less powerful against H2 than tests that utilize knowledge as to the form of the heteroskedasticity. However, when knowledge as to the form of the heteroskedasticity does not exist, the BAMSET test is the only one that can be used. To increase the power when knowledge of the variable (say xi) that is related to the heteroskedastic disturbances is known, it has been suggested by Sutcliff [1972] that the residuals should be reordered by the variable ET before the grouping is made. Recall that this can be done with BLUS residuals if one discards the observations of the vector xj that correspond to the observations omitted in calculating the A matrix. I GQN_- The last group of tests are two non-parametric tests. The first of these was developed by Goldfeld G Quant [1965] for cases in which no assumptions about the distribution of the disturbance term can be made. Hewever, this test still requires knowledge that a variable, say ET, is monotonically related to the heteroskedastic disturbance terms u1,...,un. The procedure requires first that the regression model be estimated using OLS. Second, the variable x5 is cr‘ 5%. p: ‘__.<— “6.. H‘: A I 83.5 ,. fl? ‘ U (Into (II I if U: o ’ K/I 25 ordered by increasing magnitude (decreasing magnitude if xi is inversely related to the heteroskedastic disturbance terms) and the OLS residuals are reordered to conform to this ordering. Third, the number of peaks (a peak is defined as |fij|<|fij+1|) occurring in the reordered residuals are counted. By using a table provided by Goldfeld G Quant [1967], the cumulative probability that heteroskedasticity is present can be determined. Under the null hypothesis, there will be a small number of peaks, whereas under Hz, the number will be large. Some observations of this technique are in order. First, it has been found [Goldfeld S Quant, 1967] that for small sample sizes, n < 10, the procedure is biased because the OLS residuals are not mutually independent. Second, just as with all the other tests (including BAMSET with the reordering procedure), it is necessary to know which variable is monotonically related to the heteroskedastic disturbances. Third, given that OLS residuals are themselves hetero- skedastic under H0, it is surprising that for larger sample sizes, n > 10, the test is not biased. Finally, while it would rarely be inapprOpriate to use this test for heteroskedasticity, it should be selected only when.the disturbance terms are not distributed normally. Since if the disturbance terms are normally distributed, other tests exist which prove more powerful at correctly rejecting the null hypothesis. RECURSIVE-N - The last non-constructive test to be discussed was designed by Hedayat G Robson [1970]. 'With this non-parametric test, the peak tables provided by Goldfeld G Quant are also used. This test is exactly the same as the GQN test which was just reviewed with the exception that recursive residuals are used instead of OLS residuals. 26 This test offers the advantage of not being biased even for small sample sizes because the n-k recursive residuals are mutually independent. It must once again be stressed, however, that this test, just as the GQN test, is a non-parametric test and hence should.be used when the distribution of the disturbance term is unknown. Constructive Tests .As previously mentioned, constructive tests for heteroskedasticity are most often viewed as being less general than non-constructive tests because they usually require more precise a_pgigri_information about the functional form of the heteroskedastic disturbances. For example, some of the most popular assumptions about the functional ferm.of constructive tests are: 2 _ E(ui ) - o xij , (l.lla) 2 _ 2 2 E(ui ) — o xij , (l.llb) E(uiz) = 02 (a + b ij) , (l.llc) 2 _ 2 2 E(ui ) - o (a + b Xij) , (l.lld) 2 2 E(ui ) = o E(yi) , and . (l.lle) E(uiz) = oz E(yiz), i=1,...,n. (l.llf) Glejser [1969] divided these assumptions into two types of heteroskedasticity, pure and mixed. Pure heteroskedasticity is defined as E(uiz) = ozf(zi), i=1,...,n, where f(zi) represents a fUnction in some variable 21 which passes through the origin, whereas mixed heteroskedasticity is defined as E0112) = o2(f(zi) + a),i=l,...,n, that is, the heteroskedastic disturbance term.has an intercept term. 27 .According to this convention, only equations (l.lla) and (l.llb) represent pure heteroskedasticity. Though the assumptions are more rigid, constructive tests do offer two advantages over non-constructive tests. First, the relation between a single independent variable and the disturbance term need not be monotonic. Second, since in constructive tests an estimator of the heteroskedastic variances (call it giz) is defined, the hetero- skedasticity can be corrected either by dividing the model by Si and reestimating using OLS or by reestimating the model using GLS (generalized least squares) and employing the values of 312 on the diagonal of the estimated variance covariance matrix. Three constructive tests, all formulated in terms of a basic regression model, are described in this study. Ordinary least squares estimators for the model's parameters are used in two of the tests, while in the third maximum likelihood estimators are used. The estimates Obtained from all the tests are then tested either individu- ally or in a group. PAR§_- The first estimation technique (that has since been used as a testing procedure) was develOped by Park [1966]. Before that time, it was assumed that if the variable xj were related to the heteroskedastic disturbances, u1,...,un, the relation was specified by E(uiz) = o2 xij’ i=1,...,n. In order to ease the restrictiveness of this assumption, Park suggested that when xj is known to be the cause of the heteroskedasticity, it should be assumed that the 2 o 2 _ . E(ui ) - o xij’ 1 Park then posited that the value of a could be estimated by formulating 1,...,n. (1.12) a regression model. By taking natural logs and removing the expected 28 value Operator, he obtained the model 2 2 . 1n u1 = 1n 0 + o 1nxij + 1n vi, 1=1,...,n, (1.13) where V1 is distributed as x2 with one degree of freedom. Park then suggested replacing the unobserved dependent variable In ui2 by its 2 OLS predictor 1n fii . When this proxy is used, model (1.13) becomes ln £12 = 1n oz + a 1n xij + ln'wi, i=1,...n, (1.14) wherewi is distributed as scaled x2 with one degree of freedom where the scaling factor is 2 ECwi}/E(2wi) miio = mii T = n- Cy2) (IE—E) ’ I—n— . and where mii is the i'th diagonal element of the matrix M(=I - X(X'XIX'). Estimating the model using least squares, Park 2 and o. These estimators would then enable obtained estimators of 1n 0 the researcher to correct the heteroskedastic model. In carrying this technique one step further, others (for instance Goldfeld G Quant [1972])have indicated that if one denotes d as the OLSestimate of o and Oa as the estimated standard error of a, the ratio Ry could be defined as o R = —:—- Y a. (I This ratio is approximately distributed as student's t with n-Z degrees of freedom. under HO’ a = 0, whereas under H2 of the type hypothesized, o f 0. Three points must be made. First, this process still requires knowledge of the single variable causing the heteroskedasticity. The test does not, however, require that a monotonic relation exist between the variable and the disturbance terms. 29 Second, 1n £12 is a biased predictor of In uiz. Recalling that E(§;§f) = oZM and denoting mi as the i'th row of the matrix M, one finds that Ban 1112) = Biminplzi # Eon nil) Third, it must be pointed out that when one estimates model (1.14) by the method of least squares and assumes, as Park did, the classical assumptions that the disturbance terms are distributed N(¢, 021), four specification errors are committed. The first of these errors is that of incorrectly assuming a normally distributed disturbance term (recall that the disturbance terms are distributed as loge scaled x2 with one degree of freedom). This error, however, does not affect the properties of the estimators of the lnoz or a, but rather affects the tests of significance (that is a t-test or an F-test). Hence, the t-test proposed to test H0 vs. H2 could be biased. It has been found by Srivastava [1958], however, that a t-test is robust against considerable non-normality; therefore, the procedure might prove reliable. This is especially true since the disturbance terms are distributed as loge of a scaled X2 with one degree of freedom which is a two-tailed distribution. The second specification error is that of a non-zero mean vector. The expected value of the i'th element of this vector is E(ln wi) = E(ln mi 2) 7‘ o where l_l_l"i is the i'th row of the M matrix. Since wi is based on the matrix M (=1 - X(X'X)_X'), 1n (wi) is not independent of the variable In (xi) and hence the estimate of a will be biased. In addition, the estimate of In 02 will be biased unless all of the non-zero variation in In (wi) can be absorbed by the estimate of o. 30 The third specification error is that of heteroskedasticity. This error will cause the estimated variances to be biased and hence make the estimators of 1n 02 and o inefficient. Therefore, the proposed t-test will prove more conservative than it would otherwise be. Also, it should be noted that since the dependent variable is heteroskedastic under HO, the null hypothesis will be rejected by the test a disprOportionate number of times. The last specification error is non-independence. Like the misspecification of heteroskedasticity, non-independence causes the estimated variance to be biased; hence, the estimators of 1n 02 and o are inefficient and the t-test is unduly conservative. WOrse yet, however, is the fact that the non-independence in the disturbance terms adversely affects the t—test procedure another way. If the ratio calculated is to be distributed as student's t, the numerator and denominator must be independent. Unfortunately, when the disturb- ance terms are not independent, the numerator and denominator of this ratio are not independent; thus, the t-test procedure must again be questioned. Since there is no evidence that the t-statistic is robust against non-independence, the question arises as to whether this procedure is valid. The question is considered further on in this study. FIMRL- In this procedure, suggested by Rutemuller G Bowers [1968], a likelihood ratio test is utilized. It has the advantage, unlike the previous procedure, of having an asymptotic distribution theory that is well defined. 31 Given the heteroskedastic model y. 1 = 81 + 82 X12 + ... + Bk Xik 4' Vi, 1:1,...,n, Y_~ NCfl’ v) (1.15) oi 0 where ‘V = . , Rutemuller G Bowers proposed an 9 '02 - d estimation method whereby 012,...,on2 and 81,...,Bn could be jointly determined. They posited that if the variances were a function of some variables El”"’5n (typically these variables would be independent variables from the model 1.15) whose exact functional form was known (say f(§l,...,gn)), the parameters in the function f(-) and parameters §_could be jointly determined. Because Rutemuller 8 Bower's procedure requires knowledge about the function f(o), it will be assumed, for illustrative purposes, that f(-) is a quadratic in a single variable, that is E(viz) = oz(oo + olxij+ oxz 12].), 1= -l,. (1.16) They then suggested transforming model (1.15) into the homoskedastic model 2 W+alx1j+azx1j \[°‘1":jo*°‘ O‘zxij i = 1,...,n, (1.17) where u_~ N(¢, 021), under H0. Since this model cannot be estimated using ordinary least squares, they recommended using maximum likelihood. Setting up the likelihood function 1 - - Ll ex? -% (Vi ’ 81 B2 Xiz'“ Bxxil) 11 /fi'\/f; + alx j + o2 x. 132 IO + alxij + azxijz u :13 32 . 'I‘ r they deflned 9: and B as the estimators that maximize L1. Likewise, they denoted L0 as the equation (1.18) when o1 and o2 are constrained to equal zero (this is equivalent to OLS estimation of model 1.15) and defined do and é_as the estimators that maximize L0. Finally, they defined the likelihood ratio 2+ as Being a likelihood ratio, —2 loge 2+ is asymptotically distributed as x2 with, in this case, 2 degrees of freedom (the number of degrees of freedom always equals the number of extra parameters included in L1). Under the null hypothesis of homoskedasticity, the additional parameters in model (1.17), ml and o2, are equal to zero and do = 02 (the model's variance), whereas under the alternative hypothesis of heteroskedasticity of the form hypothesized, oz and o3 are not equal to zero. Therefore, including the polynomial is found to increase the model's efficiency. An alternative test formulation of this same test has been suggested by Goldfeld & Quant [1972]. They hypothesized that the estimators of could be tested individually by using a t-test. This procedure would, of course, enable an experimentor to differentiate between pure and mixed heteroskedasticity. It must be realized, how- ever, that since gf is only asymptotically distributed normally, the test proposed would not have a student's t distribution; hence, the test statistic would not be exact for small sample sizes. This revised procedure might, nonetheless, pose only minimum difficulties under H0 since there is evidence [Srivastava, 1958] that a t-test is robust against considerable non-normality. 33 TWO final points concerning this test must be made. First, Rutemuller G Bowers suggested that if the exact functional form f(-) is not known, one should use the regression model itself as a proxy fer the unknown function. This procedure would, using the likelihood ratio test, result in -2 loge 2+ being distributed as x2 with k-l degrees of freedom. .Also, the t-test procedure (Goldfeld G Quant's suggestion), although only an asymptotic test, might be useful in determining which variable is causing the heteroskedastic disturbances. Second, Rutemuller 8 Bowers' procedure, though well defined, tends to be more difficult to implement than any other test for heterosked- asticity. There are two reasons for this; first, a good maximum likelihood (hill climbing) computer program is needed, and second, since the estimation is accomplished through an iterative procedure, the process is more costly and time consuming than are other testing procedures. GLEJSER - In this test, the last constructive test to be examined, OLS is used to estimate the parameters in the heteroskedastic model. The test, put forth by Glejser [1969], was designed to detect and correct fer heteroskedasticity that is a polynomial in some variable. It should, however, be noted that priOr knowledge about the degree of the polynomial and about the identity of the variable is required befOre the test can be used. For illustrative purposes, the form of the heteroskedasticity will be postulated as E(uiz) = 02(o1 + ozxij + 0133(1)?)2, i=1,...,n. (1.19) With the disturbance terms taking on this form, Glejser suggested that a regression model be used so that ol, oz, and o3 can be estimated. 34 Using u; as a proxy for ui2 and taking the positive square root of equation (1.19), he formulated the model [Oi] = (011 + ozxij 4' aSXIj) Vi, i=1,...,n, (1.20) where vi, i=1,...,n, are distributed are scaled x with one degree of freedom with the scale factor equal to VEEEEZ where mii is the i'th diagonal element of the matrix M. He then suggested estimating the model using OLS. Finally, he suggested calculating a set of t ratios (t2 and t3) defined as o. 1 . t“ = 3':— 9 1:293: a. 1 where Si is the OLS estimate of ai and.O ;. is the OLS estimate of the 1 standard error of Si. Although he indicated that the exact distribution of these ratios was unknown, he suggested that they might be approxi- mately distributed as student's t (with n-3 degrees of freedom in this case). .Assuming that his suggestion is true, a standard t-test could be performed. Under the null hypothesis of homoskedasticity, oz and as are each equal to zero and cl = o (the standard deviation of the disturbances), whereas under H2 of the form hypothesized, oz and o3 are different from zero. Glejser's model and testing procedure also enabled him.to determine easily whether the heteroskedasticity was of the pure or of the mixed variety. He suggested that if HO was rejected, the type of heteroskedasticity could be determined by testing the additional hypothesis of whether o1 is equal to zero (pure heteroskedasticity). To test this hypothesis, a t ratio (similar to t2 and t3 above) would be calculated and if it is again assumed that the ratio is approximately distributed as student's t (with n—3 degrees of freedom in this case), a standard t—test can be performed. 35 A.number of observations can now be made. First, it must be remembered that one must have a_priozi_knowledge about the degree of the polynomial and about the identity of the variable causing the heteroskedasticity in order to use the test. According to Glejser, however, using the wrong degree of the polynomial presents little difficulty as this error has only a small affect on the test's power. Second, because Glejser uses a t-test in each coefficient to test for H0 vs. Hz, the correct a level is difficult to obtain. The reason for this is that since the probability of a type I error in using individual t-tests is the union of the probability of committing a type 1 error in testing each coefficient, the correct alpha level is difficult to Obtain. However, when.an F-test procedure is instead used, this problem is circumvented since when more coefficients are being tested for significance, the degrees of freedom are correctly varied. Third, since [Oil is used as the dependent variable, a biased predictor of the heteroskedastic disturbance is being used. This is easily perceived by once again recalling that E(§_§D = 02M, andmii is the i'th diagonal element of the M matrix, EclfiiI) = ran—o ii at Ecluil). i=1,...,n.» The fourth point is that just as with PARK's test, Glejser's estimation of model (1.19) using OLS while assuming the classical assumptions causes him to commit four specification errors. The first error is that of a non-zero mean vector. In model (1.19), the expected value of the i'th element of v_is _ 2 Eh’i) ‘./‘“ii°i ° 36 Since this vector will probably be uncorrelated with xi and x§(=[xi§]), only the constant term will be affected. Under H0, its expected value will be n 2 1/2 E(o1) £1 (mii O ) //<1 i: = o [cmnil/Z + + (mm)1/2]/n Because mii < l andm11 + ... + In.nn = n-k, one can say that -k 0 > E(o1) > Efi—-o. This bias will, of course, affect any test of significance on the constant term. The second specification error is that of incorrectly assuming a normally distributed disturbance term. As mentioned in the section on Park's test, this will cause the tests of significance to be biased. HOwever, as previously mentioned becauseof Srivastava's [1958] findings that the t-test is robust against considerable non-normality, this Specification error might cause only minor difficulties. It should, however, be further noted since these disturbance terms are distributed as scaled x with 1 degree of freedom which is only a one-tailed distribution, it must be considered more "non-normal" than two-tailed distributions. Hence, one should expect the Park testing procedure (disturbance terms having a Z-tailed distribution) to be a more exact test under H0. The third and fourth specification errors are those of heteroske- dastic andnonindependent disturbance terms. As is true in the case of the Park procedure, these errors cause considerable difficulty. First, both errors will cause a loss of efficiency thus making both the t and the F tests prOposed too conservative. Second, the fact that the dependent variable is heteroskedastic under HO (recall OLS residuals, 37 EQgg') = 02M)causes a disproportionate number of rejections under the null hypothesis. Third, because the dependent variables are not mutually independent, the t tests break down for lack of independence between their numerators and denominators. Hence, the validity of this testing procedure must be carefully checked. Summazy .All tests for heteroskedasticity require either some knowledge of or an hypothesis regarding the form taken by the heteroskedastic disturbances. The amount of infOrmation required differs drastically, however, ranging from knowledge of a monotonic relationship to the exact form of the relationship. Table 1 summarizes these assumptions, indicates fOr which tests the assumptions of normality can be drOpped, and restates all the relevant observations. .As can be seen from Table 1, non-constructive tests offer the advantage of not requiring as:much.a_p£igri information as constructive tests. The latter, however, have the advantage of providing a correc- tive procedure for the problem of heteroskedasticity. What would be Optimal is a test whiCh would combine the advantages of both sets of tests. Since preliminary evidence exists [Glejser, 1969; Ramsey 6 Gilbert, 1972] whiCh indicates the exact functional form of the heteroskedastic disturbance need.not be Specified for a constructive test to detect the presence of heteroskedasticity, a very general constructive test might be formed. In addition, any such general test might very well not require the specification of a single variable which is causing the disturbances to be heteroskedastic. Rather, the test might only require that the 38 Table 1: Summary of Tests for Heteroskedasticity Assumptions Concerning Tests Heteroskedasticity Observations l 2 3 4 S 6 7 A B C D E F G H Nonconstructive GQP X X X THEIL + X X X RECURSIVE-P X X X BAMSET X + + X 8/0 GQN X X X 0 RECURSIVE X X X R Constructive PARK X X X 0 FIML X X X X N GLEJSER X X X 0 X Applicable to test Applicable, though not originally suggested + No assumptions according to original formulation. Any linear function of the independent variables. The disturbances are monotonically related to a single known variable. The disturbances are monotonically related to the order in which the observations are taken. The disturbances are a function in some power of a known variable. The disturbances are a quadratic in some known variable. The exact form taken by the heteroskedastic disturbances is known. \lO‘U‘I #MNH Normality of disturbance terms not required (nonparametric test). Exact test. - Asymptotically exact test. Testing procedure is not exact because OLS residuals are used. Biased estimate of O1 is used in test. Time consuming alternative procedure. Discarded p observations resulting in loss of information. Residuals used : 0-OLS, R-Recursive, B-BLUS, N-None. 39 variable(s) causing the heteroskedasticity are present in the model. The test itself might then approximate the correct functional fOrm taken by those variables if the disturbance terms are in fact heteroskedastic. Such a procedure would not require the vast amount of information now needed and hence would be of tremendous use to the average researcher. One final point must be made. .Although there is often a lack of knowledge about which variable is causing the heteroskedasticity in a specific situation, none of the currently available testing procedures is designed to deal with such a situation. Unfortunately, researchers have incorrectly devised a way to circumvent the problem. When the cause of the suspected heteroskedasticity is not known, it is not uncommon for the researcher to select some test and to use this test with first one independent variable, then another and another until the test indicates the presence of heteroskedasticity. It cannot be emphasized enough that this technique is entirely incorrect. First, this E§;R22 technique actually violates the assumption that the vari- able causing the heteroskedastic disturbances is known. Hewever, more importantly, this tedhnique usually will lead to an incorrect conclusion. A Tb illustrate this, one could take the example of the hypothesized model 1‘ 80+ 8151 + B23—32 +3; I!” N(¢, V), where V indicates an unknown variance-covariance matrix, andwherex1 and 52 are mutually independent vectors. Having no preconceived hypothesis as to the cause of the suspected heteroskedastic disturbances, the researcher decides to use one of the standard tests 40 for heteroskedasticity to determine which variable is causing the problem. Carefully setting the probability of incorrectly rejecting the null hypothesis (type 1 error) of homoskedasticity at .05 (=o level), the researcher is ready to begin testing H0 vs. H2. Using the test, first with x1 and then with x2, the researcher concludes that the heteroskedasticity is caused.by variable x2 and that the probability of type I error is .05. Using this procedure has led the researcher, as it usually does, to an incorrect conclusion. The probability of type I error occurring is the probability of its occurring when x1 "35 tested plus the probability of its occurring whenx2 was tested minus the probability of its occurring in.both of the tests. Hence, Pr (ope I) Pr (Type I + Type I ) 2El ’52 Pr (Type 1X1) + Pr (Type 1X2) - Pr (Type IE1E2) .05 + .05 - .0025 .0975 (The independence of x1 and x2 was assumed so that the calculation of the intersection would be possible.) Therefore, if one wants to have a probability of type I error equal to .05, the a level for each test must be set at about .025. Hence, if a homoskedastic model had 20 independent variables and each were tested to ascertain whether or not it was causing heteroskedasticity, the probability of incorrectly rejecting H0 would be very, very high. 1.4.4 Studies Comparing Tests for Heteroskedasticity Since there are nine tests designed to detect heteroskedasticity, a researcher is faced with a difficult choice as to which test he should 41 use in any particular situation. As has already been.mentioned, the amount of apriori information possessed by the researcher determines to some extent which test(s) he can use. However, in.many cases after this first elimination process has been gone through, there still remain.a number of different tests from which to choose. The researcher must thus use another criterion on which to base his test decision. That criterion might be that the most desirable test is the one which has the highest probability of correctly rejecting H0 (power) given a specified alpha level (the probability of incorrectly rejecting H0).1 In five studies, this criterion has been used to compare various tests for heteroskedasticity with one another. Since, however, no two regression models are exactly alike, no comparative study can furnish the researcher with the complete solution appropriate to his particular prOblem. In these studies, however, the various tests have been compared under different conditions, that is, with different sample sizes, alpha levels, and ferms of heteroskedasticity. By making conclusions regarding the performance of specific tests under general categories of conditions, the experimentors proposed to establish certain broad criteria for the researcher to use in choosing a test fOr his particular situation. There are two basic types of study that compare tests fer heteroskedasticity. The first and most common type of study uses a sampling experiment. In such an experiment, the probability of correctly rejecting Ho (the test's power) is determined through the use of a repetitive sampling process. This procedure is analogous to 1Still another criterion might be the robustness of the remaining tests to other Specification errors. HOwever, only the robustness of the BAMSET test has been analyzed [Ramsey 6 Gilbert, 1972], and hence such a criterion cannot be made. 42 determining the probability of selecting a blue ball out of a box With three blue balls and two red balls by repeatedly selecting a ball out of the box, recording its color, and returning it before the next ball is drawn. The probability of drawing a blue ball would then be the ratio of number of times a blue ball was selected number of times the experiment was repeated To use this procedure in discriminating among the various tests for heteroskedasticity, the person conducting the experiment (hereafter referred to as the experimentor to differentiate him from the researcher who Will use his findings) formulates a regression model, such as E(yi) = a + Bxi i=1,...,n, (1.21) where the value of the vector x_and the value of the parameters a and B have been previously specified by the experimentor. Model (1.21) is used to generate the expected value of a vector of dependent variables E(y). Next, the experimentor specifies a population from which to select randomly the disturbances vi, i=1,...,n. Since the experimentor vents the regression.model to be heteroskedastic, he specifies that v., i=1,...,n are independently and identically distributed as 1 N(0, Oi')’ where 0:- denotes that the.i'th population variance is 1 . related in some'wayt specified by the experimentor, to the i'th observation of the independent variable x. The experimentor then selects n random observations from this population. Defining y1 = E (yi) + Vi’ i=1,...,n he generates n observations of independent variables y1,...,yn. Following this, he applies each of the tests for heteroskedasticity that is to be compared to this vector of observed dependent variables, 43 y, The experimentor next randomly selects another sample of n observa- tions of V1 from the specified population and again calculates yi. Once again.he applies all the tests to the new vector of independent variables, y, Repeating this procedure N times, he can determine the power of each test by calculating the ratio of (number of times that tfie test rejected H0). Given that the alpha level of all these tests is the same, the test with the greatest power would be selected as being the best test to use given heteroskedasticity of the form hypothesized. The difficulty with this procedure is that, of course, the power that is calculated for each test is often dependent on the specific model which the experimentor formulated and on the values of x_and a and B which he chose. Even more importantly, this procedure requires a very large number of replications so that the probability of choosing an unrepresentative set of samples is very low. An alternative way of calculating a test's power and a way that eliminates the repetitive sampling procedure has been suggested by Imhof [1961]. This method requires that the disturbance terms be independently and identically distributed as normal with a specified mean.and variance. Also, it requires either the specification of the distribution from which the values of the vector x_can be drawn or for the values of the vector x_to be exactly specified. After the experimentor has satisfied these conditions, the exact probability of correctly accepting H2 is calculated for each test by computing the prObability that each quadratic form will occur. Although this procedure eliminates the need to sample repeatedly from the pOpulation of the disturbance terms and hence the possibility 44 of drawing a sample that is biased (that is, that the sample could have been drawn from a distribution other than the one specified), it still requires that a sample be drawn for the vector x, .Also, this technique cannot be applied to all the tests reviewed, but rather only to those tests that define a statistic that can be expressed as a convolusion of independent quadratic forms in normal variables. Only three of the tests presented meet this requirement. Of the five studies comparing the various tests for heteroske- dasticity,a sampling procedure is used in four while the direct calculation of the power by Imhof's method is used in the fifth. Unfortunately, in only one ofthese five studies are more than three tests compared. Each of these studies will be reviewed in the chronological order in which they were undertaken. This section will then conclude with a series of remarks which can be applied to all of the comparative studies. Goldfeld G Quant I - The first comparative study was undertaken by Goldfeld & Quant in 1965. In that study, they compared the two tests for heteroskedasticity which they had develOped (referred to in this study as GQP and GQN) by using a sampling experiment. In this experiment they generated their dependent variables yi by the regression model yi=oo+o1 xi+ui, i=1,...,n, (1.22) where the disturbance terms ui were independently and identically distributed as N(0, l). The xi's were drawn from a uniform.distribu- tion with ameanux and a standard deviation of ox“ They used their two tests to discriminate between the null hypothesis that the dependent variables were generated by the model 45 1 . versus the alternative hypothesis that they were generated by the model 0 = a + a X0 + u. i = 1 0.. 0 Y1 0 l 1 1 ’ ,n Since the true model was the alternative hypothesis, it followed that the null hypothesis should be rejected. To compare their two tests fer a variety of situations, Goldfeld G Quant generated the dependent variable using model (1.22) and two different sample sizes, n = 30 and n = 60. They also used 15 different combinations of values for u and x o . For each sample of the x's, 100 replications of the experiment x were made. In addition, since a central number of observations are omitted in the GQP procedures, each hypothesis was tested by using the GQP procedure five times. No observations were omitted the first time, but four additional observations were omitted each subsequent time the test was used. The power of each test was then calculated fer eadh experiment. ' Goldfeld G Quant's results indicated that the power of both of their tests increased as the sample size increased and as the ratio of 35-increased. They also found that the power of their parametric testx(GQP) increased and then decreased as an increasing number of central observations were omitted; they concluded that the Optimum number of observations to omit, p, was between one-third and one- quarter of the sample size. Finally, as one would expect, it was found that the nonparametric test (GQN) had less power than the parametric test (GQP) for any particular experiment. HOwever, it was 0' also Observed that as the ratio 55-increased, the nonparametric test's x power increased relative to the parametric test's. 46 Glejser - The next study was reported in 1969 by Glejser. After prOposing a new test for heteroskedasticity (referred to in this study as GLEJSER), he felt that a comparison should be made between his test and the pOpular parametric test of Goldfeld G Quant. To make the comparisons he used a sampling experiment. In his experiment, a vector of dependent variables was generated by the model y. = 80 + 81 xi + u1 f(xi), i = 1,...,n, (1.23) where the ui's were independently and identically distributed as N(0, 1). In his study, eight functional forms, f(-)'s, were used to generate the heteroskedastic disturbances. The values of xi were chosen from three different normal distributions with a mean of 50 and standard deviation of 5, 10, and 30 respectively. Finally, each model was tested using three different sample sizes; they were n = 20, 30 and 60. Thus, 72 cases (8 x 3 x 3) were studied by Glejser. 100 replica- tions of each case were used to determine the power of each test under the various alternative forms of heteroskedasticity. Since Glejser's test is a constructive test for heteroskedasticity, Glejser had to specify the functional form taken by the heteroskedastic disturbance. He decided to hypothesize that the heteroskedasticity was a linear function of either xi/z 1/2 and xi or x- and x-1 depending on whether f(xi) is a function of a power in X1 or in iE-respectively. Of course, he pointed out that generally, in practice, this information would not be known. .After thus specifying the functional form used in his test, Glejser was able to test the significance of each of the estimated parameters by using a two-tailed t-test. Since, however, Glejser's 47 testing procedure is not exact, he found that in using a two-tailed t-test on a homoskedastic model, a nominal alpha level (= probability of type 1 error) of 11% was needed to reject the null hypothesis 5% of the time. Hence all 72 cases were examined using his test with a nominal alpha level of 11% so that the probability of type I error would be .05. After Glejser completed his study, he made some observations about his findings. First he concluded that generally, his test compared favorably with the parametric test of Goldfeld G Quant's. He also concurred with Goldfeld 8 Quant's findings that the power of both tests increased with sample size. Next, he discovered that his test could not detect the presence of mixed heteroskedasticity when it in fact existed. Finally, he found that because his two regressors (xi and xi/Z or xil and xil/z) were highly correlated, the test's power was generally unaffected by using just a single regressor. Ramsey 8 Gilbert - The third study is similar to that of Goldfeld G Quant's in that the experimentors, Ramsey 6 Gilbert [1972], compared two of their own tests with one another. They compared the BAMSET procedure using first BLUS and then OLS residuals under the null hypothesis and under the alternative hypothesis of heteroskedasticity. A sampling experiment was used to compare the two procedures. To generate the vector of dependent variables under the alternative hypothesis, the model yi = 1.0 + 2.0 xi, -.8 xiz + ui /i725, i = 1,...,n, (1.24) where the ui's are independently and identically distributed as N(0, l), was used. Ten values of x1 and x2 were obtained from a table of random numbers. These ten numbers were then replicated two, three, 48 and five times to generate sample sizes of n = 20, 30 and 50 respectively. In a basically similar way, a homoskedastic model was generated. Realizing that a sample of disturbance terms, unrepresenta- tive of the population from which they were drawn, would adversely affect the results, Ramsey 6 Gilbert replicated each experiment 1000 times. TWO surprising results were obtained. First, since it is well known that OLS residuals are heteroskedastic under the null hypothesis, Ramsey 6 Gilbert were surprised to find that with the BAMSET procedure the residuals were found to be homoskedastic. This meant that the percentage of times that HO was incorrectly rejected corresponded to the alpha level. Secondly, they were surprised to find that when the alternative hypothesis was correct, using OLS residuals in the BAMSET procedure always proved more powerful than when the procedure was applied using BLUS residuals. They offered no explanation for either of these results. A possible explanation for both of these findings will, however, be offered by this author later on in this study. Goldfeld 6 Quant II - The final comparative study using the sampling experiment approach was again conducted by Goldfeld G Quant [1972]. This is, to date, the most extensive comparison of tests for heteroskedasticity made. Goldfeld G Quant compared four different tests fer heteroskedasticity (PARK, GLEJSER, GQP, and FDML). They generated the vector of dependent variables by using the model 7 yi = 2 + 2 xi + u1 /a+b xi + c xi, 1 = 1,...,n, where the ui's are independently and identically distributed as N(0, l). The parameters a, b, and c are given various combinations of 49 values (7 combinations in all); the xi's are independently distributed as either uniform or log normal. (All seven combinations Of values a, b, and c were tested using the uniformally distributed xi's while only two cases were examined using the log normally distributed xi's.) All nine cases were then compared using three sample sizes n.