A Monte Carlo Study for Swamy s Estimate of Random Coefficient Panel Data Model

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1 A Monte Carlo Study for Swamy s Estmate of Random Coeffcent Panel Data Model Aman Mousa, Ahmed H. Youssef and Mohamed R. Abonazel Department of Appled Statstcs and Econometrcs, Instute of Statstcal Studes and Research, Caro Unversy, Caro, Egypt. Apartcularly useful approach for analyzng pooled cross sectonal and tme seres data s Swamy's random coeffcent panel data RCPD model. Ths paper examnes the performance of Swamy's estmators and tests assocated wh ths model by usng Monte Carlo smulaton. The Monte Carlo study shed some lght nto how well the Swamy's estmate perform n small, medum, and large samples, n cases when the regresson coeffcents are fxed, random, and mxed. The Monte Carlo smulaton results suggest that the Swamy's estmate perform well n small samples f the coeffcents are random and but does not when regresson coeffcents are fxed or mxed. But f the samples szes are medum or large, the Swamy's estmate performs well when the regresson coeffcents are fxed, random, or mxed. Key words: Random Coeffcent Panel Data Model, Mxed RCPD Model, Panel Data, Monte Carlo Smulaton, Poolng Cross Secton and Tme Seres Data.. Introducton Econometrcs commonly uses Tme Seres Data descrbng a sngle enty. Another type of data called Panel Data whch means any data base descrbng number of ndvduals across a sequence of tme perods. To realze the potental value of the nformaton contaned n a panel data see Carlson 978, Hsao 98, 3, and Baltag 8. ` When the performance of one ndvdual form the panel data s nterest, separate regresson can be estmated for each ndvdual un. Each relatonshp, on our model studed, s wrten as follows: y x ε,,,3,..., t,,3,..., T, where denotes cross-sectons and t denotes tme-perods. The ordnary least squares OLS estmators of and wll be best lnear unbased estmators BLUE under the followng assumptons:

2 A: E ε A: E ε ε IT A3: E ε ε, for all j. j These condons are suffcent but not necessary for the optmaly of the OLS estmator, see Rao and Mra 97. If assumpton s volated and dsturbances are eher serally correlated or heteroskedastc, generalze least squares GLS wll provde relatvely more effcent estmator than OLS, see Gendreau and Humphrey 98. If assumpton 3 s volated and contemporaneous correlaton s present, we have what Zellner 96 termed seemngly unrelated regresson SUR equatons. There s gan n effcency by usng SUR estmator rather than OLS, equaton by equaton estmator, see Zellner 96, 963. Suppose that each regresson coeffcent n equaton s vewed as a random varable, that s, the coeffcents and are vewed as nvarant over tme and varyng from one un to anther. So, we are assumng that the ndvduals n our panel data are drown from a populaton wh a common regresson parameter, j, j,, whch s fxed component, and a random component v whch wll allow the coeffcents to dffer from un to un,.e. A4: j j v j, for,,.,, j,. where e Model can be rewrten, under assumptons to 4, as: y x e, v x v ε,,,,, t,, T, model s called Random Coeffcent Panel Data model examned by Swamy 97, 97, 973, 974, Kelejan and Stephan 983, Hsao and Pesaran 4, and Murtazashvl and Wooldrdge 8. Equaton can be wrten n matrx form as Y e, 3 where Y Y Y Y Y y y y, [ ] [ ] [ ],, T x x,, e DV ε, xt

3 3, D v v v v v v V,. The followng assumptons are added to the prevous assumptons: A: The vector V are ndependently and dentcally dstrbuted wh, v E and, v v E,,,. A6: The ε and v are ndependent for every and j, so the varance-covarance matrx of e s, Ω T T T I I I ee E where zeros are T T null matrces and s the varance-covarance matrx of as gven n assumpton. If assumptons tll 6 hold, then the GLS estmator of s gven by Y Ω Ω. 4 Swamy 97 showed that [ ] [ ] where s the OLS estmator of. The GLS estmator cannot be used n practce, snce and are unknowns. Swamy 97 suggested the followng unbased and consstent estmators T K εε, 6 and S, 7 where S. 8

