‘1‘. Act mflveizdx" . u c: BI . 1.95:. vial. :IISIii!((1(Irlztl litter vlrrltltpi . . .L‘rl. 1:11.011 W8 (.1 VI ‘H (97' 5 0 3 1293 006953 :1 H__ m LIBRARY 1 Michigan State’ University This is to certify that the dissertation entitled Preperties of Unit ?oot Tests with Heterogeneous and Dependent Errors presented by Kiwhan Kim has been accepted towards fulfillment of the requirements for Ph.D. d . Economics egree 1n % MP Major professor w “. 100/ Date J. eb 20, /,Q MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE 80 3’ 1997 MSU Is An Affirmative Action/Equal Opportunity Institution PROPERTIES OF UNIT ROOT TESTS WITH HETEROGENEOUS AND DEPENDENT ERRORS BY Kiwhan Kim A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1990 @0544‘qg ABSTRACT PROPERTIES OF UNIT ROOT TESTS WITH HETEROGENEOUS AND DEPENDENT ERRORS BY Kiwhan Kim The properties of unit root tests are considered when errors follow a generalized autoregressive conditional heteroskedasticity (GARCH) and a serial correlation in the variable's autoregressive representation. The standard Dickey-Fuller tests and various augmented and transformed versions of unit root tests are conducted. The primary concern is whether the proportion of rejections under the null hypothesis agrees with the nominal size of the test. We also consider the use of different spectral windows with different numbers of truncation terms in the long run variance estimate in Phillips-Perron test statistics. Finally Phillips-Perron tests based on the true values of innovation variance and long run variance are examined to help answer the cause of inaccuracies of Phillips-Perron tests. Careful cautions in interpreting the results of the existing unit root tests are suggested and further theoretical work seems to be necessary. ACKNOWLEDGEMENTS I would like to express my sincere gratitude to Professor Peter Schmidt, major advisor, for his continual encouragement and guidance throughout the thesis. I would also like to thank the members of my committee. They include Professor Richard T.Baillie and Professor Ching-Fan Chung. A special thanks must go to my wife, Guiyeum, and to two children for their understanding and sacrifices. Final thanks to my brother. iii TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . v CHAPTER I. INTRODUCTION . . . . . . . . . 1 CHAPTER II. VARIANTS OF THE DICKEY-FULLER TESTS . . 7 CHAPTER III. TESTS WITH CONDITIONAL HETEROSKEDASTICITY . 14 A. Design of the Experiment . . . . . . 19 B. Results of the Experiment . . . . . . 21 C. Conclusion . . . . . . . . . . 30 CHAPTER IV. TESTS WITH AUTOCORRELATED ERRORS . . . 32 A. MA(1) and AR(1) Error Process . . . . . 33 1. Design of the Experiment . . . . . 36 2. Results of the Experiment . . . . . 37 3. Conclusion . . . . . . . . 42 B. Lag Windows . . . . . . . . . 43 1. Lag Windows . . . . . . . . 46 2. Design of the Experiment . . . . . 49 3. Results of the Experiment . . . . . 50 4. Conclusion . . . . . . . . 52 CHAPTER v. TESTS USING THE TRUE VALUES OF 02 AND of . 53 A. Design of the Experiment . . . . . . 55 B. Results of the Experiment . . . . . . 56 C. Conclusion . . . . . . . . . . 58 CHAPTER VI. CONCLUDING REMARKS . . . . . . . 60 APPENDIX . . . . . . . . . . . . 63 BIBLIOGRAPHY . . . . . . . . . . . 102 iv LIST OF TABLES Table Page 1. Proportions of Rejections, 5% Lower Tail Test . . 63 (e, = .3, «>2 = .3, .6, .65, .69, .70) 2. Proportions of Rejections, 5% Upper Tail Test . . 64 (4», = .3, «>2 = .3, .6, .65, .69, .70) 3. Proportions of Rejections, 5% Two Tail Test . . 65 (, = .3, 2 = .3, .6, .65, .69, .70) 4. Proportions of Rejections, 5% Lower Tail Test . . 66 (6, = .1, <1»2 = o, .5, .8, .85, .89, .9) 5. Proportions of Rejections, 5% Lower Tail Test . . 67 (<1>, = .3, .6, .65, .69, .7, «>2 = .3) 6. Proportions of Rejections, 5% Lower Tail Test . . 68 (5,, 92) = (.3, .6), (.3, .65), (.3, .7) 7. Proportions of Rejections, 5% Lower Tail Test . . 69 8. Proportions of Rejections, 5% Lower Tail Test . . 7O (¢1= 03’ Q2 = 07) ' 9. Proportions of Rejections, 5% Lower Tail Test . . 71 (Q, = .1, «>2 = .85) 10. Proportions of Rejections, 5% Lower Tail Test . . 72 (, = .1, «>2 = .9) ll. Proportions of Rejections, 5% Lower Tail Test . . 73 (, = .3, Q2 = .65) 12. Proportions of Rejections, 5% Lower Tail Test . . 74 (Q, = .3, «>2 = .7) 13. Proportions of Rejections, 5% Lower Tail Test . . 75 (<5, = .1, «>2 = .85) 14. Proportions of Rejections, 5% Lower Tail Test . . 76 (9, = .1, «>2 = .9) 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. Proportions of Rejections, 5% Lower Tail Test . (00 = .01, Q, = .3, 92'= .3, .6, .65, .69, .7) Proportions of Rejections, 5% Lower Tail Test (Q0 = .01, 9, = .1, oz== o, .5, .8, .85, .89, .9) Proportions of Rejections, 5% Lower Tail Test . (a = -.8, -.5, —.2, o, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test . (a = -.5, o, .5, T = 100, 200, 500) Proportions of Rejections, 5% Lower Tail Test (p = -.8, —.5, -.2, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test . (p = -.5, .5, T = 100, 200, 500) Number of Negative Long-run Variance Estimates Proportions of Rejections, 5% Lower Tail Test (Bartlett's Lag Window, 1 and 1’, tests for a = —.8, -.5, —.2, of .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test . (Bartlett's Window, Tu and p1 tests for 9 = -08, -05, -02' 0’ 02' .5, .8) Proportions of Rejections, 5% Lower Tail Test . (Parzen' 5 Window, Tu and Tr tests for 0 = - .8 -.5, -.2, 0, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test . (Parzen's Window, 8“ and fl? tests for 6 = -.8, -.5, -.2, O, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test . (Bohman' 5 Window, Tu and T7 tests for 9 = -.8, —.5, -.2, 0, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test (Bohman's Window, p“ and p1 tests for o = -.8, -.5, -.2, o, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test . (Bartlett's Window, T and 77 tests for p = - 8, -.5, -.E, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test . (Bartlett's Window, a“ and p1 tests for p = -.8, -.5, -.2, .2, .5, .8) vi 77 78 79 80 81 82 83 84 85 86 87 88 89 9O 91 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. Proportions of Rejections,~5% Lower Tail Test (Parzen's Window, 1“ and 17 tests fOr p = -08, -05! -02, 02, 05, 08) Proportions of Rejections, 5% Lower Tail Test (Parzen's Window, g“ and p1 tests for p = -08, _05, —02, 02' 05, 08) Proportions of Rejections,A5% Lower Tail Test (Bohman's Window, 1 and 1, tests for p = -.8, -.5, -.2, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test (Bohman's Window, p“ and 97 tests for p = -.8, -.5, -.2, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test (ppt tests for 6 = -.8, -.5, -.2, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test (ppt tests for p = -.8, -.5, -.2, .2, .5, .8) Proportions of Rejections, 5% Lower Tail Test (ppt tests for 6 = -.8, -.5, T = 100, 200, 500, 1000) Proportions of Rejections, 5% Lower Tail Test (ppt tests for 6 = .5, .8, T = 100, 200, 500, 1000) Proportions of Rejections, 5% Lower Tail Test (ppt tests for p = -.8, -.5, T = 100, 200, 500, 1000) Proportions of Rejections, 5% Lower Tail Test (ppt tests for p = .5, .8, T = 100, 200, 500, 1000) vii 92 93 94 95 96 97 98 99 100 101 CHAPTER I INTRODUCTION It is commonly accepted that many economic time series are nonstationary. There are two very different models of nonstationarity, trend stationarity and difference stationarity, that have been considered extensively in both theoretical and empirical work. In the trend stationary model, a variable is viewed as being composed as a deterministic (often linear) trend and a stationary deviation from the trend. In this case the variable can be detrended by regressing it on a suitable function of time, and the residuals of the regression are considered. as a stationary series. A. trend stationary variable has a tendency to revert to the trend, following a shock, and so it is meaningful to talk about the prediction of the series, even long into the future. The difference stationary (or stochastic trend) model sees the variable as an accumulation of stationary differences. In this case, following Box and Jenkins (1970), differencing the series yields a stationary series. A random walk is a leading example of a difference 2 stationary model, but a random walk has independently and identically distributed (iid) differences, and in the general difference stationary model we require only that the differences be stationary. In this model a shock to the variable will persist over time, and.the uncertainty about the level of the variable grows larger as the horizon of the forecast increases. A difference stationary series contains a unit root in its autoregressive representation, and so a difference stationary series is also said to have a unit root. Because the distinction between trend stationarity and difference stationarity is important both theoretically and empirically, there~ has been considerable interest in determining whether common economic time series are difference stationary or trend stationary. The recent interest in this empirical question was stimulated by the work of Nelson and Plosser (1982) and Nelson and Kang (1981, 1984), who questioned the prevailing View (at that time) that economic time series were trend stationary. In particular, Nelson and Plosser applied the tests of Dickey (1976), Fuller (1976) and Dickey and Fuller (1979) to test the hypothesis of difference stationarity in a large number of annual U.S. time series. They found strong evidence in favor of the unit root hypothesis, in that the unit root hypothesis could not be rejected for 13 out of 14 series considered. 3 The Dickey—Fuller tests will be described in detail in the next chapter. Briefly, they involve the regression of the variable on its own lagged value, and testing whether the coefficient of the lagged value equals unity. The regression may or may not include an intercept or time trend. The test does not follow the usual distribution theory based on the normal or t—distributions, even asymptotically. Dickey (1976) and Fuller (1976) provide the necessary' distribution. theory, but ‘under’ the restrictive assumption that the errors in the regression are iid normal. More recent work has extended the work of Dickey and Fuller in many different directions. The proper trearment of trend (or, more generally, other exogenous regressors) in the Dickey-Fuller regression has been discussed by Evans and Savin (1981, 1984), Nankervis and Savin (1985, 1987), Schmidt (1989) and Schmidt and Phillips (1989), among others. Tests of joint hypotheses (such as unit root and intercept equal to zero) have been proposed by Eflckey and Fuller (1979) and Perron (1988). The robustness of the Dickey-Fuller tests to non-normality has been studied by Godfrey and Tremayne (1988) and Schmidt (1989). Extensive reviews of the unit root literature can be found in Dickey, Bell and Miller (1986) and Diebold and Nerlove (1989). In this thesis we are concerned with the consequences of the failure of the assumption that the errors are iid. The asymptotic distribution of the Dickey-Fuller statistics 4 has been derived by Phillips (1987) and Phillips and Perron (1988) under assumptions that allow fairly general forms of heterogeneity and autocorrelationa .An immediate implication of these.results is that heteroskedasticity and distributional heterogeneity do not affect the asymptotic validity of the tests. However, autocorrelation changes the asymptotic distribution of the test statistics and therefore must be accommodated in some way. This has been done in two ways in the literature. The first way in which autocorrelated errors are accommodated in Dickey-Fuller tests is to use the so-called augmented tests of Said-Dickey (1984, 1985) and Solo (1984). These tests add lagged differences of the variable to the regression, and are valid asymptotically for series with a general autoregressive moving average (ARMA) representation. The second way in which autocorrelated errors are handled is to find a transformed version of the test statistic which has asymptotically the standard Dickey- Fuller distribution. These tests based on transformed statistics are typically called Phillips-Perron tests, after Phillips (1987) and Phillips and Perron (1988), and are also valid asymptotically. They require consistent estimation of two nuisance parameters, as will be discussed in more detail in Chapter 4. The asymptotic properties of the Dickey-Fuller tests in the presence of heteroskedasticity and autocorrelation are 5 therefore well understood. Their finite sample properties are not well understood, however. In this thesis we use Monte Carlo methods to investigate the finite sample properties of Dickey-Fuller tests in the presence of heteroskedasticity and autocorrelation. We are primarily concerned with the accuracy of the tests, in the sense of whether the proportion of rejections under the null hypothesis agrees at least roughly with the nominal size of the test. We hope to provide some guidelines as to how well the various unit root tests can be expected to perform in practice. We also hope to identify circumstances under which they can be expected to be accurate and circumstances under which they cannot. The plan of the thesis is as follows. Chapter 2 describes the tests to be considered in this thesis and introduces some necessary notation. Chapter 3 deals with the unit root tests when errors follow a generalized autoregressive conditional heteroskedasticity (GARCH) process. The main interest is to see how the tests perform in conditions under which the asymptotic theory almost breaks down; namely, when the GARCI-I process is close to being integrated and when the intercept in the GARCH process is close to zero. Chapter 4 investigates the sensitivity of the Dickey- Fuller tests to autocorrelated errors. Schwert (1989), Godfrey and Tremayne (1988) and Phillips and Perron (1988) 6 have shown that the Dickey-Fuller tests and their Phillips— Perron corrected versions are rather inaccurate, even for very large sample sizes, when the errors are highly autocorrelated. Their experiments used estimates of the "long run variance" parameter based on the method of Newey and West (1987), which basically amounts to the Bartlett (1950) window in the spectral estimation literature. We also consider the use of the Parzen (1961) and Bohman (1961) windows, with different numbers of terms in the sum defining the long run variance estimate, to see whether the use of a different spectral window can improve the Phillips-Perron tests. In Chapter 5 we consider Phillips-Perron tests based on the true values of the innovation variance and the long run variance (which would be unknown in actual applications) to see whether the inaccuracies of the Phillips-Perron tests are due primarily to problems in estimation of these nuisance parameters, or whether they are intrinsic to the asymptotic theory for the tests. Finally, Chapter 6 contains our concluding remarks. CHAPTER II VARIANTS OF THE DICKEY-FULLER TESTS In this chapter we present the Dickey-Fuller tests and their transformed versions which will be employed in the following chapters in investigating the reliability of unit root tests. The tests based on White's (1980) covariance matrix correction, the augmented Dickey-Fuller tests (Said- Dickey (1984)) and the Phillips-Perron tests (Phillips (1987) and Phillips-Perron (1988)) are introduced, as well as the standard Dickey-Fuller tests (Fuller (1976) and Dickey (1976)). The main interest in this thesis is the performances of the standard Dickey-Fuller tests and their various modifications under different error processes. We now describe these tests. All of the tests are based on either regression (l) or regression (2) below. (1) -Yt=a+BYt-,+ut t=1, ...,T, (2) Yt=av+ BY,_,+6t+ut t=1, ...,T. Regression (1) is a regression of the variable , say "Y", on intercept and its lagged value, while regression (2) adds a time trend. In either case the unit root hypothesis corresponds to B = 1. 