= 30, 60 and 90. Finally, each experiment, 21 in all, was replicated either 50 or 100 times. After carrying out this elaborate study, Goldfeld G Quant drew three major conclusions. First, they concluded that the FIML method appeared "to be the most powerful test for detecting heteroskedasticity." [Goldfeld G Quant, 1972, p. 118]. Tangentially, they found that their suggested asymptotic t-test on the coefficients Obtained from the FIML technique was inferior to the likelihood ratio test originally posed by Rutemuller & Bowers. This result, they asserted, was due to the high intercorrelation between the parameters b and c. Goldfeld G Quant's second conclusion was that the power Of each test increased with the number of Observations; in this finding, they concurred with all previous experimentors. Finally, using four different tests, they were able to substantiate Glejser's finding that mixed heteroskedasticity is more difficult to detect than pure homoskedasticity. Harvey 6 Phillips - In the final study, Harvey 6 Phillips compared the three exact tests for heteroskedasticity (GQP, THEIL, and RECURSIVB-P). Rather than use a sampling experiment, they calculated the probability of correctly accepting the alternative hypothesis of heteroskedasticity by the method suggested by Imhof. That is, they calculated the probability of the quadratic form‘s occurring. 50 Harvey 5 Phillips compared the three tests fOr two types Of heteroskedasticity. They assumed that the variances of the disturbance . 2 = 2 2 _ Z 2 . terms ui were either E(ui ) o xij or else E(ui ) — o Xij' NOting that these variances critically depend on the distribution Of x,, they J assumed that x5 would take on four distributional forms. They first assumed the xj's were distributed normally, then log normally, uniformly and finally equally spaced. They then made their comparisons using three sample sizes of (n=) 10, 20, or 30 Observations, in either 2, 3, or 4 regressors and omitting varying numbers of central Observations. In computing the powers of the different rests under varying situations, they Observed that Imhof's method seemed erratic in the widely varying amounts of time that it took for the different calcula- tions. When the study was fully completed, however, they were never- theless able to make a number of Observations. First, as expected, it was fOund that the power of all three testing procedures increased with the number of sample Observations (n), and decreased with the number of regressors (k). Second, they were able to substantiate Goldfeld G Quant's findings that omitting a number of central Observations increases the power Of the testing procedure. In conjunc- tion with this, they also Observed that the number seemed to differ depending on the distribution Of the 55's. However, since omitting any number within the vicinity Of the Optimum.number resulted in very little loss of power, they felt that the difference due to the distribution of the xj's could be ignored. Third, they found virtually no difference in power among the three tests though the THEIL test (using BLUS residuals) usually out-perfOrmed the RECURSIVE-P 51 test (using recursive residuals). Finally, and most interestingly, they fOund that the power of all their tests varied considerably with the distribution Of the xj's with the highest power typically occurring when the xj's were distributed log normally and the lowest power when they were distributed uniformally. .A number Of Observations can be made on the comparative studies undertaken tO date. First, with the single exception of Ramsey G Gilbert's study, all Of the sampling experiments used a small number of replications (50 or 100). By using such a small number, the probability of drawing an unrepresentative sample is much higher than it would be if a much larger number of replications were made. This is especially true for Goldfeld G Quant's most comprehensive study [1972] as they occasionally repeated the experiment only 50 times. Second, the point has been made by Goldfeld G Quant [1972, p. 90] that the power Of the BAMSET procedure, reported by Ramsey 8 Gilbert, was calculated using a form Of heteroskedasticity that the test could best detect. .Although this is true, and was mistakenly not pointed out by Ramsey 6 Gilbert, Goldfeld 8 Quant's point is equally valid when applied to each of the other comparative studies. For example, though Glejser used seven different heteroskedastic models when the heteroskedasticity was generated by xi; instead of Xij’ the knowledge he incorporated into this test likewise changed. Similarly, in Goldfeld & Quant's own two studies, the power of the different tests is reported as if the researcher knew the variable that is causing the disturbance terms to be heteroskedastic. Finally, Harvey 6 Phillips' study makes the identical assumption. What must be shown is what the power of each Of the different tests is when the wrong 52 variable is thought to be causing the heteroskedastic disturbances and when the wrong functional form is used. Third, the preliminary findings given by Harvey 6 Phillips indicated that the distribution of the variable causing the hetero- skedastic disturbances affects the power of various testing procedures requires further study. It could well be that the power Of the tests is affected not so much by the distributional forms of the dependent variable as by the parameters that exactly specified the range Of those variables. Finally, all Of the testing procedures should be compared, unless they can be shown equivalent, under the same conditions. In this way, firmer conclusions can hopefully be drawn as to which test should be used, given a particular situation. 1.5 Summary In this chapter of the study, a vast amount Of information on the occurrence of a non-zero mean vector and heteroskedasticity in the regression model has been drawn together. In an attempt to clarify these two problems, a detailed discussion was given as to when and how both difficulties arise and what the effects will be. To further illuminate this area, an in depth review of the tests that have been proposed, and are now being used to detect each error, was given. Finally, different attempts at comparing the various tests for heteroskedasticity were presented. It is apparent that though a tremendous amount of effort has been put forth to test for the presence of these two specification errors, further attempts must be made. Two such attempts might be a more 53 general test for heteroskedasticity and a simpler formulation of the test for a non-zero mean vector. It is hOped this study will contribute to this goal. CHAPTER II A NEW APPROACH In this chapter two new specification error tests will be presented. Both of these tests are based on the ability Of a Power Series Expansion.MOde1 to estimate the conditional mean of the dependent variable. The first of these tests is used to discriminate between the null hypothesis Of a zero mean vector for the disturbances and the alternative hypothesis of a non-zero mean vector. Similarly, the second test is used to discriminate between the null hypothesis Of homoskedasticity (constant variance vector) and the alternative hypothesis of heteroskedasticity (non-constant variance vector). Both tests are being proposed in response to the Objections raised earlier in this study with the current testing procedures. Because Of the central importance Of a Power Series Expansion MOdel to both testing procedures, the concept Of a Power Series Expansion.mode1 will be introduced first. .After this discussion, the test designed to determine if the disturbance terms have a non- zero mean will be presented. This will be followed by a discussion of the second testing procedure, a test for heteroskedasticity. II.1 Estimation Usinga Power Series Expansion.MOdel In this section, the concept of a Power Series Expansion (POSEX) model will be introduced. It will be derived from both a univariate S4 SS and multivariate Taylor series expansion. The similarities between this model and Ramsey's RESET model will then be shown. Finally, an instrument will be suggested to replace the cumbersome expansion terms that appear in any multivariate POSEX model. 11.1.1 Development of a Power Series Expansion Medel A.Power Series Expansion (POSEX) model is an expansion Of the hypothesized model in powers of the independent variables. This model is applicable in those situations in which the conditional mean is an analytic fUnction in the independent variables. Suppose the regression model is given by y1 = f(xi) + ui, E(ui) = 0. (2.1) Consider first using a Taylor series expansion in the variable x to approximate the conditional mean expressed by the function f(x). In this case, the function f(x) is approximated by f(x) = f(a) + f'(a) (x-a) + ggg§;_(x-a)z + g3;(§)_(x-a)3 + ..., 2! 3! (2.2) where f(n) denotes the n'th derivative of the function f(-) and a is chosen for the ease of calculating f(a) and so that the fUnction is continuous between a and x. If, for example, the function f(-) were unknown, but n values Of x and f(x) were Observed, the POSEX model would be h 2 8o + 81 Xi + 82 Xi + "° + 8h xi + “1' Vi ‘ f(xi) * “i i = 1,...,n (2'3) where a = 0, f(i)(0) 8i and E(ui) = 0. This model will yield a good approximation if f(x) can be expressed by a low series expansion. 56 Estimating this model by the method of least squares, one would Obtain unbiased estimators Of the true coefficients. That is, E A. = f(j) , ' = O,...,h (8]) Jg0) J , where f(j)(0) is the j'th derivative of the function f(-) evaluated at zero, and j! is j factorial. Hence, given any value x0, an estimate Of f(xo), is §0 = éo + 81 x0 + £32 *3 + '°' + E3k X3 = f(’10)’ Although yo is an unbiased predictor of f(xo), the variance Of this predictor will increase to the extent to which x0 lies outside the Observed sample points x1,...,xn. Unfortunately, though this procedure is quite simple, it is not always applicable. Often, the function which is to be approximated is not a fUnction in a single variable but rather is multivariate. To analyze the multivariate case is conceptually no different from analyzing the single variate case. The Taylor series expansion of the m variate function f(x1,...,xm) is written as 8 3 f(X1,...,)Sn) = f(a,...,%) + [(Xl‘ul) “if-{4' ... 4' (xm-am) 53:11.] £3 a k ,———) f + ..., afm a1 ... am where ~3—-represents the partial derivative Operator with respect to l a [CH-211) —— + can-am) 3x. J xj, and fa a denotes the evaluation of the function f(-) after 1... m the partial derivatives have been taken at the points a1...am. Expanding this Taylor series (for the bivariate case, that is, m.= 2), one Obtains a +... 1.. m S7 = 8fa a _ f(xl, x2) f(al, a2) + l 2 (x1 a1) + 3X 1 afal 8.2 azfal 32 2 "‘7§§;" (x2 ‘ 32) + “‘;;1?“‘ (x1 ‘ 31) + x1 823531 32 2 ‘SEE‘SEE'Cxl ' a1)(x2 ' 32) + 2 a f, a 2 2 __1.z_.cxz-a2) 8X2 Reformulating this expansion into a POSEX model, as was done in the single variate case (model 2.3), one Obtains the model Vi = f("11’ X12) + ui = 80 + B1,10 x11 + B1,01 x12 + a x2+e 2x x +3 xz+ + 2,20 i1 2,11 11 12 2,02 12 °°° h h-l 1 811,110 x11 * Bh,(h-1)1 2X11 X12 * °'° * 1 h-l h ._ Bh,1(h-l) 2xil xi2 + 8h,0h xi2 + ui, 1—l,...,n (2.4) = ajfa a = _ _ — where Bj,ik ;;i—t;fiz , a1 a2 — 0, f(0, 0) - 80 and E(ui) - 0. 1 2 unfortunately, there are (h+l)(h+2)/2 = t parameters to be estimated in the above model. Hence, unless one has more Observations than parameters (n > t), the procedure breaks down. One possible solution to this problem is to assume that aif ajf ai+jf ~ - - -—I-——T-= bi+° i When this assumption is made, the 1mp11cation 3x1 3 2 3 3x1 3x2 . _ 2 = 15 that B2,20 ‘ b2 81,10’ 82,11 b2 81,10 31,01’ and that 2 . . . . 82,02 = b2 81,01. U51ng this assumpt1on and denoting 811 as 81,10, 812 as 81,01 and “i as bi’ one can transform.model (2.4) into 58 Y1 = f(xli’ X21) + “1 = 8o + 811 X11 + 812 X12 + 2 0‘2 (811 X11 + 812 X12) + '°' + “h (811 x11 + 812 X12) U. 1, l = 1,...,n. (2.5) This POSEX model now has only h+2 parameters to estimate in the bivariate case and only h+m.in thelnevariate case. NOte, however, that although model (2.5) involves very few parameters,to Obtain estimates of those parameters,one must use a non-linear estimation process. TO surmount this inconvenience one could use a two-stage procedure. The first stage would specify the linear combination of the xi's to be used for each term that is Of the fonm (811 X11 + ... + 81k Xik)j’ where j is greater than one. This first stage would provide an instrument for the non-linear terms so that a non-linear estimation technique is not needed. The second stage could then provide estimates Of the h+2 parameters (in the bivariate case). When this procedure is used, the model to be estimated in the second stage would be (once again for the bivariate case) h 2 +o1qi+...+ahqi+ui, f("11’ ‘12) = 8o + 811 X11 + 812 X12 i = 1,...,n, ' . (2.6) where q; represents a linear combination of the xi's raised to the j'th power. It might be mentioned, however, that this simplified POSEX model could instead be formulated as _°_ 2 h . = f(xil’ x12) - 80 + o1 qi + o2 qi + ... + ah q1 + ui, 1 1,...,n. (2.7) by using the linear combination Of the xi's specified in the first stage for the linear as well as the non-linear terms involving the x-‘s. 1 h + 59 However, this formulation was rejected. Although both models (2.6) and (2.7) are simplified versions Of the more complicated POSEX model (2.5), model (2.6) was chosen since it maintained.more Of the essence Of model (2.5) than did model (2.7). 11.1.2 Similarities to Ramsey's RESET'Mbdel This model (2.6) is strikingly similar to the model Ramsey [1966] used in his RESET test to determine if the disturbance term has a non-zero mean. Recall that Ramsey felt that if the vector of disturb- ance terms in model (2.1) were hypothesized to be distributed as N(¢, 021), whereas they were actually distributed as N(z, 021) then the mean vector z_could be expressed as a linear function in the moments about the origin Of i, The equation he suggested was E(E) = E_= a0 + o1 i_+ o2 QFZ) + as iFS) + o4 i‘4) + ... (2.8) where i(j) = {§{,...,§g}1. Premultiplying equation (2.8) by the matrix A' (recall that BLUS residuals 9? = Afy), limiting the expan- sion to four terms, and removing the expected value Operator, he Obtained 1.; = A, = A2 = ., Aim -. .3 A5331. .4 N554) + .1 (2.9) where w;~ N(0, OZIn-k) under H0. This model is a power series expansion in the OLS predictor Of the dependent variable, i, That is, yi is the qi in model (2.6). Tb show Still more clearly the similarities between Ramsey's model (2.9) and the POSEX.model (2.6) Ramsey's model (2.9) will be refOrmulated using the POSEX technique. When model (2.1) is rewritten as 60 u1 = yi - Bl - 82 xiZ - ... - Bk Xik’ i=1,...,n, it is clear that the disturbance terms ui are a function of the dependent variable yi and the hypothesized independent variables Xil"°"xik' Under the null hypothesis that model (2.1) generated the dependent variable yi, the yi's are a linear function Of Xi1’°"’xik’ hence, the ui's can be written as a linear fUnction in the variables Xil""’xik' However, under the alternative hypothesis that the E(u) = z_= 0, the yi's may be any (generally non-linear) function of both the k variables Xil""’xik that the researcher hypothesized in model (2.1) and Of a set of m variables zil""’zim.that the researcher mistakenly did not hypothesize as being part Of model (2.1). Hence, the vector u_must be written as a non-linear function both of the k hypothesized variables x1,...,x and of the m erroneously excluded variables 21,...zm. Because the m variables zi1,...,ziJm are erroneously excluded from the hypothesized model (2.1), they cannot be identified. Hence, the ui's must be approximated by a function in the variables Xil"°"xik' This function in xil""’xik’ if it is analytic, can itself be approximated by a power series expansion mode in Xil"'°’xik' This series of two approximations yields the POSEX model +... + “i ; f(xil”"’xik) ; Bo * 811 x11 " 81k xik o‘2(B11 X11 + + B1k x1192 + " “h (811 x:11 + + 81k Xik)h ” Vi’i=1"°"n where E(vi) = 0, which is a k variate extension of the bivariate model posed in equation (2.5). Once again this model requires a non- linear estimation process tO estimate the h+k parameters. TO solve 61 this estimation problem Ramsey suggested using §i as an instrumental . . j variable for each term that is Of the form (811 x11 + ... +Blk Xik) , where j is greater than one. The variable yi was chosen since it provided a linear combination Of xil""’xik based on the relation between the dependent variable yi and the independent variables xil""’xik' When this instrument is used, the model becomes u. é ”2 1 8o + 811 X11 + °'° + 81k Xik * “2 Y1 + °°° + Ah . ah yi + V1’ 1 = 1,...,n, (2.10) where Vi’ i=1,...,n, are independently and identically distributed as N(0, oz) under H Multiplying model (2.10) by A', one derives the 0. model 2* = Avg: (:2 AviCZ) + a3 A'ifi) + a4 A'i(4) ... ‘1’ (2.11) where w_~ N(¢, OZIn-k) under H0 and h is set equal to 4. Hence, Ramsey's model (2.9) has been Obtained by using a POSEX model fOrmulation. 11.1.3 A Suggested Instrument In using the method Of instrumental variables to simplify the POSEX model so that the linear estimation techniques can be used, the researcher must choose an instrument which is highly correlated with the term that it is replacing. However, unless the correlation between the two variables is exactly one, using the instrument reduces the accuracy Of the approximation. Hence, since the vector;y used.by Ramsey is A A X-= 8o + 81 BS1 + "' 1 Bk 5k = Xfi) 62 if the vector of estimated parameters is not a multiple of the vector of parameters {811, . . . ,Blk}', that occur in the expansion term (Bugg-1 + ... + Blkfik)cj)’ then using the instrument yfij)‘will reduce the accurate Of the POSEX model. As an alternative to using the instrument i, one could choose (e13:l + ... + ekxk), which is another linear combination Of 51""’§n’ which might, in general, be more highly correlated with the term.(811 x1 + ... + 81k 5k). Since correlation is a measure of how two groups Of variables vary with respect to one another, the coefficients e1,...,ek should be chosen by examining the variance within each of the vectors x1,...,xk. However, the variance within a vector is not the only impOrtant characteristic to be taken into consideration. The coefficients e1...ek must also reflect the scale of each Of the vectors xl,...,xk. For example, if the sample variances of each vector are identical, the vector that has the smaller elements (the smallest mean) should be given more weight. The rationale for this might not be immediately apparent, but an example will clarify the point. Three observations are drawn from two populations resulting in the samples (990, 1000, 1010) and (10, 20, 30). The sample variance is 100 in both cases. HOwever, the variance Of 100 results in a 2% variation (=T%%5-° 100) in the sample points in the first sample and a 100% variation (=%%-° 100) in the sample points in the second sample. Hence, since the variation in a variable, not the variance Of a variable, is the important characteristic, the coefficients e1...ek must be chosen by a method that takes into account both the variance and the mean Of each vector Of independent variables. 63 One such technique is the method of principal components.2 TO use this technique, one first forms the k x k matrix Of squares and cross-productions (X) from the vectors x_,...,x , that is 1 £1 2 = (‘2 ) (Zfif"§k) Ek Second, one finds the eigenvalues and eigenvectors of the matrix 2. Selecting the largest of these eigenvalues and denoting the eigenvector el‘ associated with that eigenvalue as ( ) , one can define the vector ek p (the first principal component of the matrix X) as p_= e1 x1 + ... + ek xk. This vector p_is calculated in such a way that whichever vector 31”"’§k has the most variation (reflecting both.mean and variance) has the largest coefficient. The one with the second most variation has the second largest coefficient, etc. Finally, this vector possesses the statistical property of being the best linear predictor Of the vectors xl,...,xk. This is easily shown by noting that no other normalized, linear combination Of the variables x1,...,xk has a greater variance than does the vector p, Therefore, no other normalized, linear combination of the xfs contains as much of the variability that is in the vectors xq,...,xk than does the vector p, Hence, no other combination can zIt was first suggested that principa1 component analysis be used in conjunction with Ramsey's RESET test by Professor Dudley wallace. His suggestion is greatly appreciated. 64 predict the vectors x1,...,x better than the vector p, This is not to say, however, that the coefficients e1,...,ek have any greater probability Of equalling the parameters Bll""’Blk (parameters from the term.to be replaced) than did the estimates 81,...,Bk (the estimates used to calculate Ramsey's instrument Q). It only says that given no knowledge as to the vector {811,...,81k}' no vector of weights {w1,...,wk}' = w;will produce a vector X thhat contains more variation than does the vector X e_= p, Therefore, since the unknown variability of the dependent variable is what is trying to be captured, no other vector can do a better job than the vector p, Hence, if a POSEX.model is used tO approximate an analytic multivariate function, no single instrumental variable should, in general, provide as good an approximation as that Obtained by using the vector p, Hewever, since the vector p_is more difficult to calculate than the vector i, any decision as to which should, in general, be used becomes more difficult. A.sampling experiment will be conducted later in this study to provide some insight into what- ever trade-Offs might exist between the two instruments. It is, however, evident that a POGEX model can.be formulated to approximate, to varying degrees of accuracy, any analytic univariate or multi- variate function. Hence, besides providing the foundation for the two specification error tests which will be next presented in this study, one hopes that this technique might be adapted to further uses by other researchers. 65 11.2 POSEX Test for a Non-zero Mean If one hypothesizes the model Y1 = 31 + 32 xi2 + ... + Bk Xik + ”1’ i=1,...,n, (2.12) where it is supposed that the vector Of disturbance terms 2 = (u1,... ,un)' is distributed as N(¢, 021), the null hypothesis of a zero mean vector is that the E(u) = 0 whereas the alternative hypothesis is that the E(u) = _z_ #- 0. This forrm11ation is used in three tests which have been developed to test for the null versus the alternative hypothesis (recall the tests developed by Ramsey, Ramsey 6 Gilbert, and Ramsey E Schmidt referred to earlier). However, the hypothesis space can be similarly divided by yet another criterion. Rewriting model (2.12) in matrix notation, one Obtains x = X a + 9., 2 ~ No, 021) (2.13) where y and u are (n x 1) column vectors, _8_ is a (k x 1) column vector, and X is an (n x k) matrix Of rank k. If X is independent Of 1_1_, y is distributed as u with a mean of Xe; that is, y is distributed as N(X_B_, 021). Given this formulation, the hypothesis space can be divided exactly as before by basing the division on the mean Of the vector y. The null hypothesis then would be that the My) = XE. (referred to in this section as H0), instead of E(u_) = 0, and the alternative hypothesis would be E(y) 7‘ X8 instead Of E(u_) 7‘ 0. Using this formulation Of the hypothesis space is convenient since the y's, unlike the 3's, are observable. This procedure totally eliminates the need to select a predictor for the disturbance terms. Using this formulation of the hypothesis space, a POSEX model will be developed which will estimate the conditional mean of the dependent variables yl, . - - .Yn- 66 11.2.1 Fomlatinithe POSEX Model and Testing Procedure Under the null hypothesis, the ['5 are generated by model (2.13) , whereas under the alternative hypothesis, the y's are generated either by some other function (non-linear) Of the hypothesized variables 51, . . . ,x or by some function (maybe linear) of the hypothesized variables x1“ . . ,xk and of the erroneously excluded variables _z_1,. .. ’Em' Hence, under H0, yis a simple linear function Of 51,...,x while under H1, y is some unknown function of 391,... ,xk and 51,. . . ’Em' Under H1, depending on the number of omitted variables m (1 0) and on their relation tO 51,” . ,xk (the necessary relation will be investigated later in this section), the vector y_ can be approximated by the variables 51,. . . ,x_k. Also, under H1, the unknown function can be approximated by formulating a POSEX model in the variables 331,. . . ,x . Using the POSEX technique, which was previously described and illustrated in model (2.5) , one formulates the model X: f(51"Wl‘k) 1 ‘1‘ B11 1 812 ’52 + + (2) 81k X—k + 0‘2 3 + 013 3(3) + 014 3(4) + E (2.14) where E(u_) = 0, and where a four-term expansion is used (same as in Ramsey's RESET model). Because two instruments have been proposed as the vector q, i (the OLS predictor Of y_) by Ramsey and p_ (the first principal component of the matrix X) by Wallace, both instruments in turn will be used. Later on in this study, they will be canpared to determine which provides the better instrument. Model (2.14) must now be examined under H0 so that a test for discriminating between H0 and H1 can be formlated. Under Ho that model (2.13) generated and dependent variable y, model (2.14) becomes 67 (2) (3) (4) E(X)=81+32_)£2+...-I'Bk)_(k+O-g_+0-9-+O-9L xg + Q - 9 Hence, to test the null hypothesis that (2 .13) generated the vector y, one only need test the hypothesis that 012 = = 0. If the o3 = 0.4 parameters (12, as and (14 are found to be jointly equal to zero, H0 is not rejected, whereas if oz, 013 and o4 are found to be jointly different from zero, H0 is rejected. This hypothesis is easily tested by using an F-test for the included variables 3(2), 3(3), and 3(4) [Goldberger, 1964, pp. 174-175]. The procedure is to estimate model (2.13) and model (2.14) by the method Of least squares. Denoting £1 and £12 as the OLS predictors Of the vector y_ from model (2.13) and model (2.14) respectively and u as the OLS residuals from model (2.14), one calculates the ratio (1222 Xlxl)/(n-k-3). 'u This statistic is distributed as F with 3 and n-k-3 degrees Of freedom under H0 because the ratio can be rewritten in terms of two independent quadratics in the normally distributed disturbance term 2. In examining the F statistic under the alternative hypothesis an interesting Observation can be made. Denoting wi as the portion Of the 'true' model that remains unexplained by the hypothesized model, the quadratic ratio becomes (2' + 31') Q1 (2+ E) (21 + 11') Q2 (3 + 11) where u is the disturbance term and Q1 and Q2 are the appropriate quadrations. Since this is a ratio of two non-central xz's, this 68 ratio can be greater than or less than one. Therefore, a two-tailed F-test should be used. A well defined test statistic has been developed using this formulation. Hence, no matter what the sample size, the distribution Of the statistic is known. Also, unlike Ramsey's and Ramsey G Schmidt's test procedures,with this formulation, the use of predictors of the disturbance term 3 and the calculation Of the matrix M or A' are avoided. Therefore, if a researcher uses the new formulation of setting up a POSEX model to explain the vector y_, he avoids both of the difficulties associated with Ramsey's and Ramsey 8; Schmidt's testing procedures. Finally, it has been pointed out that the apprOpriate test is a two tailed F test and not a one tailed test as was mistakenly used by the previous authors . 11.2.2 Comparison With Previous Testing Procedures It is interesting to note that formulating a POSEX model to improve the estimate Of the conditional mean of y can, under certain conditions, be shown equivalent to Ramsey's and Ramsey G Schmidt's testing procedures which determine whether a disturbance term has a non-zero mean. Of course, as has previously been shown, the hypothesis space can be equivalently divided by setting up the null and alternative hypotheses in terms of the vector Of disturbance terms 3 or vector of dependent variables y. Assume that the hypothesized model is x=Bl+82£2+ +Bkrk+2=xé+a where u is assumed to be distributed N(0, 021) under H0. Setting up a POSEX model to estimate the conditional mean of y, and using X 69 (the OLS predictor of y Obtained after the hypothesized model is estimated as the instrument , one Obtains “(2) .(3) ~(4) X==B111312352“"B11<’—‘1<“°'2X ” “3X +"'4Y- *1”: .(2) «(3) AC4) X§_+o2X_ +0132: +0141 +11, (2.15) where w is assumed to be distributed NM, 021) under H0. If model (2.15) is premultiplied by the matrix A' (recall that the BLUS residual vector 3* = A'y, where A'X = 0, A'A = In-k and AA' = M = (I-X(X'X)_X')),the model becomes ' * , '.(2) '.(3) '«(4) ' Ay=u =A X§+Ay a2+Ay a3+Ay_ a4+Aw ~(2) ~(3) ~(4) X 0.2 + X as + X (14 + A'w (2.16) 2 where A'w is distributed as N06, 0 . . In-k) under H0 and where ,«(31 ~(J) A y — y . Model (2.16) is the model in which Ramsey tested dz = a3 = o4 = 0 and hence Obtained his RESET test for the disturbance term 1_1_'s having a non-zero mean. Likewise, if model (2.15) is premultiplied by the matrix M (recall that the OLS residual vector 1; = My = (1-X(X'X)—X')y_), the model becomes “X = 13.: MXE. 4" Ming? + Micsg'g + “imam + M1”. = Miczg‘z " Miags + Mi“; Mi ' (2.17) where W. is distributed Nw, 02M) under H0. Model (2.17) is Ramsey G Schmidt's model whereby they were able to test for the disturbance term u's having a non-zero mean by testing if dz = o3 = o4 = 0. Hence, since Ramsey's and Ramsey G Schmidt's models can be Obtained from the POSEX model by prermiltiplying the POSEX model by either the matrix A' or the matrix M, respectively, all three models are mthenmtically equivalent. Furthermore, since all three tests 70 use an F-test to determine if the parameters o2, o3 and o4 are different from zero, all three tests are likewise mathematically equivalent. Therefore, since neither of the previously reviewed tests Offer any advantage over formulating a POSEX model and yet both offer the disadvantage of compelling the researcher to calculate either the matrix A' or the matrix M, there is no apparent reason to use either Ramsey's or Ramsey G Schmidt's testing procedure. 11.2.3 Examination of the New Testing Procedure Under Hl As has been mentioned, two basic errors can cause the vector of disturbance terms u_to have a non-zero mean; likewise, the same two basic errors can cause the vector of dependent variables y_to have a conditional mean other than Xe, The first error occurs when the wrong functional form of the regressors or regressand is used in the hypothesized model. The second error occurs when a number of relevant independent variables are omitted in the hypothesized model. The new testing procedure will be examined under both these errors. Incorrect Functional Form of Either the Regressors or Regressand If a researcher hypothesizes model (2.12) whereas the dependent variable y'is actually generated by the model y= f(xl,...,x_k) + ‘L (2.18) where v.is distributed as N(D, 021) and f(-) is some function other than the one hypothesized in model (2.12), a specification error has been committed. NOte that although this specification error is caused by incorrectly hypothesizing the functional form of the regressors, a similar error can be caused by incorrectly hypothesizing the functional form of the regressand (see section 1.2). Hence, only the former error will be examined in this study. 