4 4 ote that s the mean square error from the OLS regresson of Y on, and / S s the sample varance-covarance matrx of. Substute 6, 7, and 8 n, we get the feasble generalzed lest square FGLS estmator of as follows: [ ] [ ], 9 and the estmated varance-covarance matrx for the RCPD model s [ ],, Var Ω Swamy 973, 974 showed that the estmator s consstent as both and T and s asymptotcally effcent as T. Because v s fxed for gven, we can test for random varaton ndrectly by testng whether or not the fxed coeffcent vectors are all equal. That s, we form the null hypothess H :. If dfferent cross-sectonal uns have the same varance,,,...,, the conventonal analyss of covarance test for homogeney. If are assumed dfferent, as postulated by Swamy 97, 97, we can apply the modfed test statstc F * *, where y *. Under H, s asymptotcally ch-square dstrbuted, wh K - degrees of freedom, as T tends to nfny and s fxed. If the regresson coeffcents n model 3 contan both random and fxed coeffcents, the model wll be called Mxed RCPD model. The Mxed RCPD model s smply a specal case of the RCPD model where the varance of certan coeffcents, whch wll be consdered as fxed coeffcents, are assumed to be equal to zero. Thus equaton 9 stll apples to estmaton after certan elements of the matrx are constraned to equal zero.

5 . Smulaton Desgn A Monte Carlo smulaton was conducted to study the behavor of certan estmators and tests n small, medum and large samples. The smulaton was desgned prmarly to nvestgate estmaton and hypothess tests for the RCPD model dscussed before. The settngs of the model and results of the smulaton study are dscussed below. The values of the ndependent varable x, were generated as ndependent normally dstrbuted random varates wh mean µ and standard devaton. The values of x were allowed to dffer for each cross-sectonal un: However, once generated for all crosssectonal uns the values were held fxed over all Monte Carlo trals. The value of µ was set equal to zero and the value of was set equal to one. The dsturbances, ε, were generated as ndependent normally dstrbuted random varates, ndependent of the x values, wh mean zero and standard devaton ε. The dsturbances were allowed to dffer for each crosssectonal un on a gven Monte Carlo tral and were allowed to dffer between trals. The standard devaton of the dsturbances was set equal to eher, 3, or and held fxed for each cross-sectonal un. The values of and T were chosen to be,, and to represent small, medum and large samples for the number of ndvduals and the tme dmenson. The parameters, and, were set at several dfferent values to allow study of the estmators under condons where the model was both properly and mproperly specfed. Also, test of hypothess for randomness was examned to determne the observed level of sgnfcance and to obtan an dea of the power of the test. Fve dfferent combnatons of and are used as gven n Table. ote that a varance of zero smply means that the coeffcent s fxed and equal over all cross-sectonal uns. These models wll be estmated usng Swamy's estmators n order to study the behavor of the coeffcent mean estmator under msspecfcaton of the model and to study the behavor of the tests for randomness of coeffcents. Table Values of Coeffcent Means and Varances Used n the Smulaton Model Var Var 3 4 There are 4 expermental settngs for the smulaton, and, Monte Carlo trals were used for each settngs. The results were recorded n Tables through 6, wh each table consstng of three panels, numbered I through III, for the dfferent samples szes,, and. And each panel from ths panels correspondng to three settngs of the dsturbance standard devaton, 3, and. Each of the tables provdes the results for a partcular scheme of generaton of the regresson coeffcents.