8 (a) The standard Dickey—Fuller p“ test is based on the statistic T(B-1), where B is the OLS estimate of B in regression (1), while the 2, test is based on the same statistic except that it uses the OLS estimate B from regression (2). Similarly, the Dickey-Fuller Tu and TT tests are based on the t-statistics for the hypothesis 6:1 in regression (1) and (2) respectively, that is T = (fi- 1)/sd(fi) where sd(B) is the standard error of the OLS estimate 3. (b) White (1980) presented a covariance matrix estimator which is consistent in the presence of heteroskedasticity. Although the asymptotic theory for this covariance matrix correction does not extend to the case of a unit root in the variable or in its variance, we expect that the White tests will help in the conditional heteroskedasticity error model. We denote these tests with A the suffix ".w"; that is, Tuuw and.1,.w. This correction is not available for the Sn and 2, tests, based on the estimated OLS coefficients, because these tests do not use an estimate of the variance of B. (c) We consider the augmented Dickey-Fuller tests (Said-Dickey(1984)), including either one or two lagged changes in Y in the regression. Said and Dickey(1984) proposed unit root tests using an approximation of an autoregressive-moving average model by an autoregression. 9 They argue that an unknown ARIMA(p,1,q) process can be approximated by an autoregressive model if the lag length (2) in the autoregressive model increases with the sample 1“ [i.e. 2 = size T, at a controlled rate less than T 0(TV3)]. However, this rule obviously does not specify the precise value of z for a given T. Said and Dickey use the Dickey-Fuller regression t test for a unit root (B = 1) in the "augmented" model c (3) AY = a + BYFJ + E rz.AYv.‘+ u“ where AY, = lit-flit,1 . The t-statistic for the hypothesis B = 1 in this model has the same limit distributions as those tabulated by Dickey (1976) and Fuller (1976) when 2 ~ m in (3) as T ~ m, at the rate given above. We denote the tests based on the augmented models with the suffixes ".sdl" and ".sd2", respectively (e.g. Tu.sd1 is the t-statistic when one lagged AY is added to the regression (1) above). (d) Under weakly dependent and heterogeneously distributed errors, the limiting distributions of the regression coefficient and the t-statistic are represented in terms of standard Brownian motion in Theorem 3.1 in Phillips (1987). The limiting distribution of the test statistic 2 depends on the variance ratio (mi/oz). Here cu is the 2 "innovation variance" and a is the "long run variance", 10 defined as follows: T (4) of = lim(l/T) 2 var(ua, T-«n t=1 r (5) c3 = lim(1/T)var( 2 ug. T-oo t-l By using consistent variance estimators SN}, 8“? (for 02) and Sf, S2 (for of), Phillips (1987) and Phillips and U Perron (1988) provide transformed versions of the test statistics which have the same asymptotic distribution as the standard Dickey-Fuller statistics. When the innovations are independently and identically distributed, of:= 02, transformation of the test statistics would be unnecessary. Here 2(3u), Z($T), Z(Tu) and Z(TT) denote the transformed A statistics for 3“, 2“,1, and.T1, respectively. The new test statistics they propose are the following. 1' (6) Z(3,,) = T034) - (1/2) (étf 4,2) / {T'Z 21m - 37V), . t= T 2 A A (7) up.) =- (is-1) { 2 (YM - 9-.) 1 - (1/2)(s,,2 - sf) t=I T [8.41” 2 (Y. WW)". t=1 A 2 _ 4 T A 2 -1 a T A A A . where 8,, — T Z ut + 2T 2 ws, 2 utum (ut 1s OLS 1 residuals from regression equation (1).), 11 — _ 4 y, — T 2 Yp,, standard error of regression equation (1), S B = OLS estimate from equation (1). (8) u?) = TUE-1) - <1/2)(§.,2 - Efm". (9) 2m) = (B-l) / {stdcnm} - <1/2)(s.,2 - 8.3) (gm Mu? ,4, where c,== first diagonal element of the matrix (X')()'1 (X denotes the Tx3 matrix of explanatory variables in equation (2).), B = OLS estimate from regression equation (2), T l T ~ 2 _ -1 ~ 2 '1 ~ ~ ~ 0 5:: — T E ut + 2T 2: ws 2 ut 11,,s (ut is t=1 s=1 t=s+1 the OLS residuals from regression equation (2).). Sfr= standard error of regression equation (2), T T M = (1 - T'2)T’2( 2 Y,"’) - 12(T’5)( 2 “dz + t=1 t=l i t 12(T" + T'5)( 2: tY,)( 2 Y.) + (4'1"3 + 6T" + {=1 t=l T 2T‘5)( E Y,)2. t=1 12 We consider the so-called Phillips-Perron tests (equation (6)-(9)), using either one or two terms in the estimation of the long run variance 02. We denote these with the suffixes ".ppl" and ".pp2", respectively. Like the augmented Dickey-Fuller tests, the Phillips-Perron tests are designed to handle residual autocorrelation. The regression equations considered in this paper are equations (1) and (2); that is, one with an intercept and the other with an intercept and a time trend. The intercept is sometimes referred to as representing "drift", because when B = 1 a nonzero intercept implies a deterministic linear trend in Y. In discussing the effects of drift, it is important to make a distinction between the data generating process and the regression equation that generates the test statistics. The asymptotic and finite sample distributions of Dickey (1976) and Fuller (1976) were calculated under the assumption that the data generating process is a random walk without drift. But as Schmidt (1989) pointed out, using the Dickey-Fuller critical values is invalid when the time series process is a random walk with drift. Schmidt extensively tabulated the critical values for different values of sample size and drift. Nankervis and Savin (1985) also provided the distribution of the t-statistic for the case of unit root with drift. However, when the regression equation includes a time trend, 13 the distributions of test statistics based on that equation do not depend on the value of the intercept. Therefore, for time series that contain drift, tests based on regression (2) would be a safer choice than tests based on equation (1). All simulations in this paper are carried on IBM PC compatible using double precision on Lahey FORTRAN complier. The simulations are based on 10,000 replications for each choice of parameters. Normal random deviates were obtained from the routine GASDEV (using subroutine RAN3) of Press, Flannery, Teukolsky and Vetterling (1986). The required time to compute for a given choice of parameters on a 16 MHz 80386 PC ranged from 1.8 hours for T = 100 to 18 hours for T = 1000. CHAPTER III TESTS WITH CONDITIONAL HETEROSKEDASTICITY In this chapter we consider the unit root tests of Dickey (1976), Fuller (1976), and Dickey and Fuller (1979, 1981). The tabulated distributions for these tests assume that the errors in the variable's autoregressive representation are normal white noise, but we consider the case that they are conditionally heteroskedastic. Phillips (1987) has derived the asymptotic distribution of the Dickey-Fuller statistics under assumptions which allow for quite general weakly dependent and heterogeneously distributed errors. The asymptotic distribution of the Dickey-Fuller statistics depend on two nuisance parameters 2 L, and 02), the innovation variance and the long run (0 variance, that were defined in the last chapter. In the presence of autocorrelation, the Dickey-Fuller statistics do not have the same asymptotic distribution as they would 2 2 without autocorrelation because 0' is not equal to 0”, and adjustments to the tests are necessary. In the case of heteroskedasticity, however, such adjustments are 2 unnecessary. This is so because a2 and au are equal (as 14 15 long as they exist), and the standard Dickey-Fuller distribution is asymptotically correct. Thus the sensitivity of the Dickey-Fuller tests to heteroskedasticity can be only a small sample problem. This problem has been examined by several authors, including Godfrey and Tremayne (1988), Peters and Veloce (1988) and Phillips and Perron (1988). Godfrey and Tremayne investigated the performance of Dickey-Fuller tests and Phillips-Perron tests in the case of heteroskedastic errors. Their heteroskedastic innovations were generated by first obtaining independent drawings from a chi-squared distribution with 25 degrees of freedom, and then by scaling an independently drawn standardized Gaussian variate with the square root of drawings from the chi-squared distribution. The Dickey-Fuller tests are quite accurate even for small sample sizes. The Phillips-Perron tests performed well for T = 250 but poorly for T = 25. Peters and Veloce (1988) also provide Monte Carlo evidence for the case in which there is a conditional heteroskedasticity in the form of a generalized ARCH(GARCH) model of the type proposed by Bollerslev (1986). In their experiments, the Dickey-Fuller tests, the augmented Dickey-Fuller tests and the Phillips-Perron tests all reject too often when the null hypothesis is true, but not by a large amount. We consider Dickey-Fuller tests in the presence of conditional heteroskedasticity of the GARCH form. The 16 GARCH model was developed by Bollerslev (1986) as a generalization of the ARCH model of Engle (1982). The GARCH model explicitly recognizes the difference between the conditional and the unconditional variance. In the ARCH process the conditional variance is specified as a linear function of past sample innovations, whereas the GARCH process allows the lagged conditional variances to enter as well. Specifically; if we letln denote the error in the Dickey-Fuller regression, and if we let h.t denote its variance conditional on information available at time t-l (IP,), the error model is specified as follows. (10) nt = Inch. + 8.11.42 + 8211.4)”2: (11) r, ~ IN(0,1), (12) utllm ~ N(0,h,), (13) h, = (>0 + §,u,.,2 + 62h,,,. For ¢2== 0 the process reduces to the ARCH(1) process, and for 9,:= 92== 0, ut is simple white noise. If °o > 0 and ¢,+-¢g < 1, the unconditional variance of the ut exists and equals <50 / (1 - , - (>2). The first aim of this chapter is to examine the robustness of the Dickey-Fuller tests to an integrated variance. The necessary condition for the variance process to be integrated, in the GARCH(1,1) model, is that 0, and 02 sum to one. Because the integrated GARCH (IGARCH) model 17 has an infinite unconditional variance, the assumptions of Phillips (1987) are not satisfied, the limits defining the 2 2 parameters a' and au do not exist, and we should expect the distribution of the Dickey-Fuller statistics to be affected, even asymptotically. (Although we will not consider it in this paper, Hansen (1988) provides another model called the heteroskedastic cointegrated model (HCI). The HCI model is quite flexible and does not need to be specified explicitly, as do the ARCH models.) We conjecture that the accuracy of the Dickey-Fuller tests in the presence of heteroskedas- ticity will depend on how close the variance process is to being integrated. The GARCH model has proved particularly useful in describing asset price fluctuations and in modelling time- varying risk premia, and many researchers have found a near- integrated variance process in this model. For example, Engle and Bollerslev (1986) apply the GARCH model to a time series of weekly data on the exchange rate between the U.S. dollar and the Swiss franc from July, 1973 through August, 1985, for a total of 632 observations. Other examples can be found in Baillie and Bollerslev (1989), Domowitz and Hakkio (1985), Diebold and Nerlove (1989), Engle, Lillien and Robins (1987), Bollerslev, Engle and Wooldridge (1988), and Diebold and Pauly (1988). The second important case to be considered here is the case which the intercept (To) in the GARCH process has the 18 value of zero. Nelson (1988) has shown that in the IGARCH (1,1) model with no drift ”’0 = 0) , h, and ut converge to zero almost surely. We will therefore call a GARCH model with §0== 0 "degenerate". This is not a proper model for modelling changing volatility and the persistence of shocks to volatility unless one's economic model calls for variance to die out asymptotically. Here we conjecture that the performance of the Dickey-Fuller tests under GARCH errors will depend on the closeness of the intercept (To) to zero. We investigate these two conjectures with a Monte Carlo experiment. Our errors will follow the GARCH(1,1) process, so we can control the closeness of the process to being integrated or degenerate by varying the parameters in the GARCH process. For the Dickey-Fuller tests based on t-statistics, we also consider the test statistics which employ the heteroskedasticity-consistent covariance matrix estimator proposed by White (1980). The asymptotic theory for the White correction does not apply to the case in which the variable or its variance process is integrated. Furthermore, the asymptotic theory for the White correction requires the existence of the fourth moment of the error, and this condition is violated for some of the values that we consider for the GARCH parameters. (The condition for the existence of the unconditional fourth moment is 2 2 . (14) 3Q, + 20,62 + 2 < 1 , 19 see Bollerslev (1986)). The augmented Dickey-Fuller tests and the Phillips- Perron tests are also considered in this experiment. A. Design of the Experiment In this experiment we will consider the Dickey-Fuller 2“, 21, Tu and T, tests. We will also consider their Phillips-Perron and Said-Dickey extensions, with z = 1, 2 in each case. Finally, for the tests based on t-statistics (Tu and T7) we will consider the White covariance matrix correction. Our data generating process will be an integrated process with GARCH errors. Thus the data generating process will be of the form of equation (1), with B = 1. Without loss of generality we set Yo== 0. We assume no drift; that is, a = 0. Non-zero drift is potentially relevant, as argued by Schmidt (1989), West (1988) and others, but will not be considered in this chapter. As specified in equation (12), the error ut is distributed as N(0,h,)‘with the conditional variance ht generated by equation (13). We set the initial variance h0 = 1. This specific choice ofl%,does not affect the performance of the tests, because the test results are invariant to different values of h0 as long as {>0 is changed proportionally so that the ratio of (00,/1%) is held constant. We therefore need to vary only four parameters 20 in the course of our experiments; 4’0! ,, {>2 and T. Three types of experiments are considered. The first set of experiments (Tables 1-6) varies the extent to which the process is close to being integrated by varying the sum of (c,+<1>2). This is done both by varying {>2 for fixed 0, and vice-versa. In these