71 Therefore, when a researcher hypothesizes model (2.12) while the dependent variable actually has been generated by model (2.18), the specification error which has been committed is usually that of incorrectly specifying the functional form of either the regressors or regressand. If the researcher suspects this error, he may want to examine whether a POSEX model would better explain the conditional mean of y_than would model (2.12). If the POSEX model does better explain the conditional mean of y, the researcher knows that model (2.12) was misspecified. The POSEX model that the researcher would fOrmulate is (2) (3) (4) 811+812§2+ °'°181k’—‘k+°'23 ”'39- +<"43 *3 X8 + Qg + E (2.19) X. where E was assumed to be distributed N(0, 021) and where q_is used to represent either the instrument y_(OLS predictor of y;obtained after estimation of model (2.12)) or the instrument p_(the first principal component of the matrix X). If the estimate of g_is statistically different from zero, the null hypothesis that model (2.12) generated the vector ijill be correctly rejected. The probability of this test's correctly rejecting Ho depends largely on the function f(-) and on the instrument 9 chosen. First, as previously stated, f(o) must be analytic, since a non-analytic function cannot be expressed as a power series expansion. Second, since the POSEX model prOposed involves a four term expansion, one must be able to approximate f(o) using only a four term expansion in 3 the variables §1""’—k' If f(-) can only be approximated using 3Although four terms has been suggested in this study, any number of tenms may be used. There is, however, a trade Off since adding more terms changes the number of degrees of freedom involved in the proposed test. 72 more than a four term expansion in x1,...,xk, the POSEX model which is given in the equation (2.19) will provide a poor approximation; hence the testing procedure suggested will prove unreliable. Hewever, since most of the standard non-linear fUnctions are analytic, and since a good approximation of most standard analytic functions can be obtained using as few as two or three expansion terms (for example, the exponential, logorithmic, and sinosoidal functions are all approximated in three or fewer expansion terms [Thomas, 1966]), these conditions should generally cause no difficulty. Finally, the probability of correctly rejecting H0 will also vary in accordance with the correlation between the instrument 3 and the expansion terms which it replaces. That is, since g_is an instrument (representing either y, the OLS predictor of y_obtained from the hypothesized model, or p, the first principal component of the matrix X), this statement simply means that the test's power varies with the quality of the instrument used. In summary, if a model is misspecified because the functional fOrm of x1,...,x is incorrectly hypothesized, the power of the suggested test depends on two factors. The first factor, the functional form of f(') which generates the vector y, does not generally cause difficulties. The reason for this is that the fUnctional forms of f(o) generally thought probable are both analytic and easily approximated by using a power series expansion (two examples are the exponential function and the logorithmic fUnction). The other factor responsible for causing a loss in the test's power is the instrument chosen to replace the expansion terms. It is felt by this investigator that the first principal component p_ 73 will, however, in general, provide a reliable instrument for the expansion terms. It must be recalled, nevertheless, that the OLS predictor of y, i, has been successfully used as an instrument for the expansion terms by Ramsey. This instrument has the advantage of being more easily Obtained than p, This investigator feels, however, that y_will, in general, be less highly correlated with the expansion terms than will p, and hence be less reliable. Any final conclusion as to which of the two instruments is the more reliable must of course be postponed until they are actual compared in a sampling experiment. Omitted‘Variables Assume that once again a researcher hypothesizes model (2.12) y_= 81 + 82 x2 + ... + Bk 5k + u = X B + E; where it is assumed that u1~ N(0, 021), whereas the model that actually generated the dependent variable y_is X=X§+3151+°°'+.Z.m5m+K=XB+Z§_+.YJ (2.20) where v_~ N(0, 021). Model (2.12) is misspecified because m independent variables, El"°°’£m’ have been omitted. If the researcher suspects that he has inadvertently omitted some variables, he can formulate a POSEX model to explain the conditional mean of y, If, in a statistical sense, the POSEX.model explains the conditional mean of y better than does model (2.12), the indication is that the model (2.12) is misspecified. In the POSEX model which the researcher would use to explain the condition.mean of y, the variables xl,...,x would be used in the expansion. It must be remembered that the researcher suspects 74 that he may have erroneously omitted some variables; however, he does not know the identity of the variables which he may have omitted. The POSEX model thus formulated would be _ 2 3 X-' 811 * B12 52 * "° + 81k 5k + 0‘2 SF ) + “3 3‘ ) 1 0'4 9- '1 =X§+Qa+u can where w_is assumed to distributed N(0, 021) and where q_represents either the instrument y_or the instrument p, 1f the estimates of g_ are found to be statistically different from zero, the model hypothesized as generating the vector y_(model (2.12)) is found to be misspecified. Needless to say, the probability of this test's correctly rejecting HO depends on the relationship between the variables erroneously omitted and the instruments used in place of the expansion terms. Since the idea that the power of the test depends on the instrument chosen has already been discussed, further elaboration is not needed here. Rather, this section will focus on how the test's power is affected by the characteristics of the variables omitted. In order that the analysis which follows will not be unnecessarily complicated, it will be assumed that only one variable is omitted erroneously from the hypothesized model (2.12).. Assume that the model which actually generated the vector of dependent variables y_is X=81+82£2+'”+8k)—(k+65+y- = x §_+ 2_6 + v_ (2.22) where v_~ N(¢, 021), and z_is a non-stochastic vector. Since model (2.22) and model (2.21), used to test whether the null hypothesis is misspecified, differ only in their second terms, 75 the second model's ability to discriminate between H0 (g_= 0) and H1 (g_# 0) is directly related to the proportion of the vector 2_ that lies in the space spanned by the matrix Q. Although this cursory Observation is somewhat illuminating, a more indepth analysis is required. Since the instrument q_(either p_or i) is a function of x1,...,xk, the omitted variable can be Characterized as one of three types, depending on the omitted variable's relation to the variables x1,...,x . The first type Of omitted variable is highly correlated with the variables xl,...,xk, the next type is uncorrelated with them, and the final type is moderately correlated with them. I To simplify the analysis of each type of omitted variable, all of the variables xl,...,xk and z_will be orthogonalized. This linear transformation yields the k + l vectors El""’§k’ §k+l corresponding respectively to the vectors £1”°"§k’ E: Thus, the vector §k+1 contains only that part of the vector z_which is not already explained by the variables §l""’§k' The three cases of z_to be analyzed, having either high, low, or medium correlation with x1,...,x will correspond directly to the vector §k+l containing either little, a great deal, or moderate amounts of additional information. The reason for this inverse relation between the amount of information contained in the vector §k+1 and the correlation between z_and x1,...,xk is that the latter measures the amount of linear relation between z_and El""’§k while the former contains the amount of information remaining after any linear relation has been removed. For example, when the vector z_is highly correlated with xl,...,xk 76 (that is, a large portion Of the information embodied in z.is also contained in xl,...,xk), and the linear information is removed through orthogonalization, the vector §k+1 will contain very little information. Hence, high correlation between 2 and x1,...,xk will imply very little additional information in the vector §k+1' Finally, before each type of omitted variable is analyzed in turn, it is important to stress that only the linear relation between E and x1,...,x has been eliminated. Hence, there is no implication that vector §k+l 1s 1ndependent of the vectors 31""’§k’ but only that §k+1 1$ uncorrelated w1th the vectors §1""’§k' Omitted Variable Highly Correlated with the Matrix X In the case Of this type of omitted variable, if g were correctly added to the hypothesized model (2.12), the model would be highly multicolinear. When the variable 2, however, is erroneously omitted, efficiency will be lost, but the loss will be small. Unfortunately, though, there is always a cost involved when a specification error is made. In this case, the estimates of the parameters 81,...,Bk, in the hypothesized.model (2.12) will be biased. As previously mentioned, the amount of the bias associated with each estimate depends on the correlation between the variable associated with that parameters and the variable 2. If a POSEX.mode1 is used to determine whether model (2.12) is misspecified when the omitted variable z_is highly correlated with the variables xl,...,xk, the probability that the POGEX.model will better explain the conditional mean of y_is very small. The reason for this is that so little information is left in the vector §k+1 77 that even if it were explained by the POSEX model, it still may not provide a statistical improvement over the hypothesized model. Finally, even though the testing technique being suggested does not offer a very high probability of correctly rejecting H0 when the omitted variable is highly correlated with the included variables x1,...,xk, the cost of such an error is low. .A small loss in efficiency will occur and biased estimates of 81,...,Bk will be obtained. However, even if the omitted variable z_had been correctly included, the model would have been.multicollinear; hence, the matrix X'X would be ill-conditioned, so that the estimates of 81,...,Bk and 02 (the model's variance) have relatively large standard errors and the estimates are very sensitive to small perterbations in the values of the regressors. Therefore, the incorrect omission Of the variable z_is relatively inconsequential even though the omission cannot be detected by the POSEX test. Omitted variable Uncorrelated with the Matrix X In the case of this type Of omitted variable, the vector z_is virtually identical to the vector §k+1' When the variable 2, which is uncorrelated with x1” . . ,xk, is erroneously omitted, two difficulties arise. First, since z_is uncorrelated with 51""’§k’ only the constant term B1 will be biased. The amount of the bias will equal E'= .2 zi/n; hence, the expected value of the estimator 81 will be Hal):1 81 + 21 Also, since none Of the variation embodied in z_is used to explain the conditional mean of y, the hypothesized model (2.12) will be inefficient. If a POSEX model were able to explain the conditional mean of y;better than the hypothesized.model (2.12), one of two things could 78 occur. First, if the vector z_is independent of as well as uncorrelated with the vectors xl,...,xk, the POSEX model will have virtually no power. The reason for this is that since z_is independent of x_,...,xk, and as g'is a linear combination of M} .,xk, the vector 2 is independent of the vector q_as well as of 3.2)9F? and dgF4) Hence, in the POSEX model (2. 21), since the vector z_is independent of the vectors El” . "—k’ qF2)qF3)q(4)the POSEX model adds nothing to the hypothesized model. In the second case, however, the POSEX model will improve upon the hypothesized model if z_is not independent of xl,...,xk. Generally, however, in economic data, if the variable z_is uncorrelated with the variables 321,. . . ’E-k’ it is also independent of 51,...,x . Hence, the analysis of this case will be postponed until the next section. Therefore, when z_is uncorrelated with the variables x1,...,xk, the POSEX.mode1 again proves to be of little use in detecting the error. Hewever, once again, some consolation can be taken in the fact that when an uncorrelated variable is omitted, only the constant term and the estimate of the variance will be biased. Omitted Variable Somewhat Correlated With the Matrix X In the last case, which is the most common, the vector z_is neither uncorrelated nor highly correlated with the vectors xl,...,xk. Hence, in this case, because z is correlated with x1,.. .,xk, the estimates of 81,...,Bk are biased, however, since z_is not highly correlated with xl,...,xk, the estimators of 81,...,Bk are not efficient. Therefore, this type of omitted variable can cause all of 79 the estimators in the hypothesized model, 81,...,8k, 82 (model's variance) to be biased. This most troublesome type of omitted variable, however, is the case where a POSEX.model might better (in a statistical sense) estimate the conditional mean of y1 than did the hypothesized model. In this case, unlike that in which z_was highly correlated with x1,...,xk, the vector §k+1 still contains some information; hence, a POSEX model can.improve on the hypothesized model by estimating the variation in the vector §k+l' Also, unlike the case in which z_was uncorrelated with x1, . . ‘. ,x (and hence maybe independent), 2k +1 is not necessarily independent of the vectors E1’°°"X either squared, cubed, or quadrupled. Therefore, the ability of the POSEX model (2.21) to provide a better estimate of the conditional mean of y_ than did the hypothesized model (2.12) depends on how great a portion of z_and hence §k+1 lies in the space spanned by gfié)gf§)and 3(4) First, since g_is a linear combination of the vectors x1,...,xk; then qFE)qF§)and'gF4Are functions of the vectors x1,...,xk squared, cubed, and quadrupled, respectively. In addition, since only the linear relation between the vectors 51,...,x and the vector z_has been removed from the vector 3_ (resulting in the vector §k+1)’ it is not unreasonable to expect that 3(3)q(3:)and 3(42night be able to explain still more of the variation given in.the vector §k+1‘ The reason for this is the point that has been stressed over and over again; since only the linear relation between the vector §_and the vectors 51""’§k has already been explained by the hypothesized model (resulting in the vector §k+l)’ there is no reason to assume that a relation between ak+1(or z) and 80 th (2) (2) (3) (3) (4) (4 t e vectors x1 ,...,xk , or x1 ,...,xk , or 51 ""’§k oes no exist. If such a relation does exist, given that gF§)qF?)and qF4gre combinations Of these vectors, the variation in the vector §k+1 might be better explained. If the POSEX model provides a better estimate of the conditional mean Of y_because it uses part of the variation in the vector §k+1’ then the estimate of the vector of parameters g_will be statistically different from zero. Hence, model (2.12) will be found to be misspecified because the POSEX model better explained the conditional mean of y. Even though it at first appeared as if the third type of omitted variable would cause the most difficulties, it has been demonstrated that a POSEX model can be used more effectively in this case than in the other cases. Of course, the probability Of correctly rejecting H0 depends heavily on that portion of z_which is spanned by the vectors 51""’§k squared, cubed, and quadrupled. It has been shown that the probability of correctly rejecting H0, when a variable 5 has been omitted by using an F-test and a POSEX model most certainly depends on the relationship between the variables 51’°'°’X and variable E: It appears as if the power of the procedure is the highest when z_is moderately correlated with the hypothesized variables xl,...,xk. If z_is uncorrelated with the hypothesized variables, z_is most probably independent Of them and hence independent of any linear combination of them. If z_is too highly correlated with x1,...,xk, then little improvement in explaining the conditional mean of y_can be ascertained by using a POSEX.model. 81 11.2.4 Summary In this section, a new testing procedure fOr determining whether a model has been.misspecified has been obtained. It is shown to be equivalent to two current testing procedures, but is also shown to Offer the advantage of being more easily fOrmulated and carried out. When the new test was examined under both common causes of the specification error, it was suggested that the test would be more powerful when an instrument highly correlated with the non-linear term is used. In addition, it was discovered that when the error is caused by incorrectly formulating the functional form of either the regressors or regressand, the power of the test increases if the correct function is analytic and can be approximated easily. Also, finally, when the error is caused by omitting a variable from the hypothesized model, the power is related to the correlation between the omitted variable and the hypothesized variables, the highest power being obtained.when the correlation was moderate. 11.3 POSEX Test to Determine if the Disturbance Terms Are Heteroskedastic Given the hypothesized model X=81+82§2+"'+Bk§k+3=xs'+9— (2.23) where all the vectors are n x l and where u_is assumed to be distributed N(¢, 021), a number of tests exist that will compare the null hypothesis (H0) of homoskedasticity with the alternative hypothesis 0H2) Of heteroskedasticity. .All of these tests, however, require a great deal of a_p£igri_information regarding the variable, presumably x1,...,xk, that is related to the heteroskedastic disturbances. Since, however, a POSEX model can approximate any 82 analytic function, it is possible to use a general POSEX model to estimate the variance of each of the n disturbance terms. .A test can be developed which will determine whether the POSEX model is better able to explain the conditional variance of the vector of disturbance terms u. Under the null hypothesis of homoskedasticity, no group of variables (or model) will be able to explain the constant variance of the disturbance terms. Hence, if a model is able to explain the variances of the disturbance terms, the variances are not homoskedastic and the null hypothesis should thus be rejected. Finally, since a model to explain the variances is used as the basis of the proposed test, this test will be a constructive test. That is, in the case in which the null hypothesis of homoskedasticity in model (2.23) is rejected, a procedure will be offered which will enable the researcher to transform model (2.23) into a homoskedastic model. Before the model and test are developed, however, an estimator of the unobserved variance of u1,...,un must be selected. The POSEX model will then be developed. Next, it will be shown how the POSEX model can be used to reestimate the parameters in the hypothesized model. .Also, it will be demonstrated how a priori information can be included in the POSEX model. Finally, a number of different ways of estimating the POSEX.model will be suggested. Included with each of these suggestions will be a test to determine if the disturbance terms are either homoskedastic or heteroskedastic. 83 11.3.1 Estimators of the Variance of ui 2 unfortunately, the variance of each disturbance term, 01""’On’ is not observable. Hence, a number of estimators of 01""’G§ have been obtained. Denoting 81 as the i'th least squares residual obtained from model (2.23), the first estimate of 01 used (by Park and by Glejser, by Goldfeld & Quant, and by Ramsey 8 Gilbert) was 812. Unfortunately, under the null hypothesis of homoskedasticity, E(u_uf) 021, the least squares residuals are heteroskedastic, E(ué') = 02M = 02(I-X(X'X)_X'). Hence, since the diagonal terms of Ecu Rf) are E(uiz), one finds that the expected value of the n estimates, £12,...,&n2 are 02m11,...,ozmhn, where m.ii is the i'th diagonal of the matrix M. Therefore, ui2 is a biased estimate of 2 Ci even under H0. Uhder the alternative hypothesis that model (2.27) is 2 heteroskedastic, however, the estimates fii become weighted averages of the true variances. Since the diagonal elements of the matrix fig} are u12,...,uk2, this weighting scheme is most clearly demonstrated by taking the expected value of the matrix uuf under the alternative hypothesis that the -! oi fl my) = =v. L” .°’2‘ One finds that DIAG [Ecfijqj DIAG [5mm] DIAG [ME(uuf)M] DIAG [MVM] DIAG A DIAG DIAG z . m' G. .002 m ' m. o . mn1 1l 1 . n1 1n 01 L1 1 .4 Since M is symmetric (mij = mji), the j'th diagonal element becomes n 2m? ji oi. Hence, if one defines M(2)as being the squared elements 1 l of the matrix M (i.e., {mij2})’ then DIAG Eng) = Mm DIAG [V] = Mm {oinuofiu (2.24) Therefore, since under H2, each estimate £12,...,£n2 is a weighted sum of oi,...,o§; £12,...,&n2 are biased estimates of oi,...,ofi. . “ ‘ 2 . . 2 2 Hence, Since u1 ,...,un are b1ased est1mates of Ul”"’0n under both H0 and H2, it is perplexing to account for the findings of Goldfeld G Quant (using their non-parametric test) and Ramsey G Gilbert (using the BAMSET procedure with OLS residuals). They fOund, by using sampling procedures, that the probability of type I error corresponded to what was theoretically expected and that the probability of type 11 error was modest. Of course, the results could have been due to the specific models used and hence to the structure of the matrix X- HOwever, this investigator does not find this explanation at all adequate. Rather a theorem based on the 85 matrix M, will be stated and proven (in Appendix A), and another explanation offered in place of the one just mentioned. (In addition, three interesting corollaries to this theorem are also stated and proven in.Appendix.A but will not be used in this study.) Theorem: Regardless of how the vectors x1,...,xk are obtained (stochastic or non-stoChastic) the diagonal elements of the matrix M will have a maximum squared variation of §%%E%%-< §-, where squared variation of t1,...,tn is defined as £(ti - t)z/n-l. This theorem.provides a vehicle for understanding the findings of Goldfeld 8 Quant and of Ramsey 8 Gilbert. They both Observed that under H0, when OLS estimates Of CI""’O§ were used to test Ho versus Hz, the probability of type I error corresponded to the nominal alpha level at which the test was used. This finding implied that the OLS estimates were homoskedastic under H0. It has, however, always been assumed that the matrix M has unequal diagonal elements since E(ui) = 02mii’ i=1,...,n. Hence the implication that the estimates £12,...,fin2 were homoskedastic seemed difficult to accept; however, the theorem.proven in this study provides a plausible explanation for this finding. It indicates that regardless of how the variables are chosen, the maximum squared variation of the diagonal elements ofiM is never greater than —§}-. TherefOre as n + w the squared variation-+ zero regardless of the matrix X. Further, even with small sample sizes, the variation is minimal if the number of parameters is small. Hence, although OLS residuals may not be homoskedastic when the disturbance terms are homoskedastic, they may appear to be, especially if n is large or k is small. 86 .As previously mentioned, their second finding was that under Hz, the probability of type 11 error was reasonably small. This implied that the OLS estimates are heteroskedastic even though each estimator 2 is a weighted sum of each true variance, 01""’0n' The type 11 error which was found is consistent with the theorem and corolaries proven in this study. Even though each of the terms, uiz,i=l,...,n, is a weighted sum of oi,...,o§, the weights are such that the greatest weight given ui is that associated with oi. This is evident if one recalls that ‘2 _ 2 2 2 2 2 2 E(ui) - mil 01 + + mii oi + + min on. (2.25) Since, however, mil + ... + min = mii :_1, because M'is idempotent, the portion of the weight given to each variance is 2 2 2 Iii}. “Lil min (2 26) m.. , ... , m.. , ... , m.. . . 11 11 11 Since this series consists only of positive numbers which sum to one, and since the mean Of the diagonal elements of Ehe matrix M is Big-, the series in equation (2.26) is dominated by fi%%-= mii' Therefore, 11 the weighting scheme given in equation (2.25) favors the term 01' Hence, when the variances 01""’°fi are unequal (heteroskedastic), the estimates fii,...,&§ are also unequal. Therefore, the finding of Goldfeld G Quant and of Ramsey 8 Gilbert that the OLS estimates are heteroskedastic when 012,...,ofi are unequal is correct. As a result, even though £12,...,£n2 are biased estimators for 01"'°’°n2 under both H0 and H2; under H0 they are generally homoskedastic while under HZ they are generally heteroskedastic. 87 One final difficulty regarding the OLS estimators of oi,...,ofi still exists. Since u1,...,un are not mutually independent, u12,...,un2 are not mutually independent. This lack of independence, it will be recalled, caused certain difficulties in Glejser's and Park's testing procedures. In large part to solve this problem of independence, Ramsey [1969] suggested another estimator. Since the BLUS residuals, ui,...,ufi_k are mutually independent, he suggested that the n—k mutuall independent estimates u*2,...,u* 2 be used to test model y l n-k (2.23). These estimates also have the desirable property of being unbiased under HO. When it is recalled that BI = A'u, where A' is chosen such that A'X = 0, A'A = In—k’ AA' =IM, and that the diagonal elements of ufuf' are ui2,...,ufiz, it follows that DIAG [E(A'u_u_' A)] DIAG [E(ufuf') DIAG (A' 021A) DIAG (o2 A'A) 2 DIAG (o In-k) II A Q N N V G n-k However, under H2, u1*2,...,un*2 are biased estimators of 01"°"°§‘ In fact, given that 2 01 fl E(uu') = '. =‘V, it is found that DIAG [E(u*u*' )1 88 = DIAG [E(A'u u'A)] = DIAG [A'VA], __ .1 .2 ¢_. ._ .1 all 0000000 aln 01 all. 0 Oa1,n-k = DIAG Z I ' I Z . . 2 . . hEl,n-k 'an-k,n LE_ on] kiln 'an n-kJ .. 2 _] _. ._ 6'11 O1 aln On a11°'°a1 n-k = DIAG 3 I ° a 02 a o2 a a __n-k l 1' n-k n n4 __1n n n-k] ’ 2 2 2 2 “‘ all 01 + ...... + a1n on (2) = DIAG I = (A )' DIAG[V] 52 02 + + éZ O2 n-k l l "' n-k n nJ if.A(2) is defined as {aij2}‘ Hence, under Hz, the squared BLUS residuals are a weighted sum of the true unobserved variances o A'A = In_k. weights is dominant. 2 2 1,...,On. . . . . * 2 . which are assoc1ated W1th each squared re51dua1 ui , sum to one Since estimates any one of the variances oi,...,oi. it is conceivable for the squared BLUS residuals to be homoskedastic. . 2 The weights ai1 ,. a 2 "’ in ’ HOwever, unlike the squared OLS residuals, none of these Hence, no squared BLUS residual actually Therefore, under H2, Ramsey G Gilbert's observation that the BAMSET procedure used with OLS residuals was more powerful against H2 than was the same procedure using BLUS residuals can now be explained. Since the squared BLUS residuals, under H2, are each an apparently equally weighted sum of the true variances 01"' heteroskedasticity is masked. 2 01,000,0n 2 .,on, the extent of the Consequently, OLS estimates of are, under H2, more heteroskedastic than are the BLUS 89 estimates of 01""’O§' Therefore, the BAMSET procedure can more easily detect heteroskedasticity when OLS estimates are used. Although the OLS estimates of oi,...,ofi are less biased, under H2, than are the BLUS estimates, they are still biased. To offer a solution for this problem, Rao [1970] and Chew [1970] independently develOped Minimum Norm Quadratic Estimators (MINQUE). Given that Mfiz) is defined as {mij2}’ both Rao and Chew suggested that when M(2) is non-singular, the MINQUE estimator 32 can be defined. The vector of estimatorsé2 is defined as 0W2)—'DIAG(§u'), where E;is the vector of OLS residuals. These estimators are unbiased under H0 and Hz. This will first be shown under H2. As stated before (2.24), under H2, E(uu') = V = : '. and E[DIAG(1A_1u')] = MZ DIAG(u_ 3'). Hence, one obtains E(Qz) = (uh-momma (1')] = (NZ)— (MZ) E(DIAG (g on 02 ~ 1 = D1AG(E(u u')) = (z ) '2 O n The estimator E? can similarly be shown to be unbiased under H0 by 7 4 just replacing oi,...,o§ by the constant variance 0 . .Although this procedure offers unbiased estimates of oi,...,oi under H0 and H2, it does have three drawbacks. First, the n 2 n are not independent. This is obvious since n estimates 3%,...,o estimates are Obtained by using a linear transformation of the OLS residuals which have a rank of only (n-k). Second, though the 90 MINQU estimators are unbiased under HO and H2, they also may be nega- tive. To solve this problem, Rao 8 Subrahaniam [1971] have suggested that when &i is negative, either a small number or a different estimate of of be used in place of Si. Although this is a solution to the problem of negative estimates, the resulting estimates are now neither unbiased (under H0 or Hz) nor MINQU. Hence, this investigator feels that the cost of correcting the negative MINQU estimates is greater than the cost of leaving the estimates negative. The third problem with this procedure is that Mn) is not always non-singular. Mallela [1972] has, however, found a necessary and sufficient condition for the matrix MCZ) to be non-singular. The last set of estimates of the variances OI" "’Orzr are obtained from studentized residuals. Define the i'th studentized residual is ui = xii/fin; , where ui is the i'th OLS residual Obtained after estimation of model (2.23) and where mii(#0) is again the i'th diagonal element of the matrix M. The studentized estimator of the variance of the i'th disturbance term of, a: is defined as .2 . 2 oi = ui ; this estimate is unbiased under HO. When mi is defined as the i'th column of the matrix M and it is recalled that mimi = mii’ because M is idempotent, one obtains -2 _ .2 E(oi) - E(ui) = Hui/”'11) _ 1 " ‘ 1?.“ £011) 11 = —-1—- E(m'. u u' m.) 111.. -1 —— —1 11 = —l— m'. o I m 111.. —l l 11 _ 2 1 . _ 2 "'11 2 — 0 I'm— —1 Ill-i - 0 n7:- 0 91 .Also, since the i'th studentized estimate can be written in quadratic fOrm.in the normally distributed disturbance terms 2, N BIC> - N _ 1 ' 1 — U ml m. —- —1 —1 m.. a“; Q- 11 11 u 1— 6% is distributed as x2 with one (=trace Qi) degree of freedom. HOwever, there are two problems with the studentized estimates of oi,...,o§. First, the n estimates oi,...,ofi are not distributed independently. Since 6% can be expressed as a quadratic form in the normally distributed disturbance terms, 82 is independently distributed of o§(ifj) if and only if the products of the two 2 quadratics are identically zero. Hence, if Qin # 0, oi is not independently distributed of o?’ = _L . .1. 1 = ——-le——-m! m. m. m l Inii ”31 -e1 —1 —j -J mo. = fi‘:%%"ga 93 f 0' ii jj 3 Therefore, a; and o; (i#j) are not independently distributed of one another. Second, just as with the OLS estimator, the studentized estimates are biased under H2. However, also, just as with the OLS estimates, the weighting scheme is such that the expected value of the i'th studentized estimate is 2 2 2 m. m.. m. Bo?) =—1-1— o§+ ”35—1 +£1.11 .3. (2.27) 1 mii ii ii The weights are: 2 2 2 "3.1.1 “in {“12 m. a ... ,m , ... ,m i1 ii ii 92 Since, as before, the sum of these weights, which are all positive, is one, and the i'th weight m1? = mii has an expected value of EfiE-, 11 the i'th weight dominates the series. Thus, if the variances oi,...,o: are unequal (heteroskedastic), the estimates oi,...,o§ will be unequal, though not unbiased. Four estimates of the variances of the n disturbance terms have been suggested. They all have some disadvantages. The MINQU estimates are the only ones unbiased under both Ho and H2. The OLS and studentized estimates are similar to one another, except that the studentized estimates are unbiased under HO whereas the OLS estimates are only homoskedastic. The BLUS estimates are the only ones that are mutually independent. This investigator has decided to use either studentized or 'MINQU estimates because they are both unbiased under H0. BLUS 2 n were not chosen, though they are unbiased estimates of 01""’0 under H0, because of the bias that they contain under H2. Finally, OLS estimates were not chosen because they have no apparent advantage over studentized estimates. 111.3.2 The POSEX Model .As has been previously mentioned, either some assumption or E.EIiQ£1 knowledge about the heteroskedastic error terms is necessary 2 ,on to be estimated. If estimation were for all the variances 01”" to be attempted without such knowledge or assumptions, the estimation process would break down. The researcher would be attempting to estimate n + k parameters (81,..., Bk’ oi,...,o§) with only n Observations. Obviously, if one has a choice, it is more desirable 93 to incorporate a_prigri_knowledge about the variances 01"'°’O: than make assumptions that may or may not be correct. However, it is not unusual for a researcher to be confronted with a model that is suspected of being heteroskedastic although no knowledge is available as to which variable is causing the heteroskedasticity. In this case, some assumption is necessary if estimation is to be made. Present methods of estimation require that one make an assump- tion about the variable(s) that are causing the disturbances to be heteroskedastic and about the functional form that these variables take on. In contrast, if a POSEX model can be used, many of these assumptions can be drOpped since a POSEX model estimates any analytic function in a known set of variables. Hence, in using a POSEX model, the only assumption necessary, if no knowledge exists, is that the heteroskedastic disturbances be an analytic function of the independent variables specified in the model. In develOping a test based on the POSEX model, this assumption is less restrictive than any constructive or non-constructive (with one exception) test now being used. Of course, if knowledge does exist, the POSEX model should be changed to reflect that knowledge. This process will be examined later in this section. In order to develOp a POSEX model without much.a_priggi_ information, one must assume that the variances are some analytic function of the independent variables from the hypothesized model (2.23), 51""’§k' The variance of the i'th disturbance term.can be written as 2 _ 2 ._ E(ui) - o f(xi1,...,xik), 1—1,...,n. (2.28) 94 When this assumption is used, the POSEX model must approximate the analytic function f(°) in a four term power series expansion. The equation formulated would be 2 _ 2 ; E(E)‘O f(51,...,£()-811+812£2+ ooo+81k_x_k+ “2 9(2) + “3 9(3) + 0‘4 9(4) , (2.29) where g_denotes either the instrument y_(OLS predictor of y_obtained after estimating model (2.23)) of p_(the first principal component of the matrix X). Finally, when the expected value Operator is removed and either the studentized estimates éi,...,&§ di,...,Ofi are used as the instrument for the unobserved variances 01""’°§’ equation (2.29) becomes or the MINQU estimates oz_ “1 ‘ (811 + 812 X12 i=1,...,n, 2 3 4 * B1k Xik * “2 “i I “3 “i + “4 qi)wi’ + .0. 02 . 02 ..2 where g_ denotes e1ther the vector g_ or the vector g_, and where W1,...,W' are identically distributed as x2 with one degree of n freedom under H0. Note that the disturbance term is not added onto equation (2.29) but is multiplied by the model. Recalling that under °2 0 H0, o1,...,oi are each distributed as scaled x2 with one degree of freedom, model (2.29) is used to estimate the scale factors. Under HO that E(uiz) = o2 for i=1,...,n, only 811 will be significantly different from zero. The null hypothesis of homoskedasticity will be accepted if 812 ‘ whereas if any of the estimates are statistically different from zero, then H0 will be rejected. 95 Under Hz that E(uiz) = OIf(Xil"°"Xik)’ for i=1,...,n, the coefficients 812,...,81k, o2, a3, and o4 should be jointly different from zero. Of course, the probability that the estimates will be statistically different from zero depends on a number of factors. Three factors have been mentioned previously; they are whether the instrument g_is correlated with the expansion terms it replaces; whether f(-) is analytic; and whether f(-) is approximated by a low order expansion. One other factor which will influence the probability that the coefficients will be statistically equal to zero is hOW'Well the estimators Q? or g? approximate the unobserved . 2 2 var1ances, 01,...,on. 11.3.3 Estimation of the POSEX Model and Testing for Heteroskedasticity» The conditional variance of the i'th disturbance term is given by the POSEX model 02- 2 3 4 “i ’ (511 + 812 X12 + + 81k xik I “2 “i " “3 “i + “4 qi) “’1 i = 1,...,n, (2.30) where wi, i=1,...,n, are identically distributed as x2 with one degree of freedom under H0. The parameters 811, 812,..., Blk’ o2, as and o4 must now be estimated to determine if heteroskedasticity of the fOrm.hypothesized, is present. Maximum Likelihood Estimation The first estimation procedure to suggest itself is maximum likelihood. HOwever, since the disturbance terms, wi, i=1,...,n, are identically distributed as x2 with one degree of freedom, under Ho, this procedure breaks down. The reason for this is that a x2 96 distribution with one degree of freedom is an unbounded function; thus, no maximum exists. Estimation.Using Ordinary Least Squares The second method to suggest itself is the method of least squares. Denoting the estimates of the parameters as 811,..., Blk’ o2, as and o4, one finds that under H0, A _ 2 E (811) "' C E(Blz) = -°° = E(Blk) = E(az) = E(a3) = E(d4) = 09 whereas under Hz, the expected value of the estimates of 812”°”Bik o2, as, and o4 are jointly non-zero. The E(811) under Hz depends on whether heteroskedasticity is mixed, E(R #0, or whether the 11) heteroskedasticity is pure, E(811)=0. Because the hypothesis space is divided depending on whether 812"°"Blk’ o2, o3, and o4 are different from zero or not, an F test for the included variables 52""’§k’ 9?, 3;, and g? is suggested. HOwever, two difficulties exist with the suggested F test. The first difficulty is that the dependent variables 3i,...,8§ are each distributed as o2 x2 with one degree of freedom under H0. Hence, the statistic calculated by using the F test procedure is a ratio of quadratic forms in.ng§;normally distributed variables. Therefore, the statistic is not distributed as F. Research carried out by Donaldson [1968], however, indicates that an F distribution appears to be robust against nonenormality. He discovered, by using a sampling experiment, that statistics which are a ratio of quadratics in.variables distributed as either log normal or exponential (Pearson type III distributions) are approximately distributed as an 97 F distribution. This finding was true for sample sizes greater than 4. Of course, the approximation became less and less accurate the farther out on the tails the comparison was made. Since a distribu- tion.with one degree of freedom is a Pearson type III distribution, it would not be surprising if the statistic calculated using the F test procedure were approximately distributed as F. Second, since neither dependent variable (MINQU estimates or studentized estimates of 01""’O§) is composed of elements that are mutually independent, the disturbance termS'w1,...,wh are not mutually independent. Once again, Donaldson's findings can cast some light on the problem- He discovered that non-independence between the numerator and denominator of his quadratic forms helped to explain why the statistics, which he calculated using variables distributed other than normal, were distributed as F. To apply Donaldsom's findings to the current situation, it should be noted that the O O 11 non- independent estimates , oi, . . . ’Orzr can theoretically be expressed as n-k independent estimates by some linear transformation of the n estimates. Denoting this transtrmation by the (n-k) x n + + matrix B, the (n-k) independent estimates 01"'°’°§-k are defined as E? = B 3?. Using this formulation, the statistic calculated by using the F test process can be expressed as a quadratic in.n:k independently distributed variables 3i,...,§§_k. However, when so expressed, the quadratic forms are no longer independent. Hence, the findings of Donaldsom are now applicable. Given those findings, the lack of independence between‘w1,...,wh might enhance the robustness of the statistic, defined by the F test procedure, to non-normality. 98 Therefore, even though unbiased estimates of the parameters in model (2.31) can be obtained using ordinary least squares (regardless of the fact that the disturbance terms are distributed asymmetrically), the normal tests of significance break down. However, given the findings of Donaldson, the statistics calculated might still be distributed approximately as F. Indirect Maximum Likelihood Estimation The final estimation procedure suggested circumvents the problem that the disturbance terms, w1,. . . ,wn's, are not mutually independent. This is accomplished by formulating a model which uses both the k parameters 81,... ’8k and k + 3 parameters, Bll’°‘°’Blk’ (12, OS and (14. The model to be estimated is: y. X.2 X. . :1: 81 %._+ 82 +_1_.