6 3. Monte Carlo Results Tables through 6 are set up to show the followng nformaton: The coeffcent mean estmators or the estmators of the fxed coeffcents, and, that are computed as n equaton 9. The values shown n the frst row of each panel of each table are the averages over all, Monte Carlo trals at a partcular settng. Table Results of RCPD Estmaton When ~, and ~, T ε Bas of I. MSE of % egatve Varance Estmates %Rejectons H : Bas of II. MSE of % egatve Varance Estmates %Rejectons H : Bas of III. MSE of % egatve Varance Estmates %Rejectons H :

7 Table 3 Results of RCPD Estmaton When ~, and ~, T ε Bas of I. MSE of % egatve Varance Estmates %Rejectons H : Bas of II. MSE of % egatve Varance Estmates %Rejectons H : Bas of III. MSE of % egatve Varance Estmates %Rejectons H : The estmated varance of each coeffcent, Var k, averaged over, trals, s shown n the second row. The estmates are computed as the dagonal elements n equaton 7. The bas value of the coeffcent mean estmators, and, are computed as bas where s a vector of coeffcents mean estmators and s a true vector of coeffcents mean. The bas values shown n the row three of each panel. ^ 7

8 Table 4 Results of RCPD Estmaton When and T ε Bas of I. MSE of % egatve Varance Estmates %Rejectons H : Bas of II. MSE of % egatve Varance Estmates %Rejectons H : Bas of III. MSE of % egatve Varance Estmates %Rejectons H : The Mean Square Error MSE of coeffcent mean estmators that are computed as ^ ^ MSE [ k Var k bas k ] where Var k s the estmated varance of the coeffcent mean estmator, and s computed as the kth dagonal element of the varance-covarance matrx gven n equaton. The MSE values shown n the row four of each panel. 8

9 Table Results of RCPD Estmaton When and ~, T ε Bas of I. MSE of % egatve Varance Estmates %Rejectons H : Bas of II. MSE of % egatve Varance Estmates %Rejectons H : Bas of III. MSE of % egatve Varance Estmates %Rejectons H : It s possble to obtan negatve estmates of the coeffcent varances, K, when equaton 7 s used to compute the varance-covarance matrx. The percentages of negatve varance estmates shown n the row fve of each panel. The sxth row of each panel records the percentage of rejectons of the null hypothess H : k, for k and, at a nomnal % level of sgnfcance. The ch-squared statstc n equaton s used to perform the test. 9

10 Table 6 Results of RCPD Estmaton When ~, and T ε Bas of I. MSE of % egatve Varance Estmates %Rejectons H : Bas of II. MSE of % egatve Varance Estmates %Rejectons H : Bas of III. MSE of % egatve Varance Estmates %Rejectons H : As a gude to nterpretng the tables, let us consder Table as an example. When ε and T small samples, the averages mean and varance for over all, Mote Carlo trals are and respectvely. ote that the true coeffcents values for mean and varance are and, and the values of bas and MSE for are. and.8. And the averages mean and varance for are. and 4.99 respectvely. Whle the true coeffcents values for mean and varance are and. Whle the percentage of negatve varance estmates for and s zero. ote that ths percentage should be zero. And the percentage of rejectons of the null hypothess H : k for and s. Ths means

11 that the randomness test s performng as desgned even n small samples. As the varaton n the dsturbances ncrease, from ε to ε 3, the estmators get worst. Increasng both the number of ndvduals and the tme seres data wll make the estmators better. 4. Concludng Remarks From Tables tll 6, several observatons concernng the RCPD estmators and the test statstcs for the randomness test can be made: - The Swamy's estmators performs well when the coeffcents are random, even though the samples are small T. The bases, true coeffcent estmated coeffcent, of the Swamy's estmators of and decrease when the tme seres observatons and the number ndvdual uns gettng large. From Tables and 3, the bas and MSE are dong better n small and large varaton of the parameters. In general, the Swamy's estmate perform best when both coeffcents are random. - When the coeffcents are random, a small number of negatve varance estmates occurs for the small sample sze, the negatve varance estmates does not appear n medum and large samples. 3- When both coeffcents are fxed, Table 4, and the sample sze s small, the RCPD model s napproprate and a large number of negatve varance estmates occurs as suggested n Delman 98. Thus, the appearance of negatve varance estmates would suggest the possbly that the coeffcent be treated as fxed. 4- When both coeffcents are fxed and the samples szes are medum or large, the RCPD model s approprate and the negatve varance estmates wll not appear. - When one of the coeffcents s fxed and the sample sze s small, the Swamy's estmators wll not perform as well as mght be expected. The appearance of negatve varance estmates, n Tables and 6, would suggest msspecfcaton occurrence n the assumptons. But f the samples szes are medum or large, the Swamy's estmators perform well. 6- The test for randomness performs well overall. The best produces a hgh percentage of rejectons of the hypothess H : k s when the coeffcents are random and a low percentage when the null hypothess s true. 7- As the varaton n the dsturbances ncreases relatve to the varaton due to the explanatory varable, the performance of the Swamy's estmators deterorates. Ths s also true for the power of the test for sgnfcance of the coeffcent means. 8- The behavor of Swamy's estmators s not affected by the changes n the parameter mean but s affected by the changes n the parameter varance. Ths concluson apples to the three models RCPD, Fxed, and Mxed RCPD models.

