APPLICATION OF A F ERST ORDER AUTOREGfiSSWE MODEL TO SOME ECONOMIC EQELATIONS Thesis {or Hie Degree of M. S. EECEEGAR SWTE DRIVERS?” John Y. Lu 1957 ,0 313?"? ".1!" “ 5? WI ‘ I; i f a nL’PLICsLTlUI‘J UP A erT URDLJH AU’I‘OEUSUMQ‘JIVE MUDL'L TU o'UME ECONOMIC I'LEIATIUND‘ By John Y. Lu A THESIS Submitted to the College of Agriculture of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of I~LA5TEu OF SCIENCE DEPARTMENT OF AGltICUL‘l‘ UHAL EONUMICS 1957 3-." " [‘1' I '1‘. ,~" . . .. _- i“ Appmved by rib '3 L1 —_‘:k x v r' [fig—if L W's}, L ;‘\ \ \. \ H \— t/ \‘f K \t‘ \l v’ \ \ .- Q" h. ‘- ' Q._‘\‘.‘ \\ v. “‘1‘ {)3 ‘3 John Y. Lu ABSTRACT In this study, examples of the estimation of supply'and demand . relations for agricultural products were examined with a view to obtain some indications of the importance of inefficiencies in traditional least squares procedures when disturbances are autocorrelated. A sta- tistical test of autocorrelation was applied to a number of previously— fitted relations. A modified least squares procedure was used to re- estimate parameters of those relations for which the test rejected the null hypothesis of serial independence of disturbances. The test applied was the Uurbin—Jatson test of serial independ— ence of disturbances. The modified least squares method suggested was based on a first order autoregressive model. In this model, disturb- ances are no longer regarded as statistically independent but are con- sidered to be generated by a first order autoregressive process, (ut = f’ut_1 + Vt, where u and v denote elements of non-independent and independent disturbances respectively, and P is a constant which is often called the autocorrelation coefficient.) In all, there were nineteen regression equations tested by the Durbin4Watson test. Only in five cases, the hypothesis of zero auto- correlation at the 10% significance level was accepted. In ten cases, the test was indeterminate; and in four cases, the null hypothesis was rejected. Regarding the results of applying a first order autoregressive model to those cases where a significant autocorrelation of disturbances John Y. Lu was established, the sum of squared residuals from a regression line was substantially reduced, as expected, from.that obtained from the usual least squares estimates; and different values of parameters were obtained. When the autocorrelation coefficient (’ is set equal to +1 in a first order autoregressive model, it is called the first difference model. Since the first difference model has been frequently'fitted by economist to eliminate some of unfavorable effects of positive auto- correlation, it was also applied in this study to those cases where a significant positive autocorrelation had been established to see what effects it would have on estimates of regression coefficients and the sum of squared residuals. In three cases, the sum of squared residuals was increased; and in one case, it was reduced. However, there seemed to be a tendency for the regression coefficients estimated from.the first differences of variables to be a closer approximation of the true values of regression coefficients when there was an indication of posi- tive autocorrelation of disturbances. AC KILUuLJLUGH'h-QI‘H‘S The author wishes to express his sincere appreciation to Professor Clifford Hildreth for suggesting the problem and rendering valuable help and encouragement throughout the development of this thesis. The author is grateful to Professor L.L. Boger who have provided the opportunity for the author to continue his studies. The generous policy of the Department of Agricultural Economics in extending assis- tance to students from outside the United States is most gratifying to the author. A major portion of compmutations was carried out by Mrs. Iantha Perfect and the other members of the computing staff. The author would like to take this opportunity to thank them. The author also wishes to express his thanks to Professor Ben French and Earl Partenheimer who have read the thesis and gave many helpful corrections with regard to the author's English. Professor French was kind enough to help the author select the material for this study. TABLE or comma-rs Chapter Page I IIqu ltUUUCIB IUI‘V . . . . . . . . . O . . . . . . . . . O . . . 0 O . . . . . . O . . . . . O . 0 . O C C O l S c O m o 1. Study . . . . C C O C O O O O . O O C O . O . O O . C . O . . . . . . O 0 O O 0 C . l mediate Objectives ......OOOOOOOOOOO.....OOOOOOOOOOI 2 II (fl-1E LErliOD 01“ MJ‘HJYJIS .........OOOCOOOOOOOOOOIII0.0.0.... 5 Selection of Some Empirical Studies .................. 5 Significance Test of Serial Independence of Disturbances 0.00.0...000..........OOOOOOOOOOOOOOOOOOO 5 Least Squares Estimates of Regression Coefficients ... 6 First Order Autoregressive Model ..................... 8 3mm ......OOOOOOOOOOOOOO0.00000000000000IOOOOOOOOOll III REVIE‘IJOF‘SUIVEW:PED PAPh‘m ......OOOO0.00.00.000.000... 1-3 IntmduCtion ......OOOOOOOOOI......OOOOOOOOOOOOOOOO... 13 The Durbin-Watson Test of Serial Independence of DiSturbanceS 0.0.0.0..............OOCOOOOO0.0.0.000... 13 Cochrane and Orcutt's Study of the Broblem of Auto- correlated Disturbances by the Sampling Method . .... . . 15 Application of a First Order Autoregressive Model in Estimating a Supply Curve of Female Labor during the Vlorld Flax. II .0OOOOOOOOOOOOOOOOOOIOO......OOOOOOOO 16 Gurland's Study of Effects of Certain Types of Spec- ification Bias Concerning Autocorrelated Disturb- mces 0.0.0.0.........ICOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 19 Some of Watson's Findings on the Performances of Regression Analysis Containing Autocorrelated Disturbmces ................CCOOOOOOCOOOOOOO0.0...... 19 Chapter I Page IV AerICATlUN OF THE DUHBIN_WATSON TEST AND A FIRST ORDER AUTUEEGRESSIVE MODEL TO SOME PEEVIOUSLY FITTED KEWSIUIQ I‘lUDflD‘ ......OOOCOOCOOOOOOO00.00.00.000090000 23 Introduction ....................................... 23 Results of Applying the Durbin4watson Test ......... 23 Meinken's Wheat Mbdel .......................... 23 Frenches' Apple Study .......................... 25 Hoos' Tomato Study ............................. 27 Hoos' Lemon Study .............................. 27 Foytik's Demand Study for California Plums ..... 28 Linstrom and King's Studies of Slicing Cucumbers and Green Peppers .................... 29 Quackenbush and Shaffer's Study of Demand for Ice Crea-Tn 0......OOOIOOOOOOOOOOOO0.0.0.0...0.0.0 30 Hoos' Study of the Pacific Coast Cannet Fruits . 30 Shear and Hoos' Study of California Bartlett i)eavrs COOOOOOOOOOOOOOOOOOOOOOO......OOOOOOIOOOOO 31 Regression Coefficients and Sum of Squared Residuals Estimated from.First Order Autdregressive Models ... 3h Meinken's Demand Equation for the world Nheat .. 35 Linstrom and King's Demand Equation for Green Peppers O.........OOOOOOOO......OOCOOOOOOOOOO... 38 Quackenbush and Shaffer's Demand Equation for ICC crew .....OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO h]- Hoos' Demand Equation for Pears ................ Ah Foytik's Demand Equation for Plums ............. 1&7 Chapter Page v SUM-bud AND CuruuLUo‘luNo‘ 51 Bibliography ................................................... 55 Appendix Computing Procedures Supplementary to Chapter II ...... 58 Table LIST OF TaSLES Application Of The Durbjn-Watson T851; ecoooooooooooooooooo Estimates of Regression Coefficients and Sum of Squared Residuals for Different Assumed values of F’ : MeinKen's World Wheat Demand Equation (lb) ......................... Estimates of Regression Coefficients and Sum of Squared Residuals for Different Assumed Values of (> : Linstrom and King's Demand Equation for Green Peppers (6a) ........ Estimates of Regression Coefficients and Sum of Squared Residuals for Different Assumed Values of P : Quackenbush and Shaffer's Demand Equation of Ice Cream (7a) .......... Estimates of Regression Coefficients and Sum of Squared Residuals for Different Assumed Values of f! : Hoos' Demand Equation of Pears (9d) ............................ Estimates of Regression Coefficients and Sum.of Squared Residuals for Different Assumed Values of p : Foytik's Danand Equation for Midseason Plunls (5b) 00000000000000... Regression Coefficients and Sums of Squared Residuals Estimated from the Ordinary Least Squares Model, the First Order Autoregressive Model, and the First Difference Model ......................................... 38 A7 50 52 LIST OF FIGURES Figure Page 1 Variation of Sum.of Squared Residuals and Regression Coefficients as f) changes: Meinken's Wheat Demand qulat’ion (lb) 0.00.00.00.00...O...0........OOOOOOOOOOOOOOOO 37 2 Variation of Sum of Squared Residuals and Regression Coefficients as f>changes: Linstrom.and King's Demand Equation for Green Pepoers (6a) ........................... AD 3 Variation of Sum of Squared Residuals and Regression Coefficients as ,0 changes : Quackenbush and Shaffer's Demand Equation for Ice cream (7a) ........................ 43 h Variation of Sum of Squared Residuals and Regression Coefficients as (3 changes:Hoos' Demand Equation for Pears (9d) 000.000.000.0000000000000.0000000000000000009000 [8‘6 5 Variation of Sum.of Squared Residuals and Regression Coefficients as f’ changes: Foytik's Demand Equation for Ifidseason lens(5b) 00000000000000.0000...0000000000... [59 Cl’ihfil‘ifih I INTRODUCTICR Scope of Study Generally speaking, the ultimate goal of studies line the present one is to contribute to the improvement of statistical analysis of the economic relations deteqmdning movement of price and quantity. RTthe sta— tistical analysis of such economic relations could be more accurate and reliable, the significance of its application to both public and private policy would be considerably enhanced. In this thesis, the author reports the results of some empirical investigations into one of the many problems that may arise when some of the assumptions on which statistical analysis is based are unrealistic. These assumptions may be roughly claSSified into several categories. l) an investigator may assume that he has completely and appropriately spec fied all the relevant variables in his model. 2) He may assume that the variables are measured relatively free from.error. 3) He may assume that the variables can be described by a system of single equation (i.e., the system that contains only one dependent variable.) a) He has to make an assumption with regard to a form of algebraic equation to be fitted to 1 data. 5) he may regard that cisturoances are statistically independent, (i.e., no autocorrelation is involved.) l Disturbances are random.variablcs expressing the diflercnce oetween the observed values of the dependent variables and their expected values. On the other hand, residuals are differences between observed and calculated values of the dependent variables. When any of the above assumptions cannot be.maintained, it tends to cast doubt on the usefulness of statistical analysis. In the present study the author lorcuses his attention on the problem of autocorrelated disturbances. In order to concentrate on the problem, he has ignored the dilficulties arising from the inadequacy of other assumptions at present. However, it should be obvious that for'the purpose of estimating struc- tural parameters it is necessary to find a.method of dealing simultaneous- ly with all other difficulties. Immediate Objectives Since it has been known that much efficiency is lost by current methods of estimation and prediction if autocorrelation of disturbances exists, the author tries to see the extent to which autocorrelation is involved in some current fbrmulations of economic relations. It is hoped that this attempt will call the attention of economists to the seriousness of the problem. It has been suggested that to regain the lost efficiency some modification of the usual least squares method of estimation is required. Therefore, a.modified.method of the usual least squares regression is applied to several empirical studies, in which there is evidence to sue- pect that autocorrelation of disturbances exists, to estimate structural coefficients. The modified method employed in the present study is a first order autoregressive modelii It is the author's intention to find out 2 In this model disturbances are no longer regarded as statistically independent but are assumed to be generated by a first order auto- regressive scheme, “t = f’ut_1 + Vt , where u and v stand for non- independent and independent disturbances respectively, and 1° denotes a constant called the autocorrelation coefficient. Sometimes it is called a first order Markoff model. how much improvement in estimating structural coefficients of these models can be made even by a slight relaxation of the assumption with reapect to disturbances as is done in applying a first order autoregressive model. Some economists have often applied the first difference model3 when they encountered positive autocorrelation of disturbances. They claim that the first difference medel will eliminate some of unfavorable effects of autocorrelation. However, one statistic ' 1‘ recommends that one must be cautious in applying the first difference model. He reports that the application or the first difference model does not always result in an increased efficiency of estimation. These different opinions as to the advisability of the use of the first difference model are cheeked in the present study. In short, the usual least squares estimates of regression coefficients and sum of squared residuals are compared with those Obtained from the first order autoregressive mooel and with those Obtained from the first difference model. Finally, in the procedure suggested in this study (1.8., the application of a first order autoregressive model in estimating structural coefficients) it is necessary to minimize the sum of squared residuals. Like any other problan of maximum and minimum, one must ascertain whether the minimum obtained is the absolute minimum or a relative minimum within 3 The first difference model is the same as a first order autoregressive model in which 1" is set equal to *1. See tne footnote in the previous page. ‘ h G.S.Watson gives an extensive discussion of the problem of autocorrelation in his Ph.D dissertation, Serial Correlation , Department of Applied Statistics, North Carolina State College, Raleigh, N .C. an interval examined. In other words, one must determine if a multiple- minimum situation esists . CHm’Tiilfi II rm; swabs or nNALYS L51 In this study the author proposes to test empirically some of the new ideas developed by statisticians and economists in regard to the problem of autocorrelated disturbances. For this purpose some interesting empirical economic models were selected. In all of the models selected the usual least squares method of estimation and prediction was employed. Some efforts were made in the course of selection of these models so that difficulties arising from the inadequacy of the statistical assumptions, other than the one concerned with the problem of autocorrelated disturbances, might be of secondary importance. The selection was also limdted to studies for which original data were available. These studies include work done at the U.S.D.A., and the California, North Carolina and Michigan Agricultural Experiment Stations. Significance Test of Serial Independence of Disturbances In investigating the extent to which autocorrelation of disturb- ances is involved in a group of the selected empirical studies, it seems natural to apply a statistical test to them. Another