experiments we set the initial ‘variance be equal to the long-run variance (when it exists) by setting {>0 = h0(1 - <5, —Z) with h0 = 1. Since only (0 / ho) matters, this is equivalent to fixing 80 and varying h0 as (e, + 02) varies, except in the integrated case. In this setup, as ( , + 02) approaches one (as the process become nearly integrated), Q8 approaches zero (the process becomes nearly degenerate) too. This mixes two different types of parameter changes, but it automatically happens if we set the initial variance equal to the unconditional variance. Our second set of experiments (Tables 7-14) fixes 0, and 92 and varies To. This fixes the degree of near integration and varies the degree of near degeneracy, and it therefore implicitly varies the relationship between the initial and the unconditional variance. Our third set of experiments (Tables 15-16) follows the first set in varying the sum (0, + 92), but these experiments hold Q0 constant. This fixes the intercept and varies the degree of near—integration, and it also implicitly varies the relationship between the initial and 21 the unconditional variance. The results of the experiments are the percentages of rejections for a number of tests, though we also accumulated frequency distributions and moments for the various test statistics. Our main interest is in the performance of the Dickey-Fuller tests (2”, fifl.i1 and T1). However, the tests using White's (1980) heteroskedasticity-consistent covariance matrix, the augmented Dickey-Fuller tests (Said- Dickey (1984)) and the Phillips—Perron tests (Phillips (1987) and Phillips-Perron (1988)) are also considered. B. Results of the Experiment In the first set of experiments we vary (93 + 02) while setting 90 = h0(1 - 9, - (>2) . In these experiments we set the initial variance equal to the unconditional variance (when it exists), and this implies that the progress becomes nearly integrated at the same time that the value of To approaches zero. As the degree of near-integrated changes, the corresponding change in the value of Thlhappens automatically. Table 1 reports the proportion of rejections under the null hypothesis for a 5% lower tail test, for T = 100, 0,:= .3 and @2:= .30, .60, .65, .69 and .70. The results generally confirm our expectation that the accuracy of the tests should depend upon how close the variance process is to being integrated. For the smaller value of fig the tests 22 reject too often, and the rejection rates are significantly different from 5%, but the differences from 5% are not really very large. For example, when 02:: .30, the proportions of rejections are .059, .069, .059 and .064 for A A A e e 1' p“ and p1, respectively. However, when the var1ance ‘1) u, or, process is close to being integrated, the overrejection problem becomes more amd more serious for all tests. For example, the proportions of rejections for the Tu tests are .059, .074, .081, .114 and .412 for «>2 = .30, .60, .65, .69 and .70, respectively. Also proportions of rejections for Q, = .3 and <12 = .70 are .412, .353, .294 and .243 for i“, T,, 3“ and 2,, respectively. The Phillips-Perron tests and the Said-Dickey tests behave in a similar manner, although the problem is a little less severe for the 3 tests than for the T tests. For instance, for ¢2== .70 the proportions of rejections for'Tfi.pp1 and Tu.sd1 are .417 and .404, respectively, while, the proportions of rejections for Zu.pp1 and 3,-Sdl are .298 and .300, respectively. There is not much difference in the performance between the ppl and pp2 tests, or between sdl and sd2 tests. The White covariance matrix correction helps the performance of the tests but does not correct the overrejection problem completely. For instance, the proportion of rejections for the Tu.w test when §2== .70 is .121 which is much lower than the proportion of .412 for the unmodified standard Tu test, although it is still well above the nominal size of 23 the tests, 5%. The same pattern occurs for the ; T when 02:: .7. The proportion of rejections for'T7.w is .101 which is much lower than .353 for T7. Similarly, for the near-integrated case of °2‘= .69, the proportions of rejections are .114 and .068 for Tu and.Tu;w respectively. The fourth moment existence condition (14) requires 02 < .606 when Q,:= .3, but the performance of Tuzw and Tfuw for §2== .65, .69 and .70, which violate the fourth moment condition, does not show any discontinuity in the performance of the tests. The tests using the White covariance matrix correction are less severely affected in the integrated case compared to the other tests. The proportions of rejections for Tu.w and T,.w is still better than the others for (¢,== .3, ¢2== .3), (.3, .60), (.3, .65) and (.3, .69), but not by a large amount. In the integrated case the performances of the standard Dickey- Fuller tests, the Phillips-Perron tests and the Said-Dickey tests deteriorate rapidly compared to White's tests. For example, the proportion of rejections for'Tu deteriorates from .114 when 92:: .69 to .413 when @2== .7, but only from .068 to .121 for TV“ Although we have calculated results for upper tail and two tail tests as well as lower tail tests, we will discuss only the lower tail tests in detail, since the upper tail and the two tail tests are typically of little interest. 24 Table 2 and 3 report the proportions of rejection for the upper tail and the two tail tests, respectively, when @,== .3 and 92‘= .30, .60, .65, .69 and .70. Table 2 shows a slight overrejection for (.3,.3), (.3,.6), (.3,.65) and (.3,.69) for all tests and underrejection for the integrated variance error process. There is a serious problem only in the integrated case. For example, the proportion of rejections of T for (.3,.69) is .056, but .007 for u (.3,.70). The White covariance matrix correction, the Phillips-Perron tests and the augmented Dickey—Fuller tests do not help at all. They do not improve the performance of the tests compared to the standard Dickey-Fuller tests. Table 3 generally shows the same pattern as Table 1. For large values of 92 all tests reject too often and when the variance process is close to being integrated, the overrejection problem becomes serious. The White correction tests work much better than the Phillips-Perron tests and the Said-Dickey tests, especially for the integrated variance process. Table 4 reports the proportions of rejections for the lower tail test with T = 100, Q,== .1, and er= O, .50, .80, .85, .89 and .90. We are again manipulating the nearness to integration of the variance process, but compared to the case of Table 1 the process has smaller value of 0,. All the results reported in Table 1 apply here, with only minor modifications, but the problem is much less severe. For 25 example, even in the integrated case (02 = .90) the proportions of rejections are only in the range of .09 for T, tests and .07 for the 3 tests. The White covariance matrix correction is not very useful for the parameter values in Table 4. It performed well only in the integrated case and for the T“ test, and for most Qz‘values it actually made the performance worse. The fourth moment existence condition holds here for all except the two largest values of 92. Again we can not find any discontinuity in the performance of the White tests due to the failure of the fourth moment existence condition. The T,.sd tests improve the performance by a small amount, although all other Phillips-Perron tests and other Said— Dickey tests do not help at all. It can be pointed out that T1.sd2 performs better than T,.sd1 for all parameter values in Table 4. Table 5 is once again for T = 100 and for the lower tail test. Now we set ¢2== .30 and let Q,== .30, .60, .65, .69 and .70. As Q, is increased in this experiment the variance process is closer to being integrated. The same patterns that were evident in Table 1 are still evident in Table 5. For large values of Q,the tests reject too often. The proportions of rejections are increasing as the variance process is closer to being integrated. In this integrated case the results in Table 5 are substantially worse than the corresponding results in Table 1. For 26 example, the proportion of rejection for Tu'with @1,= .70 and ¢2== .3 is .888, whereas the same test with §,== .3 and §2== .70 is .412. Again Tuuw and szw perform relatively well and the proportions of rejection stay between .074 and .053 except in the integrated variance case. The proportions of rejections of'Tuuw and T1;w in the integrated process case are well above their nominal size; however, their performances are still much better than the Phillips- Perron tests and the augmented Dickey-Fuller tests. The Phillips-Perron tests and the augmented Dickey-Fuller tests do not help at all, except that Tu.sd2 and T7.sd2 do somewhat better than the other tests in the non-integrated case. Table 6 reports results of experiments which we increase the sample size. For the three (9,,02) pairs (.30,.60), (.30,.65) and (.30,.70), we set T = 100, 500 and 1000. Since the tests are asymptotically valid except in the integrated case, the performance of the tests would improve as the sample size increases, at least for the first two (,,2) pairs. This turns out to be so, but the rate of improvement is extremely slow. For example, when (43,02) = (.30,.60) the proportions of rejections for the Tu test are .074, .070 and .061 for T = 100, 500 and 1000, respectively. In the integrated case the tests become worse rather than better as the sample size increases, 27 suggesting the need for nonstandard asymptotics in the case of an integrated variance. In any case, it is clear that in determining the accuracy of the tests the sample size is not an extremely important factor, at least over an empirically relevant range of sample sizes. Tables 7-14 report the second set of experiments. We fix 0, and 92 and vary 50. Thus we are fixing the degree to which the process is nearly integrated, and varying the value of 00. We are also implicitly varying the relationship between the initial and the unconditional ‘variance. In Table 7, we fix Q,== .3, ¢2== .65 and h0:= 1.0 and vary To from 0 to 100. The proportions of rejections for the various tests generally decrease with the increase in 00. As the values of To increase, the proportions of rejections are decreasing toward the nominal size. However, the proportions of rejections remain larger than the nominal size. The percentages of rejections remain near .08 for the most of tests even with 80‘: 100. The tests perform reasonably well for large ‘0 but the overrejection problem becomes extremely severe as Torapproaches zero. We can see that the proportions of rejections of Tu.w and T1.w approach the nominal size most closely when ‘0‘: 100, compared to the standard Dickey-Fuller tests, the Phillips— Perron tests and the augmented Dickey-Fuller tests. For instance, the proportion of rejections of‘Tuzw when To = 100 is .058. 28 Table 8 shows the percentages of rejections for the integrated case; i.e., ¢,== .3 and @2:= .7. The results are very much similar to the results discussed for the near- integrated case in Table 7. There is a serious overrejection problem as the value of ‘0 approaches zero, but the degree of seriousness is actually less here than in the near-integrated case in Table 7. Tables 9 and 10 provide similar results when (0,,92) (.1,.85) and (0,,92) = (.1,.9) repectively. The same pattern occurs here as in Tables 7-8 (when Q,== .3). The tests overreject when 00 is small, and by an amount that increases as the value of ‘0 approaches zero. The interesting point here is that the degree of overrejection depends upon the magnitude of 9,.and the overrejection problem is less severe in the integrated case rather than the near-integrated case. For example, Table 9 (§,== .1, Qz:= .85) and Table 10 (Q, = .1, ¢é== .9) have less severe problems than Table 7 (9,== .3, 92== .65) and Table 8 (0,== .3, 02‘: .7) respectively, sometimes even an underrejection problem shows up for a large value of 00 in Table 10. Tables 11-14 increase the sample size from T = 100 to T = 1000 for the same values of 90 (0, .01, 1.0 and 100) and for the same sets of other parameters (9, = .3, QZ== .65), (<3, = .3, <1», = .70), (0, = .1, 42 = .85) and (4, = .1, <12 = .9). In the case when ¢o== 0, increasing the sample size causes the performance of the tests to deteriorate, with the 29 exception of the tests using the White covariance matrix correction. Actually'Tu;w and.T,;w are better off with the increase of sample size when To is equal to zero. When 00 is not equal to zero, the increase in the sample size generally improves the accuracy of the tests, although this does not always hold true. In cases of improvement, the rate is often very slow. Increasing the sample size also tends to make the White correction tests reject too seldom for large values of To. The interesting finding in the second set of experiments is that the important factor in determining the performance of the unit root tests in the presence of GARCH(1,1) errors is the intercept parameter *0 in the variance process. A serious overrejection problem happens when ‘0 is very small. This problem is even worse the larger the value of T, is and the larger the sample size is. The performance of the tests is not much influenced by how close the variance process is to being integrated. The final set of experiments (Tables 15 and 16) confirms the findings of the second set of experiments. In our first set of experiments we set Qo==t%(1 - 93 - 42). In this setup as (TH + 02) approaches one, to simultaneously approaches zero. Therefore the poor performance of the tests can be caused by either of these factors. However, from the results of the second set of experiments it is clear that the important factor in these experiments is the 30 change in.Qo, not the change in (§,-+ 02). This evidence is also supported by the results of the third set of experiments. In the third set of experiments we fix 90 and 4,, and we vary (0% + 92)'by varying 02. In Table 15 and 16 we pick T = 100, ho== 1.0 and §o== .01. According to the results of Tables 7-10, ¢0== .01 should correspond to a moderate degree of overrejection for the Dickey-Fuller tests. It turns out to be so. In Table 15 we set §,:= .3 and vary 92 from .3 to .7. Unlike the first set of experiments, the proportions of rejections do not change very much as (0,-+ 52) approaches unity. For example, when §,== .3 and with 90 fixed at .01, changing 02 from .65 to .70 hardly changes the percentages of rejections for the various tests. In Table 16 we set @,== .1 and vary Q2 from 0 to .9. We can also find a similar patterns of small changes in the performances of the various tests in Table 16.4 For example, when Q,== .1, changing 02 from .89 to .9 results in only small changes in the proportions of rejections for the various tests. It appears that the extent to which the GARCH error process is nearly integrated is not as important as the extent to which the magnitude of 00 is close to zero (that is the extent to which the variance process is nearly degenerate). 