—+ + Bkrl—Z—i- ui, i=1,...,n, (2.31) o. o. o. o. 1 1 1 1 where ul, . . . ,un are independently and identically distributed N(0, oZI), under H0, and where 0+ _ 2 3 7F i ‘,/811 1 B12 X12 1 °°' 1 81k xik+ “2 “1 1 “3 “i 1 “4 “i ' To estimate this model, a maximum likelihood procechire must be used. The maximum of the likelihood function L, under H2, is defined as L2, L = II + 6-1/2 11.2 2 ._ V211 o. 1 1-1 1 where 31 is as defined above and ui is obtained from model (2.31) . The estimates of the parameters that maximize L2 will be denoted as 81,--°:Bk9 811,---,81k, oz, 03, and 0.4. Under H0, 99 81k = dz = as = o4 = 0 whereas under H2, 812,...,81k, o2, o3 and a4 are jointly non—zero. To test the hypothesis of H0 vs. Hz, a likelihood ratio statistic can be used. Under HO that the variances are homoskedastic, the maximum of the likelihood function L is defined as L 1 2 _ n 1 Vi ‘ B1 1 82 X12’°'°’Bk xik) L0 1 i=1 VII oz exP "'2 2 O 03 If one then defines the likelihood ratio statistic i as ' 2:2, 2 it follows that -2 loge 2 is distributed as x2 with k + 2 degrees of freedom. This testing procedure is basically the one suggested by Rutemuller G Bower. However, a POSEX model is used to explain the variance rather than a model composed of the independent variables x1,...,xk. This last estimation and testing procedure does not contain any of the problems which were associated with the previous two procedures. However, this estimation procedure is more easily implemented in theory than in.practice. 111.3.4 Further Observations on the POSEX Procedure POSEX Mbdel and a Reestimation Procedure If it is found that the estimate of the parameters 812,...,81k, o2, a3 and o4 in the model (2.30) are statistically different from zero, H0 is rejected. Since model (2.30) estimates the conditional 2 mean of 01""’°§’ n estimates of 01""’°n can be Obtained. The 100 estimates, denoted as 01"°°’°§ can be used to reestimate model (2.30) and thereby increase the efficiency of the estimates of the regression parameters. TWO methods of reestimation exist. The first method is to transform model (2.23) into the model y. X. x. 01 01 Oi Ci 1 where ui, i=1,...,n are assumed to be independently and identically distributed N(0, oZI). Ordinary least squares can be used to reestimate model (2.32). The second method is to use Aiken's Method of Generalized Least Squares. Using this method and denoting (’12 '- 61. Q Q: a '3 , the new estimator for B, §= ()0 9—10-10 :2— X- These two methods will yield identical estimates of the parameters 81,...,Bk and Oz. . HOwever, it should be pointed out that since 8i,...,8§ are estimates, it is possible for them to be negative. If this is the case, model (2.32) cannot be used to reestimate model (2.23) unless the negative estimate is removed. This investigator suggests using the absolute value of the estimate when the estimate is negative; when this is done, the magnitude of the estimate variance remains the same and the square root can be taken. However, when this is done the estimates 81,...,8k and o2 will no longer be identical to those Obtained by the method of Generalized Least Squares. 101 Incorporating a priori Information into the POSEX Medel Depending on the a_p§igri_knowledge about the heteroskedastic disturbances, the POSEX model (2.30) can be varied in many ways. Three different types of a_priori_information will be presented here. First, a set of m(:_l) variables El"°°’5m is thought to be included with the variables xl,...,x in the unknown analytic function, f(-). Second, only a set of m(:_l) variables 21,...,zm is thought to be in the unknown function, f(-). Third, only a set of m(:_l) variables 31,...,zm is known to be causing the hetero- skedastic disturbances in some known way. In the first case, the POSEX model could be formulated to include the variables 31"'°’Em' This would change model (2.30) to O2 “1 1 (“11 1 B12 X12 1 °°° 1 81k xlk 1 61 211 1 °" 1 2 3 4 5m_zim.+ o2 qi + as q1 + o4 qi) wi (2.33) where w1,...,wh are each distributed as x2 with one degree of freedom under H0, and where the instrument qi is a linear combination of 51,...,zm as well as xl,...,xk. In the second case, the POSEX model could be formulated to include the variables zil""’zim.bUt not to include xi1""’xik' Hence, the POSEX model would be 02 _ 2 3 4 oi — (zil 61 + ... + zim.6im.+ o2 qi + as qi + o4 qi) w: (2.34) 1 where'w1,...,wh are identically distributed as x2 with one degree of freedom under H0, and where qi is a linear combination of the variables Zil”"’zim and not the variables Xil""’xik' In the third case, a POSEX model will not be formed since the exact functional form involving the variables El’°°"5m.is known. 102 To illustrate this, it will be assumed that the function involving the variables El”°"5m is a quadratic of the second degree. The model to be examined would be: 02 2 01 = (611 zi1 + ... + 6 z. + 8 z. + ... + 6 ) w: 2 m 1m 21 11 m 2im 1 (2.35) where wl,...,wh are identically distributed as x2 with one degree of freedom under H0. If m,is large, the squared terms could be replaced.by an instrument; however, if the knowledge embodied in model (2.35) is correct, introducing the instrument will reduce the probability that the model will be able to estimate the heteroskedastic disturbances. Similarities Between the POSEX Procedure and Other Constructive Testing Procedures Using the POSEX model building technique presented in this study, any of the current constructive testing procedures can be deduced. To illustrate this contention, assume that it is known that a single variable 55 in the form of a second degree quadratic is causing the heteroskedasticity. Using this information, Glejser's model can be obtained. If the model is estimated using OLS and either a t or F test issued to test if the coefficients are statistically significant from zero, Glejser's testing procedure has been Obtained. Similarly, Rutemuller G Bower's model and Park's model can be deduced using the concept of a POSEX model, a_p£igri information, and the different estimation procedures suggested. Thus, using the POSEX formulation and a_priori knowledge 103 as to the heteroskedastic disturbances, one can deduce all of the constructive testing procedures. 11.3.5 Summary In this section, a POSEX model was suggested to explain the variance of the disturbance terms when heteroskedasticity is presumed present. It was required only that the unobserved variances oi,...,ofi be a fUnction of the independent variables from the hypothesized model. Since the variances OI’°"’°: are unobserved, four different estimators of the variances were discussed. It was shown that although squared OLS estimates are biased, they are, nevertheless, homoskedastic under very non-restrictive conditions. Two estimators were then chosen to estimate the unobserved variances. Two possible ways in which to estimate the POSEX model were suggested. .A testing procedure for distinguishing between H0 and H2 was associated with each of these estimation procedures. Finally, some extensions of the POSEX procedure were suggested. CHAPTER III HYPOTHESES AND EXPERIMENTAL DESIGN ,A large number of hypotheses have been made in the two previous chapters of this study. Unfortunately, since there are an infinite number of different models that can be specified, none Of these hypotheses can, in general, be proven correct. Rather, each hypothesis must be carefully examined using a very carefully selected subset of model specifications. If an hypothesis is not refuted in the models chosen, it will then be assumed that it can be generalized as being valid for other similarly specified models. However, as the new models differ more and more from the models that were chosen for examination, the probability that the generalization will be invalid increases. In contrast, it should be noted that if an hypothesis is shown invalid for the models specified, the hypothesis has been proven invalid in general. Another difficulty still remains in testing the hypotheses made in this study. Since all of the hypotheses concern various test statistics, a method must be found whereby the probability of type I and II errors can be determined for each test. HOwever, since the distribution of'most of the test statistics discussed in this study is not known, a sampling eXperiment, similar to others that have been discussed, will be used to analyze the various statistics. 104 105 This chapter will be divided into two sections. First, all of the hypotheses made in this study will be restated and briefly explained. Second, a sampling experiment will be presented that examines various tests for a misspecified.mean and heteroskedastic disturbance term. 111.1 Hypotheses It is most convenient to divide these hypotheses into two groups. The first group contains hypotheses that are applicable to tests for a misspecified conditional mean. The second grOUp comprises those hypotheses that are applicable to tests for heteroskedastic disturbance terms. 111.1.1 IMisspecified Conditional Mean Five broad hypotheses are made in this study regarding tests designed to determine if a.model has a misspecified conditional mean. Since the reasoning behind each hypothesis has been previously given, each of the five hypotheses will only be stated in this section of the study. 1. Under the null hypothesis, Ramsey's test, Ramsey G Schmidt's test, and the prOposed test will each have a probability of type I error equal to the alpha level at which each test is conducted. 2. When a variable is omitted from the hypothesized model, the probability that Ramsey's test, Ramsey & Schmidt's test and the prOposed test will each correctly reject H0*will increase as the correlation between the omitted variable and the included variables increases. .At some point, however, this trend will reverse itself and as the correlation increases past this point, the probability of correctly rejecting HO will decrease. 106 When the wrong fUnctional form of either the regressors or regressand is used, the power of all three tests will be an increasing fUnction of two factors. The first factor is whether the correct functional form is analytic. The second factor, which only becomes important if the function is analytic, is the accuracy with which a Taylor expansion in four terms can approx- imate the correct function. Under the alternative hypothesis that the conditional mean of the vector y_is misspecified, the power of all three testing procedures 'will be an increasing function of the number of sample Observations (n). Under the alternative hypotheses of a.misspecified conditional mean, the power of the proposed test will be greater than the power of either Ramsey's test or Ramsey G Schmidt's test. 111.1.2 Heteroskedastic Disturbance Terms Ten broad hypotheses are made in this study regarding tests designed to determine if an hypothesized model is heteroskedastic. Since, as before, the reasoning behind each hypothesis has been previously given, each of the ten hypotheses will only be stated in this section of the study. 1. under H0 of homoskedasticity, the only tests that will have a probability of type I error equal to the alpha level will be those tests that define a statistic whose exact distributional form is known. However, all other tests will have a probability of type 1 error approximately equal to the alpha level at which those tests are examined. Furthermore, that approximation will become increasingly accurate as the alpha level increases. 107 The probability of any test's correctly rejecting H0 will be an increasing fUnction of the amount of correct a_p:ipgi_information available. The power of all the tests fOr heteroskedasticity will increase as the number of sample observations (n) increases. The power of the various tests for heteroskedasticity will, in general, be independent of the distributional form of the variable causing the disturbances to be heteroskedastic. The tests for heteroskedasticity will not display an increased probability Of type I error when the independent variables are not drawn from a fixed distribution even though this choice Of independent variables insures that the diagonal elements of the matrix M are not equal. The power of the POSEX.model and testing procedures to determine if a.model is heteroskedastic will be a decreasing function of the number of terms needed by a Taylor series expansion to approximate the functional form (taken by the disturbance terms) to some level of accuracy. The power of the POSEX model and testing procedures will be, in general, increased if the instrument p (first principal component of the matrix X) is used for the expansion terms versus the instrument y (the OLS predictor of y). The power of the POSEX model and testing procedures will, in general, be increased when E? GMINQU estimates of g?) is used as the predictor of E? versus when é? (studentized estimates of g?) is used. 108 9. The BAMSET tests with OLS residuals will have a higher probability of correctly rejecting H0 than the same tests using BLUS residuals. 10. If the same amount of a_pgiggi_information is incorporated into all of the testing procedures, the POSEX model and tests will have the highest probability of correctly rejecting H0. 111.2 Sampling Experiment In order to test these hypotheses, the probability of type 1 and type II errors must be calculated under various model specifications. Because the finite distribution of most of the test statistics is not known, these probabilities are most easily calculated by using a sampling experiment. Hence, in the first part of this section, a general sampling experiment will be outlined. Each of the following two parts of this section will, in turn, be concerned with using this experiment to examine various alternative tests for either a.misspeci- fied conditional mean of the vector y_or heteroskedastic disturbance terms. Both of these parts will have the same format. First, each of the alternative hypotheses to be generated will be discussed with their relationships to one another expressly pointed out. Next, the various tests to be examined under the null and various alternative hypotheses 'will be selected with special attention paid to justifying this selection. .After these two parts, a final summary of the experiment and of all the models that are examined will be given. 111.2.1 General Design of the Sampling Experiment In conducting this experiment, the basic procedure will be to test if a model that is hypothesized to estimate the conditional mean of a sample of variables, y1,...,yh, is misspecified because either the 109 conditional mean has been misspecified or the disturbance terms are heteroskedastic. This procedure will be repeated on 1000 independently drawn samples of first 30, then 60, and finally 90 dependent variables, y1,..., n’ (n = 30, 60 or 90). The percentage of times that each specification error test rejects H0, that the hypothesized model is correctly specified, will then be recorded for nominal alpha levels of .01, .05, and .10. The first four sample moments of each test statistic will also be calculated. In this way, by defining 17 different pOpulations of dependent variables from which the 1000 samples of y1,..., n’ (n = 30, 60 or 90) are chosen, the testing procedures under examination can be compared. Each of the seventeen pOpulations is defined by specifying the conditional mean of the dependent variable and by adding on a disturbance term that has a.mean of zero and a.specified variance. These population definitions will be referred to as the 'true' models. Sixteen of these 'true' models are specified differently than is the hypothesized model. Hence, it can be observed how the power of the various testing procedures varies under different specification errors. These sixteen models will be eXplained and examined, in turn, later in this section of the study. At this point, only the first 'true' model will be examined. It is yi = 50 + 5xil + 5x12 + ”1’ i = 1,...,n (3.1) where u1,...,u.n are independently and identically distributed as N(0, 2500). The variables x11,...,xn1, are independently drawn from a uniform distribution with end points of 0 and 100 (mean of 50 and pOpulation variance of 833.33). In contrast, variables, x12,...,xn2, 110 are independently drawn from a log normal distribution with a mean of 20.327 and pOpulation variance of 413.197. The population.parameters of the second variable guarantee that the Pr (0 f-Xiz for i = 1,...,n. Hence, with a probability of .99, both variables §_100) > .99 cover the range, 0 to 100. Since, however, the two variables come from two different independent pOpulations, they are independent of one another. One drawing Of 90 Observations was made for each Of the two variables. These Observations are divided into 3 groups of 30 observations each. Hence, when n = 30, the first group will be used; when n = 60, the first and second are used; and when n = 90, all three are used. .All 90 of these observations together with various sample statistics for either n = 30, or 60, or 90 are given in Appendix B. The conditional mean of the dependent variables Obtained from all seventeen 'true' models will be estimated using the hypothesized.model yi = 80 + 61x11 + 82xi2 + V1’ 1 = 1,...,n. (3.2) where v ..,vn are assumed to be independently and identically 1,. distributed as N(0, o2). Hence, when the 'true' model is model (3.1), the hypothesized model will be a correctly specified.model. In this way, the probability of type I error can be calculated fOr each of the tests examined. Similarly, since each of the specification error tests is also used when the hypothesized model (3.2) is misspecified, the probability of type 11 error can be calculated. Only one difference exists in the basic procedure just outlined when any of the sixteen remaining 'true' models are used. When the 'true' model has a conditional mean other than that specified by the hypothesized model, only tests for a misspecified conditional mean 111 ‘will be examined. Likewise, when the 'true' model generates dependent variables that are heteroskedastic, only tests for heteroskedasticity will be examined. .Although in using this procedure, the interrelation between the various specification errors is not brought out (a study by Ramsey 8 Gilbert, 1972, does make this comparison), this procedure was necessary to save computer time and money. Finally, in order to simplify the discussion of the sixteen remaining models, three new variables will be defined. They will be denoted by the vectors x5, x4, and x5, respectively. The variables x13,...,xn3 will be drawn from a normal pOpulation with a mean of 50 and a variance of 400. These pOpulation parameters ensure with a probability of .99 that X13""’Xn3 will lie in the range Of 0 to 100. Once the sampling is made, the observations x13,...,xn3 are never redrawn. Since 53 is drawn from a pOpulation independent of the populations from which x1 and x2 are drawn, x3 is independent of both 31 and 12- The second set of variables x14,...,xn4, is a sum of the first three variables. The i'th observation of x4 is defined by xi4 = 5.428 loge xil + 7.71 loge xi2 + 3(xis - 50). 20 (3.3) This variable is defined in such a way as to have a moderate correla- tion with either x1 or x2. Note that x14,...,xn4 W111 also be correlated with various powers of either x11,...,xn1 or x12,...,xn2. The third and final additional variable, X15""’Xn5’ is defined to be highly correlated with either x1 or x2 The i'th Observation of x5, defined in terms of x1, x2, and x3 is = .5428 xi + .771 xi + 3(xis - 50) . 20 X15 1 2 (3.4) 112 The E(xis) is 42.81 and the variance of x1 is 500.148. The population 5 correlation coefficient between x5 and x1, and between 55 and x2 is .70. Also, because of the way x5 is defined, the coefficient of determina- tion obtained by regressing 55 on x1 and x2 is 0.98. A listing of all three variables appears in Appendix B. Also, in Appendix B, corresponding to the sample sizes of 30, 60, and 90 are the sample means, variance covariance matrix and correlation matrix of the var1ables, x1, 52, x3, x4 and x5. 111.2.2 Sampling EXperiment to Examine Tests that Discriminate Between H0 Versus H1 Of the sixteen remaining 'true' models (that is, the models that actually generate the dependent variable) to be used in this experiment, six were generated so that the hypothesized model (3.2) will misspecify the conditional mean of the dependent variable. These six 'true' models are divided into two categories. The first category consists of three models designed so that the hypothesized model (3.2) mistakenly omits a relevant variable. The second category, consisting of the remaining three models, is designed so that either the regressors or the regressand of the hypothesized model has the wrong functional form. These two groups of models will be discussed in turn. variable Omitted from the Hypothesized Model To generate a pOpulation of dependent variables that omits a relevant variable from the hypothesized model (3.2), a model ., i = 1,...,u, (3.5) Y1 1 8o 1 B1 X11 1 B2 X12 1 8321 1'11 where u1,...,un are independently and identically distributed as N(0, o2), is used. In using this 'true' model, both sets of variables 113 x11....,x1n and x12,...,xn2 are defined as before. In each of the three 'true' models that use this basic form, a different set of var1ables z1,...,zn is used. These different variables are the variables x5, x4, and x_ that 5 were previously defined. The three 'true' models will then be yi = 50 + 5xi1 + 5xi2 + 5xl3 + ui, i = 1,...,n, (3.5a) yi = 50 + 5xil + 5x12 + 5x14 + ui, 1 = 1,...,n, and (3.5b) y1 = 50 + 5xi1 + 5x12 + 5xis + ui, i = 1,...,n, (3.5c) where in each model u1,...,un are independently and identically distributed as N(O, 2500). These three models (3.5a), (3.5b), and (3.5c) each generates a dependent variable that causes the hypothesized model to be misspecified because of an omitted variable. However, the omitted variables are related to the included variables in different ways. The first omitted variable is independent of the included variables, the second is moderately correlated with the included variables, and the third is highly correlated with the included variables. Hence, a relation between correlation and the power of the various tests can be Obtained. Incorrect Functional Form of the Hypothesized Model Three models are designed to cause the hypothesized model to be misspecified because an incorrect functional form is used. The models are designed so that the correct functional forms are increasingly difficult to approximate with a four-term Taylor series expansion. The three 'true' models will be defined as: 114 Y1 = exp (2 + .OSXil + .OSXiZ + Zui), i = 1,...,n, (3.68) y1 = 81.0 Xiio Xiéo Gui, i = 1,...,n, and (3.6b) -2 . yi = exp (-(-.25 + .02xi1 - .05xi2 + 'Sui) ), 1=l,...,n, (3.6c) where in each model u1,...,un are independently and identically distributed as N(0, l), and x11,...,xn1 and x12,...,xn2 are as previously defined. Because model (3.6a) is an exponential model, it will be the most accurately approximated (of the three models) by a Taylor series expan- sion. The second model (3.6b) is analytic; however, since it is a multiplicative function, it is less accurately approximated than model (3.6a). Finally, since in the neighborhood of zero the last function is discontinuous,model 7 is a non-analytic function in x11 and x12 and hence cannot be approximated using a Taylor series eXpansion. Note that each of these three models is written such that the hypothesized model has the incorrect functional form of the regressors. HOwever, the first two of these models can be reformulated so that the hypothesized models will have the incorrect functional form Of the regressand. written in this way, the two models become loge yi 2 + .05x.L1 + .05xi2 + Zui, i1= 1,...,n, and (3.6a) loge yi l + xil + xiz + ui, 1 = 1,...,n, (3.6b) where in each case u1,...,un are independently and identically distributed as N(0, 1). Since an incorrect functional form of the regressand can equivalently be eXpressed as an incorrect functional form of the regressors, Only one of the errors need be examined in flfissflfly. 115 The three models (3.6a), (3.6b), and (3.6c) each causes the hypothesized model (3.2) to be misspecified because of an incorrect functional form. However, the functional fOrms are chosen so that they are not approximated equally accurately by a Taylor series expansion in four terms. Hence, a relation can be determined between the power of the various tests and the degree of accuracy by which a Taylor series eXpansion of four terms can approximate the 'true' functional form. The Tests Compared Three testing procedures (Ramsey's test, Ramsey 8 Gilbert's test and Ramsey 8 Schmidt's test) have been used in the literature to determine if the conditional mean of the disturbance terms has been misspecified. The distributions of two of the resulting test statistics are known (Ramsey's statistic and Ramsey G Schmidt's statistic), while the distribution of the third is unknown. Hence, since Ramsey G Gilbert's testing procedure offers no advantage over the other two tests and offers the disadvantage of defining a test statistic that has an unknown distribution, their test will not be examined in this study. Using a POSEX model, two additional tests have been developed. They both determine if the conditional mean of the dependent variable has been.misspecified. The two tests differ, however, in the instrument used to replace the expansion terms in the POSEX model. In one version, the vector p (obtained from the first.principal component of the matrix X) is used as the instrument, while in the other version, the vector y_(the OLS predictor of the dependent variable y) is used. 116 This implies that four tests should be examined in this study. However, it should be recalled that Ramsey's and Ramsey 8 Schmidt's models and tests have been shown to be mathematically equivalent to using a POSEX model and test with the instrument y, Hence, only one of the three tests should be used. Because it is mathematically easier to formulate and calculate the test statistic, the POSEX test, with the instrument 2, has been chosen. Therefore, the hypothesized.model ‘will be tested for a misspecified conditional mean of the dependent variable only by using both POSEX testing procedures. 111.2.3 Sampling Experiment to Examine Tests that Discriminate Between H vs. H 0 2. Ten 'true' models remain to be defined. .All of these models are used to examine the different tests designed to determine if an hypothesized model is heteroskedastic. Hence, each of these 'true' models is designed so that the hypothesized model will be miSSpecified because it was incorrectly assumed to be homoskedastic. Since in each Of the models the heteroskedastic disturbance terms are generated in a different way, a relation can be found between the power of the various tests and the form taken by the heteroskedastic disturbance terms. These ten models can be divided into three groups. In the first group, the heteroskedastic disturbance terms are a simple fUnction Of one variable. In the second group, the disturbance terms are a non- linear function of one variable. Finally, in the third group, the disturbance terms are a function Of a variable whose mean and'variance is conditional on some other variable. ll7 Heteroskedastic Disturbance Terms are a Simple Function of One Variable Six 'true' models are generated which have the disturbance term as a function of a single variable. All of these models are of the form y1 = 50 + 5xil + 5xiz + zi ui, i = 1,...,n, (3.7) where u1,...,u are independently and identically distributed as N (0,1) and x11,...,xn1 and x12,...,xn2 are as previously defined. The variable zi represents one of six variables that will cause the \hypothesized model (3.2) to be heteroskedastic. These six variables will be chosen for their relationship to the hypothesized model. In the first two models, the variables used for zi are xi1 and x12 respectively. Since both of these variables are included in the hypothesized model, the heteroskedastic disturbances (generated by the first two 'true' models) are a function of a variable that the researcher can.identify. However, the variables differ from one another since they are drawn from different distributions. The third and fourth 'true' models are generated when either the variable x13 or the variable xi4 is used as zi. Recall that X13 is drawn from a normal pOpulation that is independent of xi1 or x12, whereas xi4 is generated so that it is partially correlated with x11 and x12. Hence, while the first two models generate heteroskedastic disturbances that are a function of a variable that the researcher can identify, the third and fourth models generate disturbances that are either independent of those known variables or are only partially correlated with them. In the fifth model, zi will be replaced with a function of the index i. The particular fUnction is 19%111.. .Although this particular 118 variable will be (like x13) independent of x11 and x12, it represents the type of variance that increases over time. NOte, however, that normally when the heteroskedasticity is generated by a function of time, the independent variables are also highly correlated with a time index. Since in this case, i is independent of xi1 and x12, this particular form of heteroskedasticity will be a more difficult type to detect than the normal type. Rather, the model generated using xi4 conforms to the more typical occurence of the variance's increas- ing over time since it is partially correlated with xil and.xiz. The sixth and last model of the group replaces zi with E(yi). This form of heteroskedasticity has been suggested by Theil [1951]. The pOpulation correlation coefficient between E(yi) and x1 is .817 1 while between E(yi) and xiz it is .576. All six Of these models correspond to various types of heteroske- dasticity. The first two represent heteroskedastic disturbances caused by a variable included in the hypothesized model. The third represents heteroskedastic disturbances that are generated independently of the model's variables, while the fourth represents disturbances that are partially correlated with those variables. The fifth represents disturbances that are related to some indexing scheme that cannot be identified. Finally, the sixth represents the case where the heteroskedastic disturbances are generated by the dependent variable. Heteroskedastic Disturbances that are a.Non-Linear Function of One variable TWO of the 'true' models to be generated have heteroskedastic disturbances that involve a non-linear function. Both of these models are of the form 119 yi = 50 + 5xi1 + 5xi2 + f(zi)ui’ i = 1,...,n, (3.8) where u1,...,un are independently and identically distributed as N(0 , l) and x11,...,xn1 and x12,...,xn2 have been previously defined. The function and variable f(zi) represent two different non-linear functions in a variable denoted as 21. The two functions, both analytic, can be approximated using a Taylor series expansion with different degrees of accuracy. 1 The first analytic function, f(zi), has been suggested by Goldfeld G Quant [1972]. It is a second degree quadratic in the variable x11. This function can be quite accurately approximated with a Taylor series expansion of just two terms. The function will be 2 1/2 (500 + 10xi1 + xi1 . The function f(zi) to be used in the second model is also an analytic function; hence it can be approximated with a Taylor series expansion. However, this approximation requires more expansion terms in the Taylor series to achieve the same accuracy as is achieved with the first function. The function is 75 + 50 SIN E(yi). Heteroskedastic Disturbances that are a Function of a variable with a Non-Constanthean This last group of 'true' models are quite different from any of the previously defined models. First a new variable, denoted as the vector 56’ must be generated. The i'th Observation of this variable is drawn from a uniform distribution with end points of 0 and l.Si (population mean of .75i and variance of .1878i2). Since each observation is drawn from.a population with a different mean and 120 variance, the vector x6 has a non-constant mean. The sample drawn appears in Appendix B together with various sample statistics. The first model generated with the variable x16,...,xn6 is yi = 50 + 5xi6 + 5X12 + “1’ i = 1,...,n (3.9) where u1,...,u.n are independently and identically distributed as N(0, 2500). In contrast, the other 'true' model in this group is generated with a non-constant variance. It is yi = 50 + 5X16 + 5X12 + X16 ui, i = 1,...,n, (3.10) where u1,...,un are independently and identically distributed as N(0, 1). In testing both of these models, the hypothesized model is yi = 80 + 81x16 + Bzxiz '1' vi, 1 = 1,...,n, (3.11) where v1,...,vh is assumed to be independently and identically distributed as N(0, o2). Hence, model (3.11) is correctly specified when model (3.9) is generated and is incorrectly specified when model (3.10) is generated. Both of these 'true' models represent forms of models previously examined; a homoskedastic model and a heteroskedastic model in an identifiable variable. HOwever, since the variable xi6 is used in both of these models, there is a significant difference between these two models and any of the previously generated.models. In the case of these two models, one of the independent variables is drawn from a population with non-constant mean and variance. Under the null hypothesis this Will cause the diagonal elements of the matrix M to be more unequal than previously and hence cause the OLS residuals to be more heteroskedaStic than previously. However, if the theorem 121 proven in this study is correct, the OLS residuals should still appear to be homoskedastic under HO, and heteroskedastic only under H2. Tests Examined that Discriminate Between Hoand H2 One final decision.must still be made before the experiment is examined; that is, which of the tests fOr heteroskedasticity reviewed and suggested is to be used to test if either of the two hypothesized models (3.2 and 3.11) is misspecified. A total of nine tests are currently being used in the literature to discriminate betweenHO and H2. They have been denoted in this study as GQP, THEIL, RECURSIVE-P, GQN, RECURSIVE-N, BAMSET, PARK, GLEJSER, and FIML. However, all of the tests do not have to be examined if the findings of Harvey 8 Phillips are referred to. It should be recalled that they found that the tests denoted as GQP, THEIL and RECURSIVE-P (recall that these test procedures were identical except for the predictors of CI""’°§ used) had virtually identical power under a large number of alternative hypotheses. Hence, there seems to be no reason to compare the three tests again. Consequently, only the more commonly used test, GQP, will be examined in this experiment. Similarly, since the two testing procedures GQN and RECURSIVE-N are identical except for the prediction of o2 l” are used, only the more widely accepted procedure, GQN will be used 2 ..,on that in this experiment. HOwever, in contrast, since Ramsey 8 Gilbert [1972] found indications that the BAMSET testing procedure can be used even.more successfully with OLS than with BLUS residuals, this procedure will be examined using both sets of residuals. The two tests will be 122 differentiated by suffixing BAMSET with O for OLS residuals and T for BLUS residuals (developed by Theil). Hence, seven of the current testing procedures will be examined in this experiment. In this study, two different testing procedures have been suggested to discriminate betweenHO versus H2. Both of these procedures used a POSEX.model to explain the unobserved variances UI”"’°§' However, because the model could be estimated in two different ways, two different testing procedures were suggested. It should be recalled that when the POSEX model was estimated with.OLS, an F-test was suggested to test H0 versus H2, whereas, when full information maximum likelihood (FIML) was used to estimate the model, a likelihood ratio test was suggested. These two tests bring the number of tests to be examined in this study to nine. When these tests were used in a cursory examination, it was discovered that the theoretically expected results were not being obtained with some of the tests. .All of these tests were formulated with models that were estimated using a.maximum likelihood procedure. Since estimation by maximum likelihood requires an iteration convergence procedure, it was found that the theoretically expected results could only be obtained by increasing the number of iterations. This result 'was not entirely unexpected since Rutemuller & Bowers [1968] fOund they needed 15 iterations to converge using the FIML technique. HOwever, since in this experiment the hypothesized model is examined for heteroskedasticity 33,000 times (11 different populations of dependent variables are estimated with 3 different sample sizes and each is replicated 1000 times - 11 x 3 x 1000), increasing the number iterations needed for each examination becomes very costly. For 123 example, it was found in the preliminary study that the 2 tests that use iterative estimation (FIML, POSEX using FIML.) required 6 times the amormt of computer time than the 6 tests that do not use iterative estimation. Hence, it was decided that neither of the tests which use a maximum likelihood procedure would be examined in this experiment. The information lost by not examining these two tests could prove to be very small. One of the tests that was drOpped from the experiment was based on the POSEX model. However, since one test still remains that is based on the POSEX model, the POSEX procedure can still be carefully examined. The second test that was drOpped is the procedure developed by Rutemuller 8 Bower [1968] and denoted as FIML in this study. Much evidence already exists on this technique. For example, Goldfeld G Quant [1972] discovered that when the correct form of the heterosked- asticity was known, the FIML testing procedure had a higher probability of correctly rejecting Ho than any other test. In contrast, they also found that when the form of the heteroskedasticity was not known, the FIML testing procedure seemed to lose this advantage. Therefore, since the FIML testing technique takes much more computer time than other testing procedures (in the preliminary examination it took 15 times as long as the other tests) yet offers no gain in power when the form Of the heteroskedasticity is not known, it appears as if the test has a comparative disadvantage to other tests when a_ p_r_i_gr_i. information does not exist. Hence, since in this experiment it is assumed that no a m information exists as to the form of the heteroskedasticity, very little information will be lost by dropping the FIML testing technique . 