objective of the present study is to analyze the effects of substituting a first order autoregressive model for the usual least squares model in some empirical studies. Before carrying out this substitution, it is desirable to decide 1 This method of analysis was outlined to the author by Professor Hildreth. on those studies where autocorrelation of disturbances at the significance level concerned can be established. For these reasons, the application of a significance test of serial independence of disturbances is considered necessary . Although there are several tests of this kind available, the Durbinpwatson test2 is con- sidered most satisfactory for the present study. Other commonly used tests such as von Neuman's ratio test are not strictly appropriate because they are designed to be applied to observed variables. When the Durbinewatson test indicates significant autocorrelation of disturbances in a least squares regression model at the 10% significance level, that model is chosen for further investigation (i.e., to substi- tute a first order autoregressive model for the ordinary least squares model.) When the tests are indeterminate or do not indicate significant autocorrelation of disturbances, no further steps are taken at present. The choice of the 10% significance level as a dividing line is rather arbitrary. Since the number of the studies tested by the Durbin— Watson test is limited, the author has to choose the 10% level as the significance level concerned so that there will be sufficient number of studies to which a first order autoregressive model may be applied. Least Squares Estimates of Reggession Coefficients Before a first order autoregressive model applied in this study is introduced, it is desirable to discuss briefly the derivation of Wrbin, J. and Watson, 6.3. "Testing for Serial Correlation in Least Squares Regression I," Biometrika, Vol. 37, pp.409-h28, “Testing for Serial Correlation Least Squares Regression ll," Biometrika, Vol.38, pp 0 159’17 8 o regression coefficients estimates by the usual least squares method. This discussion probably will help a reader to see how the two models are related. observations are available on a dependent variable, yt, t=l,2,..,T, and certain fixed variables ztk’ for t=l,2,....,T, and k=l,2,.......,K. yt is assumed to depend upon ztk in the following manner: yt-fl'lztl't’lrzztz‘t .............. +7rkztk+ut , (1) where ut is an element of independent disturbances with mean zero. For the T observations, the system.may be written in vector form: yl - 21-1 oooooooo.......le [I]. '0' 111 I O 0 o . (2) yT 21:1 0 o o o o o ZPK L’WK “‘1‘ The column vector drl is a set of regression coefficientgto be GEK estimated in the above system. If the ul, u2,......, “T are independent and have a common dis- tribution with finite variance, then best linear unbiased estimates of the regression coefficients can be obtained.by minimizing the sum of squared residuals. If the u1,...., uT are normally distributed the resulting magnitudes are also maximumelikelihood estimates. Matrix equation (2), above, may be written more compactly r-z'rr+u, (3) where the matrix 2 is of order'T x.K and rank K. Let the sum of squared residuals to be minimized be denoted by S. 3= (Y-ZfHI-Z’N) = Id — 2 Tf'Z'Y + w'z'z’n’ . (1.) When the first derivative of S is taken with respect to'1T', and is set equal to zero; a set of normal equations are Obtained. - -ZZ'Y -r 22'211’ .. 0 via» sits - z'z’ll' .. 2'! = o . (5) Hence the estimator “if is as follows: if - (2'2)‘1rice-quantity relationships both at the local (Michigan) and national levels. In the present study, only the demand and supply functions at the national level were investigated. On the demand side, French obtained three different equations from slight'”different sets of variables. He chose the first one6 for aCtI—lal use in the estimating model since it imposes less arbitrary restrictions on the coefficients. All three equations were estimated by K 5 French, 8.0. The Long-Term Price and Production Outlook _f_q£ Apples 3'._n_ Ellie United States §_n_c_1_ Michigan. Michigan State University Agricultural ‘rment Station, Technical Bulletin 255, April, 1956. 6 lbid., 13.7. 26 . single equation model of least squares regression. The use of the single mOdel seems to be justifed because disposable income, the volume of com- peting fruits, and total production may be considered as predetermined for all practical purposes. However, a question may be raised with regard to the quantity of apples actually marketed being treated as predetermined, since it may vary with a rise and fall in prices. French got around this difficulty by using net production per capita excluding export as an out put variable, and also omitting those years in which a significant amount was not harvested .7 The coefficients of the demand equation were estimated from annual data for the 20—year period 1930—1953, excluding the three war years. Since the first demand equation (2a) was used by French for prediction, it was tested by the Durbin—Watson test. The test indicated no significant autocorrelation at either the 5% or 10% level. The regression coefficients of the supply function (2b) were also estimated by the single equation model.8 It seems to be appropriate to L13e the single equation model in this case because a considerable length 01‘ time elapses from planting to harvesting, and therefore, the supply may be regarded as a function of lagged price. The test indicated that there was also no significant autocorrelation in this equation at the 5‘23 and 1013 levels. No further analyses were carried out in this apple Study at present. x n w— ( French reports that the ommission of those years did not result in any sff—gnificant difference in the regression coeffici<=nts when compared W1th those obtained from all years included. 8 French, op.cit ., p.15 . 27 3. Thr third case was hoos' report on the market situation of tomatoes and tomatoes products in Californiafl He used a single equation model to explain variations of California f.o.b. price of canned tomatoes for the years 1926-27 through 1953-51. excluding 1941-47. Among several single equations formulated, Hoos selected the euqation (l) in p.46 (de- noted 3a in this study) as the best statistical demand function because of its Simplicity in interpretation and trade use. It was tested by the Du‘Z'C‘bI’IJL-Watson test and the result was inconclusive at both the 5x5 and 1075 Significance levels. Therefore, the equation (3a) was not further analyzed. 1;. Another of Hoos' statistical demand studies examined was con- Cemed with market situations of California lemons .10 A single equation me"Tahod was used to estimate a demand function. Years analyzed were from 1921-22 through 1948-49 (excluding the war years.) Hoos considered demand equations of summer lemons (ha) and winter lemons (Ab) separately because they reflect different market characteristics. The test was inconclusive When applied to the summer lemons demand function, and it indicated no Significant autocorrelation at either the 5;} or 10% level when applied to the winter lemons demand function. Hence no first order autoregressive mdel was applied to either of the equations. §\ H003, Signey, Tomatoes and Tomato Products: Economic T__r____end ai__1_d _f_._g_._b_._ Erica RelationshiE. California A ricultural Experiment Station, Mimeographed Report No. 185, March, 1956. 10 l“1005,.3. and Seltzer, H.E. Lemons and Lemon Products: Changing Economic R-\vel_fationshipel 12‘21-22. California Agricultural Experiment Station Bulletin 729. 