31 C. Conclusion In this chapter we have conducted a Monte Carlo experiment to investigate the reliability of the various Dickey-Fuller tests when the errors in the variable's autoregressive representation follow a GARCH(1,1) process. Specifically we are interested in whether the proportions of rejections under the null hypothesis (having a unit root) agrees with the nominal size of the test. The standard Dickey-Fuller tests tend to reject too often in the presence of conditional heteroskedasticity. The problem seems to be less severe for tests based on the value of B (2“ and 2,) than for tests based on the t- statistic (Tu and T1). White's (1980) covariance matrix correction helps the performance of the Tu and TT'tests most of time, but does not solve the problem entirely. Unsurprisingly the Phillips-Perron tests performed poorly throughout the tests. This is as expected because the Phillips-Perron corrections are designed to handle autocorrelation, not heteroskedasticity. The same pattern of the results also occur for the augmented Dickey-Fuller tests (Said-Dickey (1984)). Again we might expect these poor results because this test is also designed to handle with autocorrelation, not heteroskedasticity. In choosing the parameter values for our experiment, we emphasized how close the GARCH variance process equation 32 (13) is to being integrated (§,-+ 0,==1) and how close to zero the value of To is. We found that the various Dickey- Fuller tests were seriously unreliable when the variance process has the value of 0b close to zero or very nearly so. This was so more or less regardless of how close the variance process was to being integrated. CHAPTER IV TESTS WITH AUTOCORRELATED ERRORS This chapter consists of two sections. The first section investigates the reliability of the various unit root tests in the presence of serially correlated errors. The second section uses three different lag windows (that is, Bartlett's, Parzen's and Bohman's lag windows) in the estimation of the long run variance to see how the Phillips- Perron tests perform for each different lag window. Also ten different lag truncation values (from one to ten) are tried in the estimate of the long run variance to see if there is a predictable pattern in the performance of the tests, and hopefully to be able to make some practical recommendation on the choice of the lag truncation value in applied work. A. MA(1) and AR(1) error processes Fuller (1976), Dickey (1976), Dickey and Fuller (1979, 1981) and Evans and Savin (1981,1984) proposed unit roots tests based on a first order autoregressive (AR(1)) model, and Said and Dickey (1984) allowed a more general ARIMA 33 34 (p,l,q) progress. All of these authors assumed that the innovations are iid normal. Independence and homoskedas- ticity in the error process are quite restrictive assumptions. It is more realistic to ease these strong assumptions to allow various forms of serial dependence in the error structure. The most common autocorrelated error specification has been the firsr order autoregressive (AR(1)). (15) u == put, + 6,, t = 14 ..., T, where for stationarity Ipl < 1, and the 6t are random variables with E(€t) = 0, E(€E) = a: and E(6g%) = 0 for tfs. Although the AR(1) error model has been popular, there are many instances when moving average errors are justified by economic theory (for example, Nicholls, Pagan and Terrell (1975)), and, in addition, models with MA or ARMA errors may often be a better representation of the data generating process. For MA(1) errors, we have (16) u, = e, + ae,_,, t = 1, T, where again E(6,) = O, E(e£) = of, E(6g%) = 0 for tfs, and l6| < 1 for invertibility. This weakened error process was assumed by several authors, including Schwert (1989), Phillips and Perron (1988), Peters and Veloce (1989) and Godfrey and Tremayne (1988). All of them assumed MA(1) errors in the data generating process. Schwert (1989) showed that tests for unit roots 35 developed by Dickey and Fuller are sensitive to the assumption that the data are generated by a pure AR process. Specifically, the distributions tabulated by Dickey (1976) and Fuller (1976) can be very different when the error in the data generating process has an MA component. Furthermore, the tests of Said-Dickey (1984) and Phillips (1987) do not seem to work well when the MA parameter is large, even though they should be correct asymptotically. In particular, the Phillips-Perron tests do not come close to their asymptotic distributions even for rather large sample sizes. Godfrey and Tremayne (1988) also addressed the question of whether or not unit root tests are robust to the failure of the assumption of iid innovations, and whether or not the statistics of Phillips (1987) and Phillips and Perron (1988) designed for such situations perform satisfactorily in finite samples. They allowed MA(1) errors with the moving average parameter equal to .5. They found that the Phillips-Perron tests may need a rather large sample size to be reliable. The aim of the first section of this chapter is to confirm the foundings of Schwert (1989), Godfrey and Tremayne (1988), and Phillips and Perron (1988) and to complement their results in the sense that we allow AR(1) errors in addition to MA(1) errors. 36 1. Design of the Experiments In the experiments we will consider tests based on the same two regressions as before; that is, equation (1) and (2). Tests considered here are the standard Dickey-Fuller tests, the White covariance matrix correction tests, the Phillips-Perron tests, and the augmented Dickey-Fuller tests. These tests were explained in Chapter 2. Our data generating processes are MA(1) errors and AR(1) errors. Otherwise they follow the regression equation (1) with B = 1. Without loss of generality we set Yb = 0. We assume no drift; that is, a = 0. Most of our experiments are done with a sample size of T = 100. Sample sizes of T = 200 and T = 500 are considered for selected parameters. The MA parameter is set equal to -.8, -.5, —.2, .2, .5, and .8. The same values of p are considered for the case of AR(1) errors. The "base case" in the tables means that there is no AR or MA component in the data generating process, so that in the base case the errors are iid. Each experiment is replicated 10,000 times to create the sampling distribution for the test statistics. Although we have results for frequency distributions and moments, we only report the proportions of rejections for the lower tail tests. The lower tail tests are our main concern because the usual alternative hypothesis is that the process is stationary (B < 1). 37 2. Results of the Experiment The main concern of the experiments is how close the proportions of rejections for the various tests are to the nominal size, 5%. Tables 17-20 report the results for MA(1) and AR(1) error process. Table 17 gives the proportions of rejections under the null hypothesis for a 5% lower tail test for T = 100. It shows that MA errors affect the performance of the tests very much. The rejection rates are negatively related to the MA parameter. All the tests considered reject too often for negative values of 6 and reject too seldom for positive value of 6. The degree of departure from the nominal size is much higher for the negative values of 6 than for the positive values. The proportions of rejections of all the tests for 6 = -.8 are close to 1.0. Thus MA errors definitely affect the performances of the tests. Although we do not report them here, the 5% quantiles for the test statistics reflect an extreme skewedness of the distributions toward negative values. For example, the 5% quantile for $1.pp2 reaches to -105.87 for 6 = -.8. The Phillips-Perron tests show much the same pattern as the uncorrected tests: that is, the percentages of rejections decrease with an increase in the value of the MA parameters. The Phillips-Perron corrections seem to improve the performance of the tests, but the degree of improvement is very small. For example, the proportion of rejections for 38 Tu.pp1 when 6 = -.2 is .105 compared to .132 for T“ and the proportion of rejections for Tu.pp1 when 6 = -.8 is .985 compared to .997 for T“. Because of the serial dependence 2, we do not expect that the in the error structure, 02 76 0,, Phillips-Perron tests will have the same distribution as the standard Dickey-Fuller tests. However, the minimal improvement in the performance of the Phillips—Perron tests compared to the uncorrected tests is an unfortunate result. The augmented Dickey-Fuller tests perform well compared to the other tests. For example, the Tu.sd2 and T1.sd2 tests perform somewhat better. However, the percentages of rejections are still very different from nominal size for large negative MA parameters. Schwert (1989) pointed out that the error caused by the approximation of an ARIMA process by the AR(1) process is large for MA parameters greater in absolute value than .8. The proportions of rejections for the Said-Dickey test range from .643 for Tu.sd2 to .999 for Su.sd1. The tests based on the White covariance matrix correction perform worse than the uncorrected tests for negative MA parameters, and better for positive MA parameters. It is understandable that the Phillips-Perron tests and the Said-Dickey tests, which were designed for residual autocorrelation, outperform the White tests designed for heteroskedasticity. It is interesting that 39 the Said-Dickey tests outperform the Phillips-Perron tests for all values of the MA parameters, although Tu.sd1 and T7.sd1 still clearly show the overrejection problem for 6 = -.8. In particular, the performances of Tu.sd2 and T,.sd2 are reasonably good. For example, the proportions of rejections for Tu.sd2 are .049, .047 and .038 when 6 = -.2, .2 and .5, respectively. The proportions of rejections for the base case are close to the nominal size as expected. The rejection rates for the base case should be close to those of the Dickey-Fuller tests because there is no serial correlation in the error structure, and it turns out to be so. In Table 18 we increase the sample size to T = 200 and T = 500 to investigate the effects of the increase in the sample size on the performance of the tests when 6 = -.5 or .5. The results for all tests for the base case converge to nominal size. When 6 = -.5 the increase in the sample size actually deteriorates the reliability of all the tests with the exception of the Phillips-Perron tests. For example, proportion of rejections for T“ with T = 100 is .571 compared to .622 with T = 500. Although the increase in the sample size improves the performance of the Phillips- Perron tests, the proportions of rejections are still well above the nominal size. For instance, the proportion of rejections for Tu.pp1 when 6 = -.8 with T = 100 is .457 compared to .415 with T = 500. When 6 = .5 many of the 40 tests' performances deteriorate as the sample size grows, but by a small amount. It can be pointed out from Tables 17 and 18 that the most important factor determining the reliability of the tests is the degree of the dependence in the error structure, not sample size. In the following we discuss the effects of AR(1) errors on the performance of tests. We consider same parameter values with T = 100; that is, p = -.8, -.5, -.2, .2, .5 and .8. As mentioned earlier, several authors have done Monte Carlo experiments of the unit root tests for the case of MA(1) erros, but no one has done these experiments for the case of AR(1) errors (as far as we know). Remembering that the AR(1) is a popular error specification, especially for annual data sets, the results of these experiments are interesting. Table 19 reports the rejection rates under the null hypothesis for a 5% lower tail teSt for T = 100. The test results show a similar pattern as in the MA(1) error case. The tests reject too often for negative values of p and reject too seldom for p = .2 or .5. The degree of overrejection or underrejection is usually less severe than in the MA(1) error case. However, for the AR(1) case the underejection problem for the 2“ and 3, tests when p = .5 or .8 is more severe than in the case of MA(1) error with 6 = .5 or .8. Again the Phillips-Perron tests do not seem to work well. They do better than the standard Dickey-Fuller 41 tests, but the degree of improvement is minimal. However, the Said-Dickey Tu and T, tests perform well. The proportions of rejections of‘Tu.sd and T,.sd stay around 5% for all AR parameters. The rejection rates range from .046 to .059. Su.sd and.3,.sd are better than the other 2 tests, but still reject too often for negative AR parameters and reject too seldom for positive AR parameters. Comparing the Said-Dickey tests with one and two lagged Ay's added to the regression equation, the tests using two lagged Ay's have a tendency to reject less often than those using one for Tu and T7, and vice versa for Tu and 37. However, the Phillips-Perron tests with 2 = 2 always reject more often than the tests with E = 1 regardless of the choice of tests and the value of the parameters. The White correction worsens the performance of the various tests for negative values of the AR parameter and improves their performance slightly for positive values of the AR parameter. It is an unexpected result that the White tests work better than the Phillips-Perron tests when p = .2 or .5. For example, the proportion of rejections of’Tu.w when p = .5 is .044 compared to .028 for Tu.pp1. In Table 20 the sample size is increased to T = 200 and T = 500 with p = -.5 or .5. All tests generally improve as the sample size grows, but the degree of convergence to the nominal size is slow and the proportions of rejections are still well above 5% when p = —.5 and below 5% when p = .5. 42 The augmented Dickey-Fuller T tests are fairly reliable when p = -.5 or .5. 3. Conclusion The simulations in this section confirm Schwert's (1989) results. Including MA(1) or AR(1) errors in the data generating process can make the critical values tabulated by Dickey (1976) and Fuller (1976) very misleading. Generally the proportions of rejections for the various tests are well above the nominal size for negative MA or AR parameters and are below the nominal size for positive MA or AR parameters. The White tests (designed for heteroskedasticity) do not seem to work well, which is understandable. However, it is an unexpected result that the Phillips-Perron corrections do not improve the performance of tests significantly, because they are designed for serial dependence in the error structure. The Said-Dickey augmentations certainly help the performance of the tests, especially in the AR case. In particular, the proportions of rejections of the Said-Dickey augmented Tu and T1 tests are close to the nominal size. The Phillips- Perron tests seem to be more sensitive to model misspecification than the Said-Dickey tests. The performances of all of the tests depend more critically upon the degree of dependence in the error process than upon the 43 sample size. These simulation results give rise to a warning against the careless use of the Dickey-Fuller tests in economic time series when the errors are dependent. There is an obvious need for tests that are more reliable in the presence of dependent errors. B. Lag Windows Phillips (1987) and Phillips and Perron (1988) provided transformed statistics which have the same asymptotic distribution as the usual Dickey-Fuller statistics, even given autocorrelated or heteroskedastic errors. The transformed statistics require consistent estimates of two 2 parameters, the innovation variance 0u and the long run variance 02, defined (as in Chapter 2) by r r (17) 0fr= lim (l/T) E var(ua, c3 = lim (l/T) var( 2 ug. T-oo t=1 T'“D t=1 2 Consistent estimation of 0u is trivial. Phillips suggests 2 as a consistent estimate of 0' the estimator Sufi defined by t T r _ .1 A 2 .1 A A (18) S“ -—'r E'u + 2T 2 2 u,u,_T t=1 7.1 t37+1 Here "2" controls how many autocorrelations are used in St,2 and will be called the lag truncation value. Phillips (1987, Theorem 4.2) shows that consistency of S“? requires 44 2 e w as T 4 m but at a controlled rate (e.g. 2 = 0(TV4)). This procedure assumes that there may be strong dependence among adjacent events, but events separated by a long distance are almost independent. Therefore, terms such as E(unnq) with long lags r > 2 can be ignored in the consistent estimator, Su?