124 Seven tests remain to be examined in this experiment. Since it is assumed that no a_p:ip§i_information exists as to the form of the heteroskedastic disturbances, many versions of the different tests are used. In the four non-constructive tests (BAMSETT, BAMBETO, GQP, and GQN), for example, the Observations can be reordered by a variable that is suspected of causing the heteroskedasticity. Since, however, no information is available, each of the tests will be reordered in turn by using one of the independent variables Of the hypothesized model or by using y_(the OLS predictor of y). In addition to these three versions, each of the tests will also be used without reordering. In this way, four different assumptions as to the form of the hetero- skedasticity are being made. These different tests will be designated by suffixing the test's name with the variable that was used.for reordering or by N for no reordering. Thus GQP becomes GQPXl if the test is reordered by the vector x1; GQPX2 if reordered.by x2; GQPY if reordered by i, and GQPN if no reordering occurs. Likewise, in the two constructive tests (PARK and GLEJSER) that are currently used in the literature, assumptions will also have to be made. Since no information exists as to the form of the hetero- skedasticity, it will be assumed in Park's test (denoted as PARK) that the disturbances are of the form a 2 2 E(ui) = 21 o . Since the variable Z1 is unknown, this variable will be assumed to be, in turn, one of the independent variables of the hypothesized set or yi (the OLS predictor of yi). These three versions will be denoted as PARKXl (PARKX6 when 56 is used instead of xi), PARKXZ, and PARKY. 125 In Glejser's test (denoted as GLEJSER) it will be assumed that the heteroskedastic disturbances are Of the form 2 _ 2 2 2 E(ui) - (80 + 81 21 + 82 21) o . Since once again Z1 is not known, it will be assumed, in turn, that 21 is one of the independent variables in the hypothesized.model or yi (the OLS predictor of yi). Each of these different versions will be denoted as GLEJSERXl (GLEJSERX6 when'x6 is used in the hypothesized model instead of xi), GLEJSERXZ, and GLEJSERY. The estimated coefficients 81 and 82 will be tested for significance using an F-test as was suggested earlier in this study. These assumptions together with the assumption necessary for the non-constructive tests expand these six tests into 22 tests. It is important to note that the 'true' models were designed so that each of the assumptions made in the 22 tests would be exactly correct in at least one instance. In this way it can be determined how the power of each of these tests varies when a correct versus an incorrect assumption is made as to the form of heteroskedasticity. .Although it has been shown how assumptions can be incorporated into a POSEX model, in using the model in this experiment it will only be assumed that the heteroskedastic disturbances are a function of the independent variables in the hypothesized model. Hence, when the model is hypothesized to be a function of x1 and x2 the POSEX model designed to test for heteroskedasticity becomes ,2 (2) (3) (4) 9- 15018115115123‘21129 113-cl 1143 11’ (3.12) 126 where E? is a predictor of oi,...,o§, q_is an.instrument for the expansion terms and y_is assumed to be distributed as N (p, oZI). .A similar model could, of course, be fOrmulated when the hypothesized model is a function of x6 and x2. Since OLS is being used to estimate this model (3.12), an F-test will be used to determine if the para- meters 811, 812, 72, Y3 and Y4 are significantly different from zero. This test will be denoted as POSEXHl to indicate that it is a POSEX model designed to test heteroskedasticity and estimated using OLS (the first of the two estimation procedures earlier suggested). Since it has been suggested that either i (the OLS predictor of y) or p (which uses the first principle component of the matrix X) to be used as the instruments for the expansion terms, each will be used in turn To differentiate between the two instruments, the acronym POSEXHl will be suffixed by'Y if y_is used as the instrument or P if p_is used. Similarly, since it has been suggested that either p? (studentized predictors of OI’°"’°:) or E? (MINQU predictors of CI"°"°§) be used as the dependent variable in model (3.12), each will be used in turn. As before, to differentiate between their use, either an S (studentized) or M (MINQU) will suffix the acronyms POSEXHlP and POSEXHlY. In this way, fOur versions of the POSEX test designed fOr heteroskedasticity and estimated using OLS (POSEXHl) will be examined. They will be denoted as POSEXHflPS - POSEXHl using the instrument p_and studentized predictor, POSEXHlPM - POSEXHl using the instrument p and MINQU predictors, POSEXHIYS - POSEXHU using the instrument i and studentized predictors, 127 POSEXHlYM - POSEXHl using the instrument y and MINQUE predictors . Hence, 26 different versions of the 7 different tests are to be examined in this experiment. Through the use of the different versions Of each test, it will be possible to determine the relationship between the power of each test and the version of each test used. This will be especially enlightening when the different versions of each test are the result of different assumptions as to the form Of the heteroskedasticity. 111.2.4 Summary In this section, a sampling experiment has been designed to examine tests that determine if a model has a.misspecified conditional mean or has heteroskedastic disturbance terms. The basic procedure used in the experiment was then presented: First, a population of dependent variables is defined. Second, a sample consisting of n (set first at 30, then at 60 and finally at 90) Observations, is drawn from this pOpulation. Third, the hypothesized model is estimated with the first sample of n observations. Fourth, specified tests are used to determine if the hypothesized.model is misspecified and the results are recorded. By repeating this process 1000 times, one can determine the percentage of times that a given test indicates that a.model is misspecified. This percentage then corresponds to either the prObabil- ity of type I error (if the hypothesized model is correctly specified) or the power of the test (if the hypothesized model is misspecified). In the second and third parts of this section, 17 different pOpulations of dependent variables are defined. The models that generate each of these populations are summarized in Table 2 below. 128 TABLE 2: MOdels that Generate the Dependent Variable Dependent'Variable l yi = 50 + 5xi1 + 5xi2 + 50ui 2 y1 = 50 + 5xil + 5xi2 + 5xis + 50ui 3 y1 = 50 + 5xi1 + 5xiz + 5xi4 + SOui 4 y1 = 50 + 5xil + 5xi2 + 5xi5 + SOui 5 y1 = exp (2 + .05xi1 + .05xi2 + Zui) _ 1.0 1.0 1.0 u- 6 Y1 ‘ e X11 x12 9 1 7 yi = eXp(—(-.25 + .02xil - .05xiZ + .Sui)-2) 8 yi = 50 + 5xil + 5xiz + xilui 9 y1 = 50 + 5x11 + 5xiz + xizui 10 y1 = 50 + 5xil + 5xiz + Xi3ui 11 yi = 50 + 5xil + 5xi2 + xi4ui 12 y1 = 50 + 5xil + 5xiz + E(yi)ui 13 y1 = 50 + 5xil 4» 5xi2 + 1201 i - 2 l/2 l4 yi - 50 + 5xil + 5xiz + (500 + 10xil + x11) ui 15 y1 = 50'+ 5xil + 5x12 + (75 + 50 Sin(E(yi)))ui 16 y1 = 50 + 5xi6 + 5xi2 + 50ui l7 y1 = 50 + 5xi6 + 5x12 + xiOUi u1,...,un are independently and identically distributed as N(0, l). The variables x1, 52, x3, x4, x5, and;6 are listed in Appendix B together with the relevant sample statistics. 129 The conditional mean of the dependent variables generated by the 17 models will be estimated using two hypothesized models. The first 15 pOpulations of dependent variables will be estimated using the hypothesized model Y1 1 8o 1 B1 X11 1 82 X12 + Vi. i = 1,...,n. (3.13) where v1,...,vn are assumed to be independently and identically distributed as N (0, oz). The conditional mean of the remaining two populations of dependent variables will be estimated using the hypothesized model Y1 = 80 + 81 X16 + 82 X12 + vi, 1 = 1,...,n (3.14) where v1,...,vh are again assumed to be independently and identically distributed as N (0, oz). Each Of these hypothesized models will be tested to see if it is misspecified. The hypothesized model used for the first 7 populations of dependent variables will be tested for a misspecified conditional mean. The first population should prove to be the only set of dependent variables that is correctly specified. Two tests will be used to determine this. The first is denoted as POSEXMP (POSEX test for a misspecified conditional mean using the vector p_as the instrument) and the second as POSEXMY (same test as before except that the vector y_is used as the instrument). The hypothesized models used for the remaining 10 pOpulations of dependent variables, together with the population defined by model 1, will be tested for heteroskedastic disturbance terms. The first and fifteenth.pOpu1ations should prove to be the only sets of dependent variables that are correctly specified. This will be done by using the 26 different tests which are listed in Table C1 (the first table in Appendix C). CHAPTER IV RESULTS OF SAMPLING EXPERIMENT AND OBSERVATIONS ON THE MAINTAINED HYPOTHESES In this chapter, the results of the sampling experiment outlined in the last chapter will be given. These results consist of reporting the estimated parameters of the hypothesized model, examining the percentage of times the various tests reject the null hypothesis (power), comparing and contrasting the experimental results between models in the same group, and commenting on the hypotheses stated in the previous chapter. To facilitate the discussion of these results, the two hypo- thesized.models will be restated and the groupings of the six 'true' models reviewed. For models 1 through 15, the hypothesized model is yi= 80+ 81 Xii+ 82x12+vi’ 1: 1,...,n, (4.1) while for models 16 and 17, the hypothesized model is y1 = 80 + 86 x16 + 82 x12 + vi, 1 = 1,...,n. (4.2) In both cases, it is assumed that vi, i=1, . . . ,n,. are independently and identically distributed as N (9, oz) and that n is equal to first 30, next 60, then 90. The 'true' models were divided into six groups for convenience. They were (1) a model that corresponded to the hypothesized model (4.1), (2) models that included a variable not in model (4.1), (3) models that had a different functional form.than 130 131 (4.1), (4) models that were heteroskedastic due to a simple function of one variable, (5) models that were heteroskedastic due to a non- linear function of one variable, and (6) models that included a variable which had a conditional mean. Each of these six groups of models will be discussed in one of the three sections of this chapter. The first section will consider the correctly specified model (model 1); the second, the models with a misspecified conditional mean; and the third, models that are heteroskedastic. In analyzing the results of the experiment on each group of models, the estimates of the parameters of the hypothesized model are given first. Following this are the results of the specification error tests applied to each of the models within the group and a summary of these results. So as to avoid needless repetition, some standardized notation will be introduced at this time. 1000 estimates of the parameters 80, 81 (86), 82 and oz are obtained for each of the seventeen models. For each of these models, the arithmetic average of the estimates of each of the four parameters is denoted as Eb, Ei (E6),'§é and 32. The variance of each of the estimates of 80’ 81 (B6) and 02 is denoted by V(BO),'V(81), CV(B6)),'V(BZ). These variances are calculated using the standard algorithm, A 1000 ’ _ 2 We) = .2 (Bi - a) /999. 1=l .Also, since the hypothesized model is estimated using OLS, an estimate of the variance of $0, El (E6) and a2 is obtained for each of the 1000 times the model is estimated. The average of each of these estimated variances is denoted as 32 (go), 32 (El) (32(R6)) and 32 (£2) respectively. In addition, and F statistic is calculated to determine 132 if the hypothesized model explains the conditional mean of n yi, i=1,...,n better than does the sample mean §'= Z yi/n. The i=1 average of these F statistics for each of the seventeen.models is denoted as F2 IV.l Hypothesized Model is Correctly Specified The first group of models consists only of one model. In this case, the hypothesized model is correctly specified. The 'true' model is yi = 50.0 + 5.0xi1 + 5.0xi2 + SOui, i=1,...,n, (4.3) where u1,...,un are distributed independently and identically as N(0, l). The estimates of the parameters of the hypothesized.model are shown in Table 3 for samples of 30, 60, and 90. It is evident from this table that the estimated parameters become increasingly accurate as the sample size increases from 30 to 90. This is especially true for the estimate of the variance of the disturbance term. It should also be noted that the estimated variance of each of the parameters (32(Bi)) decreases as sample size increases, that is, there is a gain in efficiency with increasing sample size. This gain in efficiency is also clear from the fact that the sample variance of each parameter (V(Bi)) decreases as the sample size increases. Finally, it should be stressed that the estimated variance of the estimates of each of the parameters is extremely close to the sample variance of each of the parameters. In addition, with only one exception (sample size 90, parameter 80), the difference between the estimated variance and sample variance becomes smaller as the sample size increases. 133 www.mmm ovv.vmvm wom.mwv NNH.vva oom.NmH omw.movm m .Nm owmmo. OVHmo. ommm.mNH Hmvno. mammo. mmvm.me vmmm. NNQH. mnao.avo mflmv > mHomo. onmmo. mvom.HMH omomo. mmomo. mom©.mmfi owun. mwomo. wvmm.hmc mHMU mm mmm.v mmm.¢ NoH.om coo.m omm.¢ nvm.mv ovo.m omm.v www.mv Hm N H o N H o m H o mpouoememm om oo om oNHm oHdEmm Homo: onwHoomm xHuooheou m :H whopoewpmm wo momeHumm ”m mHm ,_0 Third Symbol a = 0.10 0—7 ~c: o —X X X -—fl: an :n __G A 0 -—x x“ x -nei a: van --0* <> .HHo -—x x x -fl} a: ——an _n“ A ‘7“ A 1’“ G A_ -x x x HF— m fil A Schematic Diagram of Test Results for Model 1 136 each of the 3 percentage levels being the same). The results shown in Figure 1 can now be analyzed. Since the hypothesized model is correctly specified, the estimated alpha level (the percentage of times each test was observed to reject H0) should correspond to the nominal alpha level at which the test was made. Hence, the first 9, x, and mLfOr each test should be approximately aligned with the 1% rejection level; the second 2, _x_, and mwith the 5% rejection level; and the third 9, x, and‘m with the 10% rejection level. Both tests for a miSSpecified conditional mean (POSEXMY and POSEXMP) conform to these criteria. The largest deviation from the expected result occurs with the test POSEXMP at the .10 alpha level. In this case, the percentage of rejections is approximately 11%, a deviation of 1% from the eXpected result. The results for the tests for heteroskedasticity are much.more varied. In order to analyze the results, it is useful to set up a confidence interval about each of the nominal alpha levels. In doing this, one presumes that the nominal alpha levels are correct so that the probability of a rejection is known. Using the binomial distribu- tion, one obtains the standard deviation of the number of rejections at each nominal alpha level. The standard deviation is 3 for the .01 alpha level, 7 for the .05 and 9.5 for the .10. Since a binomial is approximated by a normal distribution, i 2 standard deviations from the nominal alpha level will be used asa95% confidence interval. When this procedure is followed, the tests whose estimated alpha levels lie outside the confidence intervals with the most regularity are the POSEX tests for heteroskedasticity and both of Goldfeld G 137 Quant's testing procedures. Of these, the tests that lie the furthest from the nominal alpha levels are POSEXHlYM, POSEXHlPM and GQPN. ‘With all three tests, the estimated alpha levels average over 10 standard deviations away from the nominal alpha levels. This difference is large enough to cast serious doubts on the tests' validity. Interestingly, the GQPN test procedure (Goldfeld & Quant's Parametric test with no reordering) displays the greatest amount of divergence from the expected result. This is surprising since the test defines a statistic with a known distribution and hence the estimated alpha level should approximate the nominal alpha level at which the test was made. For samples of 60 and 90, the estimated alpha levels of the remaining two POSEX tests are within 2 standard deviations of the .10 nominal alpha level. However, as the nominal alpha level decreases to .05 and to .01, the number of standard deviations between the estimated alpha level and the nominal alpha level increases. This result, although unfortunate, was not unexpected since the testing procedure used defines a test statistic that is only approximately distributed as F. It was also known [Donaldson, 1968] that this approximation becomes less accurate the farther out on the tail the comparison is made. In contrast, estimates for two of the three tests based on Glejser's method are not within a 95% confidence region for a nominal alpha level of .10, while for the lower nominal alpha levels, the estimates are within the region. Since Glejser indicated that a nominal alpha level of .11 should be used to obtain a 5% rejection level, it is surprising that a relatively high degree of accuracy is obtained 138 when a nominal alpha level of .05 is used. No explanation can be given for this result although it should be pointed out that while Glejser used a t test on gagh_included variable, an F test on the jgint_effect of the variables was used in this study. The estimates of the alpha levels of the BAMSET tests (eight of them) were never more than three standard deviations from the nominal alpha levels used. Since half of the eight tests were defined using OLS residuals and the other half using Theil's BLUS residuals, this agreement between the estimated and nominal alpha levels confirms the findings of Ramsey E Gilbert [1971] that the test can be used with either set of residuals. In contrast, it is extremely surprising that the estimates of the alpha levels obtained for both the Goldfeld G Quant parametric and nonrparametric testing procedures were so frequently outside of the 95% confidence interval about each alpha level (44 out of 72 times). Since both of these procedures define a statistic with a known distribution, it was expected that these results would always lie within the confidence limit. Equally surprising is the small number of times the three Park testing procedures lay outside of the confidence regions (1 out of 27 times). The estimated alpha levels diverged from the nominal alpha levels less frequently in this test than did any other test examined. Since the statistic is only approximately distributed as t, this accuracy was unexpected. However, as previously mentioned, it was anticipated that the estimated alpha levels in the PARK procedure would agree with the nominal alpha levels more frequently than would the estimates obtained from using the GLEJSER procedure. 139 The only general comment that can be made applies to the sample size used in each test. It appears that as the sample size increases, the percentage of rejections generally approaches the alpha level at which the test was made. However, there were exceptions even to this, most notably POSEXHIPM, POSEXHIYM, GLEJSERYZ, and GQPY. In general, it appears that if the three tests that lie the furthest outside of the confidence interval are discarded (POSEXHlPM, POSEXHIYM, and GQPN), the overall results are reasonable. When the sample size is small and the alpha level is large, the estimates of the alpha levels obtained by using Goldfeld G Quant's testing procedures lie the furthest outside a 95% confidence interval about .10. HOwever, as the sample size increases, the difference between the nominal and estimated alpha levels decreases. .At the lowest alpha level examined, .01, the estimates of the alpha level Obtained using the POSEX procedures lie the fUrthest from the nominal alpha level of .01. The nearest agreement between the nominal and estimated alpha levels were obtained by using either the PARK or BAMSET testing procedures. IV.2 Hypothesized Models with a Misspecified Conditional Moan Two of the model groups are examined in this section; the group that includes a variable not in the hypothesized model and the group that has a different functional form than the hypothesized model. .After each of these groups has been analyzed, the section will end with a discussion of the hypotheses made in Chapter III that pertain to tests for a misspecified conditional mean. 140 IV.2.1 Misspecified Conditional Mean.Due to an Omitted variable There are three hypothesized models that omit a relevant variable. In each case, the 'true' model is y1 = 50.0 + 5.0xil + 5.0xiz + 5.021 + 50ui, i=1,...,n, (4.4) where u1,...,un are independently distributed as N(O, l) and Z1 denotes the variable omitted from the hypothesized model (4.1). It should be recalled, however, that the three models differ in the degree of correlation between the variable omitted from the hypothesized model and the variables included in the hypothesized model. In the first case, the omitted variable is independent of the included variables (drawn from an independent normal distribution with a mean of SO and variance of 400); in the second case, it is moderately correlated with each included variable (xi4 is defined as _ 3(x. -50) x14 - 5.428 loge xi1 + 7.711 loge xi2 + :3 , i=1,...,n); and in the third, it is highly correlated (.7) with eaCh included variable (xis is defined as x. = .5428 x. + .771 x. + 3(XiS-SO) , 15 11 12 -——7fir———- i=1,...,n, hence the coefficient of determination between xis and both xil and xiz is .98). The estimates of the parameters of the hypothesized models are shown in Table 4 for each of the three sample sizes. It is Obvious from Table 4 that the estimates of 80, 81’ 82 and 02 Obtained do not correspond to the parameters in the 'true' model (4.4). The reason for this is that when an hypothesized.model that 141 mew.MNMH omv.HHuN owmmo. ovao. cumm.NNH NNmmo. «mmmo. HmmN.N¢H oom.m Nmo.n oHv.mm mHm.mmm vom.Nwmm owmmo. OQHmo. cumm.NNH omeo. move. mmmH.wnH ano.o oou.m voo.omH. mom.moH mw.vaNH owmmo. QQHmo. ohmm.nNH mONH. nmmH. mmwo.Hvo Nmo.m MHo.v owH.mmm N H o om oHomHhm> wouuHEO :w nqu mHoooz.:H whouosmhmm mo mouoEHumm Nvo.onn Hmo.mNNN Hmvno. NmNmo. mmvm.va mowwo. momma. Hmm¢.NHN mwN.w ono.n HVH.om Huommmwa Hmm.¢mq New.mmmm Hmwuo. mmNmo. omvm.va ONQHJ coco. vam.HON moo.o «mn.m omm.HmH whommmwm Hwo.Hm No.NonH quno. NmNmo. mmvm.va mMNv. mmom. wwv.HNoH mNm.v omn.q Hum.Nvm mnHmmmm N H o oo wa.NNN aom.¢mn~ a .Ne «Nan. NNOH. NBHO.H4© Homy> Boom. NNOH. mam.aao Homimm Hom.m mHo.N mmm.om Hm mow.aNH oom.ooom a .Nm eNma. NNOH. NNHO.H¢© HomV> amom. momH. «VOH.HmN Howl mm NNQ.N ooa.m mwo.m0H om moa.¢H oo.mHemH a .N@ amok. NNOH. “NHS.HVO HomC > New.a amoo. mm©.m~am Homo mm mHN.N oma.¢ www.mom Hm N H o muoposmhmm om dNHm desmm ”v mgm -OHxNON m-OmemN meHo. HHMva aamoo. mmooo.- ammo. om oHa.m Hoaoooma a.Nm HH.HmmON mNm.macH .QONHHNO Homo> mo.ommmH mmm.mmoH .ooqomooH Homva omo.maH omo.ou om.Nocm- om aamo.~ soomeHH a.mm a.mmHQH om.¢NHNN mooxoaeH Homc> mma.mqmm maN.m44 4c_xamm HHmUNm 4N0.mOH NHw.NaH wa.w4Hm- Hm N H o 2328me om oNHm deamm 5H3 mHoHdoz :H moooosmomd .H0 335qu H N 59? 147 TABLE 8: Percentage of Rejections of Models that have an Incorrect Functional Form 10 Sample Size 30 60 90 o (100%) Level 1% 5% 10% 1% 5% 10% 1% 5% Test Model 5 POSEMMY 2.5 7.9 12.9 85.4 89.1 90.9 66.7 72.3 POSEXMP 13.8 21.1 25.3 90.5 93.0 94.5 92.3 93.9 Model 6 POSEXMY 1.5 5.7 12.3 39.8 51.4 57.6 34.2 43.8 POSEXMP 8.6 14.5 20.6 55.6 66.1 70.7 51.7 61.3 Model 7 POSEXMY 1.6 5.2 9.8 4.8 19.1 29.7 1.0 4.9 POSEXMP 16.7 33.6 41.0 43.3 55.5 60.5 16.3 23.8 77.9 95.2 52.6 69.7 10.0 29.3 148 It thus appears as if the power function rises very quickly with respect to sample size and flattens out soon thereafter. .Also, in models 5 and 6, it should be noted that the test POSEXMP was more powerful than POSEXMY for every sample size and alpha level examined. Although there was generally a decrease in power going from models 5 to 6 to 7, the decrease in going from 6 to 7 was not as marked as expected when the POSEXMP was used. This was especially true for a sample size of 30 where the power actually increased substantially. Since model 7 is a non-analytic function (the function is not continuous) in the neighborhood of 0.0 and since a Taylor series expansion is not able to approximate a non-analytic function, it was expected that the percentage of rejections would corre- spond to the alpha level at which the test was made. These expected results were obtained for sample size of 30 and 90 when the POSEXMY test was used but were never obtained when the POSEXMP test was used. IV.2.3 Examination of Hypotheses on Tests Designed to Detect a Misspecified Conditional Mean Five hypotheses were stated in section 111.1 relating to tests designed to detect a misspecified conditional mean vector. Observa- tions on each of these hypotheses will be stated in turn. Hypothesis 1 - In both the POSEXMY (equivalent to Ramsey's and Ramsey & Schmidt's) and POSEXMP tests, the estimated alpha levels were within 2 standard deviations (95% confidence region) of the nominal levels at which the tests were made, as illustrated in Figure l. Hypothesis 2 - It was also observed, as hypothesized, that the power of each test increased as the correlation between the omitted 149 variable and the included variable would increase. However, it was also observed that this power function would decrease (as hypothesized) when the correlation increased past some point. Unfortunately, since only three points are observed on the power function, a more precise statement cannot be made. Hypothesis 3 - The third hypothesis was not completely verified by the experiment although it was observed that the power of both tests to detect the misspecified model decreased from model 5 to model 6, and to a lesser extent, to model 7. (Recall that a four- term Taylor series expansion became less accurate at approximating the correct functional form of the model as the model numbers increased from 5 to 7.) The reason that this acceptance is only partial is that for the POSEXMP test for sample size 30, the power calculated in model 7 is greater than that in either model 5 or 6; this finding is contrary to the hypothesis since model 7 is a non-analytic function. Hypothesis 4 - It was also observed that the power of both tests did not always increase as the sample size increased from 30 to 60 to 90 observations. Rather, the power increased as hypothesized only when the misspecified model either had an omitted variable with medium correlation or when the correct fUnctional form was easily approximated by a Taylor series expansion. That is, when the theory behind the POSEX tests indicates that the tests would have little power, the power is not increased by increasing the sample size. Hypothesis 5 - The last hypothesis was maintained for every alpha level, sample size, and model examined. It was continually observed that the POSEXMP test was more powerful than the POSEXMY 150 test was more powerful than the POSEXMY test (recall that this is equivalent to Ramsey's and Ramsey G Schmidt's test). HOwever, it should be pointed out that some might find the POSEXMY test more appealing because of its simplicity. Summary Hence, while the first, second, and fifth maintained hypotheses were conclusively SUpported, the experimental results did not completely substantiate the third and fourth hypotheses. However, the findings did indicate that the fourth hypothesis was true in certain important cases and that the third hypothesis seemed always to be true for large sample sizes. IV.3 Hypothesized MOdels with Heteroskedastic Disturbance Terms The remaining three model groups are analyzed in this section; the group of models that are heteroskedastic due to a simple function of one variable, the grOUp that is heteroskedastic due to a non-linear function of one variable, and the group that includes a variable with a conditional mean. As before, after each group of models has been analyzed, a discussion of the hypotheses made in Section 111.1 that pertain to tests for a miSSpecified conditional mean will be given. IV.3.1 Heteroskedasticity_due to a Simple Function of One Variable There are six hypothesized models that are heteroskedastic because the disturbance terms are multiplied by a single variable. In each case, the 'true' model is yi = 50.0 + 5.0xi1 + 5.0x2 + ziui, 1=1,...,n. (4.8) However, the variable zi differs for each model; it is xil in model 8, 151 x12 in model 9, x13 in model 10, xi4 in model 11, E(yi) in model 12, and lOOi/n in model 13. Because of these differences, the six models differ in the form of heteroskedasticity and the relation between the disturbance terms and the included variables in the model. The estimate of the parameters of the hypothesized model is shown in Table 9 for each of the sample sizes examined. In every case, the estimates of 80; 81’ and 82 are statistically equal to the true values of the parameters. Of the divergences from the true values, the greatest is about 11% and occurs in model 12 for sample size 90. This unbiasedness is even.more evident if model 12 is discarded since the largest bias in the remaining models is less than 2%. It should also be noted that in all but one case (BE, model 9), the bias in parameters 81 and 82 becomes smaller (or shows a negligible increase) as the sample size increases from 30 to 60 to 90. This does not appear to be true for 80. This result, however, is not entirely surprising since the estimates of the intercept terms have such large variances associated with them. Next, it should be noted that the estimated variance of all the parameters and the sample variance of the parameters decreases as the sample size increases; that is, there is an increase in efficiency as the sample size increases. .Also, it should be noted that with a few exceptions (most notably 82 in model 12 and in model 9), the average estimates of the variance of each parameter (32(Bi)) are extremely close to the observed variance in the parameter estimates (V(Bi)). The last estimated parameter to be examined is the variance of the disturbance term ziui. If the averages of the estimates of the variance (32) for the different sample sizes are simply compared, TABLE 9: Sample Space 30 Parameters 0 1 2 81 49.722 4.983 5.038 62(81) 916.19 .1414 1.130 V(§i) 380.05 .1683 .7760 62,F 3596.9 109.90 Bi 50.120 4.995 5.008 62(81) 103.94 .0160 .1282 V(8i) 84.734 .0129. .3322 52,? 408.06 987.59 Bi 49.174 4.989 5.055 52(61) 869.12 .1341 1.072 vtéi) 915.01 .1495 1.229 62,? 3412.1 76.51 81 49.710 4.990 5.034 62(81) 452.25 .0690 .5570 vtéi) 329.62 .0670 .6424 62,? 1775.5 213.35 81 48.649 4.881 5.275 62(81) 46735. 7.721 57.64 V(8i) 24684. 7.777 58.99 62,F 183478 2.080 Bi 49.615 4.985 5.061 52(81) 894.64 .1380 1.103 V(éi) 789.46 .1269 .8713 52,F 3512.3 75.591 152 60 0 1 2 MablB 50.273 4.986 5.002 266.68 .0702 .1106 152.32 .0869 .1804 3420.6 359.45 Mmle 50.442 4.997 4.982 57.971 .0152 .0240 88.003 .0120 .3856 743.56 1924.2 Model 10 49.666 4.997 5.007 257.872 .0679 .1069 236.115 .0689 .0572 3307.6 368.21 MOdel 11 49.923 4.995 5.002 134.88 .0355 .0559 114.82 .0341 .0886 1730.1 695.42 Medel 12. 53.323 4.908 4.925 14929. 3.930 6.190 17756. 4.267 28.97 191486 9.151 Medel 13 49.585 5.003 5.005 264.72 .0696 .1098 254.92 .0621 .1227 3395.4 363.75 Estimation of Simple Heteroskedastic Models 50.410 4.990 4.996 193.30 .0481 .0531 80.994 .0546 .0667 3670.5 572.22 50.762 4.996 4.975 61.047 .0152 .0167 86.915 .0085 .1941 1159.2 2076.4 50.032 4.997 5.000 167.811 .0418 .0462 162.448 .0415 .0450 3186.5 656.24 50.313 4.995 4.995 94.109 .0234 .0258 68.273 .0192 .0457 1787.0 1159.7 55.775 4.929 4.854 11798. 2.936 3.244 8445.2 2.653 12.80 224025 11.578 50.240 4.994 4.996 178.26 .0443 .0490 155.50 .0415 .0652 3385.0 620.86 153 extremely misleading information will result. This becomes evident if one notes that for any model and sample size, the variance of the disturbance term is _ _ 2 _ 2 2 _ 2 2 Var(ziui) — E(ziu.i E(ziui)) - E(zi ui ) — E(zi ) E(ui ) 2 2 = E(zi) = Var(zi) + (Mean(zi)) since the E(ui2)=l and where Mean and Var denote the sample mean and variance of 21. Hence, the variance of the disturbance term.depends on the sample of zi used. The expected variance of the disturbance term.for each model and sample size appears in Table 10. Whereas the TABLE 10: Variance of Disturbance Terms In Simple Heteroskedastic MOdels Sample Size 30 60 90 Mbdel 8 3677.922 3516.607 3694.970 Medel 9 432.890 978.705 .1334.608 iMOdel 10 3055.475 2936.759 3204.806 'Mbdel ll 1810.687 1764.628 1806.472 IMOdel 12 188204.8 205926.9 229139.5 Medel l3 3530.487 3431.250 3398.457 superficial examination of the estimated variances in Table 9 could lead one to conclude that the estimate of the variance was biased (especially model 9), one now finds that the estimates are unbiased ‘with the divergencies from the true values generally decreasing slightly as the sample size increases. Next, each of the six models was tested to determine if the disturbance terms are heteroskedastic. The results for each model appear on a separate figure and will be examined in.turn. The reader is referred to section IV.l for a basic explanation of all the following figures. 154 The test results for model 8 appear in Figure 2. In this model, the heteroskedasticity is caused by the variable x_. The most obvious result which can be inferred from Figure 2 is that the group of tests that assume that x2 is causing the disturbances to be heteroskedastic together with those tests that did not reorder the observations have comparatively little power. Noteworthy for their slight differences are the BAMSET tests when the observatibns have been reordered by x2. .Also strikingly obvious is the fact that the tests with the greatest power are those that assume (correctly) that the heteroskedasticity is caused by x1. Next most powerful are the tests that use the predicted value of y; which are closely followed by the POSEX tests. The difference in power among these tests appears to be very small for a sample size of 90 and increases as the sample size decreases. Generally, the results are as expected. Since the POSEX tests require less a priori_infbrmation, it was expected that they would have less power than the tests which correctly assumed that x1 was causing the disturbances to be heteroskedastic. One rather surprising finding is the extremely good results obtained by the tests that assumed that the predicted values of y were causing the heteroskedas- ticity. .Another somewhat surprising result was how well the BAMSET tests did when the observations were incorrectly reordered by x2. The results for the next model, appearing in Figure 3, are unfortunately not as definitive. The heteroskedasticity is caused by x2 in this model. Unsurprisingly, the most powerful tests overall seem to be those that assumed that the variable x2 was causing the disturbance terms to be heteroskedastic. .Although these tests as well as all the others seem to show a marked loss in power for a 155 Test Percentage Rejection 0 % 20% 40% 60% 80% 100% POSEXHIPS G "t 4’ .. X 40- mm POSEXHIYS AW“ 0 4" x X POSEXHlYM "'0 *‘r 0 X X POSEXHIPM "0 C a” X X__X m—————-mm OH 40 o BAMSETI‘Xl % —o oo BAMSETOXl % GLEJSERXI *7 0 HO. mmm PARIO(1 0““ G to x xx ‘o—oo GQPXI 4% 0* 49* 9., 1. BAMSETTXZ W): x '1‘“ H BAMSETOXZ ———..0—0—0 8" x LEGEND GLEJSEsz 58% . Sample Size 30: o——o—o PARIO<2 go ——0 5% Sample Size 60: x—-x-—x Sample Size 90: m—np—m PX2 '_°‘°‘°_ First Symbol a = 0.01 GQ W Second Symbol o = 0.05 GQNXZ XLXO‘ x 9 Third Symbol o = 0. 10 —Ju— _O——0 ”— BAMSETI’Y .. 188m BAMSETOY 0* 0—9 x. A Afimm PARKY _\.,1 0 <1, .,_ A A afi—mm PY 0P —o—ov___ GQ .. as.“ GQNY 0:: ~0— x o x "141}— H BAMSEITN —x C x O x 0 ~45? m m —o————o—-——-o BAMSETON —Ffi—x ~51} Xm GQPN fivf f0“? o—o——-———o GQNN X—x x FIGURE 2: A Schematic Diagram of Test Results for Model 8 Test POSEXHIPS posaxmrs POSEXHIYM POSEXHlPM BAMSE'ITXl BAMSE'POXl GLEJSERXI PARKXl GQPXl GQNXl BAMSETTXZ BAMSETOXZ GLEJSERXZ sz GQPX2 GQNXZ BAMSET'I‘Y BAMSE'IUY GLEJSERY PARKY GQPY GQNY BAMSE'ITN BAMSETON GQPN GQNN FIGURE 3: 156 Percentage Rejection 0% 20% 40% 60% 80% 100% o 0 =0 xe- 10 H—0» 0 Em ._C 10 0 ”Sim Ho <}——<) 3930mm 1<}1_ :{*——9 v v o .0 gov v v A A Am—H. ban—an fififié LEGEND Sample Size 30: o———o———o g§2fi§m Sample Size 60: x——Hx——Hx -o——o_o Sample Size 90: In——4n——4n x x—ex First Symbol o = 0.01 a: :n—qn _ 00 0 Second Symbol o - 0.05 x x x Third Symbol o = 0.10 -flF-—flP-——--—fll .13 *Hgo~—-o HESS —*<>e 4— H1876 113 <»——ol xxx -dmmn v H v v_x ~—o———oo 191% AC ‘iji x—fi—Xm m A-<>~ o o v v__ A A x ———<>————4) x AHA —:4mnn o—o———o v ‘m—ex A A 44mmn .