28 5. Jerry Foytik studies characteristics of demand for California Plums extensivelyfllnstead of treating California plums as consisting a homo genous product in the sense used by the theorist, he classified them into several quality-size categories, and examined the demand characteris- tics of plums with respect to each category. First, he estimated regres- sion coefficients of demand functions for three varieties of plums, viz., early, midseason, and late, from annual data for those years from 1922 ‘30 19h? excluding the war years, using the single equation least squares method. Secondly, the variations in weekly prices are related to weekly aAlction sales and a few other relevant variables. Finally, the influence of one size of plums on another size was studied. Only the varietal aspect of Foytik's study was examined in the Current study because it may be considered a typical statistical demand analysis. The application of the Durbin—Watson test to the first equation ( 5a)];2 indicated that one may accept the hypothesis that the significant a-\.11:.ocorrelation of disturbances is absent either at the 5% or 10% level. A8 to the second equation (5b)13, the test was inconclusive. In the third equation (5c)u‘, the test indicated no significant autocorrelation at the 5% level and was inconclusive at the 10% level. A first order auto- 3? Foytik, Jerry. Characteristics o__i_‘ Demand for California P_].._____ums. Hilgardia, Vol. 20, NO. 20, April, 1951, California Agricultural EXp eriment Station. 12 ibid., p.l.28, Equation (1) 13 ibid., p.1.31, Equation (2) ibid., p.1+35, Equation (3) 29 regressive model was applied to the equation (5b) and results were presented in next; section. (Originally it was not planned to analyze any equation in which the test was inconclusive. But in this case, the test was erro- neousZLv considered to be significant at first and further analysis was done - ) It is interestinp to note that Foytik himself has tested his three varietal demand equations for the autocorrelation of residuals. He applied the von Neumman's ratio test (the ratio of mean square succes- sive difference to variance) to them. His own test indicated that absence or autocorrelation in the residuals may be accepted as a suitable hy- Pothesis for all three varietal demand equations-2. 6. Linstrom and King made a study of the factors that influence prices received by growers of slicing cucumbers and green peppers in North Carolina.15 The demand function of cucumbers, which they considered as representative among many they formulated, was estimated from the Obs ervations which had been transformed to the first difference of logaritrnns.16 Since it has not been considered how to take account of aluiocorrelated disturbances explicitly in this type of equation, no inVestigation was carried out . As to green peppers, the equation (6a)l7 describing the influence of New York wholesale price on the local (Clinton, MC.) market prices E‘\ Euxcumbers and 93393 Peppers. ...... ...—l— — __ 16 ib:i.d., p.11. 17 ibid., p.1+0. was tested by the Durbin—Watson test. The coefficients of this equation were estimated from daily data for the 23-day period, June 8 - July 18, 1953 . The test indicated significant positive autocorrelation of dis- turbances at the 573 and 1023 levels. A first order autoregressive model was applied to the equation (6a). 7. Lzuackenbush and Shaffer estimated the coefficients of a demand fuIICLion \7a) for ice cream that are purchased for home consumption by t'he single equation method of least squares regression.18 Data was obtained from the 14.5.U. consumer panel. They cover 30 h-week periods from March 18, 1951 to July 11, 1953. Testing this equation by the Durbin-Watson test indicated significant positive autocorrelation at both the 5% and l-i'fié levels. Therefore, this equation was further analyzed . 8. Boos set up a sigle equation model to analyze f .o.b. prices of the racific Coast canned fruits and their relations to the major f actors affecting them.l9 Annual data were used. They covered the years 1924-25 through 1955-56 (excluding 1941-42 through 1946—47.) His three damend equations for canned cling peaches, canned pears, and canned a~13I'icots respectively were examined by the Durbin—Watson test. The test a"pplied to the equation for canned cling peaches (8a) indicated no 13‘ Quackenbush, G.G. and Shaffer, J .D. Factors Affecting Purchases 9}: kc Cream for Home Use. Michigan State University Agricultural Experiment Station. Technical Bulletin 21.9, April 1955. 19 l‘loos, S. F ..0 8. Price Relationship 3, 1255-56 Pacific C__o____ast C_a_nn__ed .Fruits. California Agricultural Experiment Station, Mimeographed Report, No. 189, July 1956. 3 O 31 significant autocorrelation of disturbances at the 55 and 10% levels.20 The test applied to the equation for canned pears (8b) indicated the similar test result.21 'The test was inconclusive when applied to the equation for canned apricots, (8c).22 9. The last one examined was "Relation between Auction Prices and Supplies of California Frech Bartlett Pears" by Moos and Shear.23 They formulated five statistical demand equations (single equation type) in an attempt to explain market characteristics of California frech Bartlett pears, and chose the Equation (5)24 as a representative demand equation (denoted 9a in this study). The test, when applied to this equation, was inconclusive both at the 55 level and 10% level. In Appendix B, they published the results of the statistical analysis of the relations of pears to its competing fruits such as plums, peaches, and oranges. They estimated the coefficients of a dozen demand equations by the least squares regression analysis. Due to time limit, only three of them were tested by the DurbinPWatson test. The test applied to the first equation (9b) in Table 11, which expresses the price of pears as a function of pear unloads; the unloads of plums; an index.of New York state factory wages; and "time“, was inconclusive. So was the test applied 20 Hoos, op.cit., p.26. 21 ibid., p.31. 22 ibid., p.36. 23 H003, D. and Shear, S .w. Relation Between Auction Prices and Supplies_ of California Frech Bartlett Pears. Hilgardia, V61. 14, No.5I—UEnuary, 19h2, California Agricultural Experiment Station. 2h ibid., p.280. 32 to the second equation (9c) in the same Table, which expresses the pricesof plums as a function of pear unloads; index of New York State factory wages; and “time". When the quantity of pears shipped is treated as dependent variable as in the first equation (9d) of Table 12, the test indicated that the disturbances are negatively autocorrelated at the 10% level and the test was indeterminate at the 5% level. In the Hoos' pear study, only the equation (9d) was examined by applying a first order autoregressive model. In the following table a summary of the Durbin-Watson test is given. Table 1 Application of the Durbin-Watson Test Equation No. of ho. of Calculated Limits of Relevant Test No. Independent Observa- Test Boundary Results Variables tions Statistic S? 192 at the dL du dL du 10% Level la h 17 0.8679 0.68 1.77 0.78 1.90 I 1b # 2 15 0.8922 0.83 1.60 0.95 1.56 R 2a 3 20 1.7789 0.89 1.55 1.00 1.68 N 2b 1 19 1.7hh6 1.06 1.28 1.18 1.h9 N 3a 3 23 1.2822 0.97 1.56 1.08 1.66 I #8 h 26 1.8586 0.91 1.66 1.01 1.78 N Ab 5 23 1.05h0 0.80 1.80 0.90 1.92 I Sa 2 25 2.3860 1.10 1.h3 1.21 1.55 N 5b # 5 25 1.2830 0.86 1.77 0.95 1.89 I So . h 25 1.6507 0.96 1.65 1.04 1.77 I be # 2 23 0.9146 1.06 1.h2 1.17 1.56 R 7a # 3 30 1.0389 1.12 1.56 1.21 1.65 R 8a 3 26 1.6580 1.04 1.5h 1.1h 1.65 N 8b 3 23 2.4885 0.97 1.5L 1.08 1.66 I Be 3 26 1.h400 1.0h 1.56 1.14 1.65 I 9a h 25 1.1625 0.94 1.65 1.04 1.77 I 9b 4 15 2.77Lh 0.59 1.84 0.69 1.97 I 90 h 15 3.0567 0.59 1.86 0.69 1.97 I 9d # h 15 3.3676 0.59 1.8L 0.69 1.97 R Abbreviations: dL : Lower mt. du : Upper limit. N I R If : Null hypothesis not rejected. : Indeterminate . : Null hypothesis rejected. : A firsr order autoregressive model has been applied. 