12 The Monte Carlo smulaton results suggest that the Swamy s estmators perform well n small samples f the coeffcents are random and but does not n fxed or Mxed RCPD models. But f the samples szes are medum or large, the Swamy s estmators performs well for the three models. Fnally, some cauton must be taken before usng the Swamy s estmators, and pretestng procedures of the randomness of the coeffcents must be made. Ths smulaton has been lmed n scope, as all smulatons must be. Hopefully wll shed some lght on performance of Swamy s estmators n panel data. References. Baltag, B. 8, Econometrc Analyss of Panel Data. 4th ed., John Wley and Sons Ltd.. Carlson, R. 978, Seemngly Unrelated Regresson and the Demand for Automobles of Dfferent Szes., Journal of Busness, Vol., pp Delman, T. E. 98, Pooled Data for Fnancal Markets Research for Busness Decson Seres. Ann Arbor: UMI Research Press. 4. Gendreau, B. and Humphrey, D. 98, Feedback Effects n the Market Regulaton of Bank Leverage: A Tme-Seres and Cross-Secton Analyss., Revew of economc Statstcs, Vol. 6, pp Hsao, C. 98, Benefs and Lmatons of Panel Data., Econometrc Revew, Vol. 4, pp Hsao, C. 3, Analyss of Panel Data. th ed., Cambrdge: Cambrdge Unversy Press. 7. Hsao, C. and Pesaran, M. H. 4, Random Coeffcent Panel Data Models., IEPR Workng Paper 4., Unversy of Southern Calforna. 8. Kelejan, H. H. and Stephan, S. W. 983, Inference n Random Coeffcent Panel Data Models: A Correcton and Clarfcaton of the Lerature., Internatonal Economc Revew, Vol. 4, pp Murtazashvl, I. and Wooldrdge, J. M. 8, Fxed Effects Instrumental Varables Estmaton n Correlated Random Coeffcent Panel Data Models., Journal of Econometrcs, Vol. 4, pp Rao, C. R. and Mra, S. 97, Generalzed Inverse of Matrces and Its Applcatons. John Wley and Sons Ltd.. Swamy, P. 97, Effcent Inference n a Random Coeffcent Regresson Model., Econometrca, Vol. 38, pp Swamy, P. 97, Statstcal Inference n Random Coeffcent Regresson Models. ew York: Sprnger-Verlag. 3. Swamy, P. 973, Crera, Constrants, and Multcollneary n Random Coeffcent Regresson Model., Annals of Economc and Socal Measurement, Vol., pp Swamy, P. 974, Lnear Models wh Random Coeffcents. n Fronters n Econometrcs Ed. P. Zarembka., ew York: Academc Press, Inc., pp Zellner, A. 96, An Effcent Method of Estmatng Seemngly Unrelated Regressons and Tests of Aggregaton Bas., J.A.S.A., Vol. 7, pp Zellner, A. 963, Estmators for Seemngly Unrelated Regressons Equatons: Some Exact Fne Sample Results., J.A.S.A., Vol. 8, pp