° Although choosing a specific value of e is an empirical matter, we can obtain a rough idea of the appropriate value of 2 by examining the sample autocovariances of ut==yg - y,,. A natural procedure would be to choose the value of 2 for which the autocovariance function is insignificantly different from zero for all lags greater than 2. Although rather crude, it is certainly preferable to choosing a completely arbitrary value of 2. However, there is an inherent danger in this approach. Because the sample autocovariances are themselves autocorrelated, the sample autocovariance function will decay more slowly than the theoretical autocovariance function, as pointed out in Priestley (1981). A problem with the long run variance estimate 8&2 is that there is a possibility that large negative serial covariances can make 2 negative. To see how likely this is we calculated the 5a number of negative values of S“? in our experiments of the last section, based on 10,000 replications. The experiments are conducted in the same manner for both AR, MA errors; that is, we use the parameter values -.8, -.5, -.2, 0,.2, .5, and .8. We also changed the value of the lag truncation 45 term 2 in S“? from 1 through 10 to see how the value of 2 affects the number of negative Suz. For example, in Table 21 the notation pplO indicates the number of negative values of S“? for the case of 2 = 10. The number of negative values of S“? increases with the value of 2 for all MA parameters except 6 = -.2 and for all positive AR parameters. The base case (iid errors) shows a similar pattern. When p = -.8 or -.5, the number of negatives is large for odd values of 2 and small for the even values of 2; however, the number of negatives gradually decreases with the value of 2. We certainly have evidence in this experiment that in some cases, for example, 2 = 1 and p = - .8 or -.5, there are too many negative numbers for the estimator 8,,2 to be useful. An appropriate weighting scheme can be used to guarantee a positive value of Suz. Thus we write 2 T 2 r s 2 = (l/T) 2 u,2 + (2/T) E E w(i) mat-.. t=l (19) t2 i=1 t-i+1 ‘ For example, Newey and West (1987) suggest w(i) = i/(2+i), i=1, ..., 2, and this guarantees a positive estimate of 02 (for any 2). Now it is true that the long run variance 02 = 2ng(0) where fu(A) is the spectral density of the error u. Thus there is a close connection between the problem of estimating 02 and the problem of estimating the spectral density function. The choice of a weighting function w(i) 46 is equivalent to the choice of a "window" for spectral density estimation. The Newey—West weighting corresponds to the well known Bartlett window. But the literature on the estimation of the spectrum suggests other windows, and they too guarantee the positive value of the long run variance estimate. In this section two more lag windows in addition to Bartlett's window are considered; that is , Parzen's and Bohman's lag windows. The first point of this section is to investigate the performance of the Phillips-Perron tests using Parzen's and Bohman's lag windows in the estimation of the long run variance. The second point is to investigate the choice of the lag truncation value (2). For each lag window we tried values of 2 from 1 through 10 to see how the value of 2 makes a difference in the performance of the tests. 1. Lag Windows We know that the true spectral density function is non— negative. Therefore, it is desirable that the estimate should share this property. Three lag windows considered here all guarantee a positive estimate of the spectral density function. The widely adopted form of the spectral density estimate is s=c+1 (20) fun) = (1/27!) 2 1(5) 11(5) cos(sA), s=-(2+1) 47 where 2 is the truncation point. {A(s)} is a sequence of decreasing weights which is called the 'lag window' and R(s) is the sample autocovariance. The sequence {A(s)) involves a truncation point 2 which controls its rate of decay. The weights given to R(s) decrease with increasing 5. In particular, suppose that 1(5) is of scale parameter form (that is, 1(5) = k(s/2), and that 1(5) is a bounded, even function. Then (a) A(0) = l, (b) A(W) = A('W): (C) k(W) = 0, w > 1. All three lag window generators considered here have the lag window of scale parameter form. Variation of the window parameter 2 simply 'stretches' or 'contracts' the function, so that 2 acts as a scale parameter. Bartlett(1950) proposed the lag window having the form (21) 1(5): 1 - (Isl / 2). Isl s e. 0, '5' > 2. This is the window used in equation (20) in Phillips (1987). This lag window applies linearly decreasing weights to the autocovariance up to lag 2 and zero weights thereafter. Parzen (1961) suggested the lag window of the following form 48 / 1 - 6(s / (>2 + 6(lsl / 2)3, Isl s 2/2, (22) A = 2((1 - (lsl / e)>3. (2/2) s Isl s e, L 0, Isl > 2. Bohman (1961) window is (23) A(s) = (1 - (lsl / 2)} cos(n(lsl / 2)} + [sin(n(lsl / 2)) / n]. The properties of the estimate of the spectral density function depend on the choice of 2 and on the form of the window. It is not surprising that, even allowing for a suitable choice of 2, different windows can lead to estimates with different properties. Priestley (1981) illustrates various criteria for judging the different windows from a statistical point of view. However, our main criterion here is that the lag window should guarantee the positive estimate of the spectral density function. All three lag windows mentioned earlier satisfy this requirement. In addition they satisfy the relationship A(1) = 0. When 1(1) 9 0, there is a discontinuity in A(s) at 1. The Parzen window's weighting function has much sharper peaks which are nearer to w = 0 than Bartlett's lag window. As Neave (1972) pointed out, the total weight in the range w > 1/2 is only 18.75 percent in Parzen's window and 23.9 percent in Bohman's window, compared to about 50 49 percent in Bartlett's window. Parzen's window is satisfactory on the basis of the arguments above. Bohman's window has not received as much attention, but it is clear that Bohman's window has properties similar to Parzen's. More discussions about the spectral analysis and lag window generators may be found in Priestley (1981) and Dhrymes (1974). 2. Design of the Experiment Once again the two types of regression equations considered are one with a intercept and lagged dependent variable and the other with a time trend added; that is, equations (1) and (2), respectively. The data generating process is equation (1) with B = 1, having first order moving average or first order autoregressive error processes with the same set of parameters considered previously (-.8, -.5, -.2, .2, .5 and .8). The base case is Gaussian white noise; i.e., p = 6 = 0 as before. Only the Phillips-Perron tests with different values of 2 are examined here, but the standard uncorrected Dickey—Fuller tests are also reported for purposes of comparison. Here the value of 2 is varied from 1 through 10. As before, the suffix 'ppl' indicates the Phillips-Perron test with 2 = 1, and similarly for other values of 2. All simulations are replicated 10,000 times for sample size T = 100 with each different choice of parameters. We are mainly interested in comparing the three 50 different lag windows in the performance of the Phillips- Perron type tests. We are also interested in how the lag truncation value used in S“? affects the performance of the tests. Although the investigation of the sample autocorre- lations of utmfill help in selecting an appropriate choice of 2, the choice of 2 is an empirical matter. As Phillips (1987) pointed out, we expect that small values of should be sufficient because the sample autocorrelations of first differenced economic time series usually decay quickly. 3. Results of the Experiments Tables 22-27 report the proportions of rejections under the null hypothesis for 5% lower tail test, for each value of the MA parameters (-.8, -.5, -.2, .2, .5, .8) and for the base case. Similarly, Tables 28-33 report the same results of the experiments for AR parameters (-.8, -.5, -.2, .2, .5, .8). As pointed out earlier, thePhillips-Perron tests perform better than the standard Dickey-Fuller tests when the error process is MA(1) or AR(1), although the degree of improvement is small. However, as the last columns of the Tables 22-27 show, for the base case, the performance of the Phillips-Perron tests deteriorate somewhat as the value of 2 grows. In general, however, the value of 2 in S“? makes little difference in the performance of the tests. The performances of the tests depend mainly on the parameters of 51 the error process. The direction of the effect of 2 on the performance of the tests varies. For some parameters larger 2 improves the test, but for other parameters vice versa. Usually the best value of 2 is around five, depending somewhat upon the lag window used. Parzen's and Bohman's lag windows generally seem to work somewhat better than Bartlett's window does, but the difference between them is minimal. There is a tendency for Parzen's and Bohman's lag windows to need a larger value of 2 than Bartlett's lag window to get the best performance in each test. Usually Parzen's and Bohman's lag windows need 4-6 terms compared to 3-4 terms for Bartlett's lag window. The performance of the tests depends little on the kind of lag windows employed in calculating Sag. This is understandable because the difference between the lag windows is the weight applied to each of the sample autocovariances. For example, the weights given to each sample autocovariances for 2 = 10 are shown in the following table. 2 1 2 3 4 5 6 7 8 9 10 Bartlett 1.82 1.64 1.45 1.27 1.09 .91 .73 .55 .36 .18 Parzen 1.91 1.68 1.35 .99 .65 .38 .19 .08 .02 .00 Bohman 1.92 1.72 1.43 1.11 .79 .50 .28 .12 .04 .00 When comparing lag windows, there is a small variation among the first three weights of each lag window. After that, 52 Parzen's weights decrease more rapidly than the other lag windows. The Bartlett window's weights appear to decrease at the slowest rate. Parzen's and Bohman's window add more weights on the adjacent events and add very little weights on the events which are separated by the long intervals, for example 1.91, 1.92 for 2 = 1 and .00 and .00 for 2 = 10, repectively. However, the weight of Bartlett's window for 2 = 10 is .18 which is large when compared to Parzen's and Bartlett's window. Parzen's and Bohman's weighting scheme fit better to the assumption 2.1 (d) in Phillips (1987), which controls the extent of the dependence in the process u This explains why Parzen's and Bohman's lag windows ,. perform somewhat better than Bartlett's window. However, it has to be remembered that the difference in the performance between three lag windows is small and the overall performances of all three windows are poor. 4. Conclusion None of the lag windows employed in the experiments of this chapter improves the reliabilityof the tests significantly. Parzen's and Bohman's lag windows generally perform slightly better than Bartlett's window. Although choosing the value of 2 is an empirical matter, it was shown in the experiment that a value of 2 around five would be a reasonable choice for empirically relevant sample sizes. CHAPTER V TESTS USING THE TRUE VALUES OF 02 AND 0,,2 The limiting distributions of T(B-1) and t-statistic given in Theorem 3.1 in Phillips (1987) depend on the ratio 2 of the unknown nuisance parameters 0u and 02; that is , (24) T030) - (1/2) (W(l)2 - (o,2 / 02)) / (Iowa)2 dr). 1 1/2 (25) ’r‘, - (a / 20,) (W(l)‘2 - (a;2 / 02)) / (I W(r)2dr ) , 0 where W(r) is a standard Wiener process. Phillips (1987) and Phillips and Perron (1988) provide transformed versions of these statistics whose asymptotic distributions are the same as those tabulated by Dickey (1976) and Fuller (1976). The transformed statistics also depend on the ratio of 0f and 02. These statistics are not directly useful because of the unknown values of the nuisance parameters. As discussed in Chapters 2 and 4, a solution to this problem is 2 and 02 by consistent estimates. However, 02 to replace 0U is not an easy parameter to estimate well. In this chapter we actually calculate the true value of 03 and 02, for AR(1) 53 54 2 2 and MA(1) errors, and we use the true values of 0u and 0 instead of using consistent estimates. We therefore eliminate the possible inaccuracy of the tests arising from errors in estimation of (cjfl 02). By taking this approach we hope to determine whether the inaccuracy of the Phillips- Perron tests is due to problems in estimating these nuisance parameters, or whether it is intrinsic to the use of the asymptotic distribution theory in finite samples. We first consider MA(1) error case; that 15,1R = 6:4-96t4 with 6,., 2 2 iid(0,0:). We can calculate the true values of 0u and 0 as follows: I 2 _ . 2 _ 2 2 (26) 0h -— 11m (l/T) 2 E(u,) - (1 + 6 ) 0‘, Two t=1 r (27) 02 = lim (l/T) E ( E u,)2 = (1 + a)"- of. F0 ta We next consider the AR(1) error case; that is, ut== pup, + 6t with et ~ iid(0,0f). Here we obtain 2 T 2 2 2 (28) 0,, = lim (l/T) z E(u,) = 0‘ / (l-p ), T-oo t=l 2 T 2 2 2 (29) 0 = lim (l/T) E( 210,) = aE / (l-p) . {no t= These formulas can be found in Phillips (1987). We then use (26) and (27) for the case of MA(1) errors and (28) and (29) 55 for the case of AR(1) errors, instead of 8,,2 and 5:22 in equation (21) and (22) in Phillips(1987) and equations (6), (7), (8) and (9) in Chapter 2 (page 341 in Phillips and Perron (1988)). A. Design of the Experiment The framework of the experiment is basically the same as in Chapter 4. Two regression equations are considered here as usual. (a) Y a + BYP, + u,, t (b) Y, = a + BYp, + 6t + u,. Obviously each corresponds to a regression of Y on lagged Y, but they differ as to whether they include a time trend. The data generating process is (1) with a = 0 and B = 1. An MA(1) error is generated by equation (16) and an AR(1) error is generated by equation (15). The parameter values 6 = -.8, -.5, -.2, .2, .5, .8 and p = -.8, -.5, -.2, .2, .5, .8 are used in the data generating process, with T = 100. The same experiments have been done with larger sample sizes, T = 200, 500 and 1000, for the parameter values of -.8, -.5, .5 and .8. Here again only the lower tail test results are reported. All simulations are based upon 10,000 replications. 