3h Regression Coefficients and Sums of Squared Residuals Estimated from.First Estimategfifrom.h First Order Autoregressiye Model, When the Durbin-Watson test was applied, a significant autocorrela- tion of disturbances was established at the 10% level in four cases. They are (lb) Meinken's demand equation for the world wheat market, (6a) Linstrom and King's demand equation relating local (Clinton, N.C.) green peppers prices to the New York wholesale prices, (7a) Quackenbush and Shaffer's demand equation for ice cream.bought for home use in Lansing area, (9d) Hoos' demand equation relating the variations of unloads of pears to wholesale prices of pears and plums. A first order autoregressive model,as explained in Chapter II, was applied to these four cases. Results of the computations were tabulated and diagrammed. One of Foytik's demand equation for mid-season plums (5b) was also analyzed and reported in the same manner as the above four cases. Originally it was not planned to apply a first order autoregressive model to those cases where the Durbin-Watson test indicated inconclusive results. The Foytik's demand equation belongs to this category. However, an error was made when the Durbin-Watson test statictic for the Foytik's equation was first calculated, and it was considered that the significant auto- correlation of disturbances was established. subsequently, a first order autoregressive model was fitted. It is felt that the results are interesting enough to be presented here. In each of the five cases, one may notice some discrepancies between the original regression coefficients estimates reported in each author's study and the regression coefficients estimated from assuming F’= O in 35 a first order autoregressive model. (This is the same as the traditional least squares model. ) This is probably due to a difference in the number of observation used in estimation. The first order autoregressive model as applied in the present study are based on the observations whose number is one less than those from which the original regression coefficients eStiJnat es were obtained. 25 1b) Meinken's World wheat Demand Equation26 Since the Durbin-Watson test indicated significant positive auto- correlation, values 0f Fassumed in a first order autorexressive process Were taken from the interval, 0 to +1. i‘he first approximation to the Value of ('5 that would minimize the sum of squared residuals under the assurnption was 0.7. By plotting the sums of squared residuals against .0 , it was realized that the minimum value of the sum of Squared residuals w01-116. most likely be between F’ = 0.6 and 0.7. Repeating the same pr‘OCedure for several different values of P between 0.6 and 0.7, it “as found that the sum of squared residuals would be minimized when P Was set equal to 0.67. One can see from Table 2 and Fig. 1 that although the first difference model would increase the sum of squares of residual from that of the original regression model, the values of parameters seem to a~EDI’Dr'oach the true values. 35“ In a first order autoregressive model, lagged values of variables fire introduced. Hence the number of observation is reduced by one. bee the Appendix. 26 geinken, op.cit., p.141, Equation (8), w -m-O.36SW+1.11W. Th 36 The regression equation obtained from the first order autoregressive model is as as follows: PW - 162 - 0.046 Sw_+ 1.35 Iw where P : Average wholesale price of wheat at Liverpool, England, per bushel. Sw : World production of wheat plus stocks about August 1, excluding Russia and China but including net exports from Russia, million bushels. Iw : Index of wholesale prices of A5 raw materials in England (l910—lh = 100). ' It is suitable to use the above equation for prediction of long tenn price. However, if one has observations on this year's independent variables and wants to predict next year's dependent variable, it would be better to use the equation of the following form. This second equation is an alternative way of expressing the first one.27 Pw - 51. .. 0.67 egg .. 0.01.6(3w — 0.67 53;) + 1.35(I., - 0.67 r5) where 33" S: and I:_stand for lagged values of the respective variables. In the following diagram, the sum of sqmared residuals and the regression coefficients estimates are plotted as functions of /° . Data used are presented in the table. R 27 See footnote 3 in p.9. 37 \l 0 CI. 0. l. n o a m p. mam. H mm 1%. we 8 E . 5 O l O r. Meinxen's Wheat Model (lb). Variation of Sum of Squared Residuals Line and Regression Coefficients Varies. Table 2 Estimates of Regression, Coefficients and Sum of Squared Residuals as P Varies: Meinken's Wheat Demand Equation (1b). Autocorrelation Regression Coefficients Sum. of Coefficients Squared Constant P.w Iw Residuals WW?) 0 1AA.92 -0.0371 1.155 301.h37 0.1 ‘152.81 -0.0387 1.150 287.958 0.2 160.26 -0.0A03 1.150 276.633 0 .3 166 .69 -0 .OA19 1 .159 266 .720 0.6 170.39 -0.0h33 1.181 257.h33 0.5 170.69 -0.0hhh 1.22h 2A8.016 0.6 166.59 —0.0A53 1.291 239.185 0.65 163.29 -0.0h56 1.330 236.281 0.66 162.61 -0.0A56 1.338 236.05h 0.67 161.93 -0.0h57 1.3h5 235.97h 0.68 161.28 -0.0h57 1.353 236.070 0.69 160.65 -0.0h58 1.360 236.3h8 0.7 160.07 -0.0658 1.367 236.185 0.8 158.55 —0.0h63 1.399 257.801 0.9 170.71 -0.0h70 1.345 313.158 1.0 183.82 —0.0478 1.245 390.126 63) Linstrom and king's Demand equation for Green Peppers in North Carolina.28 Applying a first order autoregressive model in the similar manner as in the previous case, the first approximation to;3 was found to be 0.9. When a diagram was drawn as before, it was suspected that F'would be somewhere between 0.8 and 0.9. After further computations, the sum.of 38 squared residuals was found to be the smallest when f’was set equal to 0.88. —.— 28 Linstrom and King, op. cit., 13.1.0, Equation (5.1), x1 . —0.3117 + 0.7111. x2 .. 0.0000 x3. 39 The sign of the coefficient of.X2 which.was derived from the first order autoregressive.model does not agree with Linstrom and King's assumption that the local and the New York wholesale market prices of green peppers move toward the same direction. This probably is due to the fact that the disturbances are generated by'a more general scheme than a a first order autoregressive scheme, or some relevant variables were over— looked when Linstrom.and King formulated their model. The regression equation derived from the first order autoregressive model is I _. + X1 0.6773 0.0383 X2 0.00005 X3 or alternatively stated, u .. - .. ‘11- + - 9 x1 0.0813 0.8813: 0.0383 (x2 0.88 x2) 0.00005(x3 0.se Xfi) where X1 : The daily weighted average price for peppers at Clinton, N.C. in dollars per bushel. X : The simple Average daily price paid for North Carolina 2 peppers of good quality on the New York wholesale market in dollars per bushel. X3 : Suppy on the Clinton Market in bushels. 140 n O ...... w as; em6 r0( gr mar. 8 dmw m e D... S on awe Hie drr .180 Svr 2% m da/ .8 n grnwo Hm u m: was mmm 51w nMWSD ms med. ......mm aiK if umd vow 0.5 41 Table 3 Regression Coefficients and Sum of Squares - Residuals for Different Values of f3. Linstrom.and King's Demand Study of Green Peppers (6a). Autocorrelation Regression Coefficients Sum of Coefficient Constant X2 X3 Squared .Residuals 0 -0.1716 0.7043 -0.000019 8.098 0.1 -0.1352 0.6904 —0.000017 7.576 0.2 -0.0734 0.6679 -0.000015 7.174 0.3 0.0274 0.6324 -0.000011 6.861 0.4 0.1870 0.5760 -0.000006 6.580 0.5 0.4295 0.4877 0.000002 6.237 0.6 0.7656 0.3559 0.000013 5.699 0.7 1.1262 0.1846 0.000027 4.880 0.8 1.2294 0.0242 0.000042 4.009 0.84 1.0683 -0.0168 0.000046 3.787 0.85 0.9927 -0.0241 0.000046 3.752 0 .86 0 .9103 -O .0300 0.000047 3 .727 0.87 0 .8049 -0 .0346 0.000047 3 .711 0.88 0.6773 -0.0383 0.000048 3.705 0.89 0.5229 -0.0399 0.000048 3.711 0.9 0.3351 -0.0406 0.000048 3.728 0.95 -l.7185 -0.0437 0.000048 3.962 1 -1.7470 0.0932 0.000040 5.193 7a) Quackenbush and Shaffer's Demand Function for Ice Cream. Purchased for Home Use in Lansing Area. 29 Going through the same procedure as described in the previous two cases, the first approximation of $‘ turned out to be 0.4. After some_ more computations, 0.41 was selected as a closest approximation to F . 