°-——°t;——° v v A<>— :c: 10 v V-x A A ——<>§, v v —A A 4~§F--i&-dn -——4>————<>——- —x x x ne—e -1n—-———an V V A A J\—m v—m -——<>—————————-o —x x x ~flP—v nb-——————n1 - ——x x x 4%: nr————a1 : V V V A iian “ ur' m -o-——o-——<) -—- x——+x _{n— m “ #45 A ab—-——4n -x:” ‘x EEO x ar—v nee nl o-;o-——o v v -0——:O———0 v v °__§_—° v x ——fiL —flF— 11%, : V : V V ' A m. ‘Ae—dme 1n : : V V v 2 o x 0 x. LEGEND n: a: n1 . o__o__o Sample Size 30: HEAT-m Sample Size 60: o_—o—o Sample Size 90: fi..4fi.i§ First Symbol o_. .H—e~ 9 Second Symbol AA“ X X Third Symbol xE—g——+x mm--m .A SOhematic Diagram of Test Results for Model 11 100% X'——X—'—X a = 0.01 o = 0.05 o = 0.10 161 Test Percentage Rejection 096 2096 40% 6096 8096 100% -—-o-————o——o POSEXHlPS A—--——-—-—x——-l)lc POSEXHlYS “A A" J" x x y A} m —o-——o—————o POSEXHlYM x x x H —o———o————-o POSEXHlPM x x—x a? M BAMSETI'Xl + £— 0 x x_x art—mm 0— ch —0 v v__ BAMSBI‘OXI A A x GLEJSERXl + 4* T x x M 49,— #0 o PARKXl A A m GQP“ A} ° ° X—mn AnTi A_n o v GQNXl A an :3“— mA —o——o———o BANBETTXZ x x x ——o—--o———o BAMSETOXZ x x A m M GLEJSERXI C D C x x——-—x PARKXZ D x D D): x "% -=m———-—m ——o———o——o GQPX2 A A m A -4'1———-m.. GQNXZ Lil-M _______—G_fi A0._.__._.__o BAMSE'ITY x x—x M c- —o—-—-—o v__ A BAMSETOY A A x GLEJSERY 4“ 4‘" 9 x A x4“ :3 —0—-—O v PARKY n a—“Xfl m GQPY AA" : x : x x ‘J‘f 4‘} 1‘0 v GQNY A 4" #mA ——m BAMSBTTN —x —— LEGEND 411— H . Sample Size 30: o—o—o BABBEI‘ON — ————x‘----—xH n Sample Size 60: x——x—-x o o 0 Sample Size 90: m..—m——-m GQPN A—mt— X fix an First Symbol a. = 0.01 GQNN XW x Third Symbol o. = 0.10 FIGURE 6: A Schematic Diagram of Test Results for Model 12 162 Observation number and n is the number of observations. The results of this model appear in Figure 7. Since the variance of the disturb- ance term increases with the Observation number, it was not surprising that the most powerful tests were those that did not reorder the observations. Once again, however, the other BAMSET tests displayed a much greater power than was eXpected. This is true to a lesser degree with respect to the GQP tests and the POSEX tests. .Also, unlike most other models, only a small increase in the power was observed in all the tests as sample size increased from 30 to 90. The results of this group of models seem to indicate that if the variable that is causing the heteroskedastic disturbance is known, the test used should reflect this knowledge. under these conditions, the GQP test seems to be the most powerful followed closely by the BAMSET tests and the GLEJSER and PARK tests. .Although the latter two are not as powerful, they have the advantage of being constructive tests. If, however, knowledge about the variable causing the problem is unknown, it appears that the tests with the greatest power are BAMSET tests with the observations reordered by the predicted value of y, This is followed by GQPY, the POSEX tests, GLEJSERY and PARKY. The last three tests have the advantage of being constructive tests. Three surprising results were observed in this group of models. First was the generally high power displayed by the BAMSET tests. Second was the typically large gain in power observed as the sample size was increased from 30 to 60 observations. Third was the unexpectedly high power displayed.when tests based on the E(y) were used.when the correct knowledge was unavailable. Test POSEXHlPS POSEXHUYS POSEXHInM POSEXHIPM BAMSETTXl BAMSETOXl GLEJSERXl PARKXl GQPXl GQNXl BAMSETFXZ BAMSETOXZ GLEJSERXZ PARKXZ GQPX2 GQNXZ BAMSETTY BAMSETOY GLEJSERY PARKY GQPY GQNY BAMSETTN BAMSETON GQPN GQNN 163 Percentage Rejection 20% 40% 60% X— —X 3>< >< LEGEND Sample Size 30: Sample Size 60: Sample Size 90: First Symbol Second Symbol Third Symbol fix a 100% x—x—-—-x 0.01 0.05 0.10 Q Q II II II A inlan—4fi FIGURE 7: .A Schematic Diagram.of Test Results for Model 13 164 IV.3.2 Heteroskedasticity Due to a NOn-linear Function Two models are examined which are heteroskedastic because the disturbance terms are multiplied by a non-linear function of a single variable. The basic model is given in equation (4.8); however, in this group of models, 21 is a non-linear function. In model 14, the 2 1/2 1 + x11) 15, the disturbance term is multiplied by 75 + 50 sin (E(yi)). These disturbance term is multiplied by (500 + 10xi while in model two fUnctions differ from the last group of models in two ways. First, the heteroskedasticity generated in these models is mixed (has a non-zero intercept) while in the previous group it was pure. Second, the heteroskedastic disturbances generated in this group are more complex than in the previous group since more Taylor series expansion terms are needed to correctly identify the function in this latter grOUp. The results obtained from estimating the hypothesized model appear in Table 11. .AS‘With the previous group of models, the estimates of 80’ 31, and 82 obtained are statistically equal to the parameters given in.model (4.8). HOwever, as with previous models, while the estimates of 81 and 82 converge to the true values as the sample size increases, the estimate of 80 does not. In contrast, all three parameters show a significant gain in efficiency as the sample size increases from 30 to 60 to 90 Observations. Finally, it should be noted that although the estimated variances seem to be very volatile, this is once again due to the samples of x1 and x2 used in this study. When these differences are taken into account, the variance converges to the expected variance as the sample size increases. 165 oww.oom non.mnmo wonoo. ovwo. mmov.vnm oHQH. mmHmo. nmom.mom Noo.m omm.v omw.om nmm.v¢v mmN.oonv mHmwo. owooo. mnom.mHH mowoo. HoHoo. mHmm.mvm omm.v omm.v Nvo.om N H o co HHmn.wnH Hmn.nwwo mammo. omNH. mem.mnm vHNN. mOVH. mmnw.mmm NHo.m omm.< www.mv mmwmmmmmH HHm.mnm mmm.nmvv omHN. wwOH. omvw.uHN HmvH. memo. omuH.mvm moo.m omm.e mnH.om «H Howe: N H o oo mam.ma oaao.~wwm a.Nm omm.~ «mom. noo.HmmH HHmV> vam.H NHmN. vo¢.wQVH mevmm wmo.w mwm.a oHo.ma um mNa.om oaH.NHoa a.mm cNao.H mocm. oaca..wm Humv> H omaa.fl meH. mma.aaHH H.mc~m H mao.m Nam.a mom.ma .m N H o mhmuoemhwm on onam oaasmm mHowoz.uHummwoxmououoz pmoch-coz mo :oHmeHuwm HHH mHm A -Hanaa 5m1 a V V r A A A V V V ffi A A 099 m x V V A A A” ‘_V’ AC x x—x "I ilnl! <}AA —Hc 4~o x x—x v v ox --—x 44}, AH—Cge Ho ' v v A :1“ A V : : x x—x am——amn AA n} :22" ifi -<>—~—<>——o -ao———ao——+x LEGEND -—flb———flP———€n ro—o Sample Size 30: o—-—o——o __fi;:::3§:fi§n Sample Size 60: x+——xH——x Sic v4~c v .0 Sample Size 90: In——4n——4n W“ First Symbol o = 0.01 W Second Symbol a = 0-05 w—‘X Third Symbol a = 0-10 A Schematic Diagram of Test Results for Model 168 The SIOpe of this function is 2xil + 10 which is a monotonic function in the region in which xi1 is restricted (O §_x. §_100). Therefore, 11 the advantage the POSEX test appeared to have is minimized and in some cases removed altogether (the Goldfeld 6 Quant tests and the BAMSET tests). In conclusion, it should be recalled that this model was included in this study because it was a more complex function (involving two terms and a non-zero intercept), not because it was thought to be nonrmonotonic in the area of interest. Instead, the following model was designed to fill this gap. In Figure 9, the test results for model 15 are reported. In this model, the POSEX tests have the lowest power closely followed by the other constructive tests (GLEJSER and PARK). Although this result was suspected for the latter two tests, it was unexpected for the POSEX tests. HOwever, in retrospect, it should have been expected. To understand why this is, one must carefully examine the function used; it is: 75 + 50 Sin(E(y)). Since Sin x ranges between -1 and +1, it was expected that this function would range between 125 and 25. However, because E(yi) is conditional on xil and x12, both of which range between 0 and 100 with a probability greater than .99, the E(yi) can range from 0 to 1050. [This means that the function oscillates between 25 and 125 a total of 167 times. Therefore, since only a maximum of 90 points are observed on this function, the fUnction is not clearly defined and the points appear to be random. Thus, the POSEX tests, as well as all the other tests, do not reject the null hypothesis of homoskedasticity as often as would be expected. 169 Test Percentage Rejection 0% 20% 40% 60% 80% 100% POSEXHIPS % POSEXHIYS POSEXHIYM “WE POSEXHIYS $51—50 —-O—-—O BAMSETI‘Xl x x x —o———o—o BAMSETOXl x x x GLEJSERXl £13m PARKXI g—Ogfi—Lx GQPXl x9 J: x 3 x}? m GQNXI g :fiv :x BAMSE'I'I'XZ ——%——x:——x: BAMSETOXZ -— ——x GLEJSERXZ "39%: : : PARKXZ :mH :-————x GQPX2 x—jx— °—‘ " ° GQNxz x£°"°_x ——9——°¢—‘9 BAD/BETTY .. an .. an“~m “0+9 BANSETOY .. .. m .. m GLEJSERY fiofio - PARKY xix-ix: GQPY -—-xc : ——Cx -O—O-——;——O v GQNY ...“,L a. m “ LEGEND BAIVBE'I‘I'N —x X-—- .1 Sample Size 30: o——-o——o ' Sample Size 60: x———x—-x BAMSETON — X X Sample Size 90: m——-m—m J. 0 First Symbol a = 0.01 GQPN _Xar: X_m ‘fixn1 Second Symbol a = 0.05 " Third Symbol a = 0.10 GQNN x——°°x—-—xo FIGURE 9: A Schematic Diagram of Test Results for Model 15 170 Finally, since the constructive tests reject the null hypothesis of homoskedasticity less often than do the non-constructive tests, it appears as if the non-constructive tests are slightly more sensitive to the alternative hypothesis than are the constructive tests. This is probably because the distribution of the non-constructive test statistics is known exactly (or asymptotically) while the distribution of the constructive test statistics is known only approximately. HOwever, it should be emphasized that while the loss in power incurred in using the constructive tests is slight, these tests possess the great advantage of providing the researcher with estimates of the heteroskedastic variances. IV.3.3 Mbdels that Involve the Variable x5 There are two models which involve the variable x6. One of these models is homoskedastic while the other is heteroskedastic because the variable 56 is multiplied by the disturbance term (see Table 2). In both cases, the model hypothesized is yi = 80 + 86 xi6 + 82 xi2 + vi, i=1,...,n. These models differ from all of the previous models in that the variable x6 is drawn from a uniform pOpulation conditional on the observation index i. The distribution which xi6 (i'th Observation) is drawn from is (0, i.5i). Hence, each observation of x6 is drawn from a different distribution. The results of estimating the hypothesized model appear in Table 12. It is once again obvious that the estimates of 80, 86’ and 82 are statistically equal to the 'true' parameters' values of SO, 5, and 5 in both models. .Also, a gain in efficiency is noted for the 171 www.mmHH Hmm.mvma Hn©.mnw mam.mmoa mommo. KOCH. 0wmv.Hm 00050. ONOH. moflm.qv HNmNC. nmmo. sawm.mm wovmc. mmvo. nmww.¢m ©00.v oco.m Hac.cm w00.v HHc.m mmo.0v NH Hmflor. 00m.mom 0m©.m0vm OOH.mmm mov.vw¢N Umeo. mmomm. mmmv.0m mummo. wvcfi. Hmm0.0mH m0mmo. anomo. m0HH.mw 0qawo. mHOM. 0000.mma 000.? 000.? mw0.0fi moo.m ma0.m nmN.0v OH #0602 N o c m o 0 00 co 9% macmflpm> wcfl>ao>cH maowo: mo :oHmeHpmm mao.mo omomo. mvmwo. oac.m vmm.cm comm. omen. avo.m (‘3 “NH mqm mmmwc. emou.um AimVNm . a. aco.m aco.oq . Hmm.qan a.mc mmaa. cumh.msm «Has» mmma. Ammm.omm .cchw «Ho.m mmo.mq cm 0 o mcoposmcxd om oNom ofleaam 172 estimated parameters 80, B6 and 82 as the sample size increases. Finally, it should be observed that the average of the estimated variance of each parameter, 32(gi), is different from the observed variance, V(éi) for the parameter 86 in model 17 for all three sample sizes. The results of testing model 16 for heteroskedasticity appear in Figure 10. Since this is a homoskedastic model, the percentage of rejections for all the tests should correspond to the alpha level at which the test was made. However, since in this case, the vari- ables x16,...,xn6 are drawn from n different pOpulations, the diagonal elements of the matrix M will vary more than they will in the other homoskedastic model examined. Hence, the expected value of the OLS predictors, fii,...,fin, of the time variance 02 will vary more than in the other homoskedastic model examined. Therefore, on the basis of this information, it would seem reasonable to suspect that tests fOr heteroskedasticity which use OLS residuals will incorrectly reject the null hypothesis a disprOportionate number of times. However, as was shown earlier in this study, since the maximum squared.variation in the diagonal elements of the matrix M is :(2:E :_%, the OLS predictors of the variance (which are a function of the diagonal elements of the matrile), although not constant, actually display little variation under the null hypothesis of homoskedasticity. Thus, it should instead be eXpected that all of the tests for heteroskedasticity will reject this model as often as they rejected the other correctly specified model (model 1, Figure 1). The test results substantiate these expectations. The single exception is the test GQPY'which rejected the null hypothesis a much greater percentage of times than it did in testing model 1 (the null Test POSBXHIPS POSEXHlXS POSEXHIHM POSEXHIYS BAMSETTXI BAMSETOXI GLEJSERXI PARKXl GQPXI GQNXl BAMSETTXZ BAMSETOXZ (31.5.1513sz PARKXZ GQPX2 GQNXZ BAMSETTY BAMSETOY GLEJSERY PARKY GQPY GQNY BAMSBTTN BAMBBTON GQPN GQNN FIGURE 10: 173 Percentage Rejection 1% 5% 10% 15% 20% 25% if v ‘00 v 0 1'“ A _fi A m {hr v C v 0 d" A m 1‘ In G —O —() x x x m m 4m #0 o 4.) x x x a: m :m —o of o —x x x m an in -—*}—— 1%}, so -flt m~ eean. o .0. o ——x x x -m m m —or oes 0 -—an *an In ... A A —-fl'i m m 4x} 4C} o :gF——m» x X . m -——oiei a} :0 -X X x —H‘fi m ‘1“ -fl1 m2, —-fll A i0 V x A0 -fl} HF—i Am G; 4%} *0 ——x x x —m fiTF m "flh H‘fi A fl A ——o Aeaoi .40 X X X . ‘ -—an an in LhGhND _G 0 __o . —-————- x x x . —-m m m Sample Size 30 0—0—0 A Sample Size 60: x+——x+——x IQ X U—A X0 Sample Size 90' 1n-—an——4n —'n‘r m m . ‘ o G. n 0 First Symbol a = 0.01 2%; X em Km Second Symbol a = 0.05 35%? {; o v Third Symbol a = 0.10 l §fi fl A C v {‘r v 4.) v .. A m* A m _ 4} Gr ., o —31g :7: A m A —0 V —GV vfl —m A 44% Am —C;, 4*} 43‘, A A A ne———————qn :n m A 11:7 A m A Hean n: A an A A Schematic Diagram of Test Results for Model 16 174 model). The rest of the tests generally rejected the null hypothesis about the same percentage of times as they did when model 1 was tested. There are, of course, some occasions where, fOr a specific alpha level and sample size, different results are obtained (for example, GLEJSERZ, sample size 60, alpha level .10; and PARKXZ, sample size 30, alpha level .10), but no general pattern was visible. Also, because the tests suffixed by X1 are now using a different variable, x6, there were some minor differences in the percentage of rejections for sample size 30, but by sample size 90, these differences had vanished. The results of the heteroskedastic model involving 56 appear in Figure 11. There are many marked differences between these results and the results of model 8 (heteroskedastic in x1) which appeared in Figure 2. The most striking difference is that the tests which do not reorder the observations show an extremely large gain in power ‘with the percentage of rejections about tripling. The only other major increase in power is observed at all three alpha levels for the POSEX tests when only 30 observations were used. Interestingly, the PARK, GLEJSER, and GQN tests all show a decrease in power for all three alpha levels when the 30 observations category is used. This decrease is especially acute for the tests when the expected value of y.is thought to be causing the heteroskedasticity. In general, none of the tests, except those without reordering, showed any change in power for either sample size 60 or 90. Test POSEXHlPS POSEXHlYS POSEXHlYM POSEXHlYS BAMSETI‘Xl BANSETOXl GLEJSERXl PARKXl GQPXl GQNXl BAMSE'ITXZ BAMSETOXl (31.13.1515sz PARsz GQPX2 GQNXZ BAMSETTY BANBETOY GLEJSERY PARKY GQPY GQNY BANSE’I'IN BAMSETON GQPN GQNN FIGURE 11: 175 Percentage Rejection 0% 20% 4095 60% 80% 100% go *o—a oX x+x eammn C; 0 73 xex fo——————{»———o x xx eammn II} o x0 xx ~—mmm e1} 09 Aadfifi 4mm 0: c; o ulllllll n V, 4_0_f o xe—xx Aammn :& *amnn or x o——i§ :namn -——<>————<>———ag v A III—A ’11:; m rC—zv ‘°———:° v A m A 1‘13 m LEGEND ooo . ---—- — - —X Sample Size 30: o——o—-—o xx—ex Sample Size 90: 1n——an——an First Symbol a = 0.01 —x~—x+—x Second Symbol a = 0.05 oo___o Third SY'mbOI a = 0.10 xxex mmm ex} aeeo—————o xxx eamnn <>—~ *4} *0 x :nmn -——4>—————<»————o xxx 4—1mmn cy—4>————4> x xx llullll V’ 0 0 xxx llllllll ——o-———<> v <> ., ., A :ml——4n4=an 9mmm {x a—{>—<> 9 00xxx *mmm 3" v 4‘0 1:? x A flr—————dn———fin .A Schematic Diagram of Test Results for Model 17 176 IVL3.4 Examination of Hypotheses on Tests Designed to Detect Heteroskedasticity Nine hypotheses were stated in section 111.1 relating to tests designed to detect heteroskedastic disturbance terms. .A number of comments, observations, and findings pertaining to those hypotheses will now be given. Hypothesis 1 - In testing the correctly specified model (model 1), it was observed (from Figure 1) that the tests POSEXHlYM, POSEXHlPM, and GQPN rejected the null hypothesis many more times than hypothesized. Although this finding was contrary to the hypothesis, it was especially unexpected in the case of Goldfeld G Quant's parametric test. Since the distribution of the GQPN test statistic is known, it was expected that the estimates of the alpha levels would be very close to the nominal alpha levels at which the tests were made. Instead, it averaged over 10 standard deviations away from the nominal alpha levels. In general, it was not observed, as hypothesized, that the tests which were within a 95% confidence region (:_two standard deviations) about the nominal alpha levels were those with a test statistic with a known distribution. Rather, the tests that were within the confidence region.most regularly were the PARK and the BAMSET (asymptotic distribution of the test statistic is known) testing procedures. The tests that were outside the confidence limits most regularly (if the three extremely inaccurate tests are discarded) for high alpha levels were both of the Goldfeld G Quant procedures and fbr small alpha levels, the two remaining POSEX procedures. HOwever, it was observed, as hypothesized, that the estimated alpha levels, in general, converged toward the nominal alpha levels at which the tests were made. 177 Hypothesis 2 — The experimental results substantiated the hypothesis that the probability of any test's correctly rejecting H0 is an increasing function of the amount of a_prigri_information available. It was further observed that when a simple function of some variable was causing the heteroskedasticity (models 8 through 13), the tests that used this information were the most powerful (the results can be seen in Figures 2 through 7). It was, however, also observed that when the heteroskedasticity was caused by either x1 or x2, only a small decrease in power resulted from using the same tests with y_instead of either x1 or x2. This observation was predictable since y_is a weighted sum of x1 and x2 and therefore embodies both correct and incorrect information. It was further noted that when the a_prigri_information also concerned the functional form of the heteroskedastic disturbances, a notable increase in power was observable. This observation was made on.mode1 14 since it is heteroskedastic because of a quadratic function of xi. In this model, the tests that used x1 still showed the highest power. However, when the test results that used x1 were compared with those of model 8 (simple function of'xl), a marked decrease in power was observed. It should also be noted that since the GLEJSERXl test assumes (correctly in this case) that the 2, this test correctly showed the E(“iz) = (80 + 81 X11 * B2 X§1)° smallest decrease in power while the PARKXl test showed the largest decrease in power because it incorrectly (in this case) assumed that the E(uiz) = Ail 02. Therefore, without exception, the results indicate that the most powerful test for heteroskedasticity is the one incorporating the most correct information about the heteroske- dastic fUnction. 178 Hypothesis 3 - The experimental results also indicated (as hypothesized) that if a test was observed to have any notable power when a sample size of 30 was used, this power increased as the sample size was increased. However, in many cases, it was also noted that the gain in power was minimal as the number of observa- tions was increased from 60 to 90. Presumably, this was because the power was already approaching 100% and hence only a small increase could be made. Hypothesis 4 — Recalling that since the basic difference between model 8 and 9 is the distribution of the variable causing the hetero- skedasticity (the variables cover the same range with probability of .99), any noticeable differences in the percentage of rejections should be primarily due to the different distributional forms. In comparing the two models (Figures 2 and 3), the tests using x1 in model 8 must be compared with the tests using x2 in model 9 since these are the variables causing the heteroskedasticity in each model. HOwever, when this comparison is made, no appreciable differences can be observed (the tests on model 8 seem to have a slight edge for sample size 30 but for sample sizes 60 and 90, the tests on model 9 show more power). NOnetheless, a comparison of the POSEX tests in two models seems to indicate clearly that these tests have more power when x2 is causing the heteroskedasticity. {A comparison of the tests that do not reorder the sample observations reinforces this result. HOwever, since these tests are not expected to have any power, this increase in power is itself surprising. It should also be noted that' the tests which assume that E(y) is causing the disturbances to be heteroskedastic are more powerful when x1 is causing the disturbances 179 to be heteroskedastic. This seems to be because the E(y) is more dependent on x1 since it has a larger mean and variance than.x2. HOwever, these differences in the mean and variance are caused by the fact that the variables have different distributions and hence must be considered. Therefore, no clear pattern emerges. .Also, since the differences in power disappear as the sample size increases, this investigator feels that the distributional form of the variable causing the heteroskedasticity is not as important as are the para- meters of that distribution (e.g., mean, variance, or range over which the variables vary). Hypothesis 5 - The next hypothesis concerns the probability of type I error when the model includes a variable that is drawn from a distribution with a non~constant mean and variance. In this case, it should be recalled that since all the variables are not drawn from fixed distributions, the diagonal elements of the matrix M vary more than in the previous models examined. Hence, the OLS residuals will be more heteroskedastic than those residuals obtained from the other homoskedastic model. However, since it has been shown that the squared variation of the diagonal elements of the matrile is always small, the OLS residuals should appear to be homoskedastic. The three tests which would be affected if this hypothesis is wrong are the GLEJSER, PARK and BAMSET tests. The test results of model 16 appear in Figure 10. If these results are compared with the results Obtained from testing model 1 (the homoskedastic model consisting of variables drawn from fixed distributions whose results appear in Figure 1), one sees that all of the tests (with the exception of the test GQPY) reject the null 180 hypothesis approximately the same percentage of times in both models. It was further observed that any divergences that do exist become insignificant as the sample size is increased from 30 to 60 and then to 90 observations. Hence, the experimental results substantiate the claim that the OLS residuals are nearly homoskedastic even when a variable in the model is drawn from a non-constant distribution. Hypothesis 6 - Unfortunately, since the nonemonotonic function used in this study (model 15) could not be pr0perly defined by the small number of observations available, the power of the POSEX tests could not be determined for a nonrmonotonic function. Therefore, it could.not be determined how the power of the POSEX tests relates to the complexity of the function causing the heteroskedastic disturbances. Hypotheses 7 a 8 - The next two hypotheses predicted the probable relationship between the four POSEX tests. HOwever, it was observed (that all four tests had virtually identical power. In some cases, one of the tests would show a slight advantage, but no general pattern could be detected. However, it.must be remembered that since the tests POSEXHlPM and POSEXHIYM were fOund to give poor estimates of the nominal alpha level under the null hypothesis, the other two tests are recommended. .Also, since the test POSEXHIPS requires the calculation of principal components and.yet offers no power advantage over POSEXHIYS, it appears as if one should use the POSEXHIYS test for its simplicity. It is interesting to note that it was hypothesized that this test would have the lowest power. Hypothesis 9 - In contrast to the last findings, the test results substantiated this hypothesis. It was observed, with rare exception, 181 that the BAMSET tests were more powerful when OLS rather than BLUS residuals were used. The few exceptions occurred when the wrong variable was used to order the Observations. Although in no case was the difference in power very great, the result nevertheless substantiates the claim that the BLUS residuals to some extent mask the heteroskedasticity. It should also be noted that regardless of the residuals used, the test estimates of the alpha level under the null hypothesis were within a 95% confidence limit of the nominal alpha levels used. Hypgthesis 10 - The last hypothesis indicated that if the same amount of §_prigri_information is built into all the tests, the POSEX tests would have the most power. Although the POSEX tests can be altered to include E 221231 information, this was not done in this experiment. Therefore, this hypothesis really states that given.no information, the POSEX tests will be the most powerful. One version of each test will be compared to the POSEX tests. The version used ‘will assume that y_(or E(y)) is causing the heteroskedasticity. i was chosen rather thangg1 or x2 since it is a linear combination of 51 and x2 and hence is more general than either x1 or x2. The one other alternative would be to use each test first with x1 and then with x2; however, since the correct alpha level cannot be determined, this procedure was not undertaken (presumably an alpha level of .10 could be obtained by using each test at approximately the .05 alpha level). First, the tests POSEXHIPM, POSEXHlYM, and GQNY will be discarded from the comparison because of the large biases displayed in testing the null model. Of the remaining tests, the largest bias 182 is by GQPY for high alpha levels (distribution of the test statistic known) and by the two POSEX tests for low alpha levels. In comparing the remaining tests (Figures 2 through 9 and 11), it was discovered that the most powerful tests were the BAMSET tests. These were followed by the GQPY test, the POSEX tests, and the GLEJSER and PARK tests. Although it was not found that the POSEX tests were the most powerful given no information, they are the most powerful of all the constructive tests. Summary From the above results, it appears as if only the second, third, fourth, fifth, and ninth hypotheses were strongly substantiated. The rest of the hypotheses (with the exception of the sixth which was not adequately tested) were only shown to be true in certain cases and not in general. In the first hypothesis although it was fOund that the estimates of the alpha levels for all of the tests become more accurate as the sample size increases it was not found that the tests with the smallest divergence between the estimated and nominal alpha levels were those with a test statistic that has a known distribution. In the seventh and eighth hypotheses, although it was found that all of the POSEX tests had reasonable power, none of the four tests could be found superior. Lastly, in the tenth hypothesis, although the POSEX tests were observed to have the most power out of the class of constructive tests, it was not observed that they were the most powerful in general. CHAPTER‘V SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR FURTHER RESEARCH The disturbance terms in a linear regression model must have an expected value of zero and be homoskedastic if one is to obtain estimates using ordinary least squares which have certain desirable properties. Several tests are currently being used to detect a non- zero mean in the disturbance term and others to detect heteroskedasticity. A major problem with the current testing procedures for disturbance terms with a non-zero mean is that these tests have not achieved the simplicity necessary fOr popular acceptance. The first procedure which was develOped is based on BLUS residuals and hence is computationally difficult. While a revised version of the test uses OLS residuals, which are more easily calculated, the procedure is still somewhat cumbersome since the calculation of the matrix M is necessary. A In contrast, the current tests for heteroskedasticity have achieved the necessary simplicity for pOpularity; however, they are not always applicable since they all require some §_p§iggi_knowledge about the variable causing the heteroskedastic disturbances. One group of current tests requires further that the variable be monotonically related to the disturbance terms, while the other group requires that the researcher hypothesize the functional form.taken by the disturbance terms. 183 184 TWO new testing procedures, one for each specification error, are suggested in this study. Both tests are based on a Power Series Expansion (POSEX) Mbdel which has the advantage of approximating any analytic function by using a linear combination of known variables raised to various powers. Two linear combinations are used in this study; the first principal component of the known variables and the least squares predictors based on a regression model in those variables. The proposed procedure for a misspecified conditional mean is to transform the hypothesized model into a POSEX model by adding three power series expansion terms. If this model explains the conditional mean of the vector y_statistically better than does the hypothesized model, then the hypothesized model is misspecified. This procedure achieves the simplicity previously lacking since it does not require the calculation of any special matrix and since the test statistic can be obtained by using any existing least squares program, In addition, it was proven that when this procedure is used, with the expansion term being the OLS predictor of y_from the hypothesized model, the test statistic is mathematically equivalent to the statistic obtained from the existing testing procedures fOr distur- bance terms with a non-zero mean. The testing procedure proposed for heteroskedasticity uses the same POSEX model as above to explain the conditional mean of an instrument for the unobserved variances. The instrument used is either studentized or Minimum Norm Quadratic (MINQU) predictors of the unobserved variances because they are both unbiased under the null hypothesis. If this model explains the conditional mean of 185 the vector of predicted variances statistically better than does the sample mean of those variances, then the hypothesized model is heteroskedastic. This procedure achieves the generality previously lacking since it uses a POSEX model which approximates any analytic function and hence does not require that the functional form be hypothesized. In addition, lmowledge of the variable causing the heteroskedasticity is not required since all of the variables from the hypothesized model are included in the POSEX model. Moreover, it was proven in this study that regardless of how the matrix X is chosen (stochastic or non-stochastic),the diagonal elements of the matrix M display a minimal squared variation. Hence, the squared OLS residuals, which are a function of the diagonal elements of the matrix M, will be approximately homoskedastic provided that the disturbances are homoskedastic. In order to compare the new tests with the current tests, a sampling experiment was used. Seventeen definitions of the conditional mean were used to compare the tests under various null and alternative models. 1000 samples of the conditional mean of the vector 1 were examined so as to ensure that the samples would reflect the pOpulation from which the vector y was drawn. It was found that although both versions of the POSEX test for a misspecified conditional mean are exact, the test using principal components is more powerful. However, since the POSEX test using y is less complicated to use, a trade-off exists between the test's simplicity and its power. It was also discovered that the power of the POSEX procedures for a misspecified conditional mean varied depending on the number 186 of observations used, the correlation that an omitted variable has 'with the variables included in the hypothesized.model, and the correct functional fOrm.of the variable which is used in the hypothesized.model. The power was found to increase substantially as the sample size was increased from 30 to 60 but only moderately as the sample size was further increased from 60 to 90. If an omitted variable was causing the conditional mean to be misspecified, it was discovered that when the omitted variable was moderately correlated with the variables included in the hypothesized model, the test had the most power. Finally, as the functional fOrm used to define the conditional mean of the vector y becomes more complex, the power of the POSEX.procedure to reject correctly the hypothesized linear model decreases. The results of the tests for heteroskedasticity were more varied. .Although.most of the tests were fOund to be relatively exact under H0, both POSEX procedures using MINQU predictors of the variance and, in some cases, Goldfeld G Quant's parametric test were not exact. It was also noted that the estimated alpha levels using Park's procedure were, as expected, closer to the nominal alpha levels than the estimates obtained using Glejser's procedure. Finally, it was noted that the BAMSET procedure was always exact when either OLS or BLUS residuals were used. With striking unifOrmity, the power of the testing procedures increased as more correct knowledge was incorporated into them. In a parallel fashion, as the tests became more general, they also showed a marked decrease in power. An exception to this was the small decrease 187 in power observed in general when i, rather than the correct variable, was used as the variable causing the heteroskedasticity. Although the power varied depending on the sample size, it did not vary according to the distribution of the variable causing the disturbance to be heteroskedastic. Again, the greatest gains in power caused by increasing the sample size were made between 30 and 60, not between 60 and 90 observations. Of the POSEX procedures for heteroskedasticity, the most useful seems to be the one that uses studentized predictors of the variance for the dependent variable and i' s for the expansion variables. As previously mentioned, the two POSEX procedures that use MINQU predictors for the dependent variable are biased and hence cannot be considered. Also, since the remaining two tests have approximately the same power under the alternatives examined, the less complicated test was chosen as the more useful. Finally, it was observed that the BAMSET procedure which reordered the variables by i had the greatest overall power . However , of the procedures that offer a corrective procedure, the POSEX test generally had the most power. Since the POSEX procedure is more general than the BAMSET procedure, this result should have been expected. Both tests require that the variable causing the hetero- skedasticity be in the hypothesized model; however, the BAMSET procedure also requires that the functional form taken by that variable be monotonic whereas the POSEX procedures does not. This study has offered solutions to the problems posed at the begixming of this study. In the first instance, it has provided a procedure to test for a misspecified conditional mean that is 188 mathematically identical to the current procedures yet much less complicated to use. In addition, it has proposed a different version of the same test, based.on.principal components, which, a1though.more complicated, has a higher probability of correctly rejecting the alternative models. In the second instance, it has provided a general constructive test for heteroskedasticity that is more powerful than the current constructive procedures. In addition, it has also offered a procedure to ease the restrictive knowledge requirement that had been previously demanded of all current tests. Applying this new procedure to the BAMSET test proved to be the most powerful procedure overall under the alternatives examined. HOwever, since the BAMSET test does not provide a corrective procedure if heteroskedasticity is present, a trade-off exists between power and being able to correct for the heteroskedastic disturbances. It should, nonetheless, be reiterated that if knowledge about the variable causing the hetero- skedasticity is available, it should be incorporated into either the BAMSET or the POSEX.procedure. When this is done, the power of both tests increases substantially. In carrying out this study, further questions which require research have been generated. In examining the tests fOr a mis- specified conditional mean, it was observed that the omitted variable's relation to the included variables is of paramount importance. .A study that examined this relation in more detail would be of great use. A useful way to perform this comparison.might be to calculate the probability of the quadratic's occurring in normally distributed variables. 