56 B. Results of the Experiments The standard Dickey-Fuller, the Phillips-Perron tests and the Phillips-Perron true value tests are compared in Tables 34-39. We denote the Phillips-Perron true value tests with the suffix 'ppt'. For example, r.ppt denotes the Phillips-Perron t-statistic (equation (8) and (9) in Chapter 2 ) using the true value of (02,0=h0(1-,-2) (4’1, 4'2) (.1,0) (.1,.5) (.1,.8) (.1,.85)(.l,.89)(.l,.9) in .0472 .0493 .0558 .0568 .0651 .0913 I“.w .0664 .0673 .0658 .0649 .0696 .0768 Iu.pp1 .0490 .0528 .0578 .0592 .0676 .0925 In.pp2 .0507 .0535 .0591 .0616 .0693 .0952 Iu.sd1 .0499 .0516 .0568 .0581 .0665 .0918 ru.sd2 .0462 .0479 .0526 .0547 .0637 .0891 i, .0573 .0591 .0617 .0639 .0680 .0874 172w .0808 .0805 .0804 .0803 .0802 .0847 Ifgppl .0588 .0601 .0653 .0652 .0718 .0911 I,.pp2 .0634 .0644 .0678 .0686 .0749 .0942 1,.sd1 .0531 .0554 .0593 .0595 .0640 .0828 r,.sd2 .0477 .0498 .0509 .0548 .0601 .0784 p, .0495 .0533 .0566 .0575 .0599 .0681 pu.pp1 .0515 .0543 .0579 .0591 .0640 .0707 pn.pp2 .0553 .0572 .0602 .0612 .0648 .0742 pu.sd1 .0568 .0576 .0611 .0635 .0667 .0737 pu.sd2 .0595 .0602 .0631 .0657 .0691 .0782 p, .0536 .0541 .0561 .0568 .0575 .0658 p,.pp1 .0560 .0573 .0605 .0616 .0648 .0712 p,.pp2 .0610 .0611 .0641 .0647 .0674 .0772 p,.sd1 .0608 .0617 .0640 .0645 .0693 .0782 p1.sd2 .0679 .0692 .0727 .0744 .0745 .0835 67 TABLE 5 Proportions of Rejections, 5% u w u. A“. ppl 1).-PP?- Iu°5dl ru.sd2 fl)fl)‘l> >fl)"\> q A S A, PP1 r .pp2 A .sdl .sd2 9! If 71' Tu apppl @“oPP2 p“.sdl p#.sd2 2. 2,-PP1 p,.pp2 pf.sd1 p,.sd2 T: (.30,.3)(.60,.3)(.65,.3)(.69,.3)(.70,.3) .0591 .0650 .0591 .0601 .0564 .0518 .0688 .0739 .0710 .0718 .0642 .0532 .0589 .0600 .0611 .0612 .0642 .0637 .0675 .0680 .0697 .0754 (81, 82) .0962 .0562 .0930 .0914 .0860 .0725 .1064 .0578 .1065 .1070 .0961 .0713 .0933 .0904 .0897 .0934 .0899 .0976 .1002 .1003 .1039 .0989 .1032 .0535 .1008 .0988 .0941 .0803 .1155 .0543 .1158 .1167 .1049 .0782 .1017 .0999 .0987 .0999 .0950 .1079 .1097 .1090 .1118 .1061 100, 40 = h0(1 - 4, - 42) .1142 .0528 .1109 .1091 .1042 .0879 .1285 .0541 .1269 .1271 .1183 .0861 .1100 .1081 .1079 .1094 .1042 .1185 .1209 .1197 .1217 .1167 Lower Tail Test .8880 .3457 .8902 .8917 .8830 .8570 .8327 .2999 .8335 .8341 .8207 .7895 .8203 .8202 .8159 .8207 .8072 .7249 .7220 .7138 .7255 .7083 68 TABLE 6 Proportions of Rejections, .0858.0714.0644 .0910. 5% Lower Tail Test 40 = 11,,(1 - 4, - 42) (4,, 42): (.3, .6) (.3, .65) (.3, .7) T : 100 500 1000 100 500 1000 100 500 1000 in .0735.0698.0607 .0814.0817.0726 .4125.8765.9676 luyw .0631.0463.0454 .0617.0454.0426 .1207.1040.1004 Tu.pp1 .0746.0703.0609 .0838.0809.0714 .4174.8783.9677 Iu.pp2 .0745.0692.0601 .0844.0796.0699 .4214.8776.9687 Tu.sdl .0728.0694.0597 .0791.0782.0725 .4043.8757.9689 ru.sd2 .0661.0639.0578 .0726.0731.0693 .3820.8677.9667 if .0829.0755.0683 .0889.0912.0833 .3531.8138.9374 1,.w .0698.0422.0427 .0687.0371.0355 .1011.0535.0488 1,.pp1 .0849.0747.0682 .0932.0907.0826 .3602.8149.9386 1,.pp2 .0872.0756.0669 .0930.0894.08l4 .3632.8162.9391 1,.sd1 .0795.0732.0676 .0845.0876.0815 .3457.8113.9402 1,.sd2 .0642.0671.0614 .0693.0806.0771 .3171.8000.9352 p” .0732.0684.0641 .0780.0799.0756 .2944.8190.9469 pu.ppl .0729.0678.0633 .0796.0779.0749 .2981.816l.9465 pu.pp2 .0740.0678.0628 .0807.0785.0738 .2982.8138.9461 pu.sdl .0752.0683.0632 .0826.0783.0750 .3009.8165.9457 pu.sd2 .0782.0659.0628 .0839.0790.0731 .2941.8l30.9470 p1 .0742.0733.0655 .0790.0885.0794 .2429.7354.8934 p,.pp1 .0782.0729.0653 .0813.0878.0782 .2495.7329.8945 p,.pp2 .0808.0743.0652 .0852.0850.0774 .2553.7310.8948 p1.sd1 .0843.0744.0651 .0871.0881.0781 .2502.7339.8958 0851.0779 .2475.7273.8932 69 TABLE 7 Proportions of Rejections, 5% Lower Tail Test T= 100, 4, = .3, 42= .65, ho: 1 0 .00001 .0001 .001 .01 .l 1 10 100 .6540 .6010 .4537 .2499 .1185 .0795 .0749 .0743 .0743 .1749 .1586 .1297 .0960 .0722 .0610 .0588 .0584 .0584 .6581 .6032 .4561 .2508 .1202 .0811 .0758 .0747 .0746 .6612 .6075 .4580 .2532 .1199 .0817 .0769 .0762 .0762 .6486 .5920 .4421 .2373 .1136 .0773 .0726 .0722 .0720 .6182 .5500 .4074 .2183 .1015 .0707 .0669 .0663 .0663 .5767 .5393 .4243 .2410 .1183 .0878 .0817 .0810 .0810 .1437 .1378 .1232 .1006 .0788 .0681 .0653 .0649 .0650 .5819 .5448 .4283 .2468 .1213 .0906 .0862 .0854 .0854 .5867 .5486 .4309 .2502 .1234 .0908 .0865 .0861 .0861 .5718 .5298 .4092 .2328 .1127 .0828 .0784 .0775 .0775 .5356 .4891 .3670 .2018 .0944 .0687 .0642 .0639 .0639 .5190 .4639 .3342 .1757 .0956 .0775 .0749 .0747 .0747 .5183 .4659 .3373 .1812 .0985 .0787 .0765 .0761 .0761 .5148 .4643 .3403 .1803 .0996 .0799 .0776 .0775 .0775 .5217 .4610 .3319 .1802 .1010 .0818 .0795 .0792 .0792 .5079 .4426 .3148 .1740 .0990 .0829 .0813 .0808 .0808 .4214 .3868 .2884 .1650 .0954 .0785 .0760 .0758 .0758 .4290 .3924 .2972 .1702 .1004 .0809 .0785 .0782 .0782 .4305 .3933 .3022 .1746 .1025 .0838 .0823 .0820 .0820 .4250 .3870 .2929 .1673 .1023 .0869 .0847 .0846 .0846 .4163 .3705 .2794 .1646 .1033 .0897 .0876 .0874 .0875 70 TABLE 8 Proportions of Rejections, 5% Lower Tail Test T=100,4,=.3,42=.7, h0=1 40 : 0 .00001 .0001 .001 .01 .1 1 10 100 in .4125 .3990 .3484 .2326 .1208 .0774 .0711 .0700 .0699 1 .w .1207 .1173 .1072 .0889 .0684 .0539 .0528 .0521 .0520 T:.pp1 .4174 .4044 .3516 .2362 .1213 .0787 .0710 .0700 .0698 iu.pp2 .4214 .4095 .3542 .2396 .1234 .0804 .0733 .0721 .0721 T .Sdl .4043 .3919 .3409 .2269 .1153 .0736 .0678 .0669 .0668 T:.Sd2 .3820 .3698 .3173 .2092 .1077 .0700 .0637 .0631 .0631 1 .3531 .3441 .3064 .2082 .1165 .0828 .0775 .0766 .0766 ,.w .1011 .1004 .0980 .0882 .0701 .0593 .0558 .0550 .0550 A,,.pp1 .3602 .3510 .3109 .2159 .1185 .0843 .0787 .0782 .0782 If.pp2 .3622 .3531 .3155 .2181 .1218 .0865 .0806 .0800 .0800 TT.sd1 .3457 .3374 .2969 .2013 .1138 .0792 .0747 .0740 .0740 T,.Sd2 .3171 .3083 .2678 .1815 .0966 .0676 .0624 .0619 .0618 ~I>fi>fl> p“ .2944 .2850 .2439 .1615 .0992 .0808 .0773 .0772 .0773 pu.pp1 .2981 .2890 .2471 .1665 .1033 .0823 .0794 .0794 .0792 pu.pp2 .2982 .2885 .2483 .1652 .1043 .0855 .0813 .0807 .0806 pu.sdl .3009 .2884 .2461 .1681 .1061 .0850 .0819 .0817 .0817 pu.sd2 .2941 .2812 .2362 .1654 .1066 .0873 .0841 .0837 .0838 p1 .2429 .2358 .2054 .1416 .0946 .0779 .0755 .0753 .0753 p1.pp1 .2495 .2434 .2129 .1488 .0992 .0814 .0796 .0792 .0791 p,.pp2 .2553 .2484 .2197 .1560 .1019 .0847 .0819 .0816 .0816 pf.sd1 .2502 .2433 .2124 .1499 .1029 .0868 .0841 .0836 .0836 21.5d2 .2475 .2397 .2103 .1521 .1047 .0907 .0883 .0878 .0877 71 TABLE 9 Proportions of Rejections, 5% Lower Tail Test T= 100, 4, = .1, 42= .85, h0=1 0 .00001 .0001 .001 .01 .1 l 10 .4220 .4193 .3969 .2671 .1008 .0524 .0464 .0457 .1194 .1187 .1157 .1042 .0823 .0626 .0593 .0590 .4266 .4235 .4022 .2720 .1042 .0538 .0486 .0479 .4322 .4291 .4039 .2762 .1064 .0556 .0497 .0493 .4196 .4167 .3885 .2636 .0986 .0530 .0483 .0476 .3961 .3926 .3661 .2437 .0924 .0505 .0457 .0453 .3521 .3503 .3319 .2350 .0974 .0591 .0556 .0552 .1008 .1014 .1027 .1062 .0968 .0773 .0735 .0728 .3612 .3591 .3397 .2397 .1027 .0618 .0581 .0577 .3657 .3624 .3438 .2438 .1054 .0655 .0605 .0597 .3519 .3500 .3303 .2297 .0918 .0561 .0516 .0510 .3238 .3216 .3007 .2071 .0840 .0509 .0468 .0467 .2690 .2662 .2456 .1565 .0738 .0546 .0534 .0533 .2695 .2663 .2468 .1621 .0761 .0574 .0552 .0548 .2718 .2700 .2510 .1662 .0771 .0593 .1575 .0573 .2731 .2699 .2486 .1604 .0785 .0614 .0596 .0592 .2676 .2643 .2415 .1563 .0812 .0646 .0622 .0621 .2176 .2156 .2033 .1385 .0701 .0555 .0539 .0537 .2292 .2264 .2129 .1465 .0764 .0598 .0589 .0587 .2328 .2316 .2190 .1534 .0818 .0634 .0619 .0618 .2275 .2257 .2134 .1455 .0786 .0639 .0623 .0621 .2296 .2281 .2153 .1504 .0867 .0727 .0708 .0702 100 .0457 .0590 .0479 .0492 .0476 .0452 .0551 .0727 .0576 .0596 .0510 .0465 .0531 .0548 .0573 .0592 .0619 .0536 .0587 .0617 .0621 .0702 To A In In‘w Iu'ppl 1fi.pp2 A Iu'Sdl ru.sd2 q .w .ppl pp2 .sdl .sd2 fl)‘-\>‘i)‘l)fl>fl> a a a a 0 .3 Tu 4,.pp1 4,.pp2 pu.sd1 pu.sd2 2. 2,-PP1 4,4392 p1.sd1 p,.sd2 72 Table 10 Proportions of Rejections, 5% Lower tail T= 100, 4', = .1, 02= 0 .00001 .0001 .001 .01 .0553 .0612 .0580 .0599 .0563 .0541 .0826 .0737 .0854 .0879 .0828 .0792 .0902 .0766 .0917 .0946 .0908 .0877 .0913 .0768 .0925 .0952 .0918 .0891 .0913 .0769 .0925 .0950 .0918 .0891 .0605 .0739 .0627 .0670 .0572 .0519 .0796 .0843 .0833 .0877 .0757 .0718 .0863 .0849 .0907 .0937 .0814 .0779 .0873 .0847 .0911 .0940 .0826 .0783 .0874 .0847 .0911 .0942 .0828 .0784 .0567 .0590 .0613 .0631 .0647 .0679 .0706 .0733 .0732 .0771 .0653 .0671 .0702 .0703 .0746 .0681 .0707 .0741 .0736 .0780 .0681 .0707 .0742 .0737 .0782 .0536 .0605 .0644 .0654 .0720 .0653 .0710 .0766 .0776 .0830 .0624 .0690 .0736 .0725 .0793 .0657 .0712 .0770 .0780 .0835 .0658 .0712 .0772 .0782 .0835 .9, ho .1 .0385 .0445 .0391 .0415 .0399 .0371 .0454 .0603 .0471 .0504 .0443 .0397 .0527 .0558 .0581 .0584 .0601 .0504 .0555 .0592 .0612 .0691 = 1 1 .0365 .0420 .0367 .0385 .0364 .0337 .0426 .0556 .0447 .0475 .0418 .0373 .0510 .0552 .0570 .0569 .0593 .0504 .0561 .0607 .0614 .0699 Test 10 .0362 .0419 .0369 .0384 .0361 .0333 .0424 .0552 .0442 .0474 .0415 .0371 .0510 .0550 .0567 .0566 .0593 .0505 .0561 .0609 .0617 .0699 100 .0363 .0417 .0368 .0384 .0360 .0333 .0425 .0550 .0442 .0474 .0415 .0371 .0511 .0550 .0567 .0566 .0593 .0506 .0561 .0608 .0617 .0699 40 z T : A '3“. w Au ,ppl T: .pp2 T:.sd1 .sd2 q S )~!>‘\)‘i> q 'U ’U H T: .pp2 T .sdl T .sd2 Tu 4,.pp1 2,.pp2 pu.sd1 p“.sd2 2. 2,-PP1 2,-PP2 p1.sd1 p7.sd2 §,== 0 100 .6540 .1749 .6581 .6612 .6486 .6182 .5767 .1437 .5819 .5867 .5718 .5356 .5190 .5183 .5148 .5217 .5079 .4214 .4290 .4305 .4250 .4163 1000 .9975 .1616 .9976 .9975 .9973 .9965 .9922 .0957 .9923 .9929 .9922 .9902 .9960 .9966 .9966 .9962 .9957 .9841 .9841 .9852 .9841 .9830 .3, 73 Table 11 Proportions of Rejections, 52 =.65, .01 100 .1185 .0722 .1202 .1199 .1136 .1015 .1183 .0788 .1213 .1234 .1127 .0944 .0956 .0985 .0996 .1010 .0990 .0954 .1004 .1025 .1023 .1033 1000 .0847 .0429 .0829 .0828 .0838 .0792 .0977 .0370 .0965 .0961 .0949 .0912 .0839 .0838 .0830 .0825 .0794 .0892 .0885 .0884 .0880 .0866 ho: 100 .0749 .0588 .0758 .0769 .0726 .0669 .0817 .0653 .0862 .0865 .0784 .0642 .0749 .0765 .0776 .0795 .0813 .0760 .0785 .0823 .0847 .0876 l 1.0 1000 .0701 .0420 .0686 .0672 .0702 .0670 .0782 .0348 .0781 .0764 .0782 .0729 .0735 .0726 .0723 .0729 .0715 .0774 .0760 .0750 .0755 .0758 5% Lower Tail Test 100 100 .0743 .0584 .0746 .0762 .0720 .0663 .0810 .0650 .0854 .0861 .0775 .0639 .0747 .0761 .0775 .0792 .0808 .0758 .0782 .0820 .0846 .0875 1000 .0698 .0421 .0684 .0670 .0700 .0670 .0781 .0346 .0778 .0761 .0780 .0728 .0735 .0725 .0721 .0729 .0713 .0772 .0759 .0750 .0753 .0757 74 Table 12 Proportions of Rejections, 5% Lower Tail Test 4, =.3, 42 =.7, h0 =1 40 : 0 .01 1.0 100 T : 100 1000 100 1000 100 1000 100 1000 in .4125 .9676 .1208 .1165 .0711 .0889 .0699 .0886 14‘“ .1207 .1004 .0684 .0360 .0528 .0338 .0520 .0337 1,.pp1 .4174 .9677 .1213 .1149 .0710 .0881 .0698 .0874 Iu.pp2 .4214 .9687 .1234 .1135 .0733 .0880 .0721 .0875 Zu.sd1 .4043 .9689 .1153 .1139 .0678 .0893 .0668 .0889 ru.sd2 .3820 .9667 .1077 .1095 .0637 .0839 .0631 .0835 f, .3531 .9374 .1165 .1387 .0775 .1073 .0766 .1072 I,.w .1011 .0488 .0701 .0235 .0558 .0223 .0550 .0222 I,.ppl .3602 .9386 .1185 .1380 .0787 .1054 .0782 .1051 I,.pp2 .3622 .9391 .1218 .1365 .0806 .1057 .0800 .1053 I,.sd1 .3457 .9402 .1138 .1364 .0747 .1067 .0740 .1060 1,.sd2 .3171 .9352 .0966 .1297 .0624 .1002 .0618 .0996 p“ .2944 .9469 .0992 .1175 .0773 .0989 .0773 .0989 pu.pp1 .2981 .9465 .1033 .1176 .0794 .0988 .0792 .0983 pu.pp2 .2982 .9461 .1043 .1157 .0813 .0979 .0806 .0973 pu.sd1 .3009 .9457 .1061 .1185 .0819 .0990 .0817 .0985 pu.sd2 .2941 .9470 .1066 .1147 .0841 .0963 .0838 .0962 p1 .2429 .8934 .0946 .1298 .0755 .1098 .0753 .1095 p,.pp1 .2495 .8945 .0992 .1306 .0796 .1108 .0791 .1104 p,.pp2 .2553 .8948 .1019 .1289 .0819 .1075 .0816 .1073 p,.sd1 .2502 .8958 .1029 .1313 .0841 .1109 .0836 .1106 21.5d2 .2475 .8932 .1047 .1267 .0883 .1063 .0877 .1061 .2 75 Table 13 Proportions of Rejections, 5% Lower Tail Test 40: 0 .01 1.0 100 T: 100 1000 100 1000 100 1000 100 1000 in .4220 .9922 .1008 .0705 .0464 .0500 .0457 .0497 Iu'w .1194 .0967 .0823 .0518 .0593 .0477 .0590 .0475 1,.pp1 .4266 .9926 .1042 .0702 .0486 .0498 .0479 .0497 Iu.pp2 .4322 .9933 .1064 .0702 .0497 .0496 .0492 .0495 Iu.sdl .4196 .9911 .0986 .0688 .0483 .0495 .0476 .0492 Tu.sd2 .3961 .9917 .0924 .0674 .0457 .0483 .0452 .0483 f, .3521 .9758 .0974 .0714 .0556 .0492 .0551 .0490 1,;w .1008 .0467 .0968 .0489 .0735 .0426 .0727 .0425 1,.pp1 .3612 .9773 .1027 .0731 .0581 .0509 .0576 .0506 I,.pp2 .3657 .9786 .1054 .0724 .0605 .0510 .0596 .0508 ;,.sdl .3519 .9775 .0918 .0727 .0516 .0501 .0510 .0500 1,.sd2 .3238 .9760 .0840 .0710 .0468 .0501 .0465 .0499 p“ .2690 .9890 .0738 .0640' .0534 .0536 .0531 .0534 pu.ppl .2695 .9892 .0761 .0647 .0552 .0533 .0548 .0529 pu.pp2 .2718 .9894 .0771 .0644 .0575 .0534 .0573 .0534 pu.sd1 .2731 .9893 .0785 .0633 .0596 .0541 .0592 .0539 pu.sd2 .2676 .9894 .0812 .0622 .0622 .0537 .0619 .0538 p, .2176 .9564 .0701 .0571 .0539 .0466 .0536 .0465 p,.pp1 .2292 .9582 .0764 .0581 .0589 .0482 .0587 .0481 p,.pp2 .2328 .9583 .0818 .0597 .0619 .0494 .0617 .0494 p,.sdl .2275 .9599 .0786 .0581 .0623 .0480 .0621 .0477 p,.sd2 .2296 .9591 .0867 .0576 .0708 .0490 .0702 .0489 Ir'ppz 11.5d1 17.5d2 Tu fiu-Ppl fiuoPP2 pu.sd1 pu.sd2 2. 