29 Quackenbush and Shaffer, op. cit., p. 10, X1 - 0.1860 — 1.1134 X2 + 0.0035 X3 + 0.0037 Xh In this case, the first difference transformation of variables made the sum of squared residuals greater than the originally-fitted model. Yet the values of regression coefficients did not approach those of the true parameters. Probably it would not be appropriate to use the first difference model in this case. The appropriate regression equation obtained under the assumption of a first order autoregressive model is x1 - 0.1593 - 0.8929 x2 + 0.0052 x3 + 0.0036 x4 or 11 ' 0.0940 - 0.43 Xi - 0.8929 (X2 - 0.43 X3) + 0.0032 (X3 - 0.43 Kg) + 0.0036 (Xh - 0.43 x2) where X1 : Pints of ice cream per capita. X : Price per pint. X : weekly family income. X : Mean Temperature, Lansing. 43 r! n o ih SS) sua eb7 rn( 8e. mam dam m c s .e um...“ «.mr .1 o sz . m deli m t 3a“; cu .0 use. Ffad 6mm t ma” fs: otr ne 3.. icu afia .ndd ...vanwm . Table 4 Regression Coefficients and Sum of Squared Residuals for Different Values of f9 . Quackenbush and Shaffer's Demand Study of Ice Cream (7a). Regression Coefficient Autocorrelation Sum of Coefficient Constant X2 X3 Xh squares hesiduals 0 0.0441 -0.7378 0.0040 0.0036 0.029521 0.1 0.0588 -0.7726 0.0039 0.0036 0.027668 0.2 o .0803 —0 .8102 0 .0038 0.0036 0 .026394 0.3 0.1114 -0.8502 0.0036 0.0036 0.025743 0.38 0.1461 -0.8835 0.0033 0.0036 0.025461 0.40 0.1565 -0.8920 0.0033 0.0036 0.025134 0.42 0.1725 —0.9004 0.0031 0.0035 0.025459 0.43 0.1734 -0.9047 0.0031 0.0035 0.025468 0.5 0.2202 ~0.934l 0.0026 0.0035 0.025622 0.6 0.3062 -0.9725 0.0018 0.0034 0.026024 0 . 7 0 .4120 -1 .0001 0 .0008 0 .0033 0 .026469 0.8 0.5235 -1.0064 -0.0004 0.0030 0.027198 0.9 0.6155 -0.9827 -0.0015 0.0028 0.029036 1 0.7574 -0.9292 -0.0032 0.0027 0.034178 30 9d) Hoos' Demand Equation for Pears. When the Durbin—Watson test was applied, significant negative autocorrelation was established. It was, therefore, assumed that the true value of 73 would lie between 0 and -l. The first and second approximations to F; were —0.6 and -0.66 respectively. The first difference model was There was a subtantial increase in the sum of squared also fitted. residuals without noticeable change in regression parameters. 3O H008 and Shear, op. cit., p.295, Equation (1), ixl ‘ 163.4166? — 19.17780 11 - 15.60794 I: + 8.51542 W — 10.83552 T. 45 This case seems to suggest that one must be cautious in apply- ing the first difference model, if there is a reason to suspect that disturbances are negatively autocorrelated. The regression equation derived from the first order autoregressive model in which ,0 was set equal to —0.66 is as follows: X1 =- 173.25 - 22.459 Y1 - 17.351 Y2 + 9.366 W - 11.106 T or X1 = 287.60 + 0.6610}. - 22.459(Y1 + 0.66Yi) - 17.351(Y2 + 0.6695) + 9.355 (N +0.66%) - 11.106(T + 0.661%) where l : New York unloads of pears (in 100 tons.) Y : New York wholesale prices of pears (in dollars per 100 pounds.) Y : New York wholesale prices of plums (in dollars per 100 pounds.) New York state factory wages, June—August (dollars per week.) 2': O. T : "Time" in years (origin, July-August, 1931). 46 0 n 0 .1 ....d e rm g .M R d... @S O .0 Sh 1 a o US .06 .101 Sr ea I4.nv Cu 0%.! O F: _ )1 u d 0.89 .DW/k rm. 3 .nr wta 38 5E? fsr OtO nf ne 0.1n .160 +0.11 aft sum 30 . VCE Regression Coefficients and Sum of Squared Residuals for Table 5 Different Values of p . Hoos' Demand Equation for Pears (9d) 47 . Regression Coefficients Sum 0f Autocorrelat ion Squared Coefficient Constant Y1 Y2 W T Residuals 0 1610070 ~2301-ho ”M0129 80958 ' ‘100982 7814005149 -0 .1 167.91 ~23 .318 --14 .447 8.959 ~11.053 6840 .105 -0.2 170.13 ~23 .436 ~14.825 9.000 ~11.108 6008.607 --0 .3 171 . 78 ~23 .454 ~15 .308 9 .070 ~11 .144 5340 .961 -O .4 172 .81 ~23 .340 ' ~15 . 862 9 .156 ~11 .161 4841 .266 -O . 5 173 .30 ~23 .087 ~16 .450 9 .244 ~11 .155 4510 .176 —O .6 173 .34 ~22 .719 ~17 .028 9 .321 ~11 .131 4349 .346 —O .65 173 .38 --22 .504 ~17 .299 9 .354 ~11.ll3 4329 .218 -O .66 173 .25 ~22.459 ~17.35l 9 .360 ~11.112 311 .043 —O .67 173 .10 -22 .415 ~17 .404 9 .366 -11.106 4316 .735 -O .7 173 .07 -22.278 ~17 . 552 9 .382 -11.094 4359 .631 —O .8 172 .58 ~21.810 ~17.991 9 .422 ~1l.048 4345 .262 -O .9 172 .00 —21 .356 ~18.331 9 .444 -11.001 4904 .161 -1.0 171.38 ~20 .943 ~18.575 9 .449 ~10.954 5437.704 *1 .0 11.1 . 60 -21 .450 -13 . 978 10 .470 -10 .345 27040 .092 31 5b) Foytik's Demand Equation for Midseason California Plums. Originally it was thought that there was a significant positive antOCorrelation at the 10% level in the disturbances of this regression model, and a first order autoregressive model was fitted accordingly. Later when the Durbin-Watson test statistic was recomputed, an error was discOVered, and it was decided that test result was inconclusive. However, Dince a. part of computations which had been finished showed some interest— Fix Foytik. op.cit..p.z»31. Equation (2). P - 1,7271 - 0.0270 2 2 Q + 00011.2 I " 0.131 T + 0.0125 Q1 " 000311 32. 48 ing results, the investigation was continued. In applying a first order autoregressive model, different possible tnilues of f3 ranging from.-0.5 to +1.0 were inserted in the error process. The closest approximation to ,3 was 0.48. Parameters estimated from the thirst difference model were quite close to those derived from the first (31138r autoregressive model. The regression equation derived from the first order autoregressive model is as follows: P2 - 1.9649 ~ 0.0264 02 + 0.01126 I - 0.0161 T + 0.00792 Q1 ~ 0.03232 82 oz'alternatively stated, P2 - 1.0217 - 0.48 8'2:- - 0.02646 (02 — 0.48 03-) + 0.01126 (1 — 0.48 1*) - 0.0161 ('1‘ - 0.48 T-X) + 0.00792 (01 - 0.48 0i) - 0.03232 (32 - 0.1.8 85) where, P : New York-Chicago auction price for midseason varieties, dollars per crate. 02 : New York-Chicago auction sales of midseason varieties, in 10,000 equivalent crates. I : Index of 0.3. non-agricultural income payments, May-October average, 1935-1939 = 100. T : Time in years, with origin at 1921. New York—Chicago auction sales of early varieties in 10,000 equivalent crates. pf: “ : Supply of early peaches (total produc ion in 0a., S.C., N.C., and Ark.,) in million bushels. 50 Table 6 Regression Coefficients and Sum of Squared Residuals for Different Values of f9 . Foytik's Demand Equation of Mid— Season Plums (5b). w ’1 ibutocorrelation Regre531on Coefficients Sum.of 'Coefficient Constant 02 I T 01 32 Squares Residuals V'V(P) .43.5 1.5915 .0.02579 0.01086 .0.01577 0.01462 -0.02657 0.41227 1.6344 -0.02621 0.01090 -0.01558 0.01426 -0.02758 0.37650 1.6807 -0.02658 0.01095 -0.01537 0.01376 ~0.02862 0.34446 1.7275 —0.02684 0.01101 .0.01513 0.01309 -0.02960 0.31563 1.7726 -0.02699 0.01107 —0.01491 0.01230 .0.03043 0.28976 1.8146 .0.02702 0.01112 —0.01474 0.01142 —0.03109 0.26696 1.8526 —0.02696 0.01116 -0.01465 0.01053 .0.03157 0.24721 1.8864 —0.02684 0.01120 -0.01469 0.00968 —0.03190 0.23130 1.9165 -0.02670 0.01123 .0.01492 0.00892 ~0.03212 0.21974 1.9438 .0.02656 0.01125 -0.01543 0.00831 -0.03225 0.21314 1.9620 -0.02646 0.01126 —0.01600 0.00795 -0.03230 0.21235 1.9649 -0.02646 0.01126 -0.01610 0.00792 _0.03232 0.21182 1.9675 .0.02646 0.01126 —0.01621 0.00787 .0.03232 0.21192 1.9702 -0.02645 0.01126 —0.01632 0.00783 -0.03233 0.21208 1.9729 -0.02644 0.01126 -0.01644 0.00779 -0.03234 0.21229 1.9993 -0.02635 0.01128 -0.01786 0.00750 -0.03238 0.21698 2.0386 -0.02629 0.01130 -0.02065 0.00729 40.03241 0.22816 2.1108 -0.02626 0.01133 -0.02651 