189 Similarly, in analyzing the tests for heteroskedasticity, two areas for additional research became clear. First, since hetero- skedastic disturbances need not be monotonically related to the variable causing the difficulty, various non-monotonic forms should be examined. Second, since the POSEX procedure was found to be the most powerful of the constructive tests examined, the gains in efficiency made from using this procedure should be examined. . [a]! } I; 3.x l {I I lull APPENDICES APPENDIX.A THEOREM.AND COROLLARIES REGARDING THE MATRIX M Theorem: Regardless of how the vectors x ,. .. ,)_ck are obtained (stochastic or non-stochastic) the diagonal elements of the matrix M will have a maximum squared variation of k(n-k; < _1_<_ where squared variation of 2 n n-l - n t1,...,tn is defined as 2(ti - _) /(n-l). Proof: Defining mii as the i'th diagonal element of the matrix M, the squared variation (52) of the diagonal elements is 2 mi ‘ 1% “"1192 . s = n _ 1 . Recalling that M is idempotent and denot1ng n as the number of observations and k as the number of parameters, 2 mii = n - k, since the trace of an idempotent matrix equals its 1 trace. Also, since M is idempotent, no diagonal element can be greater than one (mii i l for all i), hence Zmii _<_ n - k. Since 52 is maximized if Xmii is as large as possible, the maxiimxm value taken by $2 is l -— 2 2 2 (n-k - n (n-k) _ n(n-k) - (n-kL max 5 ) (n-l) _ n(n-l) (n-kL(n-n+k) n (n- 1) k(n-k) < _13 n(n-l) -n QED 190 191 Corollary 1: Defining the coefficient of variation (V) as vcx) = S >£~" ......21'1'1; 36.8247 3.74?.U fifioié ‘Jovfimb 25.13): D .; so 89.999: 1903V1C +1.;Lu- “Donate GaOCEAO 47. ”J: 35.95.22 1i..‘:..'1/+".‘ i'Lo'f’Za "lo"~"2.i' 35:.7150 34.2/...; 3“.5€)7r_’ "o’zaerE—S 34.95" 723.334.: 250?._/¢71 370C)"1'3 9.42564 42.?29: 46.6Jt~ Au.e51” ?7.597v #9.rn:~ 1)) rxl P" I." ,7.“ \IO .1 U 3"“! I'm 1‘.) K) H r—a IN: N P.) 4" b: n x: ~J (7" NR) .9. .a‘J l‘) >6 ' .' C' . U . \l.‘ D.‘ K. I: k}. TAULE 51. X1 79.5166 14.5101 5.04679 1.53453 46.9975 12.7199 3?.1592 -.9033 1.261300 9‘1.4071 ‘1‘3.81."‘-"> 64.6811 64.5002 32.6936 57.2367 74.2425 63.0451 4.3.7501 54.7375 63.2910 92.2533 .632860 87.107C 90.5968 95.6275 33.48Q6 27.3038 6.96685 49.5913 22.3614 76.3541“ 25.2083 83.4594 54.1052 62. 2352 93 .1901 66.1216 31.7865 84.7244 4.08369 98.2598 47.5557 38.8548 52.1354 34.0346 70.8406 55.0507 36.5052 21.1924 64.7006 89.4377 (CONTIUED) X R) 13.9261 24.4277 22.9662 13.1487 16.930? 3.13-;ff 3.061" r‘, 19.14 1 2?.7*0u 16.613.) 1.61%“- 87.62654 19.9371 6. 66 17 70.5??? (‘.K 471M. b.4555: 21.“?41 16.3125 7.57759 54.7770 5.3594: 6.55JFC 3.78115 16.7571 11.5’54 11.1970 8.7775 9.1;“95 1A.50;7 ’2 Roof 7 77);, ?,’).’.\ nL+ 59.0 ”215+ 134.J53 149.50C 4.41317 9.53650 83.6951 47.32;? 5. 73 36 21. 3;91 39.36U3 3.};98 11.6923 26.1536 36.3V57 35.6161 17.1432 3.59742 10.3483 9.59186 . _ _, . . ‘1 V . ‘, Z V v ’ I ‘ ‘ t L, u . + "Q I u .. ,‘ _ J a' 7 . J ‘ . ‘i‘ 6' i , O 3 C. \ a [#4. /"7.. 073 \ : p-) |\; O unxw ('48.: , I 73.42“. 77.6L. IR .33;3 29.362H 62.720d 99.62.. 55.33JL 35.16-- 02.2975 45.~b; 2?.5Q. 67.CQ}J 39.5-;, 32.14"9 35.56UJ 37.7U.O )6.62UQ “3.28sd 26.96 W 66.66;- 33.54)U 47..J4')U 27.1209 68.2.6'11) 63.68UJ 7-. '.(-‘:’)I..' 61.09LJ 48.95U5 31.12Ud 41.2 43.76416 (_,~ 195 X4 I '3 '\'| 4L.u¢41 5.5."4 .51") .\ I- ’1‘ .1 -‘...x _‘<‘.. {11.517949 '71.. ‘TR\ 4. L 7w."15~ . -‘ _. _. 45 . .l ‘4 “r 'r: L ’0 0’4. .) 3:1.-. ..c.1 an'. :(‘f r- " 31.;7F7 53.31h7 54..AL“ 1.). I.) I... ‘- ’. ‘97.? 1“ . :‘;\'2(. 37.3573 53.9363 14.9766 “ho-511 31.6444 51.43.. 4w.1313 34.5736 21;". 1C'." + -.¢71” 37.2440 35.1974 46.5?77 64.41320 66.24v2 .e?.312fi 33.4921 41.1493 51.6112 5.)."'r-'§UJ 42.7771 46.J21u 48.5354 36.37.) 6;.1356 .37177 53.6353) 63.9655 45.968v 26.6872 39.7917 ,"57.v2(3.5 91.14: '1;.3.'.._ 99.1771 42.66LC 7..4C71 ‘16.]."161 41.751; [4 1.1.3 4 “7) ’11.262 a 4 .~7?2 39.9;12 05.8157 9.59163 97.6677 43.92v2 69.773¢ 11. 2752 21.92 45 5.96375 35.70134: 23.1Jv3 :§.5755 41.156; ).4Z5l 137.544 15v.33v Fl.8365 44.2536 78.3 )22 79.4a62 82.5653 67.3351 55.716- 31.5138 6;.Q529 au.7175 50.3465 59.6651 63.~233 18.5U54 62.2123 “2.2221. II\ C} 56.9) d3.h¥ 21.47 35.;-77 13.;57» 6~.37d4 .613765 24.3573 1d.~a6© 53.3L97 69.322» 07.1474 2.93391 Dh.d122 46.4:63 35.11v9 62.é365 55.4457 3 .9242 26.9234 43.1387 1.67396 47.J6J9 (6.5576 11.5383 11.9963 4.75614 62.7189 42.2369 7. 9432- 1!.21+61 1v.1)b3 39.1106 43.6766 27.6153 58.7477 61.4312 12.7636 44.9469 47.3226 43.3616 71.1222 55.6825 18.2928 50.5?55 98.1317 17.3499 126.544 110.679 125.5VS 'KL ‘1 ‘43 h) \1’ K.“ J33. 4- $4 36 b7 58 59 6. 61 62 63 64 65 66 67 ad 7. 71 72 73 74 7b 76 77 78 79 8. 81 82 83 84 85 86 87 86 r. 7 9- 196 )54. £5, and 56 .Means, variances, Covariances, and Correlations of for Sample Size 30. Upper Triangle, Covariance; Lower Triangle, Correlations; TABLE 82: variables 51’ §2, E5, and Diagonal variances £1 £2 53 El 879.406 -14.668 -107.068 52 -.047 110.005 60.440 §3 -.157 .250 531.116 E4 .486 .661 .506 E5 .847 .457 .165 E6 --- -.145 -.423 Nbans 52.901 17.969 50.243 £4 126.942 61.132 102.758 77.713 .833 41.629 is 449.972 85.918 68.150 131.451 320.711 43.297 56 -15.980 -102.338 109.88 13.171 197 TABLE B3: Means, variances, Covariances, and Correlations of variables El’ 52, §3, 54, 53, and £6 for Sample Size 60. Upper Triangle, Covariances; Lower Triangle, Correlations; and Diagonal, Variances. ZS1 2c-2 13 354 2$5 56 31 842.269 94.956 -53.896 152.717 522.506 --- g2 .141 535.013 -S6.046 126.489 455.630 88.495 E3 -.091 -.119 417.924 55.904 -97.784 -114.586 E4 .574 .597 .299 83.874 188.804 --- §5 .715 .783 -.010 .819 633.440 --- 56 --- .185 -.271 —-- --- 428.413 Nkans 51.714 21.064 50.188 40.997 44.875 26.228 198 Means, Variances, Covariances, and Correlations of Variables El, 52’ 53, 54, §5, and £6 for Sample Size 90. Upper Triangle, Covariances; Lower Triangle, Correlations; and Diagonal, variances. TABLE B4: E-1 £1 866.219 E; .102 53 - .114 §A .524 £5 .635 56 --- Means 53.186 52 83.888 783.908 - .001 .662 .826 .802 23.467 $3 65.669 .585 384.240 .273 .039 .194 53.109 £4 151.622 182.181 2.628 96.484 .835 41.352 55 525.011 649.839 21.540 230.648 789.233 47.429 86 62.241 -105.188 768.222 32.622 TABLE C1: Acronym POSEXMY POSEXMP POSEXHIPS POSEXHIYS POSEXHIMM POSEXHlPM BAMBETTN BAMSETTXl BAMBETTXZ BAMSETTY BAMSETON BAMBETOXI BAMSETOXZ BAMBETOY APPENDIX C TEST RESULTS Acronyms Used to Designate the Specification Error Tests Being Examined Test Power Series EXpansion (POSEX) model for a non-zero IMean Using Y as the instrument POSEX.mode1 for a non-zero Mean using P as the instrument POSEX model for Heteroskedasticity using method 1 to estimate the model, P as the instrument and.studentized predictors (S) of the variances POSEXHl, ? as the instrument and studentized predictors (S) of the variances POSEXHl, T as the instrument and MINQU predictors of the variance POSEXHl, P as the instrument and MINQU predictors of the variance BAMSET testing procedure using Theil's BLUS residuals (BAMSETT) and NOt reordering the observations BAMSETT, reordering the observations by the variable 31' BAMSBTT, reordering the observations by the variable X2. BAMSETT, reordering the observations by'?. BAMSET testing procedure using OLS residuals (BAMSETO) and Not reordering the observations BAMSETO, reordering the observations by the variable, [X BAMSETO, reordering the observations by the variable, X2. BAMSBTO, reordering the observations by T. 199 200 TABLE C1 (cont'd) Acronym GLEJSERXl GLEJSERXZ GLEJSERY PARKXl PARKXZ PARKY GQPN GQPXl GQPX2 GQPY GQNN GQNXl GQNXZ GQNY Test GLEJSER'S test using 51 as the independent variable GLEJSER's test using X as the independent variable N GLEJSER's test using 2 as the independent variable PARK's test using 51 as the independent variable PARK's test using X2 as the independent variable PARK's test using 2 as the independent variable Goldfeld 6 Quant's Parametric test with.No reordering GQP, reordering the observations by the variable X1 GQP, reordering the observations by the variable 32 GQP, reordering the Observations by'v Goldfeld 8 Quant's NOn-parametric test with No Reordering GQN, reordering the Observations by the variable 51 GQN, reordering the Observations by the variable 32 GQN, reordering the observations by Y YES? POSEXHV POSEXHP POSEXH1PS POSEXH1YS POSEXH1YH POSFXHIPN BAMSET?" BlHSETTX1 BINSETTXZ BANSETTY BIHSEOTN BAHSEOTXI BIHSEOYXZ BAHSEOTY SLEJSERX1 GLEJSERXZ GLEJSER? PIRKMI PARKXZ PIRKY 50PM .GOPX1 GQPX2 GQPY GQNN GOIXI GONXZ SONY TABLE C2: ALPHA LEVEL .01 .05 .10 6. 69. 103. 16. $9. 113. 35. 66. 135. 62. 89. 162. 43. 90. 145. 36. 82. 13k. 16. 59. 119. 7. 57. 110. 16. 59. 113. 9. 53. 107. 17. 51. 110. 7. 56. 126. 15. 71. 121. 16. 65. 116. 16. 57. 111. 9. 39. 63.- 12. 66. 110. 21. b9. 96. 9. 52. 111. 12. 57. 96. 66. 175. 29k. 33. '105. 169. 13. 76. 130. 1a. 52. 102. 22. 70. 191. 27. 73. 166. 15. 70. 17B. 26. 69. 170. 201 MEAN .110261E§01 .115229E001 .125266E001 .1261395+01 .1307025+01 .125542=401 .aizazarooi .202610E+01 .212522E+01 .203569F901 .2067175601 .2082886001 .ZZ1356E001 .208506EO01 .1159’“F§01 .1096118001 .1113u36901 ..1133765+01 .113377F601 .107166E001 .21suare.p1 .1362636001 .131263F401 .1125905401 .3066006001 .3otuooc.oi .3oaunosooi .3022005401 VARIANCE .9572658‘00 .112727E901 .18352056017 .2349305401, .263083E*01 .2006525401 ..7606~E+01 .397230E*01 .663957E+01 .3353086’01 .676753E+01 .9123658901 .6969995001 .565633E001 .169366E001 .125251E*01 .139365E’01 .590016E§01 .275930E901 .336653E001 .271906E901 .1saiuoeooi .isorzse.01' .1075335901 .256016E001 .259660Etfl1 .2563295001 .2626165901 Test Results - Mbdel 1, Sample Size 30 SKEHNESS .189819E001 .226669E001 .575683E001 .632536E§01 .667156E+01 .576790E901 .209266E901 .1ssouueoo1 .1995475.01 .1784575.01 .239099E+o1 .1604435+o1 .1904398+01 .1qzs7ae.01 .31zz7eeoo1 .2a1sass.oz .2047535001 .706667E*01 .351603E901 .636233E901 .4341335001 .2991sus.o1 .3957115401 .3162525001 .3904725400 .6631558000. .3320665000 .571659E000 KURYOSIS .912679E001 .1180618602 .635696E002 .661951E§Oz .726195E+02 .6109925‘02 .90629SE§01 .5699175401 .asasrse.oi .727osse.oz' .1248652402 .6056866001 .7527735.01 .rssorse.o1 .zoosivcooz .1sa9oa£+oz .623512E601 .7zsrsssooz .2366506602 .3190485002 .3070915ooz .1612066002 ' .2997965002 .1azssqeooz .zaqsasaoox .3134595.01 .2822568601 .3265795.ox 7:51 Poscxnv POSEXMP POSEXH1PS Possxn1vs Possxn1vn POSEXH1PH BAHSETTN sanssrrx1 annssrrxz annsertv annseoru annseorx1 nauseatxz annseorv GLEJSERX1 GLEJSERXZ GLEJSERY pARKx1 ' PARKXZ PARK! LGOPN__ GOPX1 sanz can? scan . soux1 cunxzo couv 202 TABLE C2: Test Results - Mbdell, Sample ALPHA LEVEL‘ " .01 6. 11. 61. 36. 101. 91. 14. 16. 11. 1b., 16. 12.7 12. 1a. 17. 5. a. 16. 13. 15. 35. 16. 17. 17. 9. 9. 6. .05 99. _ 98. 72. 83. 172. 169. 53. 62. so. 61. 59. 56. 59. 63. 66. 26. 39. '49. ,56. 62. 126. 67. 75.. a1: 62. 69._ 61. 66. .10 93. . 96. 110. 116.’ 222. 217. 101. 127. 108. 119. 125. 116. .109. 111. 11“. 69. ' 70. 113. 101. 116. ’231. 119. 123. 16“. 1BZ.' 155. 137. 172. MEAN. .102321E.01 .1048195o01 .111935e.u1 .113361s.01 .198511soo1 .192409E.01 .201761F.01 .216619E+o1 .2068065+o1 .214139E.01 .2137455001 .216296E001 .21097AE001 .220035E+01 .11“578E+01 .906958E600 .9813868000 '.113254£+01 .1079025+01 .1133739901 .151717r.01 .1116805601 .112133EOO1 .115960E601 .3727008001 .376500E601 .369500‘001 .367000EOO1 VARIANCE .726989E+00 .7699626+00 .1389956901 21666612401 .1754395502 .15163TE+02 .466461E+01 .4569875+01 .414177E.o1 .6628726001 .6965625+o1 .432296£+01 .466038$+01 .u5uua7s+o1 .1467945.01 .777948£+00 .960569E+00 .320761E+o1 .226612Eto1 .3091a9soo1 .269629Evoo 6330192E§00 ..3599BGE+00 .296143:.o1 .3201168001 .323221coo1 .3230335+o1 Size 60 SKEHNESS .156228E+01 .1893655901 .6611985001 .5526795001 .8777768001 .917ESBE+01 .262220E001 .165439E+01 .193303E001 .175326E+01 .2360685601 .1887775901- .2211608+01 .187671E601 .221109E+01 .245260E+o1 .2660706+01 .4807575+01 .2776316001 .313162E+01 .213166E001 .1561395601 .151661E.o1 .167213E+01 .425115£.oo 6.6330698600 .665551E000 .921509E900 KURTOSIS .562291E001 .6556606.01 .3670696.02 .5181378502 .9968126.oz .1150226003' .176910E*02 .735105£.o1 .6753106.o1 .670771eoo1 .1197032.oz .610619E+o1 .109379E.oz .753693E+01 ..992167E+01 .1307735002 .125809E002 .9538325002 .13k100E002 .1652208002 .9662385001 .6717778601 .6602008001 .799150EQO1 .319199E901 .3163696001 .3366628001 .2376065601 TEST POSEXHY POSEXMP possxu1ps 'POSEXH1YS posst1v4 posexn1pn BAHSETTN annserrx1 annssrrxz eansertv annszoru BAHSEOTX1 ennseorxz annssorv GLEJSERX1 GLEJSERXZ GLEJSERY pnnxx1 PARKXZ PARKY GOPN GGPX1 capxz copy can» caux1 60~x2 couv 203 TABLE C2: Test Results - MOdel 1, Sample Size 90 5L???“| LEVEL .C1 11. 13. 36. 13. 1C. 19. 11. 1B. 7. ‘3' o 12. 11. 11. 15. .5? 97. c6. 71. 83. 97. Q30 ‘08. 99. 59. 99. “9. “30 .1C 112. 116. 136. 133. 168. 119. 97. 98. 810 95. 10“. 9B. 195. ’1th. 121. 126. 113. 159. 96. 125. ”SAN .123059E.c1 .134666s.01 .1094752.c1 .1oe9ase.01 .122936E+01 .124322:+01 .203131s+a1 .2113435+a1 .197669E+01 .zcsu1za+01 .2:3291E+01 .2135355+31 .204»96E+01 .21371zs.11 .105396£+o1 .999167E.30 61038255001 .10313SE+01 .10666ua.81 .109727E.91 .1373965.01 .105971E+01 .109696s+01 .11csaaa.a1 .u1c9oce+a1 .u1usuosos1 .6932906+21 .4124)LE+c1 VARIANCE .7721625+00 .3115925*03 .129785E001 .1268615001 .199892E001 .2324026+01 .9179635601 .96h3333+01 .375536E051 .4154052+a1 .4136155.01 .667213Efa1 .4314795401 .435177s+o1 .12666JE+01 .5551735ooo .1082362.01 .2334055.01 .2311nncoo1 .1933682.01 .1163452.29 .1622555+oo .197911E.ao .1668652.oc .366679E+01 .356756E+91 .3364365+o1 .3696125901 SKEHNESS .1601676001 .183351aoa1 .5310865od1 .461991£+o1 .55975TE+01 '.66£03BE+01 .2063465.01 .2195382.01 .191aa9a.01 .1952596+01 .188595E+01 .231a3oe+o1 .2227295.01 .1951402.01 .222196£+a1 .236639Eo01 .245667£.n1 .33611aa.o1 .2~669~e.91 .260156s.o1 .1972435o61 .121asze+o1 .194247£+01 .1121965+a1 .6266925.0o .410921aooa .9aaz7zs.oa .332017sooo KURTOSIS .6366366001 .6595335001 .495267E.az .3773032.02 .5399592.02 .7124002.02 .9486652.01 .99261oeon1 .7761565901. .7936692.n1 .712991E.a1 .1113756.oz .1606672.02 .6604606.01 .106937E.oz .966676E.o1 .1252925002 .1949696.02 .1162696.02 .1200336.02 .9299526.01 .6220236.01 .6979656.01 .517267E.a1 .302537E001 .362673£.o1 .33171.E.61 .2656656001 Test POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP TABLE C5: % Rejection at o .01 .05 19.9 30.2 45. 62. 10.0 44.7 55.5 68. 81. Test Results - Models 2, 3 and 4 .6 15. .3 15.5 .10 \IO 12.2 0 80. 4 90.6 17.5 54.9 66.8 .6 10.6 204 Mean Mmklz Sample Size .433063 .788937 Sample Size .367349 .496730 Sample Size .798915 .518366 Mbdel 3 Sample S1ze 1.41422 1.70927 Sample Size 3.45077 3.97375 Sample Size 4.76302 5.80468 Model 4 Sample Size 1.01705 1.12064 Sample Size .962272 .998673 Sample Size 1.03153 1.01445 variance 30 .078897 .202364 60 .071884 .100863 90 .191194 .112849 30 1.39668 1.94973 60 4.37355 5.02404 90 5.83823 7.10041 30 .816278 .958701 60 .649272 .677300 90 .744858 .738918 Skewness 1.20911 .971755 1.20513 .913994 .911975 1.08354 1.62245 1.71405 1.28624 1.00875 .974417 .777068 1.83557 1.85398 1.57627 1.73510 1.79539 1.63736 Kurtosis 5.01158 4.20716 4.58735 .375351 4.28510 4.34822 6.29281 7.29449 5.74488 4.43899 4.13101 3.62015 8.09556 8.29186 5.79053 7.15953 8.04375 6.77984 Test POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP POSEXMY POSEXMP TABLE C6: % Rejection at o .01 .05 2.5 13. 85. 8 90.5 66.7 92.3 39.8 55.6 34.2 51.7 16. 43.3 16.3 7. to 21.1 89. 93. 72.3 93.9 14. 51.4 66.1 43.8 61.3 33.6 19.1 55.5 23.8 .10 12.9 25.3 90.9 94.5 77.9 95.2 12.3 20.6 57.6 20.2 52.6 69.7 41.0 29.7 60.5 10.0 79.3 205 Mean Medel 5 Test Results - Models 5, 6 and 7 Variance Sample Size 30 1.13959 5.38091 1.74522 1158.26 Sample Size 60 453.589 12039.5 159614. 526541000. Sample Size 90 296.825 48.5093 Model 6 Sample Size .845045 2.83147 Sample Size 21.3371 34.5887 Sample Size 18.0231 10.6566 Model 7 Sample Size 1.44189 3.32074 Sample Size 2.34178 5.33288 Sample Size 1.50024 2.71246 165941. 4991.84 30 .794669 75.6904 60 5034.05 16791.2 90 3290.12 286.901 30 1.32832 10.9710 60 1.67149 23.5468 90 .850417 8.79796 Skewness 2.69912 15.6630 .519493 4.77357 1.43958 4.58076 N .95879 .68327 6.51388 .73581 .85384 5.23155 2.52995 3.52249 1.27134 1.67051 1.59515 .36190 Kurtosis 11.6914 285.515 2.97602 46.0781 4.52777 29.7045 17.6064 120.388 55.2004 97.2244 43.4622 52.3475 14.8763 29.5735 5.68311 6.69945 6.70044 9.23780 YFST POSEXNY Possxnp POSEXHlPS POSEXHlYS POSEXH1YH' POSEXH1PN BAHSETTN annssttx1 aanszrrxz anuserrv annseoru BAHSEOTX1 aanseorxz BANSEOTY GLEJSERX1 GLEJSERXZ GLEJSERY PARKX1 PARsz PARKV GQPN VGQPX1 COPXZ GQPY GONN GQNX1 GONXZ GQNY TABLE C7: ALPHA LEVEL .01 22. 18. 107. 105. 87. 83. 95. 625. 86. 506. 65. 790. 70. 662. 509. 389. 371. 338. .05 65. 51. 307. 289. 245. 208. 837. 211. 763. 17k. 930. 190. 866. 798. 7. 658. 652. 38. 700. 329. 83. 963. 829. 83. 699. 123. 583. .10 102. 83. A69. 922. 362. 39k. 314. 919. 321. 860. 275. .96“. 272. 932. 904. 22. 822. 772. 86. 793. #36. 122. 987. 895. 23h. 87k. 293. 610. 206 MEAN .102399E901 .947367E+00 .237458F+01 .228981E‘01 .209107E§01 .2170636901 .3911585+01 .114777E702 .389356F+01 .100752F+02 .341522E+01 .144523E+02 .34343u=.01 .125283E+02 .632699V+01 .764928€+00 .532093E+01 '.804.29E+01 .939210E+00 .9577522+01 .2541932401 .12273aE+o1 .1846115002 .9396595401 .2614005.01 .6uusooe+o1 .363500E+01 .5928908001 VARIANCE .1667795901 .151176E+o1 1254072€+01 6268113E+91 .2120655+01 .2268535‘01 .1280905+02 .3191166+02 .1296745+02 .2923525+02 .103966£+02 .394877E.oz .1102055+02 .378959E002 .1541535102 .424896£.oo .1150945ooz .5970525+02 .19126ue+o1 .550449E+oz .692107Ef01 .2609oos+o1 .3554325.03 .131247E+03 .176076E+o1 .28956SE+01 .210786€+o1 .267950E*01 Test Results - Model 8, Sample Size 30 SKEHNESS .3315395+01 .392662E+01 .317209E+01 .2895h1E+01 .291336E+01 .3261Q3F+01 .161461E+o1 .6603585+00 .172223£+01 .8327715+00 .175233€.a1 .526245£+00 .179092£+01 .699133£+00 .213448£.o1 .16327ZE+01 .1686505+o1 .233681E+01 .259052£+01 .1958495+01 .6288395+o1 .606349E+o1 .482653E+01 .6529695oa1 .5668686+oo .2664505+00 .6666965.oo .300967E400 KURTOSIS .1898645002 .26426SE+02 .19919.E+92 .156694E+02 .163254E+02 .206503E+02 .628253E+01 .390933E+01 .719598E+01 .360774E001 .739409E001 .32751TE+01 .7601115001 .399926E+81 .1255805002 .655696E901 .916900E501 .103276E602 .1181726002 .902932E001 .325771E902 .661515E902 .470581E002 .785258E602 .329039E001 .285757E001 .333751EOO1 .297326E601 IESI POSEXMY POSFXHP POSFXH1PS POSEXHlYS POSEXH1YH POSEXH1PH BAHSETTN BAHSETTX1 BAMSETTXZ BAHSETTY BAHSEOTN BAHSEOIX1 BAHSEOTXZ BAHSEOIY GLEJSEPX1 GLEJSERXZ GLEJSERY PARKX1 PARKXZ PAPKV GQPN GOPX1 GQPX2 GOPV GQNN GONX1 GONXZ SONY 207 TABLE C8: Test Results - MOdel 8, Sample Size 60‘ ALOHA LEVEL .01 21. 27. 497. 471. 529. 549. 67. 974. 181. 952. 77. 981. 123. 963. 967. 9. 912. 849. 5. 629. 101. 20. 995. 920. 9. 650. 16. “02. .05 79. 78. 776. 760. 785. 810. 167. 993. 331. 988. 187. 998. 982. 940. 30. 923. 255. 58. 1000. 975. 79. 909. 101. 79“. .10 195. 126. 890. 879. 886. 900. 256. 998. 936. 99k. 276. 999. 357. 997. 999. “7. 996. 969. 72. 959. 391. 90. 1000. 989. 192. 978. 220. 916. MEAN .1147325001 .1129505+01 .6042315991 .398029E501 .554857’601 .545173€+01 .33011OE+01 .2592655602 .531346E+01 .225673F+02 .3922445001 .2773945602 .4469205901 .2415775602 .117274¢+82 .735137E+00 .9393365+01 ..219383F002 .867561E+00 .1672965002 .1815935001 .966902E000 .190879F+02 .795205E+01 .629700E+01 .9395006681 .9161002601 .813000'+01 VARIANCE .127505E+01 .129233E+01 .861476E601 .9785025+01 .11759BE+03 .907616E+02 .1161625+02 .8522455.02 .2441s1s+02 .790984E+02 .130689E+02 .947347E+02 .197091E+02 .877744E+02 .252891E+92 .953183E+00 .1.0323£+oz .2565976403 .1504315401 .117164E003 .861418E+00 .3837175‘00 .7195368002 .2573436502 .216616E+01 .950958E+01 .323431E001 .8197255001 SKEHNFSS .222537E+01 .224789E.o1 .5899436001 .613891E+01 .1032355.02 .106271E+02 .195489E.a1 .453889E+00 .1695992+o1 .553944E+oo .2172925.o1 .4237156+oo .209236E+01 .4372925+co .1516705.01 .417224E+o1 .11114oe+o1 .150014E.o1 .2928065501 .1343915401 .2962695901 .189226E+01 .163058E+01 .17151aE+01 .532669E+00 .363561E+00 .631665E+00 .372629E+00 KURTOSIS .1026728002 .9637215001 .679623E+02 .637762E+02 .1353542403 .1556075403 .6643255401 .349667:+o1 .681178E001 .3452845+o1 .1062115o02 .3251795.o1 .9233645401 .32136SE+81 .732911E+01 .2927326902 .572619E+o1 .650946£.o1 .152525£+oz .624060E001 ‘.1707705002 .8227165401 .8093305001 .669535E001 .3276108001 .3673995001 .323689E081 .2955155001 208 TABLE C9: Test Results - MOdel 8, Sample Size 90 ALOHA LEVEL TEST .f1 0‘35 .10 MEAN VARIANCE SKEHNESS KURTOSIS POSEXMY 35. 91. 130. .1121735101 .1319625+J1 .2242776o01 .9943955.01 POSEXMP 27. 72. 119. .1059285+01 .12843ZE+D1 .2724272+01 .1336992402 possxn1ps 012. 959. 903. .522180E+01 .9271366+01 .4319402101 .316980£.02 POSEXH1YS 000. 950. 900. .504301E.01 .805634E+01 .421537E+01 .3119046402 Posexn1vn 764. 945. 979. .51517as+c1- .1300922402 .530430£.01 .435414£.02 Posexu1pn 790. 957. 931. .538341a.01 .1657795o02 .535401E+01 .407963E002 BAHSETTN 95. 210. 3312. .375830£+c1 .1398392o02 .1700602401 .6609902o01 eanserrx1 1000. 1000. 1900. .407906£+02 .1349942403 .4745376100 .3501596o01 aAnserrxz 151. 279. 369. .4737562+01 .2055445002 .1675195001 .6269566901 BAHSETTY 995. 999. 1030. .332650£+02 .1216656603 .3929075400 .2941062401 annssoru 09. 211. 296. .362446E.01 .1301642102 .164023E+01 .6368175001 BAHSEOTX1 10t0. 10:0. 1010. .430743E+02 .1504525.03 .4614702400 .3403912401 aawssotxz 121. 242. 335. .4153935+01 .174077E+02 .179000£+01 .72637kE+01 00552077 997. 999. 1000. .353509£.02 .1356902o03 .EEZBZEEOOO .316301E+01 GLEJSERXi 10:0. 1000. 1006. .179954s+02 .341167E+02 .9949512100 .493632£+01 GLEJSERXZ 13. 42. 64. .3764055+00 .1151506+01 .3635352001 .223371E002 GLEJSERY 996. 1000. 1000. .132573E+02 .1570275402 .769243E.00 .4744992401 PARKXi 973. 996. 999. ..3682128002 .520263E+03 .145011E.01 .6676966o01 PARKXZ 4.~ 35. 91. .9093472+00 .1005272101 .36689BEOO1 .3216165002 PARKY 959. 990. 997.- .265477£.02 .223529E.03 .9256565400 .677066E001 GQPN 99. 249. 351. .1573005401 .3404272400 .2193792401 .9863995001 00Px1 23. 63. 96. .950764E+00 .2375642100 .165671E.01 .7720952o01 GQPXZ 1000. 130c. 10cc. .137294E.02 .45211ze+02 .2069696001 .1161742402 GQPY‘ 970. 990. 994. .6761942401 .134047E+02 .1659436061 .7740316o01 GQNN 4 2:. 6?. 172. .4661002.01 .2373716401- .3973112400 .332223a.01 GQNX1 599. 965. 970. .1153502o02 .6002365101 .2762755.00 .276246E001 GONXZ 20. 51. 11:. .4197006401 .341160E001 .4640292100 .2947702001 couv 746. 867. 942. .9952002401 .4096596401 .1953462100 .2974302001 ILIIT .Ill‘r ..114 3.1111. .11. . I111! 1.31.115 TEST POSEXMY POSEXMP POSEXHlPS Posexn1vs POSEXH1VH POSEXH1PH BAMSETTN 8AMSETTX1 BAMSETTXZ BAHSETTY' BAHSEOTN annssorx1 annseorxz aanseofv GLEJSERXi GLEJSERXZ GLEJSERY PARKX1 PARKXZ PARKY GQPN GOPX1 GQPX2 GQPY GQNN GQNX1 GONXZ GONY 209 TABLE C10: Test Results - Medel 9, Sample Size 30 ALDHA LEVEL .91 281. 261. 188. 551. 22. 44. “Rb. 455. 441. 413. 156; 271. 756. 267. .15 Sf. MEAN .e159915+00 .1:e7955+61 .277602£+01 .299811E+01 .276317E+01 .253184E+61 .3212645001 .434509E+01 .10669SE+82 .4364365401 .3652715701 .41384ZE+01 .121641E+02 .519414E001 .952631£+00 .7580795791 ~.1223905+01 .9198325000 .791138E+01 .165745E401 .2704585*81 .16937GE+92 .132629E+01 .293672E001 .294930E+01 .3881OOE+01 .5914?OE+01 03893006901 vnaxnwca .603627E+00 .8402482.30 .332157s.01 .44891as+01 .3846093+01 .2710922+01 .993483E.01 .151496s+02 .356451a+02 .157319E+02 .1273272402 .1414585082 .3625352402 .1529945o02 .7964622+00 .404758E002 .1003615+J1 .242649E+01 .4373632o02 .513971E601 .7687695401 .212242£+03 .2206452401 .1001552402 .2090492o01 .1572012+01 .2003412o01 .233923E+01 SKEHNESS .2112975001 .197277E+01 6116224E+01 .1454462o01 .1345022+01 .1046123401 .139421s+01 .1433412o01 .503351E+00 .153212£+01 .2294492o01 .15391cs+01 .544469E900 .1776475401 .202250£+01 .3514405+01 .1378712401 .4995352+01 .223106£+01 .298996£+C1 .5779652+01 .2066752:01 .4723912401 .560572E901 .4724792+00 .7501722400 .2830345400 .4002112400 KUQTOSIS 09691155901 .883511E001 .6603235031 .6062932.01 .602573E001 .4394542401 .7771702.01 .526979£.01 .3062756o01 .636092£.01 .1296255402 .5790066401 .3062312401 .749510£.01 .103905£.02 .2595076402 .1005312402 .441536E.02 .1090622402 ‘.17a769£.02 .5285935002 .191806E002 .653172E002 .561085E902 .298170E601 .3856365001 .318017E001 .3137005001 210 TABLE C11: Test Results - Model 9, Sample Size 60 ALPHA LEVEL TEST .01 .05 .16 MEAN VARIANCE SKEHNESS KURTOSIS Possxn7 507. 616. 668. .881998E+01 .1469662403 .3326615o01 .1967626o02 POSEXMP 599. 709. 765. .104693£+02 .1605235+03 .323278E701 .187664E002 POSEXHiPS .927.' 961. 976. .2639465.02 .200254£.04 .4260166701 .2795496402 possxu17s 951. 961. 989. .364927E+02 .4255452.04 .5576566.01 .4644716402 POSEXH1YH 963. 982. 99:. .130664s+03 .32919SE+06 .162616E+02 .4292066403 POSEXHiPH 939. 967. 960. .757646s+02 .2342426+05 .4362446401 .2664956.02 BAHSETTN 276. 446. 531. .679164E+01 .391694e+02 .1499636+01 .5525992401 BAHSETTX1 363. 540. 640. .674774s.01 .5975522+02 .1564632.01 .6006456601 aanssrrxz 994. 1020. 1002. .403239E+02 .2591155703 .6577656+00 .364270E701 aanserrv 624. 774. 621. .1546666.02 .1500636.03 .1270076401 .469406£.01 BAMSEOTN 299. 461. 558. .72007oe+01 .4266206.02 .166500E701 .5162796401 6AMSEOTX1 3E9. 546. 629. .6827126.01 .5305602402 .1609825701 .6232402401 sansaorxz 997. 1020. 10:6. .4192775+02 .257519E+03 .6937902.00 .3796616401 BAHSEOTV 633. 756. 609. .1569366402 .1592046403 .1306548701 .5063566.01 01:3senx1 2. 9. 21. .655221e.00 .454536E+00 .2450922401 .1263616402 GLEJSERXZ 965. 996. 1030. .267621E+02 .327425£+03 .1601236401 _.694736£.01 GLEJSERY 564. 713. 760. .9576446+01 .924692£+02 .1743516401 .6530956401 PARKX1 2. 27. 75. '.6364952.00 .1371106+01 .2421132401 .1025016402 PARKXZ 756. 693. 939. .1747952.02 .179961E+03 .1360945401 .5242242401 PARKY 142. 310. 439. .3345636.01 .1350352+02 .193502£+01 .6061466401 GQPN 324. 475. 570. .266479E701 .3963766.01 .25036ZE+01 .1116026402 '609x1 1000. 1060. 1000. .363119£+02 .706699E+03 .2206202781- .1052012402 00972 410. 536. 6:2. .353473E+01 .1626076+02 .4410036401 .3521962402 GQPY 733. 659. 695. .6547666+01 .4039956+02 .3096992+01 .179211E.oz GQNN ‘ 7. 75. 191. .4126005401 .2626265701 .4403916+00 .3150546401 GONX1 6. 96. 237. .4445002401 .2317296+01 .351615E700 .2915262401 60Nxz 576. 666. 966. .6962006.01 .4312872701 .2203046400 .2704535401 60N7 56. 266. 454. .542300£.01 .329937E.01 .3625936400 .2972776401 TEST POSEXMY POSEXHP POSEXH1PS POSEXH1YS POSEXH1YH POSEXH1PH BAHSETTN BAHSETTX1 BAHSETTXZ BAHSETTY BAHSEOTN 8AMSEOTX1 BAHSEOTXZ BAHSEOTY GLEJSERX1 GLEJSERXZ GLEJSERY PARKX1 PARKXZ 94447 GQPN GOPX1 GQPX2 GQPY GQNN 60Nx1 GONXZ GONY ALPHA LEVEL .C1 5800 583. 983. 992. 99C. 982. 536. 638. 1060. 969. 7th. 632. 1060. 970. . 1000. 956. 17. 993. 570.. 593. 1008. 363. 981. 2“. 67. 969. 26k. .05 6‘3. 698. 991. 10:0. “992. 656. 768. 10090. 983. 821. 779. 1070. 989. .16 6510 7&3. 831. 106?. 968. 37. 10(2. 969. 109. 1000. 869. 891. 1006. 997. 997. 152. 317. 991. 685. 211 MEAN .819753E+01 .969223E701 .430778E+02 .443163E+02 .3768345+02 .913925E+02 .13t136£+02 .165295E702 .907213E+CZ .446667E+02 .186346E+02 .167796E+02 .939406E+02 .653679E+02 .9988858+00 .636889E+02 .3185345+02 .1191315701 '.3940616+02 .9357536.01 .3587285491 .5973866+02 .210387E+01 .126443E402 .4764008401 .577800Eot1 .115720E+02 1.7414006+c1 TABLE C12: Test Results - Medel 9, Sample VARIANCE .1313175+03 .137832E+33 .8242185706 .6980855+04 .8282556709 .8537285709 .137341£+03 .180820E+83 .638147E+03 .495190E+03 .167897E+03 .193359E+03 .626225E+03 .5198255+03 .4369446+00 .1595366+04 .3916216403 .3103346+01 .4603606403 .4622736+0z .7966966+01 .168161E706 .3460706701 .94365:6.02 .293960E701 .245717E+01 .6004626+01 .3692306o01 Size 90 SKEHNESS .2669446.01 .2625416+01 .637871E701 .6265566.01 .5067296+01 .617336E601 .1562736.01 .1334326401 .2644536+00 .5095316+00 .9653566+00 .1357626761 .3669476o00 .5609035700 .1126196.01 .2466376401 .2433796401 .3551736+01 .9612346+00 .1220936401 .2470716401 .4566166.01 .2616066o01 .2647436.01 .661216E400 .2499526400 .3410146.00 .2570336400 KURTOSIS .1373346402 .1424166.02 .5474595402 .5566256402 .3592058402 .5114456902‘ .5992296401 .5065556.01 .315278E701 .3619246401 .3863475601 .5164266.01 .3547276401 .3858098601 .5516696.01 .127010E402 .1155256402 .2179646402 .4437538001 .5135845401 .1154246.02 .5095626402 .1367096402 .158652E082 .3067946o01 .3159626401 .3216066401 .3289756481 7557 posexnv POSEXM9 9055xn195 POSEXH1YS 9055x9177 9055x9199 04nss77u a4nsz77x1 9475577x2 04955777 3495507" 94nsao7x1 annsso7x2 04955077 GLEJSERX1 GLEJSERXZ 01545597 949xx1 PIRKXZ 94977 0009" 009x1 capxz 0097 GQNN cqnx1 GONXZ GONY 212 TABLE C13: Test Results - Model 10, Sample Size 30 DLPHA LEVEL .71 u. '3. 1a. 17. 19. 17. as. an. no. ‘60. 7:. 59. as. $2. 7. 21. 8. 17. 8. 220. 65. 2‘. 20. 6. 9. 17. F. '- A ...' 38. 416. 173. .1: 79. so. an. 111. 1:9. as. 259. 199. 277. 21.. 282. 233. 214. 23?. 37. 1:2. 70. 70. 13;. 97. 537. 257. 110. 81. 9240 .9525375+c0 .9286SJE+00 .11c5745+01 .1:907«E+01 .1crcqae+01 .1:62395+01 .3425505+01 .29454ua+01 .2eur7za+c1 .29533ee.o1 .3637595721 .3103735+o1 .3a1nase+01 .3211375731 .1552795+c1 .112031£+o1 ..9220345+uo .9579075+ac .1ssseoaoo1 .105599E+p1 .346206£+o1 .1791366001 .11343ssos1 .101851a+c1 .2aneoos+a1 .2753:as+o1 .3uzaoos+c1 .2836935+31 VARIANCE .75977BE+05 .6859SZE+OG .7.azcas+ao .aeaaarz+oa .e7a7soe.aa .6aasoze+oo .157100£+02 .5729045+01 .730333E+81 .a1eaa1a+01 .1125275+02 .9essses+o1 .531549£+a1 .9736935+a1 .1435265oc1 .1937355+o1 .951assaone .22837SE+J1 .392225£+31 .2322515061 .1h9836E+02 .379693£+01 .1392.5£+01 .1263.5£+01 .22613uzto1 .1913905fo1 .200652£+c1 .2053165+01 SKEHNESS. .178366E091 .1913265+01 .2593655001 .2786818701 .2661035701 .25h13fiE+61 .1727965001 .1870858+01 .16612“Eo§1 .17u3816031 .1694536901 .1898826+01 .1688116+61 .191223E+01 .h75510E701 .5379956001 .283“65£+01 .4283626+01 .3098h56+01 .382016E601 .8150235001 .b689185601‘ .3365868931 .B3MSDHEOO1 .SZBQZJEOOO .6226005000 .3362926780 .6851175980 KURTOSIS .7zu1025+01 .8125035001 .1391518902 .1517ras+oz .1303045.02 .131u355702 .seaoozeoo1 .7273225o01 .6230u35+o1 .67550u5+o1 .698737E001 .5217oue.a1 .suoar15oo1 .a121995r01 .ssuorcaooz .unzrszzooz .1011335702 .3zozauzooz .173355£+02 .2749595702 .116561E003 ...asooEooz .2071veaonz .3306uszonz .293557:.01 .2955995oa1 .zassassoo1 .30250.£on1 TEST POSEXHY POSEXMP "9056xn195 POSEXHIYS POSEXHIYH POSEXHIPH BIHSETTN BIHSETTX1 BIHSETTXZ BIHSETTV BIHSEOTN BAMSEOTXI BIHSEOTXZ BIHSEOTY GLEJSERXl GLEJSERXZ GLEJSER? PIRKX1 PIRKXZ, PIRKY' 6099 '60971 90972 6097 9099 - 697071 607072 5097 213 TABLE C14: Test Results - MOdel 10, Sample Size 60 41994 LEVEL .01 .05 .10 . 1.. .1. 4. 20. .6. 1. 12. 27. 5. 17. 35. 16. an. 67. 12. 29. 52. 55. 166. 221. 37. 121. 161. .5. 135. 291. 49. 133. 21a. 30. 213. 29.. 37. 129. .201. no. 193. '225. 66. 162. 290. 5. 3.. 79. u. 23. so. 1. 1b. 30. 7. 39. 92. 10. .3. 97. 11. 57. 108. 13.. 209. .39. 23. .65. 1.0. 33. 96. 199. 23. 97. 159. 1. 1o. 39. 8. 79. 193. 6. Mo. 99. 7. 69. 165. NE!" .757862E000 .8239785000 .6723515980 .7061225000 .901231E*08 .6556616.00 .317876E081 .281379E001 .311188E001 .2971715901 .3888595901 .2895608981 .2973626001 .3287675081 .925915E080 .7839685‘80 .809568E088 '.976267E.00 .1oo7uos.o1 .107621E901 .1697696.o1 .113161E901 .1199176.01 .1190766.01 .2563006.01 .619808E981 .316.006o01 ..oo7ooE.o1 VARIANCE .602189E*00 .9879568900 .2376978900 .300008E700 .709767E+00 .5989585900 .1052206o02 .792116£+01 .8267175701 .927220£.01 .130076E.02 .051795E+01 .016132£+01 .113SSZEOOZ .8752885500 .503111£+00 .51881ZE+00 .201652E001 .222601E*01 .Z3BZ17E001 .651659sooo .6718ZZE080 .9923306o00 .5036756080 .2567665001 .2570675001 .3188295081 .252167EOO1 SKEHNESS .165680E701 .235830E981 .213285E601 .2366336oo1 .5596506901 .5556035901 .209011E901 .1950716o01 .1661906+01 .203635E901 .1763328901 .20321.E+o1 .163099E901 .1999236+o1 .162763E+01 .2589116601 .166.576.o1 .36461ssoo1 .3231.26+o1 .25920~E+o1 .2136565901 .2233526oo1 .1995276.o1 .1939616o01 .3932326.oo .5o3oo.E.oo .8131666900 .973ouo£.on KURTOSIS .619706E081 .133163E002 o 115 “O“EOOZ .1175875902 .6397885902 .5986868082 .9286682001 .810168E981 .689676E001 .8796216001 .729995E901 .891175E901 .619358E981 .826226E901 .7103975001 .135786E002 .818886E901 .2551azeooz .1826815082 .187917E902 .932509E981 .130361E082 .9025095901 .910690E001 .292767E901 .371602E981 .310572E901 .318311E681 214 TABLE C15: Test Results - Model 10, Sample Size 90 T?ST POSEX”Y POSEXMD POSEKHIPS 905699179 905699179 905699199 BAHS‘TTN 949967771 649567792 RAHSETTY 64956079 649560771 BAHSEOTXZ 64956077 616356971 GLEJSE°X2 61635697 949971 949972 94997 6099 ' GOPX1 609x2 6097 6099 , 60971 60972 SONY ALDHA LTVEL 19. 97. 22. 29. 37. 45. 97. “U. 91. 70. .L5 29. 119. 12k. 196. 13%. 38. 229. 78. 1C9. 71. 6. 9C. 36. 20. .- ."o 69 I O 70, ‘32. 61. 76. 199. 199. 19*. 92. 86. 382. 136. 15%. 116. 22. 89. 7‘. 63. 9649 .970939:+90 .9236625050 .3582995+CO .8392h6EOJ0 .997693E+00 .91667QE+GB .282217E+01 .2899285+01 .2916796901 .32?0395+01 .326296E921 .2959935901 .2929685+Ci .3589965901 .“557305780. .8951916‘08 .8698295‘90 ‘.9995176+00 .9446616ooo .9739998900 .1545126401 .1C99635721 .1129955901 .1051656+s1 .2792005901 .3awccosoc1 .3691006.a1 .3579006981 VAQIANCE .5723035000 .65598EE003 .519986E700 .625910F’80 .733858E090 .992639E700 .755929E781 .9176725051 .821U1ZEO01 .9393815FJ1 .899862E731 .1818325002 .963890E901 .1120985’02 .9019395000 .598638Effifl .6391035‘00 .188533E901 .16922OE501 .188632EFDI .2671936000 .2355976900 .2931935709 .223803F706 .300579EOO1 .373689E001 7.3797325761 .361117E001 SKEHVFSS .1951369901 .187956°+81 .279833F+01 .3252975701 .32863957C1 .27CU125701 .17‘1736781 .2708095701 .1860177701 .17h566?+81 .1592“’E*31 .2583396+G1 .2169935901. .1662395+01 .201993=+01 .1859206781 .1678935001 .2699806901 .25E335E701 .2330365+91 .199093F901 .137569E631 .16h9965001 .132821FOD1 .3896965900 .2589109§00 .3653715480 .35C637F900 KUDTOSIS .8973OSE*81 .776282E751 .196993E+C2 .1898C96082 .1887915‘02 .1346646onz .7150976.c1 .159377E902 .7343766.01 .6699126oc1 .5929916+01 .136579E902 .97uacs6+c1 .6406016+01 .89709169t1 .7838095981 .6593566901 .129970E602 .111792E902 .150589E602 .8796765001 .692837E601 .7702625001 .619922E7C1 .2787296681 .2801086901 .276333E001 .292933E081 TEST POSEXMY POSEXMP POSEXH1PS 905679175 905699179 POSEXH1PH 64956779 649567771 6495677x2 64956777 64956079 649560771 BAHSEOTXZ 64956077 GLEJSERX1 GLEJSERXZ 61635697 949991 949992 94997 GQPN GOPX1 GOP32 GQPY GQNN GONX1 GONXZ SONY 215 TABLE C16: Test Results - MOdel ll, Sample Size.30 ALPHA LEVEL .01 5. 5. 25. 29. 19. 26. 26. 38. 20. 21. 35. 39. 39. 18. 31. 15. 39. 90. 31. 73. 110. 72. 68. 23. b3. 65. 79. .05 28. 39. 72. 77. 83. 66. 75. 97. 116. 97. 73. 106. 185. 117. 62. 107. 72. 108. .111° 1190 210. 260. 197. 166. 66. 138. 183. 179. .10 60. 63. 127. 129. 129. 118. 162. 159. 213. 172. 128. 19k. 261. 201. 131. 1.3. 137. 168. 195. 208. 332. 356. 310. 256. 168. 305. 367. 362. MEAN .8731175000 .898607E600 .1206995001 .1229925001 .120229E701 .1157035901 .236197E701 .258758E701 .286910E781 .267310E701 .229359E081 .277869E001 .3052375701 .2835835001 .1287695701 .1556865901 .1391565081 .160730E901 .176275E901 .1770805001 .231563E001 .236661E981 .197710E001 .1862075901 .3071005001 .387500E+81 .6086005001 .612800E001 VARIANCE .716926E780 .755556E900 .836688E088 .120111E081 .1162436.01 .7665676.00 .5627766.01 .6253118081 .7246966o01 .6035565701 .5779388901 .669298E701 .8030018901 .7173996001 .1674766.01 .2759526.01 .1544636791 .983527E901 .5790376o01 .7309146.01 .361728E001 .623729E781 .288819E001 .272732E901 .239630E001 .217955E001 .226285E001 .236796EIO1 SKEHNESS .2156386081 .217666E001 .210935E601 .365900E701 .366675E001 .280591E081 .20363BE001 .180996E’fl1 .169176E001 .165151E001 .253068E081 .15959269019 .1649156.01 .1613666.01 .2113605081 .282507E081 .196251E081 .7678736081 .2526066791 .4541376o01 .988269EOB1 .565901E901 .27881BEOO1 .365075E701 .686388E088 .529306EOBI .257271E088 .668862E008 90970575 .9466976o01 .992828E001 .992376E601 .2735976082 .2495656.02 .6661666.01 .0557966901 .718228E081 .6698256901 .697371E001 .161618E082 .582000E001 .725183E001 .607615E781 .185258E002 .158657EOIZ .863835EOI1 .118180E003 .109260E682 .3990675002 .682566E002 .628887E082 .156585E082 .268019E082 .309583E681 .3389886081 .257966EOO1 .301056E981 TFST POSEXMY POSEXMP POSEXHipS POSEXHlYS POSFXH1YH POSEXHiP“ BAHSETTN BAHSETTXi BAHSETTXZ BAHSETYY BAHSECVN BAHSEOTX1 BAHSEOTXZ BAHSEOTY GLEJSERXl GLEJSERXZ GLEJSERY PARKX1 PARKXZ PARK? GQPN GOPX1 GQPX2 GQPY GQNN GONX1 GONXZ GQNY TABLE C17: 216 Test Results - MOdel 11, Sample Size 00 ALPHA LFVEL .01 23. 31. 155. 157. 268. 258. 23. 75. 70. 69. 27. 69. 66. 92. 39. 72. 82. 75. 58. as. :1. ”114. 193. 218. 7. 39. 32. 70. .05 51. 107. 230. 298. 396. 368. 73. 207. 138. 231. 133. 187. 233. 183. 15“. 228. 171. 328. 396. “#1. 53. 237. 199. 302. .10 139. 161. 30°. 3‘2. #81. A60. 12°. 302.’ 286. 352. 168. 305. 27%. 33k. 293. 280. 350. 290. 2&5. 351. 285. 952. 668. 563. 135. 923. 373. “9°. MEAN .1170305001 .129059E+01 .?11570?*01 .231528€+01 .935989E901 .2977206o01 .239223?+01 .172992E+31 .3616116+01 .911293E+61 .2959952+01 .3795666o01 .3571266+t1 .9160166+o1 .1711536+01 .192981E+01 _.22679BE+01 .9391006+o1 .1961355901 .2725266901 .190399=+o1 .1662356+o1 .1920136+o1 0215003E§U1 .3616006.01 .5265006+o1 .