73,-ppl moppz p1.sd1 p1.sd2 Table 14 76 Proportions of Rejections, 5% Lower Tail Test 100 .0913 .0768 .0925 .0952 .0918 .0891 .0874 .0847 .0911 .0942 .0828 .0784 .0681 .0707 .0742 .0737 .0782 .0658 .0712 .0772 .0782 .0835 ‘1 0 1000 .4991 .0533 .5007 .4992 .4973 .4915 .4297 .0247 .4313 .4315 .4315 .4254 .3892 .3905 .3882 .3896 .3882 .3207 .3209 .3214 .3234 .3199 100 .0553 .0612 .0580 .0599 .0563 .0541 .0605 .0739 .0627 .0670 .0572 .0519 .0567 .0590 .0613 .0631 .0647 .0536 .0605 .0644 .0654 .0720 .01 1000 .0569 .0349 .0561 .0572 .0575 .0579 .0648 .0246 .0663 .0658 .0685 .0639 .0697 .0700 .0699 .0704 .0704 .0667 .0684 .0693 .0686 .0684 =.1, 42 =.9, h0 = 1 1.0 100 .0365 .0420 .0367 .0385 .0364 .0337 .0426 .0556 .0447 .0475 .0418 .0373 .0510 .0552 .0570 .0569 .0593 .0504 .0561 .0607 .0614 .0699 1000 .0409 .0276 .0410 .0421 .0416 .0407 .0530 .0216 .0537 .0543 .0545 .0521 .0640 .0632 .0637 .0635 .0636 .0630 .0636 .0639 .0639 .0636 100 100 .0363 .0417 .0368 .0384 .0360 .0333 .0425 .0550 .0442 .0474 .0415 .0371 .0511 .0550 .0567 .0566 .0593 .0506 .0561 .0608 .0617 .0699 1000 .0409 .0276 .0406 .0415 .0412 .0406 .0531 .0217 .0537 .0544 .0544 .0519 .0639 .0632 .0637 .0635 .0637 .0624 .0636 .0639 .0638 .0636 77 TABLE 15 Proportions of Rejections, 5% Lower Tail Test T = 100, h.0 = 1, 40 =.01 (81: 82) (.3,.30)(.3,.60)(.3,.65)(.3,.69)(.3,.70) .0591 .0964 .1185 .1229 .1208 .0649 .0710 .0722 .0706 .0684 .0592 .0967 .1202 .1243 .1213 .0601 .0967 .1199 .1261 .1234 .0564 .0906 .1136 .1182 .1153 .0518 .0817 .1015 .1091 .1077 .0688 .1035 .1183 .1196 .1165 .0740 .0776 .0788 .0723 .0701 .0710 .1057 .1213 .1216 ..1185 .0718 .1084 .1234 .1254 .1218 .0642 .1004 .1127 .1158 .1138 .0532 .0813 .0944 .0994 .0966 .0590 .0835 .0956 .1010 .0992 .0600 .0847 .0985 .1037 .1033 .0611 .0857 .0996 .1040 .1043 .0613 .0854 .1010 .1069 .1061 .0642 .0861 .0990 .1055 .1066 .0637 .0848 .0954 .0961 .0946 .0675 .0903 .1004 .1020 .0992 .0681 .0931 .1025 .1044 .1019 .0697 .0931 .1023 .1049 .1029 .0754 .0945 .1033 .1068 .1047 78 TABLE 16 Proportions of Rejections, 5% Lower Tail Test T = 100, h0 =1, 4,, =.01 (81,82) (.1, 0)(.1,.5) (.1,.8) (.1,.85)(.1,.89)(.l,.9) in .0472 .0493 .0885 .1008 .0671 .0553 zurw .0664 .0673 .0809 .0823 .0711 .0612 1'.pp1 .0490 .0528 .0907 .1042 .0692 .0580 1:,pp2 .0507 .0536 .0937 .1064 .0714 .0599 Iu.sd1 .0499 .0516 .0856 .0986 .0693 .0563 ru.sd2 .0462 .0479 .0772 .0924 .0652 .0541 f, .0573 .0591 .0911 .0974 .0700 .0605 I uw .0808 .0808 .0951 .0968 .0810 .0739 I7gpp1 .0588 .0601 .0938 .1027 .0733 .0627 1,.pp2 .0634 .0644 .0974 .1054 .0771 .0670 1,.sd1 .0531 .0554 .0856 .0918 .0653 .0572 17.502 .0477 .0498 .0752 .0840 .0614 .0519 , .0495 .0533 .0693 .0738 .0607 .0567 p:.pp1 .0515 .0544 .0711 .0761 .0638 .0590 pu.pp2 .0553 .0572 .0729 .0771 .0658 .0613 pu.sd1 .0568 .0576 .0735 .0785 .0668 .0631 63.502 .0595 .0602 .0751 .0812 .0698 .0647 p1 .0536 .0541 .0673 .0701 .0581 .0536 p,.pp1 .0560 .0573 .0725 .0764 .0658 .0605 p1.pp2 .0610 .0612 .0781 .0818 .0684 .0644 p,.sdl .0608 .0617 .0749 .0786 .0699 .0654 p1.sd2 .0679 .0692 .0817 .0867 .0748 .0720 fi)~i>‘i>~i>‘i>‘4> q 4 '4 A a O mRJTSS '0 p.) BASE .0497 .0683 .0533 .0531 .0492 .0472 .0525 .0805 .0567 .0613 .0517 .0488 .0506 .0524 .0554 .0561 .0600 .0500 .0547 .0611 .0593 .0717 -.8 .9965 .9971 .9848 .9825 .8802 .6425 1.0000 1.0000 1.0000 1.0000 .9799 .8094 .9985 .9899 .9862 .9778 .9370 1.0000 1.0000 1.0000 .9995 .9936 79 Table 17 Proportions of Rejections, T = 100 -.5 .5714 .6076 .4571 .4413 .2025 .0923 .7987 .8344 .7047 .6957 .3046 .1273 .6408 .4938 .4725 .4331 .3510 .8290 .6988 .6933 .6047 .4979 5% Lower Tail Test -.2 .1324 .1630 .1050 .1031 .0595 .0489 .2009 .2579 .1650 .1643 .0703 .0510 .1604 .1193 .1149 .1126 .1086 .2127 .1617 .1626 .1460 .1461 .2 .0299 .0402 .0361 .0391 .0559 .0465 .0157 .0337 .0252 .0288 .0637 .0472 .0163 .0303 .0342 .0325 .0316 .0092 .0238 .0287 .0304 .0322 .5 .0265 .0345 .0284 .0313 .0861 .0377 .0090 .0179 .0146 .0169 .1177 .0315 .0046 .0194 .0251 .0204 .0107 .0022 .0114 .0151 .0158 .0071 .8 .0270 .0343 .0273 .0290 .1138 .0311 .0081 .0161 .0128 .0148 .1675 .0200 .0026 .0169 .0227 .0172 .0034 .0013 .0088 .0121 .0129 .0025 flu-PP2 p“.sd1 pu.sd2 6. a..pp1 b,oPP2 p7.sdl p,.sd2 Proportions of Rejections, BASE 100 200 500 .0497.0499.0496 .0683.0613.0512 .0533.0505.0505 .0531.0513.0513 .0492.0502.0507 .0472.0472.0490 .0525.0581.0521 .0805.0733.0569 .0567.0600.0528 .0613.0625.0539 .0517.0533.0504 .0488.0495.0485 .0506.0516.0476 .0524.0535.0477 .0554.0557.0494 .0561.0549.0481 .0600.0569.0500 .0500.0456.0501 .0547.0495.0512 .0611.0513.0515 .0593.0515.0520 .0717.0540.0524 80 Table 18 -.5 100 200 500 .5714.5976.6216 .6076.6150.6262 .4571.4385.4153 .4413.4010.3451 .2025.2237.2331 .0923.1078.1139 .7987.8356.8492 .8344.8532.8546 .7047.6893.6405 .6957.6534.5643 .3046.3557.3624 .1273.1485.1506 .6408.6646.6834 .4938.4829.4660 .4725.4325.3834 .4331.4457.4499 .3510.3566.3539 .8290.8449.8743 .6988.6736.6618 .6933.63l7.5734 .6047.6005.6245 .4979.4852.5005 5% Lower Tail Test .5 100 200 500 .0265.0240.0251 .0345.0299.0267 .0284.0296.0305 .0313.0319.0290 .0861.0890.0930 .0377.0346.0373 .0090.0089.0086 .0179.0151.0110 .0146.0192.0199 .0169.0254.0251 .1177.1236.1147 .0315.0318.0311 .0046.0044.0041 .0194.0192.0188 .0251.0269.0267 .0204.0180.0178 .0107.0080.0073 .0022.0019.0014 .0114.0111.0148 .0151.0176.0229 .0158.0127.0147 .0071.0046.0043 81 Table 19 Proportions of Rejections, 5% 4-: 4. 'U 'U H -.8 .7981 .8276 .6086 .6838 .0506 .0460 .9580 .0603 .8705 .9137 .0518 .0486 .8444 .5872 .6998 .2336 .2404 .9669 .8380 .9118 .3292 .3448 -.5 .3627 .4038 .2243 .2500 .0505 .0475 .5651 .6273 .4007 .4464 .0514 .0484 .4287 .2309 .2686 .1577 .1645 .5972 .3701 .4391 .2210 .2349 100 -.2 .1144 .1421 .0891 .0908 .0496 .0475 .1727 .2249 .1345 .1415 .0508 .0473 .1386 .0985 .0995 .0929 .0978 .1828 .1309 .1397 .1176 .1281 Lower Tail Test .2 .0288 .0372 .0317 .0347 .0497 .0483 .0134 .0285 .0195 .0230 .0520 .0489 .0122 .0233 .0285 .0254 .0276 .0062 .0168 .0216 .0218 .0267 .5 .0349 .0440 .0280 .0253 .0520 .0487 .0094 .0162 .0079 .0085 .0544 .0484 .0006 .0031 .0070 .0029 .0036 .0001 .0014 .0029 .0017 .0020 .8 .0941 .1126 .0608 .0468 .0541 .0522 .0358 .0490 .0200 .0142 .0590 .0565 .0000 .0000 .0002 .0001 .0001 .0001 .0001 .0002 .0001 .0001 82 Table 20 Proportions of Rejections, 5% Lower Tail Test 1?); e>s>s>s>e>s> 1:51:51: 'OTS£ 'U'U ktF‘ t ~44 q ‘DWD‘D‘D‘D‘D a « 013T5£ ’U I-' -.5 .5 100 200 500 100 200 500 .3627 .3764 .3833 .0349 .0342 .0351 .4038 .3946 .3907 .0440 .0393 .0374 .2243 .1951 .1609 .0280 .0255 .0265 .2500 .2155 .1723 .0253 .0235 .0258 .0505 .0494 .0499 .0520 .0495 .0495 .0475 .0475 .0490 .0487 .0470 .0491 .5651 .5898 .5880 .0094 .0085 .0096 .6273 .6186 .6018 .0162 .0131 .0111 .4007 .3416 .2509 .0079 .0078 .0080 .4464 .3827 .2760 .0085 .0099 .0096 .0514 .0541 .0507 .0544 .0545 .0492 .0484 .0518 .0490 .0484 .0512 .0486 .4287 .4432 .4461 .0006 .0003 .0004 .2309 .2058 .1784 .0031 .0027 .0028 .2686 .2314 .1878 .0070 .0071 .0072 .1577 .1637 .1612 .0029 .0025 .0022 .1645 .1661 .1607 .0036 .0025 .0026 .5972 .6011 .6255 .0001 .0001 .0000 .3701 .3058 .2516 .0014 .0007 .0006 .4391 .3550 .2831 .0029 .0027 .0027 .2210 .1987 .1990 .0017 .0011 .0006 .2349 .2065 .2004 .0020 .0009 .0008 ll 83 Table 21 Number of Negative Long-run Variance Estimates (out of 10,000 replications) T = 100 MA: -.8 -.5 -.2 2 .5 .8 BASE ppl 0 0 36 0 0 0 0 pp2 0 0 19 0 0 0 0 pp3 0 1 10 2 0 0 0 pp4 1 0 5 5 2 2 2 pp5 10 3 9 11 11 15 14 pp6 12 4 4 6 18 28 31 pp7 34 7 3 19 52 86 101 pp8 58 10 8 18 106 167 188 pp9 78 27 8 37 166 244 275 pp10 109 38 8 40 216 348 405 AR: —.8 -.5 -.2 .2' .5 .8 ppl 6204 1328 O 0 O 0 pp2 0 1 O 0 0 0 pp3 479 27 2 0 0 O pp4 1 5 2 0 0 0 pp5 54 14 10 8 1 0 pp6 1 1 6 18 8 O pp7 10 4 17 51 38 2 pp8 3 11 19 104 114 11 pp9 4 9 40 168 195 34 pp10 5 10 44 231 345 68 84 Table 22 Proportions of Rejections, 5% Lower Tail Test Bartlett's Lag Window, T = 100 MA: -.8 -.5 -.2 .2 .5 .8 in .9965 .5714 .1324 .0299 .0265 .0270 Iu.pp1 .9848 .4571 .1050 .0361 .0284 .0273 Iu.pp2 .9825 .4413 .1031 .0391 .0313 .0290 Zfl.pp3 .9834 .4474 .1062 .0392 .0319 .0299 Zu.pp4 .9850 .4626 .1110 .0388 .0320 .0301 I“.pp5 .9872 .4799 .1148 .0382 .0316 .0299 Iu.pp6 .9894 .4960 .1181 .0381 .0305 .0289 Zu.pp7 .9910 .5126 .1225 .0375 .0291 .0277 Iu.pp8 .9928 .5270 .1259 .0367 .0290 .0271 Tu.pp9 .9942 .5396 .1303 .0359 .0279 .0263 Tu.pp10 .9949 .5507 .1328 .0346 .0271 .0255 1.0000 .7987 .2009 .0157 .0090 .0081 .ppl 1.0000 .7047 .1650 .0252 .0146 .0128 .pp2 1.0000 .6957 .1643 .0288 .0169 .0148 IT.pp3 1.0000 .7117 .1725 .0288 .0168 .0148 :1.pp4 1.0000 .7317 .1817 .0274 .0155 .0137 I,.pp5 1.0000 .7510 .1912 .0259 .0138 .0117 If.pp6 1.0000 .7696 .1988 .0237 .0119 .0104 Ir.pp7 1.0000 .7864 .2029 .0216 .0109 .0099 IT.pp8 1.0000 .8011 .2068 .0190 .0104 .0096 IT.pp9 1.0000 .8138 .2101 .0176 .0102 .0093 TT.pp10 1.0000 .8240 .2131 .0158 .0096 .0090 BASE .0497 .0533 .0531 .0548 .0552 .0556 .0556 .0566 .0556 .0561 .0557 .0525 .0567 .0613 .0632 .0640 .0634 .0609 .0587 .0572 .0547 .0531 Proportions of Rejections, MA: all. ppppl ppppz 2,. pp3 2,.pp4 4,.pp5 Au-pp6 2),.pp7 4,448 2),. pp9 pwpplo 2. 2,-ppl 4,-pp2 2,-pp3 2,-pp4 2,-pp5 2,-pp6 2,.pp7 fi,oPP8 2,.999 85 Table 23 5% Lower Tail Test Bartlett's Lag Window, T = 100 -.8 .9985 .9899 .9862 .9868 .9885 .9907 .9923 .9941 .9951 .9959 .9966 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 p,.pp10 1.0000 -.5 -.2 .6408 .1604 .4938 .1193 .4725 .1149 .4774 .1191 .4934 .1253 .5113 .1304 .5299 .1376 .5481 .1427 .5637 .1485 .5787 .1528 .5924 .1573 .8290 .2127 .6988 .1617 .6933 .1626 .7122 .1749 .7376 .1879 .7629 .1981 .7826 .2079 .8024 .2165 .8194 .2223 .8336 .2259 .8440 .2285 .0163 .0303 .0342 .0361 .0356 .0348 .0336 .0321 .0303 .0282 .0254 .0092 .0238 .0287 .0294 .0274 .0251 .0206 .0170 .0133 .0108 .0086 .5 .0046 .0194 .0251 .0263 .0265 .0254 .0234 .0210 .0194 .0165 .0138 .0022 .0114 .0151 .0141 .0123 .0104 .0081 .0048 .0035 .0025 .0020 .8 .0026 .0169 .0227 .0242 .0243 .0224 .0206 .0188 .0164 .0144 .0117 .0013 .0088 .0121 .0115 .0100 .0083 .0058 .0033 .0023 .0015 .0011 BASE .0506 .0524 .0554 .0568 .0581 .0593 .0597 .0596 .0596 .0592 .0575 .0500 .0547 .0611 .0642 .0662 .0657 .0636 .0609 .0571 .0524 .0499 Ill 86 Table 24 Proportions of Rejections, 5% Lower Tail Test Parzen's Lag Window, T = 100 MA: -.8 -.5 -.2 .2 .5 .8 BASE in .9965 .5714 .1324 .0299 .0265 .0270 .0497 Iu.ppl .9921 .5155 .1183 .0315 .0258 .0249 .0515 Iu.pp2 .9839 .4499 .1032 .0369 .0290 .0280 .0530 I“.pp3 .9805 .4274 .0997 .0398 .0326 .0309 .0541 Iu.pp4 .9801 .4254 .1010 .0407 .0333 .0323 .0545 Iu.pp5 .9815 .4337 .1041 .0404 .0340 .0326 .0551 Iu.pp6 .9826 .4450 .1070 .0402 .0338 .0331 .0552 Iu.pp7 .9844 .4590 .1104 .0403 .0336 .0325 .0560 Iu.pp8 .9863 .4727 .1136 .0404 .0331 .0321 .0562 I#.pp9 .9882 .4863 .1172 .0393 .0319 .0310 .0565 Tu.pp10 .9899 .4988 .1205 .0395 .0314 .0295 .0569 ET 1.0000 .7987 .2009 .0157 .0090 .0081 .0525 If.pp1 1.0000 .7386 .1806 .0195 .0103 .0092 .0548 I1.pp2 1.0000 .6894 .1642 .0258 .0153 .0133 .0577 IT.pp3 1.0000 .6802 .1609 .0303 .0185 .0169 .0609 11.pp4 1.0000 .6896 .1633 .0323 .0198 .0175 .0629 I,.pp5 1.0000 .7078 .1694 .0314 .0188 .0174 .0646 IT.pp6 1.0000 .7262 .1760 .0304 .0161 .0165 .0661 IT.pp7 1.0000 .7431 .1832 .0292 .0164 .0151 .0653 If.pp8 1.0000 .7602 .1897 .0276 .0149 .0137 .0644 :1.pp9 1.0000 .7758 .1963 .0261 .0131 .0119 .0634 TT.pp10 1.0000 .7882 .2023 .0234 .0122 .0112 .0613 ll Proportions of Rejections, 5% Lower Tail Test MA: fill- 2),.pp1 4,.pp2 4,.pp3 Au-pp4 4,.pp5 4,.pp6 4,.pp7 4,.pp8 2,.pp9 pu.pp10 2. 2,-ppl 4,.pp2 21,-PP3 2,-PP4 2,-pp5 73,-PP6 2,.pp7 ¢,oPP8 4,-pp9 p,.pp10 87 Table 25 Parzen's Lag Window, T = 100 -.8 .9985 .9961 .9882 .9840 .9832 .9847 .9866 .9877 .9893 .9981 .9926 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -.5 -.2 .6408 .1604 .5713 .1406 .4826 .1161 .4497 .1112 .4473 .1111 .4582 .1138 .4717 .1200 .4845 .1249 .5000 .1288 .5176 .1345 .5316 .1398 .8290 .2127 .7704 .1866 .6919 .1596 .6727 .1562 .6787 .1608 .6975 .1696 .7181 .1807 .7394 .1892 .7591 .1969 .7768 .2053 .7909 .2136 .2 .0163 .0225 .0317 .0366 .0391 .0397 .0402 .0396 .0390 .0378 .0353 .0092 .0154 .0255 .0319 .0339 .0336 .0326 .0302 .0271 .0236 .0199 .5 .0046 .0097 .0211 .0276 .0298 .0311 .0307 .0305 .0291 .0268 .0249 .0022 .0041 .0126 .0181 .0209 .0200 .0176 .0150 .0126 .0099 .0075 .8 .0026 .0074 .0186 .0255 .0286 .0291 .0292 .0283 .0268 .0246 .0230 .0013 .0032 .0094 .0157 .0174 .0172 .0146 .0130 .0107 .0077 .0057 BASE .0506 .0513 .0531 .0547 .0564 .0576 .0590 .0605 .0614 .0610 .0610 .0500 .0517 .0563 .0615 .0639 .0666 .0677 .0681 .0678 .0665 .0642 88 Table 26 Proportions of Rejections, 5% Lower Tail Test Bohman's Lag Window, T = 100 MA: -.8 -.5 -.2 .2 .5 .8 BASE in .9965 .5714 .1324 .0299 .0265 .0270 .0497 Iu.pp1 .9905 .4992 .1141 .0329 .0263 .0249 .0516 Iu.pp2 .9826 .4407 .1013 .0377 .0302 .0284 .0530 Iu.pp3 .9797 .4249 .0999 .0403 .0329 .0315 .0540 Iu.pp4 .9804 .4279 .1020 .0406 .0334 .0325 .0547 Iu.pp5 .9822 .4397 .1050 .0405 .0340 .0325 .0554 Iu.pp6 .9839 .4524 .1092 .0403 .0336 .0327 .0560 Iu.pp7 .9856 .4683 .1127 .0404 .0332 .0323 .0561 Iu.pp8 .9877 .4824 .1161 .0397 .0322 .0313 .0563 Zu.pp9 .9897 .4960 .1195 .0396 .0314 .0299 .0572 Tu.pp10 .9908 .5111 .1234 .0387 .0301 .0287 .0577 ? 1.0000 .7987 .2009 .0157 .0090 .0081 .0525 f,.pp1 1.0000 .7386 .1765 .0213 .0113 .0098 .0552 1,.pp2 1.0000 .6894 .1622 .0274 .0165 .0142 .0587 IT.pp3 1.0000 .6802 .1618 .0314 .0193 .0172 .0616 IT.pp4 1.0000 .6896 .1664 .0321 .0193 .0175 .0632 I,.pp5 1.0000 .7078 .1728 .0308 .0185 .0173 .0656 IT.pp6 1.0000 .7262 .1801 .0304 .0175 .0155 .0657 I,.pp7 1.0000 .7431 .1886 .0281 .0152 .0143 .0650 TT.pp8 1.0000 .7602 .1948 .0266 .0135 .0127 .0643 1 .pp9 1.0000 .7758 .2013 .0240 .0124 .0114 .0622 r .pp10 1.0000 .7882 .2052 .0216 .0115 .0105 .0598 Proportions of Rejections, 89 Table 27 5% Lower Tail Test Bohman's Lag Window, T = 100 MA: -.8 pu .9985 p#.ppl .9954 pu.pp2 .9867 pu.pp3 .9831 pu.pp4 .9836 pu.pp5 .9858 p#.pp6 .9874 p#.pp7 .9890 pu.pp8 .9909 pu.pp9 .9923 p#.pp10 .9939 p, 1.0000 p1.ppl 1.0000 p7.pp2 1.0000 pf.pp3 1.0000 p1.pp4 1.0000 p1.pp5 1.0000 pT.pp6 1.0000 p1.pp7 1.0000 pf.pp8 1.0000 p1.pp9 1.0000 p7.pp10 1.0000 .6408 .5508 .4686 .4464 .4512 .4650 .4785 .4951 .5131 .5291 .5448 .8290 .7494 .6830 .6714 .6857 .7069 .7313 .7519 .7721 .7888 .8051 .1604 .1340 .1134 .1099 .1115 .1161 .1230 .1278 .1328 .1392 .1439 .2127 .1806 .1573 .1581 .1642 .1743 .1856 .1960 .2032 .2126 .2193 .0163 .0248 .0336 .0377 .0392 .0406 .0402 .0393 .0385 .0368 .0346 .0092 .0174 .0271 .0335 .0346 .0339 .0315 .0286 .0243 .0206 .0176 .0046 .0119 .0240 .0286 .0308 .0309 .0306 .0298 .0275 .0254 .0237 .0022 .0050 .0150 .0202 .0212 .0194 .0164 .0137 .0111 .0078 .0058 .0026 .0092 .0207 .0266 .0292 .0293 .0290 .0276 .0251 .0231 .0203 .0013 .0038 .0122 .0166 .0183 .0157 .0141 .0113 .0088 .0062 .0045 90 Table 28 Proportions of Rejections, 5% Lower Tail Test Bartlett's Lag Window, T = 100 AR: -.8 -.5 -.2 .2 .5 .8 in .7981 .3627 .1144 .0288 .0349 .0941 Iu.ppl .6086 .2243 .0891 .0317 .0280 .0608 Iu.pp2 .6838 .2500 .0908 .0347 .0253 .0468 In.pp3 .6730 .2574 .0947 .0364 .0254 .0401 Iu.pp4 .7081 .2749 .0985 .0361 .0248 .0366 Iu.pp5 .7176 .2893 .1019 .0351 .0246 .0344 Iu.pp6 .7393 .3038 .1053 .0351 .0253 .0330 Iu.pp7 .7486 .3177 .1077 .0347 .0245 .0315 Iu.pp8 .7639 .3300 .1107 .0344 .0242 .0302 r .pp9 .7749 .3422 .1141 .0338 .0240 .0301 Tu.pp10 .7859 .3524 .1154 .0323 .0235 .0302 f, .9580 .5651 .1727 .0134 .0094 .0358 I,.pp1 .8705 .4007 .1345 .0195 .0079 .0200 I,.pp2 .9137 .4464 .1415 .0230 .0085 .0142 I,.pp3 .9135 .4643 .1510 .0234 .0093 .0113 I,.pp4 .9334 .4955 .1594 .0222 .0091 .0098 I,.pp5 .9394 .5186 .1666 .0215 .0086 .0099 I,.pp6 .9495 .5392 .1729 .0197 .0082 .0094 Z,.pp7 .9547 .5612 .1781 .0174 .0077 .0093 I,.pp8 .9601 .5796 .1808 .0159 .0074 .0092 IT.pp9 .9636 .5955 .1842 .0142 .0073 .0095 T7.pp10 .9666 .6072 .1839 .0130 .0071 .0096 Ill 91 Table 29 Proportions of Rejections, 5% Lower Tail Test Bartlett's Lag Window, T = 100 AR: -.8 -.5 -.2 .2 .5 p“ .8444 .4287 .1386 .0122 .0006 pu.ppl .5872 .2309 .0985 .0233 .0031 pu.pp2 .6998 .2686 .0995 .0285 .0070 pu.pp3 .6819 .2757 .1047 .0311 .0102 pu.pp4 .7286 .2992 .1102 .0316 .0121 pfi.pp5 .7348 .3157 .1170 .0313 .0123 pu.pp6 .7601 .3366 .1221 .0298 .0111 pu.pp7 .7712 .3542 .1269 .0277 .0107 pu.pp8 .7894 .3699 .1322 .0263 .0095 pu.pp9 .8001 .3839 .1368 .0237 .0084 pu.pp10 .8102 .3976 .1396 .0210 .0061 p1 .9669 .5972 .1828 .0062 .0001 p,.pp1 .8380 .3701 .1309 .0168 .0014 p,.pp2 .9118 .4391 .1397 .0216 .0029 p,.pp3 .9104 .4632 .1507 .0232 .0038 pT.pp4 .9345 .5015 .1624 .0220 .0037 p,.pp5 .9417 .5306 .1752 .0190 .0030 p,.pp6 .9529 .5591 .1836 .0160 .0021 p,.pp7 .9584 .5814 .1884 .0134 .0011 p,.pp8 .9630 .5991 .1934 .0105 .0007 p,.pp9 .9674 .6161 .1965 .0081 .0006 p,.pp10 .9717 .6317 .1986 .0061 .0005 .8 .0000 .0000 .0002 .0009 .0016 .0020 .0022 .0024 .0026 .0023 .0023 .0001 .0001 .0002 .0002 .0003 .0003 .0003 .0001 .0001 .0001 .0001 Proportions of Rejections, 92 Table 30 5% Parzen's Lag Window, AR: -.8 in .7981 Iu.ppl .7089 Tu.pp2 .6193 Tu.pp3 .6266 Iu.pp4 .6501 Iu.pp5 .6724 Iu.pp6 .6885 Iu.pp7 .7042 Iu.pp8 .7191 I“.pp9 .7343 Tu.pp10 .7438 f, .9580 I,.ppl .9191 I,.pp2 .8783 TT.pp3 .8859 T .pp4 .9009 11.pp5 .9135 11.pp6 .9259 11.pp7 .9340 11.pp8 .9410 11.pp9 .9483 T1.pp10 .9530 .3627 .2910 .2263 .2232 .2336 .2484 .2620 .2755 .2867 .2995 .3107 .5651 .4799 .4062 .4072 .4289 .4562 .4793 .5014 .5190 .5363 .5539 .1144 .1005 .0889 .0872 .0893 .0924 .0959 .0984 .1012 .1044 .1070 .1727 .1524 .1345 .1346 .1394 .1466 .1538 .1618 .1667 .1717 .1773 Lower Tail Test T = 100 .2 .5 .0288 .0349 .0292 .0302 .0325 .0276 .0354 .0251 .0376 .0255 .0381 .0254 .0374 .0260 .0375 .0267 .0370 .0269 .0366 .0265 .0365 .0263 .0134 .0094 .0152 .0079 .0204 .0082 .0242 .0088 .0255 .0096 .0261 .0101 .0251 .0103 .0242 .0103 .0232 .0100 .0215 .0093 .0197 .0087 .0941 .0755 .0558 .0468 .0410 .0383 .0351 .0341 .0324 .0310 .0302 .0358 .0251 .0185 .0140 .0116 .0102 .0099 .0098 .0099 .0095 .0096 l 93 Table 31 Proportions of Rejections, 5% Lower Tail Test AR: pa 2,.pp1 4,.pp2 a, . pp3 4,.pp4 2,,- p95 4,.pp6 4,.pp7 4,.pp8 A, - pp9 pu.pp10 2. 