0.00717 -0.o3241 0.24578 2.3293 —0.02623 0.01140 -0.04455 0.00712 -0.03240 0.26989 2.9728 -0.02636 0.01087 —0.09170 0.00754 -0.03254 0.30854 CHArTEUL V D‘UI'JULLLY rial) CUIJULUdlUNS Since one of the osjectives of this study is to see how important the autocorrelation of disturbances is in the selected group of previously- fitted regression models, each of the regression models was tested by the Durbin-Watson test of serial independence of disturbances. Results were presented in the previous chapter. Amons' nineteen regression equations tested, the hypothesis of zero autocorrelation in disturbances at the 10,15 significance level may not be rejected in only seven of them. In four cases, significant autocorrelation was established , and subsequently a first order autoregressive model was fitted to them. In the remaining nine cases, the test was indeterminate. Although no further test was applied to these nine cases at this time, it seems reasonable to expect that auto- correlation was a factor in some of the cases. From these results, it is evident that some of the regression models tested are highly autocorrelated, either positively or negatively. These autocorrelated disturbances may account for a part of the loss of efficiency in estimation and prediction by the current method of least squares regression. Another objective of this study is to compare the least squares eatimates of regression coefficients and the sum of Squared residuals with those estimated from the first order autoregressive model, and with those estimfited from the first difference model. For this purpose, the FElGVE-‘nt estimates of regression coefficients and sum of squared residuals for 52 each of the five cases are summarized as follows: Table 7 Regression Coefficients and Sum of Squared Residuals Estimated from the Ordinary Regression Model, the First Order autoregressive Model, and the First Difference Model. 1$quction Regression Coefficients Sum of 5 No. Squared Residuals r ( lb ) C onst ant SW Iw = 0 144.92 00371 1.155 301.437 7 P - 0.67 161.93 —0.0457 1.345 235.974 - 1 183.82 -0.0478 1.245 390.126 (6b) Constant X2 X3 ~ 0 -O.l716 0.7043 —0.000019 8.098 )0 - 0.88 0.6773 -0.0383 0.000048 3.705 ~ 1 -1.747 0.0932 0.000040 5.193 (7a) Constant X2 X3 Xh — ~ 0 0.0441 -0.7378 0.0040 0.0036 0.029521 1° - 0.41 0.1593 -0.8929 0.0032 0.0036 0.025134 ' 1 0.7574 -O.9292 —0.0032 0.0027 0.034178 (9d) Constant Y1 Y2 W T ~ 0 164.70 43.140 44.129 8.958 -10.982 7840.549 P = -0.65 173.25 -22.459 -17.351 9.360 41.112 4011.043 - -1.0 171.38 _20.943 -18.575 9.449 -10.954 5437.704 ~ 1.0 111.60 -21.450 -13.978 10.570 -1o.345 27040.092 (5b) Constant 02 I T Q1 32 IC’ = 0 1.8146 ~0.02702 0.01112 ~0.01474 0.01142 ~0.03109 0.26696 ' 0.48 1.9646 ~0.02646 0.01126 ~0.01610 0.00792 ~0.03232 0.21182 ' 1 2.9728 ~0.02636 0.01087 ~0.09l70 0.00754 ~0.03254 0.30854 53 According to the above table, in three of four cases where a positive autocorrelation was indicated the first difference transformation tends to bring the estimated parameters closer to those derived from the first order autoregressive model,although the sums of squared residuals were increased compared with the original least squares models. In the case (9d) where the significant negative autocorrelation was established, it seems to suggest that if the sign of autoccorelation is mistaken the result will be very serious. in applying»: the first difference model, one must first ascertain the sign of the autocorrelation, and must remember that even in the caSe of a positive autocorrelation the first difference transformation does not always eliminate unfavorable effects of auto— correlation of disturbances. Last of the main onjectives was to examine sum of squared residuals as a function of the assumed autocorrelation coefficients. This curve was plotted in the tOp of each of the five graphs in the previous chapter. Each curve seems to have only one minimum within the interval examined. If it can be proven in general that there is always a unique autocorrelation coefficient that minimizes the sum of squared residuals, the procedure suggested by Marshall and Hirshleifer ( briefly discussed in Chapter III) may be used to estimate regression coefficients and an antocorrelation coefficient. To find out how effective a first order autoregressive model is, one must test it against empirical data and use it for actual predictions in tol'le similar manner as one does to any other statistical estimation of °°°n0mic relations . The present study is concerned only with effects of applying 54 a first order autoregressive model to some single equation.models in which there is a significant evidence of autocorrelated disturbances. For further investigation into the problem of autocorrelation, it may be interesting to extend this kind of empirical study to economic models in which there are simultaneous relationships between the variables. Probably it is also worthwhile to look for a more efficient computing procedure because a considerable amount of time was spent in inverting a number of moment matrices by the Dolittle method in this study. Professor Hildreth suggested that efforts should be made to find distribution functions of the estimate of the autocorrelation coefficient and regression coefficients in the future. This will enable us to find confidence regions and perform tests of significance. He also suggested that the assumption of a first order autoregressive model should be checked against still more general models such as a second order autoregressive model or some other entirely different disturbances generating process. 55 BIBLIUChArHY General Bibliography Aitken. A.C. “0n Least Squares and Linear Combination of Observations," Royal Society of Edinburgh Proceedings, Vol.55, 1934-35, pp.42-48. Cochrane,D and Orcutt, G,H. "Application of Least Squares Regression to Relationships Containing Autocorrelated Error Terms," Journal of American Statistical Association, Vol.44, 1949. pp.32-61. Durbin, J. and Watson, 0.5. "Testing for Serial Correlation in Least Squares Regression l," Biometrika, Vol.37, pp.409-428. Durbin, J. and Watson, G.S. "Testing for Serial Correlation in Least Squares Regression II," Biometrika, Vol.38, pp.159-178. Gurland, J. "An Example of Autocorrelated Disturbances in Linear Regression," Econometrica, Vol.22, 1954, pp.218-227.. slein, L. Econometrics. New York: Row,Peterson and Co., 1953. Marshall, A. w. and Hirshleiier, J. The Sgpply of r‘emale Labor in vorld War II. The Rand Corporation, Santa Monica, California.“ Mood, A.M. Introduction to the Theo§y_ of Statistics. New'York: McGrawe Hill Book Co., 1950. 'Tintner, G. Econometrics. New York: John Wiley and Sons, 1952. Watson, 0.3. Serial Correlation in Regression Analysis. Unpublished Ph. D. dissertation, Department of Applied Statistics, North Carolina State College, Raileigh, N .0. field, H. Demand Analysis. New York: John Wiley and Sons, 1953. II-.Reports and Bulletins from Which Regression Equations were selected. H003, Sidney; F.0.B. Price Relationships, 1252:56 Pacific Coast Canned Fruits. California Agricultural Experiment Station Mimeographed Report No. 189, July, 1956. Hoos,S. and Seltzer, R.E. Lemons and Lemon Products:Changing Economic Relationships, 12§1-§2. California Agricultural Experiment Station Bulletin 729. “003, .S. and Shear, S. w. Relation Between Auction Prices and Supplies of’California.Fresh Bartlett Pears. Hilgardia v01..14, N0. 5, «January, 1942, California Agricultural Experiment Station. 