«965006o01 .561500E+01 VARIANCE .1116625001 .1370285001 .9335906901 .1096766+02 .1126066+03 .902393E+02 .6102666+01 .1075166902 .109737E*OZ .1329276+02 .6656776+01 .1053916+02 .103893F002 .1293076o02 .2392576001 .9295296501 .359297E601 .1092615002 .6967578001 .1026995’02 .3673996300 .9999906+00 .1079096+o1 .1339855‘01 .2357666+01 .3001766701 .339112E901 .3922205001 SKEHNESS .1930775001 .201193€+01 .6199166oo1 .5697266o31 .987773E301 .1056076.02 .2506365001 .1962606+01 .1529556.01 .17bB9BE+01 .2256796+01 .199992E+01 .1529516+01 .171977E+01 .2220.36+01 .2666906o01 .2396166o01 .3h32756001 .2276106.01 .2500506+01 .2061276.01 .1956536o01 .1911096+01 .1650366o01 .5300696+00 .39036CE000 .2226336+00 .3076136o00 KURTOSTS .8013Q3E+01 .6609795901 .67975?E+02 .5201128002 .195099E903 .1729036703 .130686E+02 .5856055+01 .5967268901 .7725?5E+01 .110996£+02 .5901995701 .5751905001 .7990016701 .118697E+02 .1ZSOTSE+OZ .136699E+02 .2h97905+02 .9778525001 .125633E+02 .9261796+01 .5850368001 .915870E601 .6999076901 .331289E+01 .296357E001 .2866675901 .2891235001 YES? POSEXMY POSEXMP POSEXH1PS 905699175 905699179 905699199 64956779 649567791 649567792 64956777 64956079 649560791 649560792 64956077 616356991 616356992 61635697 949991 949992 94997 .... ‘60991 60992 6097 GQNN . 60991 60992 6097 217 TABLE C18: Test Results: Medel 11, Sample Size 90 ALPHA LEVEL .01 3h. 39. 25k. 281. 30k. 277. 2“. 102. 171. 199. 20. 109. 17k. 200. 73. 215. 292. 15%. 135. 218.1 31. 316. 20k. 372. 22. 132. 89. 177. .05 10“. 120. #06. #37. 965. “17. 90. 285. 393. 411. 9“. 285. 376. 399. 270. 927. 503. 361. 319. “35. 138. 559. “20. 626. 67. 259. 17k. 331. .10 163. 19k. #95. 537. 560. 513. 173. “11. 509. 5h3. 158. #03. 505. 531. #29. 592. 639. 665. “31. 56B. 251. 688. 555. 1796. 105. #32. 325. 520. 9649 .1251606.01 .1378925701 .2919825701 .307566E901. .337160‘701 .325h999701 .2508105001 .955315E*01 .551052F+01 .593605E+01 .239295E001 .9580058901 .5501855001 .592629E001 .238775E001 .33h6k3E+01 .366656E001 '.3678h6E+01 .3270066+01 .4516636o01 .1411646i01 .2040336o01 .1807616901 .2199326o01 .4122006+01 .6318005901 .5693006oo1 .669000FO01 -- ., VARIANCE .139810E701 .1590055’01 .1300965+02 .122269E+02. .1696566+02 .2035706+02 .616316E001 .1917396+02 .1629306o02 .1970906+02 .5766756+01 .1557036+02 .1762666o02 .2029976+02 .2700376901 .835271E601 .7371805901 .160919E002 .1206196o02 .16.2676+02 .196619E000 .6563536000 .6066996900 .6371616+00 .3976596.01 .3620706.01 .9166925901 .3963aa6+o1 ..-. - . . .... --.. SKEHNESS .217361E001 .2311636+01 .5943766+01 .4619936+01 .0662126+01 .5669266+01 .1717576+01 .158672E001 .1069276+01 .1261156+01 .166563E+01 .1696696+01 .125120E901 .1363606+01 .1bZthE+01 .1953066+01 .2177936oo1 .2194716+01 .1619666+01 .1536358901 .1615116+01 .1290426+o1 .1629346+01 .1369BGE+01 .5014066o00 .4332766+on .2695306ooo .2093596900 KURTOSIS .102965EO02 .1172906.02 .5032036o02 .350039E002 .3296236o02 .4649666ouz .6615596on1 .7165566.o1 .4440076o01 .5930666001 .6083198901 .7930155401 .5306626o01 .5709365601 .6665516o01 .965128E001 .1104276+02 .1009166002 .5661116701 .6155766oo1 .792353E001 .693703E001 .815912EO01 .65b210EO01 .317998E001 .316970EO01 .32569QEO01 .293995E001 TEST POSEXMY 9056999- POSEXHlPS' POSEXHlYS POSEXH1YH POSEXH1PH BAHSETTN" BAHSETTX1 BAHSETTXZ BAMSETTY BAHSEOTN BAHSEOTX1 BAHSEOTXZ BAMSEOTY GLEJSERX1 GLEJSERXZ GLEJSERY 9ARKX1 PARKXZ PARKT GQPN GOPX1 GQPX2 GQPY GQNN GONX1 GONXZ CONT TABLE C19: Test ReEUlts, MOdel ALPHA LEVEL .01 12. 1k. 59. 79. 75. A3. 65. 195. 56. 177. 36. 317. 599. 277. 33. 20b. 56. 1674 .05 97. 52. 178. 186. 167. 138. 163. 666. 158. #08. 116. 588. 169. 511. “28. 63. 363. 335. 113. 358. 25“. 188. 797. 995. 61. 366. 170. 361. .10 92. 03. 269. 296. 277. 255. 215. 616. 239. 598. 209. 722. 292. 6.1. 598. 110. 499. 966. 195. #82. 385. 257. 066. 519. 207. 653. 353. 581. 218 MEAN .938872E900 .102033E+01 .189367F+01 .198691E+01 .1986555’01 .1705003fflt .309803Ff01 .630350E+01 .329111E+01 .57013HEOO1 .286670E+01 .765329E+01 .321861E+01 .6869205*01 .353631E§01 .122916E+01 ..3065206’01 0922538F+01 .1553ZSE+01 .9273216+01 .250011E+01 .1966595001 .701607E001 .3805605‘01 .3573006001 .519100F601 .9033006901 .988900E701 12, Sample Size 30 74914906 .112897E901 .127019E901 .15216~6+01 .3368765+01 .61.6556+01 .1392596+01 .6766566701 .1573656+02 .9365746.01 .1567666+02 .777957E+Q1 .207513E+02 .960267E001 .208103E*02 .66088hE+01 .152671E+01 .589633E+01 .3061925+02 .6980095601 .235728E+02 .958536E901 .4996196+01 .371003E+OZ .2556266+02 .197725E001 .2531155901 .2260176901 .3137825001 SKEHNESS .296181E*01 .319266E+01 .275682E901 .50272&E+01 .9076568*01 '.2791025701 .18h122E+01 .107005E+01 .1791255+01 .1008506+01 .209636E+01 .86221TE+00 .193191E+01 .969615E+00 .209629E+01 -.269979E+01‘ .1961776+01 .3919916901 .2950195951 .2361775901 .5266496oo1 .4671666+01 .2930156+o1 .6327906+01 .62223hE+00 .3606676+00 .hh7fi1hE+00 .176889E+00 90970575 .1592956902 .1905256+0z: .1649527802" .93133¢E§UZ' .1295256+03- .1669095+02’ .7356356902. .9660636+01 .6662566+01 .3966026o01 .1002825002 .hOhBTUE+01 .8995085701 .6062696on1 .1zsu1s6+02 .1339236+02 .1005076.02 .2600606.02 .1069536o02 .125125E+02_ .5336096.02 .952909E+02 .1660236o02 .6950926o02 .3561906o01 .2990696+o1 .2937666.01 .3224796001 TEST POSEX“Y FOSEXNP POSEXH1°S POSEXHiYS POSEXHiYH POSEXHIPH BAHSETTN BAHSETTxl 3AHSETTX2 BAHSETTY BAMSEOTN BAHSEOTX1 BAMSEOTXZ BAMSEOTY GLEJSERXi 616385992 GLEJSEPY PARKX1 PARKXZ PAQKY GQPN GOPX1 60992 6097 GQNN . 60991 GONXZ 6097 TABLE C20: Test Results, Mbdel 12, Sample 182. 36. 237. u N m o 220. 7530 .357. 718. 219 9649 .2363036+s1 .2941926701 .79596§E+31 .632527E931 .1632955+32 .179733E+.2 .2819255+31 .136236E+32 .56197#E+01 .13h212E+32 .2057955+;1 .1929SJE+62 .685195E+01 .192060E+82 .616999E+01 .355383E+017 .7&5959E*01 '.7699796+01 .257317E701 .953869E051 .1690636+01 .1960256761 .679533E+01 .35c9656+a1 .u296:06+C1 .esusaos+a1 .1933306+c1 .676323E731 VARIANCE .725911E+81 .377155E+31 .1953E5EOJ3 .238532E+33 .332829E+39 .ZZGTBhE+Jb .328713£+01 .927389E602 .1036998032 .583912£§uz .8631535151 .997883E+CZ .18h367E+02 .638159E+02 .1261QTE+32 .1995226+82 02“38“7E+32 .0333“CHE+52 .358613E031 050656“6032 .5957153+09 .197595E+31 .1533965+02 .911186EOJ1 .2399635731 .389630E001 .3606125901 Size 60 SKZHNESS .2936CHE+01 .321976E+01 .105207£+02 .7937595+01 .126820E+02 .1020905+02 .1823166901 .716592E+00 .169877E+01 .698810E000 .1990686+C1 .673236E+GO .179372E+01 .598267E+00 .207988£+01 .29313SE+a1 .1620925901 .185398E+01 .2233905001 .1369655+G1 .265692£+a1 .169890E+01 .2257635+01 .2929275o01 .3911355+00 ..139266ooc' .3962956700 .6533366901 ..1889035760 KURTOSIS .167952E002 .185626E+02 .1692362903 .1006576o03 .2399066.03 .1661766+a3 .7301995901 .3637666.01 .7330796.01 .3651466o01 .6a1uaz6oo1 .3769666oo1 .6196666+01 .3298585001 .1312156o02 .1506346o02 .769789E001 .6633536oo1 .1061366+02 .552396E001 .1966506o02 .6632666o01 .1126766o02 .1566376+az .2920196o01 .326787an1 .319999E701 .3383912001 TEST POSEXMY POSEXMP POSEXH1PS POSEXHiYS POSEXHlYH PCSEXH1°H BAHSETTN BAHSETTX1 .BAHSETTXZ BAMSETTY BAHSEOTN BAHSEOTX1 BAHSEOTXZ BAMSEOTY GLEJSEin GLEJSEEXZ GLEJSFPY PARKXi PARKXZ PARKY GQPN GO°X1 GQPX2 GQPY GQNN GONX1 GONXZ GONY 220 TABLE C21: Test Results, MOdel 12, Sample Size 90 41°93 LEVEL .01 .35 .1; 203. 316. 366. 193. 327. 382. 696. 993. 977. 969. 933. 963. 64?. 9’6. 96;. 83C. 992. 971. 117. 232. 39:. 916. 965. 969. ’52. 357. 657. 937. 971. 978. 112. 237. 327. 920. 973. 99k. 366. 556. 658. 991. 9‘0. 979. 669. 981. 997. 675. 730. 612. 9:9. 975. 965. ‘89. 927. 961. 291. 521. 655. 855. 936. 96!. 191. 266. 9:9. 039. 697. 7&3. 961. 995. 996. 67?. 8;1. 656. if. 77. 179. 352. 535. 728. 79. 175. 3EC. 56°. 727. 899. MEAN .267953E+91 .252570£+C1 .12913ZE+02 .1193:2§+32 .123319E+§2 01921995+02 05560125*61 .2129855722 .869998E+61 .2999365+L2 .55317SE+31 .2113195002 .628655E+01 .259808E+C2 .6631193+U1 .812953E+01 0192773E902 .139526E+02. .5193136+61 .1717106o02 ) .1656366+01 .239SJTE+01 .59525HE+01 .35399857C1 .h987?i£+01 .785230E+01 .5697COE+01 .9888)SE*C1 VARIANCE .10639!E+02 .838598E+01 .5263.66+03 .91827ZE+03 .503997S+03 .902313E+03 .1531855+02 .73296:6+02 .3995966+02 .1123396+03 .198983E+JZ .796557E+QZ .9352316+02 .122035E+93 .13:7756+92 .5976716+02 .769325£+02 .6792995+02 .1600526+02 .9706926+02 .9252652900 .155757E+01 .7696855031 .500815E+01 .2977815901 .957867E+01 .9938838001 SKEHNESS .275793E+01 .2565286901 .6051675001 .8735755901 .816387E+21 .63811GE+01 .1595255+G1 .588323E+U£ .1530736+01 .3297616+11 .1966?9£+01 .6039176+:c .1612126+01 .3110356+6c .1363166+01 .235916E+fi1 .2600256o01 .1252026+01 .1218115001 {.9019956056 .198253£+01 .255739E+01' .1815566901 .1698236001 .2999125+00 .5096flbE+00 .397925500C .6309775001 -.226996E¢00 KURTOSIS .1321905002 .117168E002 05007655902 .125176E903 .1C6156E903 .5609696+02 .568627E+01 .360672E901 .7726326901 .322822E901 .522239E+51 .3712375931 .8296395001 .313656E+01 .707912E601 .1137325002 .1636186082 .599786E901 .967605E+u1 .99967ZE901 .855831E*01 .182860E932 .8919586001 .6512935901 .309160E901 .358799E901 .305767E001 .395690E601 TEST POSEXMY POSEXHP POSFXH1PS 905699175 POSEXH1Y9 905699199 64956779 649567791 '649567792 64956777 64956079 649560791 649560792 64956077 616356991 616356992 61635697 949991 949992 94997 6099 GOPX1 GQPX2 GQPY GQNN GONX1 GQNXZ SONY TABLE C22: Test Results, MOdel 13, Sample ALPHA LEVEL .01 10. 12. 19. 667. 99. 65. 65. 663. 58. 79. 7h. 9. 5. 11. 12. 910. 111. 68. 36. 373. 19. 25. 23. .05 55. 5h. 92. 59. 63. 98. 731. 137. 179. 178. 899. 163. 186. 206. 29. 19. 28. 52. 60. 97. 960. 213. 99. 619. 73. 65. 87. .10 118. 112. 69. 88. 89. 68. 825. 221. 299. 256. 908. 259. 265. 296. 60. 50. 83. 65. zzi MEAN .115659E501 .1169996+01 .102277E+01 .1100255001 .111317E+01 .1003805OD1 .9633165701 .3166735901 .327685E+01 .3628805001 .1292365002 .339925F+01 .391711E901 .370135F+01 .987699F+00 .8111065+00 .9998505+00 .1000456+01' .1015526001 .101389E‘01 .2212635+02 .221356E+01 .155223E901 .1305426.01 .6061006+01 .3059006+01 .2739006901 .3105005901 VARIANCE .109230E901 .1167055701 .109293E+01 .171921E+01 .197102E+01 .123551F+01 .293126E+02 .912676E+01 .108925E702 .977366E+01 .909083E+02 .103158E+02 .1192516902 .119215E002 .6579508900 .78k7106+uo .91833ZE+00 .273QQSE+01 .299512F+01 .2301TOE+01 .8005758003 .781192E701 .297298E+01 .205917E901 .2953395401 .258811E+01 .283571E+01 .2390536+01 Size 30 SKEHNESS .199869E+01 .2591955001 .607087E+01 0627““5E901 .622150E+01 .618522E+01 '.871273E§00 .185972F*01 .179272F+01 .1607778901 .7735835700 .186290E+01 .170793F+01 .162233E+01 .1769106+o1 .2630466+01 .188939E+01 .367696E+01 .2937636+01 .2992116+a1 .65895EE+01 .973669E*01 '37357“E’°1 .3614156+o1 .357268E+00 .3992316000 .6276076+00 .330391E+00 KURTOSIS .9005855701 01‘63Q65902 06168835902 .606530E402 .606381E902 .6215165702 .393393E+01 .7782056901 .690896E+01 .619919E901 .367990E+01 .7872525901 .69371SE+01 .695077E901 .7999295*01 .135099E*02 .793913E+01 .2161125002 .133323E+02 .192197E+02 .801056E+02 .381912E702 .233861E902 .2969389002 .309629E+01 .3063S!E+n1 .3467746o01 .2615965901 FEST POSEXMY POSEXHP POSEXHiYS POSEXH1Y1 POSEXH1PH BANSETTN BAHSEYTX1 BAHSETTXZ ' BAMSETTY 64956079 649560791 649560792 64956077 016356991 016356992 61635697 949991 949992 94997 GQPN GOPX1 GQPX2 GQPY GQNN GQNX1 GONXZ SONY 222' TABLE C23: Test Results,.Mode1 13, Sample Size 60 ALPHA LEVEL .01 81. 155. 119. 982. 107. 69. 15b. 999. 117. 71. 165. h. 15. 6. 999. 28. 35. 79. 582. 10. 2. 7. .05 7h. 98. 136. 239. 162. 997. 257. 176. 311. 1000. 276. 19k. 323. 23. 38. 27. 30. 26. 32. 1000. 60. 92. 188. 993. 7b. 26. 51. .10 132. .168. 197. 298. 229. 999. 355. 271. #07. 1000. 371. 296. “22. 76. 6%. A6. 75. 57. 66. 1000. 105. 193. 265. 956. 166. 77. 151. MEAN .121661E+01 .1322585+01 .1910705*01 .230861F§01 .197916F+01 .2523835002 .92293SE+01 .39369QE001 .A99180F901 .2993615002 .040552E001 .361395F001 .519761E+01 .9136385+00 .836187E+00 .76h9795700 ..879319F+00 .7859695*00 .865776E+00 .1736825t02 .101366E601 .119951F§01 .191598E001 .905900EO01 .395800Efo1 .3052005+01 .3890005901 VARIANCE .109863E+01 .191019E701 .h90973E001 .225971E+02 .169211E002 .7627466o02 .159b995002 .1066056+02 .2072916+02 .9570518702 .172389E002 .119020E002 .2260546+02 .7762166+oo .10356~6+o1 .7625506.oo .1.91776+o1 .1363766+01 .151127E701 .1056685003 .5167266.00 .6716406+60 .772915E900 .6944976+01 .2750636+o1 .2730036+o1 .2902308001 SKENNESS .199699E+01 .23171AEO01 .617959E+01 .100802E+02 .1026626o02‘ .A316006+00 v.16957OE701 .1634696+o1 .1662676oo1 .3606676ooo .167763E701 .179678E+01 .1693116601 .1752666o01 .2913066+o1 .3uazazeou1 .2717936901 .3001066oo1 .2957565601 .208600E001, .289699E001 .251BZOE+01 .1556636+03 .1991266900 .63793hé700 .3664366.0o .599857E000 KURTOSIS .917731E701 .126703EO02 .66h69hE+02 .1539865703 .1A5596E003 .3196616901 .7163316oo1 .7667676+o1 .7013566+01 .3ouou.6+01 .70h2738001 .7636196+o1 .7036236101 .6963166001 .1h5950E+02 .Zh09815502 .139.156+02 .1665576ooz .1997756+02 .1070216702 .161607E702 .1390935902' .6819065001 .272621E701 .393538E+01 .306135EOO1 .358101E+01 TEST POSEXMY POSEXMP 905699195 POSEXHlYS POSEXHiYN 905699199 64956779 649567791 .649567792 64956777 64956679 649560791 649560792 64956077 616356991 616356992 61635697 949991 949992 94997 GQPN GOPX1 GOPXZ GQ‘Y GQNN GONXl GQNXZ GONY RLPHf LEVEL .11 ’A 1. L8. 999. F. CLE. Q7. 71. 138. 83°. 15. 6. r .13 179. 7a. 82. 1510. 1950 223' MEAN .131521S+51 .11585§E+Ci .158166E+&1 .1183635+U1 .127991€+al .1158335+81 .LJ6S3AE+32 .38693SE+C1 .398917E+C1 .521ZBZE+$1 .L62315E+62 .3980925+91 .357966E751 .5957BG£+C1 .839976E+uu .139671£+61 .998077E+CD .81G433E+CU 08#5§97E+80 .9338615+EE .1732736+62 .117929E+01 .126312£+C1 '.1533736+c1 01586335+C2 09125355+01 .93;9JCE+91 .3753335+81 TABLE C24: Test Results, Model 13, Sample 7421496: .7666136+oo .9761236+60 .128775E701 .1916526+01 .2216716+:1 .1531166+31 .1265166+03 .1363356+62 .11737:E+32 .216589E+02 .152755E+D3 .196693E002 .1237295732 .2361ABE+02 .66823?£+JO .166369E+31 .121:616+31 .13791SE+31 .1222756+u1 .17095~6+o1 .629171E732 .373372E+JC .923666£+30 .998593E+UC .625882E701 .338976E+01 .2998585+11 .3361655+G1 Size 90 59699655 .1636126+a1 .166991E701 .3537666+61 .uuezusa+01 .h23817E+91 .3559216+01 .3611966+66 01732915+51 .16259ZE+31 .1619656+01 .2879136+00 .17929~E+01 .1639126+o1 .1613725901 .zcu1rcz+n1 .2916736+01 .2611826+a1 .3991755001 .2110776+01 .3023576+n1 .179019E+01 .1820935751 .165980E701 .137h796951 .797h13E-01 .3956636+LO .395185£+00 .kh1659E+00 KURTOSIS .739981£+31 .6772885031 '02138765‘02 .3915668902 .3J26OCE+32 .2999565+JZ .3099266o01 .6667666o31 .5853835001 .6616716+01 .3916766+31 .695626E701 .5911666+01 .6562606oo1 .9326776oa1 .159668E002 .1316166o02 .2362916+32 .6305696+31 .173791E+J2 .833856E601 .999156EQJ1 .7391636601 .53599QE+31 .3622905051 .3316585001 .319889E+51 .2690786901 TEST POSEXHY POSEXMP POSEXH1PS POSEXHlYS POSEXH1YH POSEXH1PH BAHSETTN BAHSETTX1 BAHSETTXZ BAHSETTY BAHSEOTN BAHSEOTX1 BAHSEOTXZ BAHSEOTY GLEJSERX1 GLEJSFRXZ GLEJSERY PARKX1 PARKXZ PARKY GQPN GQPX1 GQPX2 GQPV GQNN GONX1 GONXZ GONY 224' TABLE C25: Test Results, Model 14, Sample Size 30 ALPHA LEVEL .01 19. 16. 79. 61. 58. 335. 56. 259. 39. #80. 51. 379. 268. 207. 167. 196. 110. 33. 219. .05 59. #6. 221. 203. 167. 183. 160. 603. 160. 523. 133. 735. 136. 630. 572. 12. “9“. 366. 35. “#3. 281. 80. 870. 630. 83. “7“. 108. 911. .10 95. 81. 351. 331. 272. 293. 2A3. 730. 99“. 7k. 562. “05. 122. 918. 721. 203. 718. 271. 670. “EAN .101316”§01 .9590675600 .209265E001 .198208E091 .182682E+D1 .188153E+01 .3322976001 .78193ZE+01 .3331BSE+01 .6980395901 .2989875501 .9614329601 .3058126+01 .865561F+01 .b52331E+01 .822192E+00 ..389127F+01 .hh79895001 .9878765+00 .9929305901 .2628235+01 .122886E+01 .9015025001 .9967585701 .350200E701 .5952805001 .369700F+01 .521600E701 74914906 .1+13795+01 .1250676+01 .2104666701 .215588E701 .1662666oo1 .1816656+01 .956273F+01 .2177596+02 .1006526+02 .292156E+02 .823809E+01 .273251F702 .8902696+01 .2615975702 .182199E+02 .5220386700 .811910F+01 .252369E+02 .1951716001 .257618E+02 .509559E701 .209753E+01 .8298995902 .271931£+02 .1831836701 .2902605901 .2005208901 .2686035701 SKEHNFSS .319925E001 .39105DE701 .30?698E+01 .321701E+d1 .32809h€+01 .310311E701 .169BASE+01 .92h617E+00 .189951E+01 .102925E701 .189979E+01 .718975E000 .186175F701 .91966CE+00 .205h53E+01 .17296SE+01 .195037E+01 .321761E+01 .285252E+01 .2911566+01 .905589E+01 .5219th+01 .3198026601 .532173E+01 .6013876+00 .288569E+00 .9708896+00 .1698855’00 KURTOSIS .179988E+02 .2031168+02 .185679E+02 .200976E702 .206607E+02 .192992E002 .656509E+01 .3959906+01 .7992865+01 .916380E+01 .8929715901 .353SZQE+01 .7939895001 .382398E+01 .118208E702 .698961E+01 .936298E+01 .197291E+02 .197406E+02 .179206E002 .300701E+02 .525181E702 .195875E+02 .568009E902 .399860E001 .3238125701 .303k76E901 .2775315001 TEST POSEXMY POSEXHP POSEXHiFS POSEXH1YS POSEXHiYfi POSEXH1PM BAMSETTN BAHSETTX1 BAHSETTXZ BAHSETTY BAHSEOTN BAHSEOTX1 BAHSEOTXZ BAHSEOTY GLEJSERXl GLEJSERXZ GLEJSERY PIRKX1 PARKXZ PAQKY GOPN GQPX1 GQPX2 GQPY GQNN GONXl GQNXZ GONY TABLE C26: 6179* L .E! .15 17. 72. 22. 7;. ‘88. 698. 333. 612. 956. 6’2. 920. 5°9. bk. 139. 961. 937. 117. 29A. ‘69. 972. #7. 158. 861. 53. 81. 195. 781. 916. 753. 928. Q. 31. 6L9. 872. A89. 729. 1. 37. 867. 69:. 8k. 215. ‘ 16. 63. 053. 996. 777. 910. k. 79. 2&9. 6‘1. 9. 0?. 197. 5‘2. 225 Test Results, Model 14, Sample EJ§L .13 128. 126. 773. 7L? 3B9. 95. 992. 958. MEAN .111136‘731 .11128~E+Cl .393236£+31 .3369JZE+01 .987838Z+01 .6767625+91 .29C56JE791 .16359SE+CZ .9317825701 .1969696902 .3102655+01 .1739165+02 .3761295751 .153557E+02 .778690E+01 .785875E000 ,.632836E+01 .8629795+01 .92874550CC .7853“6E+01 .1722665+01 .9630075700 .7314166+c1 .6642756+01 .6157306+o1 .7332646+11 .9157238931 .68978£E+C1 VAQIANCE .11198257U1 .109532E+01 .6769755+01 .7921855061 .9715322+62 .760959E+02 .9193thOD1 .551225E902 .171687E002 .5392265602 .182889E+52 .5875985032 .1hh3h857J2 .5962725+02 .1662C25782 .9931142000 .937615E+01 .997836E002 .1869885+01 .374h165+02 .6155956+3a .3616796+0u .1569195002 .’56356E+01 .2283636701 .381559E+01 .3127985001 .QCGBBOE+O1 Size 60 59699655 .2116326+a1 .2155A8C701 .5617335681 .5622166+a1 .1001596+02 .1067756+62 .1969156+01 .6296176+00 .1763985781 .7726226o60 .217789E701 .5996636oso .2201696+c1 .6766285930 .1619136+01 .3717515701 .11C9A9E001 .2062356761 .3766166+01 .1667666+01 .2722705761 .1675035901 .1636666+o1 .1606326+61 .9966146060 .326797Eona .2750166+ao .9160C8EOOG 90970515 .9950336oo1 .9350506001 .61172850J2 .569133E602 .1296658003 .1569966+03 .8718066oo1 .397919E+01 .716975E091 .3621036+01 .106999E702 .326559£+31 .100572E702 .3500216001 .7995886001 .2399538002 .595h606001 .1695626+02 .2795615002 .6767676.61 .1980308002 .697h9TEOO1 .697698E001 .6396895601 .2823165001 .3237188001 .383889E001 .321917E001 226 TABLE C27: Test Results, Model 14, Sample Size 90 41994 16V61 7657 .01 .05 .10 9649 94914906 59699655 90970515 9056997 32. 71. 131. .109710F+01 .1161156+01 .2177756+a1 .977605£+D1 9056999 23. 62. 116. .1053626+01 .1138985601 .262709E701 .13202?E*02 905699195 651. 885. 956. .b38968E+01 .7696266+01 .Q33193E+J1 .3181926002 905699175 536. 660. 946. .6253636+01 .6566516+01 .9161296+01 .3030256902 905699179 626. 877. 964. .6365466+01 .1074036+02 .5169016+01 .6073966+02 905699199 636. 661. 969. .4566916+01 .1356736+02 .5301926+01 .39792thDZ 64956779 66. 161. 266. .3276556+01 .1067516+02 .1709536+01 .661625€+01 649567791 960. .997. 1000. .2566796+02 .6562396+02 .6100196+00 .3756096+01 649567792 96. 223. 301. .3963906+01 .1503666+02 .1751936+01 .6606106+01 64956777 961. 982. 993. .2152505002 .7560936+02 .5926376+00 .3237256+01 64956079 60. 157. 269. .3176016+01 .1004696+02 .1635006o01 .618183E+01 649560791 981. 998. 1000. .2710156+32 .9669066+02 .6267566+00 .3729A9E+01 649560792 86. 195. 270. .3550396+01 .1307715702 .1661696+01 .7771666901 64956077 940. 983. 996. .2261679602 .6326796+02 .6020726+00 .3999568001 616356991 967. 997. 999. .119316F002 .2264216+02 .1096066101 .50873ZE981 616356992 15. 34. 68. .912822E+DD .1121076+01 .3516966+01 .2202276902 61635697 909. 989. 996. '.898923EOOL .1104926+02 .9560016+00 .5502766101 949991 766. 906. 957. .1366016+02 .798131E702 .1702756+o1 .83297hE+01 949992 9. 99. 76. .9566616+00 .1752266+01 .2621906oo1 .1269555002 94997 720. 877. 970. .1206566+02 .5862?ZE+02 .1094746+01 .6609966+01 6099 63. 215. 326. .1521216+01 .2765566900 .2056596+01 .8801796901 60991 1=. 53. 95. .9695536+00 .209299E000 .1506666+01 .6951166o01 60992 997. 1000. 1000. .729909E+01 .1019SZE+02 .1606736+01 .986227E001 6097 666. 965. 976. .4220736+01 .395589E+01 .1276916+01 .5632636o01 6099 18. 56. 165. .4659006901 .2961566+01 .3761766+0o .2936556+01 60991 476. 670. 636. .6559006+01 .6999066+01 .3546506+00 .327920E001 60992 20. 60. 123. .6306006+01 .3505196o01 .926889E900 .2967h58001 6097 377. 566. 761. .799300F701 .9311266o01 .2300666+00 .2979658901 227 TABLE C28: Test Results,.Mode1 15, Sample Size 30 41994 LEVEL 7657 .01 .05 .10 9649 94914966 59699655 90979575 9056997 .2. 4. 27. .7267656+oo .3973546+43 .1953526+01 .9710216+61 9056999 5. 25. 57. .6727446440 .6653436+40 .2359696641 .1269666+02 905699195 5. 26. 55. .6924566640 .6632296+oa .2225096401 .1116956+oz 905699175 16. 46. 76. .9726646+ou .6645146+oo .2701936601 .1397796+02 905699179 11. 51. 61. .9716966+60 .6637266+oo .2643566+o1 .1196555902 905699199 2. 27. 56. .6766666444 .6251326+oa .1967676+01 .8969065?01 64956779 50. 142. 226. .3062936401 .6626906+o1 .1836779601 .7717666401 649567791 53. 146. 226. .3101965901 .6667696+01 .1605176+01 .7524236601 649567792 55. 160. 251. .3314546+01 .9396416+01 .1696056+01 .6626426601 64956777 75. 169. 261. .3577906+o1 41371986702 .1565966+01 .599607E*01 64956079 66. 196. 296. .3606956+01 .1156606+62 .1791176+o1 .808829E601 649560791 53. 153. '235. .321198F+01 .9776978701 .1797966+01 .7069926+01 649560792 61. 156. 253. .3326516+o1 .1019196+62 .173017E+01 .6725675901 64956077 76. 244. 349. .371511r+o1 .1166966ooz .1633736+o1 .6291646+01 616356991 6. 16. 67. .6961526400 .6310716+no .3964626+o1 .6097936002 616356992 65. 119. 169. .1530626601 .6522506+01 .3609516+01 .ZblzhiEOOZ 61635697 2. 26. 53. ‘4913336F70? .7750556+oo .2066656+01 .9112136+o1 949991 23. 51. 96. .115038E+01 .6625606+o1 .6301756+41 .2740766+42 949992 7. 37. 62. .990795E+00 .2625166661 .5492166401 .5166936+02 94997 16. 59. 104. .1162266+01 .3215136+01 .3309076+u1 .1761636+oz ‘GOPN 119. 276. 660. .2627936+c1 .5651176+01 .5092966+01 .4655336o02 60991 127. 262. 360. .2400166+01 .7019676+01 .3316916701 .1669766402 60992 72. 159. 235. .1735756+01 .3066456+01 .6050136oo1 .3666676+02 6097 69. 130. 166. .1571936401 .3297396+41 .6671596oo1 .6161166+42 6099 21. 65. 176. .3116006601 .2262366+41 .6269656+00 .308713E601 60991 26. 94. 236. .3567006+o1 .1967666461 .6609266+00 .3016556+01 60992 17. 65. 119. .2676006401 .2335366+o1 .6439046+oo .3366996401 6097 30. 63. 236. .3596006o01 .1936726+o1 .575993E600 .3630218601 TEST POSEXMY POSEXMP POSEXH1°S POSEXHIYS POSEXHlYH POSEXHipW BAHSETTN 649567791 649567792 64956777 64956079 649560791 649560792 BAMSEGTY 616356991 616356992 61635697 949991 949992 94997 5099 GQPXl 60992 6097 6099 9 60991 60992 GQNY 226 TABLE C29: Test Results, Model 15, Sample Size 60 f‘L’Hf‘ L§J*L 4'1 1. 27. 19. he 2“. .l5 .1: WEAN ob615515+ic .657598£+50 .7166705+OE 47397595+£3 4827782E+£3 .6366526+00 .3066166+01 .3553685931 43352955+91 .3621815791 .33769QE+C1 .3650986+01 .34C7l?5+01 .3721596+C1 .195020E921 .775121E+00 08982185+00 4129798E+01 .190559E+01 .895161£+00_ 61635725+£1 .107887E+C1 .1466496+o1 .1364136+o1 .3944006+41 .thhdJanl .3329066+o1 .6955606+o1 VARIANCE 43633555066 o3707355’03 017833“E*0& .1862295703 6361151E+00 .3“389#E+00 .897592E+01 .118369E792 .1031536+02 .1216696+02 .1125056602 .1266906+02 .1C78855402 41251895+02 449697525730 .6666736+oa .6669666+00 .6166676+01 .3529966+01 .1702726+o1 .5367216+00 .6636666+06 4869351E+d3 .80631BE+00 .2677598091 o300h895001 42565395701 42779755001 SKEHNESS 41662415001 42799093701 41172305001 0125Q77E§01 03686765+01 63726485+01 42156305+01 41725796701 .1659285+G1 4172316£+01 .2088335901 .16&BZDE+51 .172996E7C1 .161696€+51 61968735901 .2669725901 4176527i+01 .958912£+01 42918166901 .2951§3E+01 42532125901 42021266901 61783315001 42977825701 6346259E+OC 45703916700 .hBB392E‘00 49263525+06 KURTOSIS .693511E701 41736076902 oh92595£601 .531606E001 4279587E+02 .300k296002 41093585902 o693387€*01 46339985701 4673681E001 49787985701 66366946001 .6912338+61 45756265001 65611315001 .6136116E902 47972535701 69279996+02 .1926625002 61952228732 .129138E002 49959295001 47399875601 6297525E002 6260522E001 .3397825701 63916968001 63137668001 7657 9056997 9056999 905699195 905699175 905699179 905699199 64956779 649567791 649567792 64956777 64956679 649560791 BAHSEOTXZ 64956077 616356991 616356992 GLEJSERY 949991 949992 94997 6099 GOPX1 GOPXZ GQPY GQNN 60991 GONXZ GQNY TABLE C30: 229 Test Results, Model 15, Sample Size 90 3L3H5 LEVEL .51 66. 119. 62. 6°. 51. 11“. 12. 59. .05 13. 13. 137. 191. 2‘3. 6“. 16. 154. .. 0". 26k. 266. 2&3. 337. 131. 29. 157. 22L. 9549 .6931895+C0 .6617156+60 .7956006+00 .7769776+66 .7667526+60 .6075656+cu .3616966+01 .3361 66+31 .3269526+c1 .1167356+01 .3529366+01 .3611336+01 .3296626+61 .6166756+91 .1191566+01 .8259335+56 ‘.1266555+01 .1966106+01 .1165666+01 .1199166+61 .1565766+01 .9624146+66 .156113E+01 .1319666+01 4633600E+01 .5337666+01 .3775306+c1 45577055+81 74914906 .3767676+cu .3956666+:a .2176136+00 .2297696630 .2314366+00 .2165138700 .1360005+82 .1637676+62 .12713ZE+32 .1652566+02 .1173266+02 .1067676+02 .1236666+02 .1522636+02 .1192005701 .939716£+00 .1029546+01 .7700266o01 .2424656+01 .2611966+01 .3076726400 .1965636+06 .5963576+06 .3577726+34 .2619266+01 43253615701 .2766766601 .3h81555781 SKEHNESS 41758565+01 .191k765+31 61257375791 4135867E+C1 41385385+61 .12h783i+31 42379685+01 o166237£+01 .2h8#055+61 414799BE+31 4197719E+Ci 41659715701 42336735+31 61519965+91 6165178E001 o1h63h1i+fi1 61568238001 635623257fl1 4229355€+01 4255261E§31 41859696+61 61265855+01 6163919E+91 41193165701 6371859E+OJ .6123fl35000 oh31656E+Ob oZfiGBCOE+DO 90970515 .677373E051 .6163206+01 .530980E761 .537775E901 .5506636+01 45299908901 .1015696+42 .6226666+61 .1345236+02 6592267E701 4888076E+01 .6096236oa1 .1211536o02 .5635396+01 .6662696401 .6163576+01 .6526666+61 .251855E+32 .9592536+01 .167099E+02 .73837SE001 '.6692726601 4795521E+01 49495555931 .298035E901 .291Z9QE501 63070975931 .ZTGQZ6E601 230 TABLE C31: Test Results, Model 16, Sample Size 30 ALOHA LEVFL 7657 .01 .05 .10 9649 74914906 59699655 90970575 9056997 15. 51. 11a. .113778E+01 .1076646+o1 .2172066+o1 .1062526+02 9056999 13. 52. 107. .1121176+61 .1098728901 .2067636+a1 .93990%E+01 905699195 43. 65. 125. .1212426+u1 .1994166+61 .4134676o01 .2761046+02 905699175 40. 80. 121. .1167526+u1 .1976556+o1 .0586315901 .3361976+02 905699179 56. 120. 165. .1529196+o1 .526159E+01 .607956E+01 .568568E902 905699199 70. 132. 176. .1551989901 .4520756+01 .993971E+01 .3671506+02 64956779 10. 97. 109. .205905E901 .4204666+01 .2040736+01 .9564306oo1 649567791 12. 52. 96. .2013926+n1 .0056016+o1 ‘.1954776+n1 .8099785601 649567792 16. 50. 1n1. .2121165901 .0559326+o1 .2179906+o1 .1001996+oz 64956777 13. 51. 95. .1996336+o1 .906899E+01 .1971466+01 .8009795001 64956079 17. 59. 105. .2136126+o1 .0720109+o1 .2159576oo1 .1035066+02 649560791 19. 56. 111. .2117616+o1 .9302796+o1 .1695736+o1_ .7706726+o1 649560792 16. 66. 123. .2190499+01 .992062E+01 .2692126+o1 .9655306+o1 64956077 12. 59. 106. .2090556+o1 .6261116oo1 .188530E901 .77561u6+o1 GLEJSERX1 7. 37. 62. .1033476+o1 .107959E+01 .2326875001 .1161616+02 GLEJSERXZ 9. no. 69. .1033676+o1 .139223E+01 .3740296oo1 .29921SE+02 0L6J5697 2.‘ 36. 74. .985382E+00 .9320996ooo .2031055001 .8739038901 949991 10. 56. 109. '.1106056+c1 .261567E601 .2631666o01 .1166666+02 949992 9. 06. 102. .1063336+n1 .2454416+o1 .3376266+o1 .219069E902 94997 15. 55. 109. .1099999+o1 .3066696+01 .3369066+01 .160099E902 0099 62. 165. 306. .2167766901 .309822Ef01 .5766976+o1 .5590795602 00991 32. 90. 157. .1371666+01 .1939295+01 .«991976+o1 .501986E902 00992 26. 67. 161. .1351626+o1 .1369556+01 .2963926oo1 .1626016+02 0097 57. 190. 216. .1799356o01 .3365666+o1 .4269296+01 .329062E+02 0099 , 29. 66. 164. .3046066+01 .2936136+o1 .9125646oun .3029536+01 00991 27. 6k. 165. .3106009+o1 .252986E901 .4275276+oo .3199366+o1 00992 16. 59. 163. .300300E+01 .2391366+u1 .9351236900 .3197908901 0097 19. 62. 166. .3oosoo6oo1 .2311296+o1 .9716576ooo .3050988001 TABLE C32: TEST POSEXMY POSEXHP POSFXH1PS POSEXHIYS POSEXHlYH POSEXH1PH BAMSETTN BAHSETTX1 BAHSETTXZ BAHSETTY BAHSEOTN 8AMSEOTX1 BAMSEOTXZ BAHSEOTY GLEJSERX1 GLEJSERXZ GLEJSERY PARKX1 PARKXZ PARKY GQPN GOPX1 GOPXZ GQPY GQNN GQNX1 GONXZ GONY 231 Test Results, Model ALPHA LEV‘L .01 8. 8. 9?. #3. 99. 97. 18. .05 9?. 92. 13. 77. 170. 169. S6. #7. 99. 61. 61. 53. ‘01. 26. 36. 61. 52. 67. 12“. 75. 67. 180. 69. 61. 69. 67. .10 103. 100. 10‘. 109. 222. 228. 98. 101. 110. 87. 111. 119. 117. 10“. 105. 63. 98. 116. 109. 121. 229. 136. 122. 263. 157. 144. 137. 199. fiEAN .1013505601 .101433E+01 .119065F901 .113912E991 .2009165901 .2000.66+01 .205277F+01 .2095685601 .2063805901 .193586F+O1 .2136D7E+31 .7137995901 .212759E+01 .202173E001 .11ouu7roo1 , .90793QE+00 .1097965*91 ..112926E901 .1064536.o1 .1165796oq1 .1513096.o1 .1130256+o1 .1139306901 .1k95665601 .3763006+a1 .369509E901 .370900E001 .372200F901 VARIANCE .7868675900 .291159E+01 .2936516+01 .2152636.02 .216959E902 .9566666+01 .9593136+o1 .9316766+o1 .3945276+01 .9967096+o1 .0666516+o1 .9791946+o1 .39%BSOE+01 .132931E901 .76105~6+00 .1063176+o1 .279257E+01 .2544536+61 .3106746+u1 .266‘T1E+00 .3323825900 .392938E+00 .636120E+00 .307999E+01 .3015996+01 .3091316001 .3119!“E+01 16, Sample Size 60 SKEHNFSS .1931596901 .195191E+01 .981522E+01 .982799E001 .105393E+02 .1089696+02 .2173856901 .2185176031 .219895E901 .1699168001 .231875E+01 .236063E*01 .2291088+01 .1937625+01 .2‘1261F+01 .2978835001 .2703395901 .2986356601 .285767E+01 .3122355901 .218121E+01 .1h87326901 .169695E+01 .2182095601 .h331025f00 .5176636900 .h18737E000 .5792526000 KURYOSIS .8957515+01 -.891259€901 .1562226o03 .15671GE+03 .1571955903 .1688298003 .9611606+01 .1039576.02 .10851.6+02 .6576966+01 .1159766+02 .1337265902 .1195066.02 .8671178901 .,137293£.oz .132995E+02 .167911E902 .159057E+02 .191309E002 7.1TOhZOE+02 .105585E+02 .6186685001 .7697596901 .1300056002 .3259978001 .3966996001 .3928356901 .3k55185901 TEST POSEXMY POSEXMP 905699195' POSEXHiYS POSEXH1YH POSEXH1PH BAMSETTN 8AMSETTX1 BAMSETTXZ BAHSETTY BAHSEOTN BAHSEOTXi BAHSEOTXZ BAMSEOTY GLEJSERX1 GLEJSERXZ GLEJSERV PARKX1 PARKXZ P‘RKY GQPN GOPXl GQPX2 GQPY GQNN GQNX1 GQNXZ GQNY TABLE C33: 41994 L .11 .65 9. 42. 6. 41. 25. 67. 26. 63. 26. 79. 29. 76. 11. F3. 4. 52. 1o. 52. 14. 61. 7. 51. 6. 99. 12. 49. 19. 61. 1o. 97. 6. 31. 1o. 49. 13. 52. 6. 56. 17. 63. 26. 99. 11. 52. 15. 65. 67. 207. 16. ‘ 93. 12. 32. 13. 33- 10. 39. Test T127. 165. 72. 179. 160. 152. '169. 2:1. 163. 149. ‘321. 96. 97. 66. 870 232 MEAN .9853BGE+00 .9956905+00 .15771GE+01 .108737E+01 .1190335901 .113199E+01 .1995925+01 .198OZZE+01 .197796E+01 .2106988+61 .200319E+01 .209565E+01 .2029225901 .217189E+01 .180776E+01 .992760E900 .1072195+01 .109787E+01 .1061365901 .1993236o01 0 13.710 35’0 1' .1464526401 .106287E+01 .191269E+01 .9131OOE+01 .9079006+01 .9610006+61 .3996036+o1 74974906 .7190596+uc .6995265900 .795908E+00 .773726E+00 .898575E+OG .8274895+00 .9361316+01 .3763196+01 .363391E901 .9326906+01 .391039E+01 .3996596+a1 .919021E+01 .966999EOJ1 .1031726+o1 .896399E+00 .1063396+n1 .2989836+01 .2197525001 .2591655+01 .1216666ooo .1626176+oé .1609226+on .2940966+oo .350139E001 .3250016+u1 .3265176+01 .3365366+u1 Results, Model 16, Sample Size 90 SKEHNESS .191397E+01 .159261Eto1 .258159E+01 .262233E+01 .269973E+01 .2622628+01 .197287E901 .166237E+61 .180753E+01 .1788225+01 .1765606+o1 .10?235£+01 .209093E+01 .2044656on1 .1653056+o1 .2064756+n1 .1661176+o1 .3365226+o1 .2496166+o1 .2792576+o1 .2156696+o1 ..1275736901 .136981E901 .106085E901 .3066735+03 .3129696+00 .9095616+00 .280209E900 KURTOSIS .998713E601 .616395E901 .1906978002 .1fl2858E902 .1505988902 .1953035902 .859271E901 .636138E+01 .6951875901 .6908998001 .6795306001 .798063E901 ..9397775931 .9551156oo1 .7393926+o1 .161725E002 .6316906.n1 .2197596+42 .1076556902 .1296236o42 .101272E902 .656997E901 .6209876001 .998992E901 .2991625901 .2932665001 .308788E901 .2796066901 TEST POSEXMY POSEXMP POSEXH1PS POSEXHIYS POSEXHlYH POSEXHiPH BAHSETTN BAHSETTXI BAHSETTXZ 64956777 649560TN 649560791 . 649560792 6495601Y GLEJSERX1 GLEJSERXZ GLEJSERY 949991 949992 94997 0099 ' 00991 00992 0097 0099 00991 00992 0097 233 TABLE 034: Test Results,'Model 17, Sample Size 3.0 ALPHA LEVEL . 1 13‘0. 150. 3060 303. 342. 339. 494. 666. 61. 261. 664; 955; 99. 3776 526. o. 06. 340; 0. 1‘0 651. 54; 996. 662. .26. 664. 13. 93; .05 238. 267. 450. 464. 490. 486. 722. 967. 207. 482. 860. 993. 226. 580. 825. 2. 218. 576. 12. 89. 931. 131. 1000. 635. 660. 661. 36. 225. .10 323. V348. 572. 583. 599. 567. 812. 988. 302. 9649 .243880E*0£ .266537E*01 .495437E‘0! .5633915‘02 .8990805901 .6956465901 .966924E901 .1756756002 .376556E’0! .6607796*0! .1241416402 ,215965E202 ,435740E20£ .3232106401 .6556206902 .4931676400 ‘.245489E‘Oi ,715422E'01 .7695356900 .1742055‘0! .238891E*02 .1525886'0! .1653736208 .1164395‘02 .6264005‘0‘ .7377OUE*0t 6267200E‘Ot .4539006001 ngIANCE .9660766451 01407675032 08975185*52 .158067E963 .58847IE*53 .1730966453 .3600966452 .4654266462 .1372666462 .2909996462 4377359E‘62 46§15535052~ .1506396462 63480235’62 .1544636462. .2916366460 .5535036451 .44673395‘52 .6511326460 .2394356461 01615515‘647 .4549596461 .2616766465 .259157E053 .2350666461 .2657736461 .2i96616461 .2164666461 SKEHNESS :3317726401 ;§63664E401 21672746402 31135466402 59086626001 :6402376401 .7636308400 33675576400 .1673306401 .1i66506401 .6343166400 .7346405400 $196759E401 .9065315400 .1966366401 22369636401 .2423395401 :2296636401 .2525396401 31676668401 .6416665‘01 '.5676336401 0563454E‘Q1 .5462388901 33513346400 '33353766400 .5536306400 .3525726400 90970515 .i661246402 .3692646402 .1914696403 .1695166403 .1204096403 .7362666402 .3665076401 .2937136401 .7630366401 .4754366401 .3656166401 .4266306401 .9150256401 .3660716401 .6765656401 .i012105902 .1069646402 .1044376402 .i163328902 .6622936401 .5679656402 .6113046402 .3756566402' .4766606402 .3449636401 .3265026401 .3163266401 .3160266401 TABLE C35: TEST POSEXHY POSEXMP POSEXP1PS POSEXH1YS POSEXHiYH POSEXH1PN BAHSETTN BAHSETTXI BAMSETTXZ BAHSFTTY BAHSEOTN BAHSEOTX1 BAHSEOTXZ BAHSEOTY GLEJSERX1 GLEJSERXZ GLEJSERY PARKXi PARKXZ PARKY 60PM GOPX1 GQPX2 GQPY GQNN GQNX1 GONXZ SONY ALPHA LEVEL .01 110. 112. 811. 816. 811. 809. 999. 1000. 151. 995. 963. 1000. 167. 999. 999. 36. 969. 903. 2. 886. 993. a... 1000. 996. 527. 758. 1. “Q3. .05 215. 218. 990. 938. 991. 937. 986i 1000. 290. 1000. 990. 1000. 302. 1000. 1000. 89. 996. 968. 29. 962. 999. 103. 1000. 998. 873. 962. 17. 30“. .10 302. 308. 973. 970. 967. 966. 995. 1000. 367. 1000. 996. 1000. 405. 1000. 1000. 132. 999. 979. 71. 963. 1000. 151. 1000. 999. 957. 990. 56. 928. 234 Test Results, Model “EAN .195895E001 .198651F+01 .762325F§01 .7668116901 .8959215901 .886169E+01 .227110F+02 .h3h898F002 .976176F+01 .330189E+02 .2h6233E+02 .9931505+02 .h9657hF+01 .38h135E+02 .199988E002 .1200h2F+01 .129019F002 '.270623E+02 .866615E000 .193193E+02 .1609696402 .107321E901 .519362E+02 .183750E+02 .873100E+01 .103110F+02 .2717005001 .8268006901 17, Sample VA°IANCE .h129335+01 .h32616E+01 .k750005*02 .5006166+02 .1165966403 .112896E403 .6955656402 .1537696403 .210668E+02 .1401716403 .1019626403 .1606225403 .2267526402 .1500696403 .9597976402 .251879E401 .3156096402 .4340016403 .1403615401 .132968E403 .1739305403 .85Z9Q3E+00 .185“55E+0“ .213989F903 .392956E+01 .5852138001 .2983396001 .387805E’01 Size 60 SKEHNESS .268692F+01 .262279F*01 .355011E+01 .3627636401 .0950996401 .5193626401 .5012139400 .28266°E+00 .181521E+01 .4635599400 .5690066400 .2465265400 .1795066401 .510028E+00 .1764156401 .2961316401 .1126756401 .1666966401 .2460955401 .1009659401 .4639659401 .2644125401 .2613025401 .303259E+01 .20296QE000. .3066106900 .5719136900 .220“5&E+00 KURTOSIS .1526235602 .1395h8E602 .2062205002 .21009BE+02 .3998999402 .99731SE+02 .3095338601 .310506E+01 .769837E401 .315910E+01 .3180888901 .3030295+01 .751509E401 .333297E401 .7798096+01 .1962h76902 .5566325401 .6226935401 .110525£+02 .9001556401 .510595E+02 .157016E+02 .195560E902 .199515E002 .273337E‘01 .2807766001 .3228255001 .27627SE+01 235' TABLE C36: Test Results; MOdel 17, Sample Size 90 TEST POSEXMY POSEXMP POSEXH1°S POSEXH1Y9 POSEXHiYWV POSEXH1PH BAHSETTN BAHSETTXi BAMSETTXZ BAHSETTY BAHSEOTN 9AMSZCTX1 BAMSEOTXZ BAMSEOTY GLEJSERX1 GLEJSEPXZ GLEJSERY PARKXi PARKXZ PAQKY GQPN GOPX1 GQPX2 GQPY GQNN GONX1 GQNXZ GONY ALCHJ LEVEL .11 13. 100C. 050. 2. 96?. 99h. 62. 19.0. 999. 709. 964. 766. ' C .4- 197. 1LC.‘ 992. 999. 397. 152:. 16.0. 697. 962. 9. 887 .1. 2(5. 219. 998. 996. lOLL. “96. 16(2. 989. E7. 999. 993. 13C. 101;. 10L;. 9‘1. 996. 31. 959. 9519' .1777515+01 .1676366401 .1C1791E+t2 .9582995+£1 .879772E901 .9332985+01 .277591E+UZ .757999E+CZ .627319E+C1 .56“423€+22 .353738£+02 .P:7518€+£Z .6923593+01 .602061E+02 .355716E+62 ..181659+08 ..1893963+32 05210935902 .7619505+03 .3135895002 ‘.117763£+02 .91k1195400 .5399666462 .2299965462 .98;3806491 .12691az+.2 .331chE401 .1G&ZBSE*52 VAQIANCE .3735135731 .2318685651 .2893255*02 .2251‘35732 .193385E‘32 6247667E+92 .146783E+03' .293265E+03 .383919E+32 .278728E4J3 .1805255+03 .3“9398€+33 .3982156+02 .3355615093 .2659775403 .115117E+J1 .3619156002 .909286E403 .119571E731 .2668715093 .653.19£432 .3675106400 .8269CCE+33 .208877E+03 .h898905001 .7931955001 .293106£*51 .5659885+01 SKEHKESS .2971965431 .292983E+31 .1555h95+01 .13559SE+61 .1“1314E+01 .16th75931 .6323135600 .3'33915000 .2219195‘01 .3121EZEOOC .h957155+20 .2970662400 .221612£+u1 .2747356400 .189879E+91 .2789755961 .71925OE+OJ '.1ZHSBCE‘C1 .2869375921 .6583765000 .2996255‘01 .1685735991 .1507066081 .19h1585+01 .315396E+08 .3392185600 .5226236+30 .2619b5E+00 90970515 .1160595402 .111597E402 .6567506401 .5635365.31 .5912616401 .6622665401 .3761016401 .2690066401 .1204635402 .2962626401 .332502E401 .2691965431 .1252915402 .2614336401 .6566326401 .19269BE+02 .39673ss.01 .9701575401 .1620185402 .3136606401 .206257E+JZ 665558h6501 .5999208001 .909522E001 .3103665‘01 .2883898901 .297920E+01 .3J737BEOO1 BIBLIOGRAPHY BIBLIOGRAPHY Abrahamse, A.P.J. and Louter,.A.A. 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