2,-PPl 73,-PP2 2,-PP3 2,-PP4 AT-PP5 2,-PP5 2,-PP7 2,-PP8 27~PP9 p,.pp10 Parzen's Lag Window, T = 100 .8444 .7368 .6045 .6150 .6528 .6803 .7010 .7214 .7357 .7520 .7656 .9669 .9203 .8526 .8682 .8910 .9109 .9256 .9360 .9439 .9514 .9573 -.5 .4287 .3317 .2327 .2247 .2405 .2600 .2809 .2964 .3100 .3265 .3449 .5972 .4850 .3785 .3822 .4167 .4515 .4800 .5080 .5321 .5548 .5741 -.2 .1386 .1185 .0974 .0931 .0955 .1005 .1056 .1097 .1157 .1208 .1244 .1828 .1534 .1314 .1312 .1379 .1462 .1564 .1664 .1751 .1833 .1882 .2 .0122 .0177 .0254 .0293 .0332 .0343 .0352 .0355 .0347 .0333 .0318 .0062 .0109 .0183 .0240 .0276 .0273 .0257 .0245 .0227 .0197 .0160 .5 .0006 .0015 .0040 .0072 .0110 .0135 .0145 .0153 .0160 .0158 .0150 .0001 .0005 .0017 .0033 .0044 .0058 .0061 .0061 .0051 .0042 .0030 .0000 .0000 .0001 .0002 .0007 .0013 .0017 .0022 .0024 .0030 .0034 .0001 .0001 .0002 .0002 .0002 .0003 .0003 .0004 .0004 .0004 .0004 93 Table 31 Proportions of Rejections, 5% AR: Tu moppl p,.pp2 2,.pp3 p,.pp4 4,.pp5 4,.pp6 4,.pp7 4,.pp8 4,.pp9 pu.pp10 2. ppppl 4,.pp2 4,.pp3 4,.pp4 4,.pp5 2,-PP5 4,.pp7 2,.pp8 4,.pp9 p,.pp10 -.8 .8444 .7368 .6045 .6150 .6528 .6803 .7010 .7214 .7357 .7520 .7656 .9669 .9203 .8526 .8682 .8910 .9109 .9256 .9360 .9439 .9514 .9573 -.5 .4287 .3317 .2327 .2247 .2405 .2600 .2809 .2964 .3100 .3265 .3449 .5972 .4850 .3785 .3822 .4167 .4515 .4800 .5080 .5321 .5548 .5741 -.2 .1386 .1185 .0974 .0931 .0955 .1005 .1056 .1097 .1157 .1208 .1244 .1828 .1534 .1314 .1312 .1379 .1462 .1564 .1664 .1751 .1833 .1882 Lower Tail Test Parzen's Lag Window, T = 100 .0122 .0177 .0254 .0293 .0332 .0343 .0352 .0355 .0347 .0333 .0318 .0062 .0109 .0183 .0240 .0276 .0273 .0257 .0245 .0227 .0197 .0160 .5 .0006 .0015 .0040 .0072 .0110 .0135 .0145 .0153 .0160 .0158 .0150 .0001 .0005 .0017 .0033 .0044 .0058 .0061 .0061 .0051 .0042 .0030 .8 .0000 .0000 .0001 .0002 .0007 .0013 .0017 .0022 .0024 .0030 .0034 .0001 .0001 .0002 .0002 .0002 .0003 .0003 .0004 .0004 .0004 .0004 94 Table 32 Proportions of Rejections, 5% Lower Tail Test .3 )fl)~1)‘1)~i) q 15% TVU N I-J q .pp3 It’pp4 Ir'pps 11.pp6 Ir'pp7 Zr'pp8 Ir'ppg q 11.pp10 Bohman's Lag Window, T = 100 -.8 .7981 .6798 .6140 .6376 .6585 .6795 .6979 .7151 .7310 .7424 .7543 .9580 .9063 .8768 .8918 .9061 .9199 .9309 .9387 .9471 .9524 .9577 -.5 .3627 .2713 .2216 .2267 .2397 .2547 .2691 .2836 .2964 .3089 .3209 .5651 .4581 .4009 .4158 .4412 .4671 .4917 .5140 .5323 .5514 .5714 .1144 .0968 .0877 .0878 .0905 .0939 .0972 .1003 .1037 .1066 .1092 .1727 .1476 .1328 .1357 .1429 .1506 .1592 .1657 .1708 .1761 .1818 -.2 .2 .0288 .0298 .0336 .0365 .0381 .0381 .0375 .0376 .0366 .0368 .0359 .0134 .0158 .0218 .0245 .0260 .0260 .0249 .0237 .0224 .0203 .0179 .5 .0349 .0293 .0270 .0251 .0256 .0259 .0265 .0268 .0266 .0264 .0262 .0094 .0077 .0081 .0089 .0099 .0103 .0108 .0100 .0094 .0088 .0083 .8 .0941 .0711 .0532 .0449 .0398 .0362 .0342 .0330 .0312 .0304 .0295 .0358 .0238 .0171 .0129 .0112 .0100 .0100 .0090 .0096 .0096 .0096 Table 33 Proportions of Rejections, 5% Lower Tail Test AR: fill moppl moppz 4,.pp3 2,-pp4 2,.pps 4,-pp6 4,.pp7 p,.pp8 4,.pp9 4,.pp10 2. 2,oPP1 2,-PP2 AT-PP3 2,-PP4 2,-PP5 b,oPP5 27~PP7 2,-PP8 2,-PP9 p,.pp10 .8444 .6998 .5974 .6330 .6626 .6887 .7134 .7320 .7474 .7629 .7766 .9669 .8990 .8497 .8781 .8986 .9181 .9322 .9413 .9496 .9568 .9603 .4287 .3034 .2227 .2296 .2482 .2703 .2898 .3067 .3221 .3415 .3567 .5972 .4542 .3701 .3936 .4312 .4651 .4961 .5248 .5487 .5710 .5893 .1386 .1127 .0953 .0938 .0976 .1026 .1080 .1143 .1197 .1240 .1280 .1828 .1463 .1297 .1321 .1413 .1522 .1615 .1719 .1815 .1880 .1927 Bohman's Lag Window, T = 100 .0122 .0188 .0268 .0310 .0339 .0346 .0353 .0348 .0342 .0321 .0307 .0062 .0118 .0200 .0257 .0280 .0263 .0252 .0233 .0200 .0166 .0141 .0006 .0017 .0049 .0082 .0128 .0140 .0151 .0160 .0159 .0158 .0140 .0001 .0010 .0020 .0039 .0051 .0062 .0064 .0055 .0044 .0033 .0025 .0000 .0000 .0001 .0004 .0012 .0017 .0022 .0024 .0029 .0034 .0036 .0001 .0001 .0002 .0002 .0002 .0003 .0003 .0004 .0005 .0004 .0003 96 Table 34 Proportions of Rejections, 5% Lower Tail Test T = 100 MA: -.8 -.5 -.2 .2 .5 .8 in .9965 .5714 .1324 .0299 .0265 .0270 Iu.pp1 .9948 .4571 .1050 .0361 .0284 .0273 Tu.ppt .8109 .1598 .0568 .0498 .0511 .0515 f, 1.0000 .7987 .2009 .0157 .0090 .0081 17.pp1 1.0000 .7047 .1650 .0252 .0146 .0128 1,.ppt .9186 .2510 .0655 .0534 .0561 .0570 p“ .9985 .6408 .1604 .0163 .0046 .0026 pu.ppl .9899 .4938 .1193 .0303 .0194 .0169 pu.ppt .1129 .0413 .0426 .0558 .0593 .0602 p1 1.0000 .8290 .2127 .0092 .0022 .0013 p,.pp1 1.0000 .6988 .1617 .0238 .0114 .0088 p1.ppt .0689 .0275 .0327 .0614 .0723 .0754 Proportions of Rejections, 5% Lower Tail Test 5 A>q>q> 3: “.ppl 4: 'U 'U {1' ~'I)"I)‘i> «~44 T36 TIC (1' H ‘0 u 4,.pp1 2),.ppt T fif’ppl pf'ppt .7981 .6086 .3759 .9580 .8705 .5532 .8444 .5872 .1427 .9669 .8380 .1608 97 Table 35 T = 100 -.5 -.2 .3627 .1144 .2243 .0891 .1053 .0561 .5651 .1727 .4007 .1345 .1670 .0660 .4287 .1386 .2309 .0985 .0527 .0451 .5972 .1828 .3701 .1309 .0477 .0400 .0288 .0317 .0521 .0134 .0195 .0580 .0122 .0233 .0580 .0062 .0168 .0697 .0349 .0280 .0659 .0094 .0079 .0936 .0006 .0031 .0752 .0001 .0014 .1162 .0941 .0608 .1171 .0358 .0200 .2306 .0000 .0000 .1395 .0001 .0001 .2755 .5 98 Table 36 Proportions of Rejections, 5% Lower Tail Test '06 rt)- -4>-1>-\> -1>-a>-I> a -\ 1: 1: 1: . '0’0 'O'U rt l—' q 'U'U q DEE WW ‘5) I: F 'I: '06 pro rt)- 9 q A '0'!) 'U'U t'I’l-J -.8 -.5 100 200 500 1000 100 200 500 1000 .9965 .9976 .9992 .9993 .5714 .5976 .6216 .6209 .9948 .9880 .9861 .9875 .4571 .4385 .4153 .3986 .8109 .7830 .6982 .5863 .1598 .1267 .0887 .0726 1.00001.00001.00001.0000 .7987 .8356 .8492 .8572 1.0000 .9999 .9999 .9998 .7047 .6893 .6405 .6319 .9186 .9341 .9048 .8379 .2510 .2010 .1264 .0932 .9985 .9991 .9998 .9996 .6408 .6646 .6834.6849 .9899 .9922 .9912 .9920 .4938 .4829 .4660 .4559 .1129 .1459 .1525 .1421 .0413 .0456 .0489 .0524 1.00001.00001.00001.0000 .8290 .8449 .8713 .8760 1.0000 .99991.0000 .9998 .6988 .6736 .6618 .6511 .0689 .1074 .1551 .1604 .0275 .0345 .0433 .0468 100 .0265 .0284 .0511 .0090 .0146 .0561 .0046 .0194 .0593 .0022 .0114 .0723 200 .0240 .0296 .0503 .0089 .0192 .0578 .0044 .0192 .0570 .0019 .0111 .0561 Table 37 .5 500 .0251 .0305 .0495 .0086 .0199 .0536 .0041 .0188 .0492 .0014 .0148 .0525 99 1000 .0265 .0319 .0515 .0076 .0197 .0512 .0043 .0199 .0557 .0017 .0143 .0490 100 .0270 .0273 .0515 .0081 .0128 .0570 .0026 .0169 .0602 .0013 .0088 .0754 200 .0248 .0285 .0507 .0076 .0161 .0580 .0025 .0173 .0581 .0004 .0091 .0569 Proportions of Rejections, 5% Lower Tail Test .8 500 .0265 .0298 .0492 .0079 .0178 .0537 .0025 .0163 .0495 .0009 .0118 .0526 1000 .0263 .0308 .0514 .0070 .0177 .0508 .0026 .0178 .0558 .0009 .0117 .0492 q) A“ I“.ppl Tu.ppt T.pp1 ,.ppt ‘i>~l)‘i) 'Q )1. 2,.pp1 2),.th A. par'ppl 2,-ppt 100 Table 38 Proportions of Rejections, 5% Lower Tail Test 100 .7981 .6086 .3759 .9580 .8705 .5532 .8444 .5872 .1427 .9669 .8380 .1608 200 .8165 .5285 .3057 .9728 .8136 .4725 .8607 .5106 .1377 .9773 .7574 .1562 -.8 500 .8356 .4045 .2069 .9786 .6741 .3381 .8754 .3992 .1086 .9853 .6341 .1409 1000 .8343 .3234 .1546 .9808 .5683 .2437 .8767 .3286 .0937 .9854 .5322 .1152 100 .3627 .2243 .1053 .5651 .4007 .1670 .4287 .2309 .0527 .5972 .3701 .0477 200 .3764 .1951 .0849 .5898 .3416 .1247 .4432 .2058 .0521 .6011 .3058 .0472 —.5 500 .3833 .1609 .0661 .5880 .2509 .0811 .4461 .1784 .0506 .6255 .2516 .0497 1000 .3809 .1454 .0609 .6000 .2198 .0672 .4463 .1654 .0542 .6338 .2190 .0483 )fl) 6 u. A“. ppl ru.ppt q , T.ppl 1.ppt ~I>~’t)~1> aII» 4,.pp1 4,.ppt 21 AT-Ppl bTOPPt 101 Table 39 Proportions of Rejections, 5% Lower Tail Test 100 .0349 .0280 .0659 .0094 .0079 .0936 .0006 .0031 .0752 .0001 .0014 .1162 200 .0342 .0255 .0575 .0085 .0078 .0777 .0003 .0027 .0687 .0001 .0007 .0776 .5 500 .0351 .0265 .0519 .0096 .0080 .0586 .0004 .0028 .0532 .0000 .0006 .0596 1000 .0355 .0271 .0533 .0085 .0065 .0533 .0002 .0026 .0568 .0000 .0011 .0540 100 .0941 .0608 .1171 .0358 .0200 .2306 .0000 .0000 .1395 .0001 .0001 .2755 200 .0898 .0576 .0913 .0319 .0185 .1485 .0000 .0001 .1012 .0000 .0000 .1519 .8 500 .0908 .0567 .0640 .0327 .0172 .0827 .0000 .0001 .0702 .0000 .0000 .0871 1000 .0919 .0562 .0595 .0310 .0151 .0662 .0000 .0000 .0643 .0000 .0000 .0674 BIBLIOGRAPHY BIBLIOGRAPHY Baillie, R.T. and T. Bollerslev (1989), "The Message in Daily Exchange Rates: A Conditional Variance Tale," Journal of Business and Economic Statistics, 7, 297- 306. Bartlett, M.S. (1950), "Periodogram Analysis and Continuous Spectra," Biometrica, 37, 1-16. Bohman, H. (1961), "Approximate Fourier Analysis of Distribution Functions," Arkiv fur Mathematik, 4, 99- 157. Bollerslev, T. (1986), "Generalized Autoregressive Conditional Heteroskedasticity," Journal of Econometrics, 31, 307-327. Bollerslev, T., R.F. Engle and J.M. Wooldridge (1988), "A Capital Asset Pricing Model with Time Varying Covariances," Journal of Political Economy, 95, 116- 131. Box, G.E. and G.M. Jenkins (1970), Time Series Analysis: Forecasting and Control, San Francisco: Holden-Day. Dhrymes, P.J. (1974), Econometrics: Statistical Foundations and Applications, New York: Springer-Verlag. Dickey, D.A. (1976), "Estimation and Hypothesis Testing in Nonstationary Time Series," unpublished Ph.D. dissertation, Iowa State University. Dickey, D.A., W.R. Bell and R.B. Miller (1986), "Unit Roots in Time Series Models: Tests and Implications," The American Statistician, 40, 12-26. Dickey, D.A. and W.A. Fuller (1979), "Distribution of the Estimators for Autoregressive Time Series with a Unit Root," Journal of the American Statistical Association, 74, 427-431. Dickey, D.A. and W.A. Fuller (1981), "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit 102 103 Root," Econometrica, 49, 1057-1072. Diebold, F.X. and M. Nerlove (1989), "Unit Roots in Economic Time Series: A Selective Survey," Advances in Econometrics: Co-Integration, Spurious Regressions and Unit Roots, forthcoming. Diebold, F.X. and P. Pauly (1988), "Endogeneous Risk in a Rational Expectations Portfolio-Balance Model of the Deutschemark/Dollar Rate," European Economic Reyiew, 32, 27-54. Domowitz, I. and 0.8. Hakkio (1985), "Conditional Variance and the Risk Premium in the Foreign Exchange Market," Journal of International Economics, 19, 47-66. Engle, R.F. (1982), "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation," Econometrica, 50, 987-1008. Engle, R.F. and T. Bollerslev (1986), "Modelling the Persistence of Conditional Variances," Econometric Reviews, 5, 1-50. Engle, R.F., D.M. Lillien and R.P. Robbins (1987), "Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model," Econometrica, 55, 391- 408. Evans, G.B.A. and N.E. Savin (1981), "Testing for Unit Root: 1," Econometrica, 49, 753-779. Evans, G.B.A. and N.E. Savin (1984), "Testing for Unit Root: 2," Econometrica, 52, 1241-1269. Fuller, W.A. (1976), Introduction to Statistical Time Series, New York: Wiley. Godfrey, L.G. and A.R. Tremayne (1988), "On the Finite Sample Performance of Tests for Unit Roots," unpublished manuscript, University of York, England. Hansen, B.E. (1988), "A Model of Heteroskedastic Cointegration," unpublished manuscript, Yale University. Nankervis, J.C. and N.E. Savin (1985), "Testing the Autoregressive Parameter with the t Statistic," Journal of Econometrics, 27, 143-161. Nankervis, J.C. and N.E. Savin (1987), "Finite Sample Distributions of t and F Statistics in an AR(1) Model 104 with an Exogeneous Variable," Econometric Theory, 3, 387-408. Neave, H.R. (1972), "A Comparison of Lag Window Generators," Journal of the American Statistical Association, 67, 152-158. Nelson, 0.8. (1988), "stationarity and Persistence in the GARCH(1,1) Model," Working Paper 88-68, University of Chicago. Nelson, C.R. and H. Kang (1981), "Spurious Periodicity in Inappropriately Detrended Time Series," Econometrica, 49, 741-751. Nelson, C.R. and H. Kang (1984), "Pitfalls in the Use of Time as an Explanatory Variable in Regression," Journal of Business and Economic Statistics, 2, 73-82. Nelson, C.R. and C.I. Plosser (1982), "Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications," Journal of Monetar Economics, 10, 139- 162. Newey, W.R. and K.D. West (1987), "A Simple Positive Definite Heteroskedasticity and Autocorrelation Consistent Covariance Matrix," Econometrica, 55, 703- 708. Nicholls, D.F., A.R. Pagan and R.D. Terrell (1975), "The Estimation and Use of Models with Moving Average Disturbance Terms: A Survey," International Economic Review, 16, 113-134. Park, J.Y. (1988), "Testing for Unit Roots in Models with Structural Change," unpublished manuscript, Cornell University. Parzen, E. (1961), "Mathematical Considerations in the Estimation of Spectra," Technometrics, 3, 167-190. Perron, P (1988), "Tests of Joint Hypotheses for Time Series Regression with a Unit Root," working paper, Princeton University. Peters, T.A. and W. Veloce (1989), "Robustness of Unit Root Tests in Finite Samples," working paper, Brock University. Phillips, P.C.B. (1987), "Time Series Regression with a Unit Root," Econometrica, 55, 277-301. 105 Phillips, P.C.B. and P. Perron (1988), "Testing for a Unit Root in Time Series Regression," Biometrika, 75, 335- 346. ‘ Press, W.H., B.P. Flannery, S.A. Teukolsky and W.T. Vetterlink (1986), Numerical Recipe : The Art of Statistical Computing, Cambridge: Cambridge University Press. Priestley, M.B. (1981), Spectral Analysis and Time Series, Vol. 1, New York: Academic Press. Said, S.E. and D.A. Dickey (1984), "Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order," Biometrika, 71, 599-608. Said, S.E. and D.A. Dickey (1985), "Hypothesis Testing in ARIMA(p,1,q) Models," Journal of the American Statistical Association, 80, 369-374. Schmidt, P (1989), "Dickey-Fuller Tests with Drift," Advances in Econometrics: Co-Inteqration. Spurious Regressions and Unit Roots, forthcoming. Schmidt, P. and P.C.B. Phillips (1989), "Testing for a Unit Root in the Presence of Deterministic Trends," unpublished manuscript, Michigan State University. Schwert, G.W. (1987), "Effects of Model Specification on Tests for Unit Roots in Macroeconomic Data," Journal of Monetary Economics, 20, 73-103. Schwert, G.W. (1989), "Tests for Unit Roots: A Monte Carlo Investigation," Journal of Business and Economic Statistics, 7, 147-159. Solo, V. (1984), "The Order of Differencing in ARMA Models," WW. 79. 916-921. West, K.D. (1988), "Asymptotic Normality When Regressors Have a Unit Root," Econometrica, 56, 1397-1417. White, H. (1980), "A Heteroskedasticity Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, 48, 817-838. ,gyirnlrrn STATE U NIV. LIBRARIES Ill llllllllllIlllllllllllllllllllllll 3221 9300605 MICHIGAN lllllllllllllll 3'l2