56 Hoos, S. Tomatoes and Tomato Produgts: Economic Trends and F.0.B. Price Relationships. California Agricultural Experiment Station Mimeographed Report No. 185, March, 1956. Foytik, J. Characteristics 2;; Demang for Californié r'lums, Hilgardia, Vol. 20, No.20, April, 1951, California Agricultural Experiment Station. French, 8.0. The Long-Term Prige and rroduction Outlook £95 Apples in the United States and Michgg' an. Michigan State University Agricultural Experimant Station, Technical Bulletin 255, April, 1956. Linstrom, LA. and King, R.A. The Demanjq for North Carolina 811025 Cucumbers and Green Peppers. Department of Agricultural Economics, North Carolina State Col1ege, A.E. Information Series No.h9, March, 1956. Meinken, K.W. The Demand _a_r_lg_ I’rice Structure £o__r Wheat. Technical Bulletin No .1136, November, 1955, United States Department of Agriculture. Quackenbush, (3.0. and Shaffer, J .1). Factors Affecting Purchases g; .1193 Cream _f_9_1; Home Use. Michigan State University Agricultural Experiment Station Technical Bulletin 2A9, April 1955. APRENDIX warm mo 31100311011113 SURRMiENTMtY T0 CHAPTER II 58 AEBENDIX The least squares computing procedures followed in estimating regression coefficients and a sum.of squared residuals by assuming disturbances to be generated through a first order autoregressive model are described with an illustration. In general, the first step in a regression analysis of this type is to compute the moments of variables being analyzed. The moment matrices are specified in Equation (12), Chapter II. It is expressed as follows : nu 1H?) - [(Z - fzamz - Fax->14 [(2 -f’z+:-‘)'(Y - Fifi-)1 . (11) Hence the moments are (Z - PZ*)'(Z -f> 2*) and (Z —PZ*)'(Y -/°Y*). If they are denoted by M(P) and N(P) respectively} (A1) may be re- written as, t (e) = mrrlmm. (A2) To refresh our memory, symbols used in (A1).are again identified here. Z and I stand for observations on independent variables and a dependent variable. They are of the following matrices. Z ' Zn 0 o o o e 0 21K 2* = 201 o e e o e ZOK O O o o 2 ZTl o o o o o Zrh 211 1’1 0 o Zr l’K/ Y a yl Iii- : yl \ 59 To faciliate the computation, 11(9) and M?) are expanded. (A3) (A4) mp) - Z'Z .. 2922-12 9 922-8128 N(<°) .. 2'! + 9(z-x1 + 7.08:) + 9222.1... When Z'Z,Z*'Z etc. , are computed a pair of quadratic of P can be obtained. From these quadratics, desired moment matrices may be derived by insert- ing 3W value P , [PK 1, in them. This procedure is illustrated with the Meinken's demand equation for the world wheat market. In the Meinken's demand equation for the world wheat market (it is called 1b in this study), They are expressed in terms of deviations from Z,Y,Z*,I* are as follows. sample means . 2 Y z 1* 3w I P“ ‘3“; 1* P5; 1925 -667 .14 36 .84 60 .64 1924 —694.79 46.10 69 .01. 27 .288.14 28.44 44.34 26 -355 .79 27 .80 46 .84 28 92 .86 22.24 19 . 51. 27 -185 .79 24.00 36 .21. 29 -54.14 9.44 21.44 28 195 .21 17.80 11.40 30 285.86 43.86 ~29.76 29 48.21 5.00 13.44 31 329 . 86 -20 .16 -34 .06 30 388.21 -18.30 -37 . 86 32 272.86 -2h016 '35016 31 [032.21 '215060 -42016 33 362 .86 -21.46 4.1 .06 32 375 .21 -28.60 --43 .26 34 93.86 -19 .16 -30.46 33 465.21 -25.90 -49.16 35 -77 .14 -15 .96 -22.46 34 196.21 -23 .60 -38 . 56 36 -304.14 -1.26 14.34 35 25 .21 -20.40 --30 . 56 37 .225.14 -1 . 56 11.01. 36 -201.79 -5 .70 6 .24 38 635 .86 -11056 -36036 37 '122079 -6 .00 5.94 z 'z 2 Ir 2 12* “me 2 {3w 1708457.7 -78163 .0 [-153751.6 [1181104.4 470979 .4 406729.53 1 1 PW [ 17628.1) [ 16051 .1.) Z-n- l2 2* II z-x- 12* 21‘3 II 21%? 1181104 .4 -93 543 .9 -150266 .3 1792902 .4 -104737 .6 -180510 .0 I -70979 .4 6719 .9 10337 .1 -104737 .6 8336 .1 13029 .9 Me: I: 16051.4] [ 21337.1] 60 Compute sums of squares and cross products of all the variables and fill them in prOper places in the brackets below the table. For instance, the 2 x 2 matrix in the upper left corner is 2'2 or Z 32 E s i . It is immediately noticed that this matrix is t tw t tw t 2 S- i s E. i t t tw t t symmetrical. One has to compute only those elements on the main diagonal and those above the main diagonal. Those elements below the main diagonal may be computed to check the accuracy of computations. The 2 x 1 vector . . . 2 . ' ‘ r r is 2* I or § audit“ . The Single element is 2t, ptw . Inserting Si 10 t. t tw the values of the sums of squares and cross products into the two quadratics (A3) and (M), the following expressions are derived. 1 NH”) -[1708457.7l4 -78163.02€i-f’2362208 464523.310}+(‘E792902.357 404737.600 6191 .412 13439 . 800 8336 .080 Me) - 153751.614} - P -256956.156 + P‘ 480510.007] 10055.394 20853.282 13029.970 For each value of f inserted, say 9 1, a pair of moment matrices Iv1( F1) and MG.) can be obtained from the above quadratics. Next step is to invert MP) and multiply this inverse withh'U‘). In carrying out this computation, it is convenient to normalize M(f’ ) and MP). The normalization used here is to convert 11(6) and IMP) into correlation matrices. A nonnalizing factor for ”ii (a typical element 1 Since the matrices in M”) are symmetrical, the terms below the main diagonal are not filled. 61 on the main diagonal of MG”) ) is simply the reciprocal of mii itself, so that all elements on the main diagonal become 1 when normalized. To find a normalizing facor of mij (a typical element above or below the main diagonal), take the square roots of mu and mjj and form a product of their reciprocals. For n1 (a typical element in M?) ), a normalizing factor used is l/IIE W) , where Y(?) is defined as, H?) a PI - ZPY'Y* +92%“! 2 (A5) This process is illustrated with the case where P is set equal to 0.1 in the Meinken's wheat model. since 4930.793 8100.370 and Y(O.l) - 14631.184, h(0.1) = [1490165.852 -62758.073] N(O.1) =[-129861.098} the normalizing factors for M(O.l) and N(O.l) are as follows: 1 for ml]. 1 .1 for H112 f1490165.852 JT490165 .852 37.90165 .852 J 4930.793 1 . 101‘ 11122 . J4930.793 J4930.793 1 . 1 lor n for n2 J1490165.852 J14631.184 J4930.793 J14631.184 The correlation matrices are obtained by multiplying each element in l«1(0.l) and N(O.l) by its respective nomalizing factor. They are denoted N N by M(O.l) and N(O.l). 2 In Meinken's case, Y(P) I 17628.091. - 1” 32102.816 + F2 21337.112 fi(o.1)=[l -O.732139b25l N(0.1‘) = -o.879472o38 l 0.953688977 Now invert 3410.1) and perform the matrix multiplication fiQO.l)-l fi(0.l) as follows: [2.155304473 1.577983808 -O.879h720381 = -0.390624254] 2.155304473 0.953688977 0.667697482 Elements in the column vector on the right side of the equality sign are normalized regression coefficients estimates. To transform.them back to the original variables, each regression coefficient estimate 7T1, i=1,2, is multiplied mm LN; . —O.39062425h x 0.099088311 = -0.038706298 0.667697482 x 1.722587729 = 1.150167h90 Therefore, estimates of the regression coefficients of the original variables 8H and I are approximately ~0.039 and 1.2 when the auto- correlation coefficient P is set eeual to 0.1. The third step is to calculate the sum of squared residuals as described in Equation (14) in Chapter II. The sum of squared residuals ‘V'V(P) is expressed as, 7:929) -- Y'Y .. 291's": + 92 we mum?) (so) or = YW) ~ 77'1“?) - In the Meinken's model where (3 is assumed to bt 0.1,‘giV20.1) is = 14631.184 - -0.0387062981 [429361.098] 1.150167490 8100.370 = 287.958 The final step is to calculate the mean or constant (fro). Here the sums of dependent variables and independent variables are espressed in terms of original observations, and T denotes the number of observations. . ___1_ - . . ‘1 W0 T +1{‘S-t-yt (1121-?th 00000000000000. 1TK thtK) In the Meinken's case, 71:, s 1%.[1551 .. (.0.0387()6298 x 65284 + 1.150167490 x 16683)] = 152.816h in summary, the repression enuation obtained under the assumption, f'= 0.1, in the Meinken's wheat model and the correspondina sum of squares of residual are as follows: U = 0.1 U* + V and its sum.of seuared residuals is 287.958. ... mf’fiflfi Ififi" A...” EQQURN UJC UHL“ Date Due Demco-293 \lltllumajlfllfltfllufilmlllllllllllllmllliUllllillll 145 9021