A Comparatve Study for Estmaton Parameters n Panel Data Model Ahmed H. Youssef and Mohamed R. Abonazel hs paper examnes the panel data models when the regresson coeffcents are fxed random and mxed and proposed the dfferent estmators for ths model. We used the Mote Carlo smulaton for mang comparsons between the behavor of several estmaton methods such as Random Coeffcent Regresson Classcal Poolng and Mean Group estmators n the three cases for regresson coeffcents. he Monte Carlo smulaton results suggest that the estmators perform well n small samples f the coeffcents are random. Whle estmators perform well n the case of fxed model only. But the estmators perform well f the coeffcents are random or fxed. Key words: Panel Data Model Random Coeffcent Regresson Model. Mxed Model Monte Carlo Smulaton Poolng Cross Secton and me Seres Data. Mean Group Estmators. Classcal Poolng Estmators.. Introducton Econometrcs commonly use me Seres Data descrbng a sngle entty. Another type of data called Panel Data whch means any data base descrbng number of ndvduals across a sequence of tme perods. o realze the potental value of the nformaton contaned n a panel data see Carlson 978 and 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 unt. Each relatonshp on our model studed s wrtten as follows: y t x ε t t 3... t 3... denotes cross-sectons and t denotes tme-perods. he ordnary least squares OLS estmators of and wll be best lnear unbased estmators BLUE under the followng assumptons: A: E ε A: E ε ε I A3: E ε ε for all j. j
hese condtons are suffcent but not necessary for the optmalty of the OLS estmator see Rao and Mtra 97. If assumpton s volated and dsturbances are ether serally correlated or heterosedastc 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. here s gan n effcency by usng SUR estmator rather than OLS equaton by equaton estmator see Zellner 96963. 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 unt to anther. So we are assumng that the ndvduals n our panel data are drown from a populaton wth a common regresson parameter j j whch s fxed component and a random component v whch wll allow the coeffcents to dffer from unt to unt.e. A4: j j v j for. j. e Model can be rewrtten under assumptons to 4 as: y x e t t t t v xt v ε t t model s called Random Coeffcent Regresson model examned by Swamy 97 97 973 974 Hsao and Pesaran 4 and Murtazashvl and Wooldrdge 8. Y Equaton can be wrtten n matrx form as Y e 3 [ Y Y Y ] Y [ y y y ] [ ] x x e DV ε x v D v v V v v v.
he followng assumptons are added to the prevous assumptons: A: he vector V are ndependently and dentcally dstrbuted wth E v and E v v. A6: he ε t and v are ndependent for every and j so the varance-covarance matrx of e s I I E ee Ω I zeros are 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 Ω Ω Y. 4 Swamy 97 showed that and estmators [ ] [ ] s the OLS estmator of. he GLS estmator cannot be used n practce snce are unnowns. Swamy 97 suggested the followng unbased and consstent and εε S K 6 S. 7 8 ote that s the mean square error from the OLS regresson of Y on and S / s the sample varance-covarance matrx of. Substtute 6 7 and 8 n we get the feasble generalzed lest square FGLS estmator of as follows: 3
4 [ ] [ ] 9 and the estmated varance-covarance matrx for the model s [ ] Var Ω Swamy 973 974 showed that the estmator s consstent as both and and s asymptotcally effcent as. 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. hat s we form the null hypothess H :. If dfferent cross-sectonal unts have the same varance... the conventonal analyss of covarance test for homogenety. If are assumed dfferent as postulated by Swamy 97 97 we can apply the modfed test statstc F * * y *. Under H s asymptotcally ch-square dstrbuted wth K - degrees of freedom as tends to nfnty and s fxed. If the regresson coeffcents n model 3 contan both random and fxed coeffcents the model wll be called Mxed model. he Mxed model s smply a specal case of the model the varance of certan coeffcents whch wll be consdered as fxed coeffcents are assumed to be equal to zero. hus equaton 9 stll apples to estmaton after certan elements of the matrx are constraned to equal zero.
. Mean Group Estmator A consstent estmator of can also be obtaned under more general assumptons concernng and the regressors. One such possble estmator s the Mean Group estmator proposed by Pesaran and Smth 99 for estmaton of dynamc random coeffcent models. he estmator s defned as the smple average of the OLS estmators :. When the regressors are strctly exogenous and the errors ε t are ndependently dstrbuted an unbased estmator of the covarance matrx of can be computed as * Cov 3 * S. For a proof frst note that under the random coeffcent model we have v 4 v v let then we can rewrte the equaton as follows 6 v ε and GM V 7 V v and. herefore GM v V 8 so
6 V v V v V v V v GM GM and GM GM E. 9 But Cov V Cov Cov GM [ ] E from 7 we can get as follows S and let S *. Substtutng nto we get * 3 and also substtutng 3 nto we get [ ] GM E Cov * * 4 tae the expectaton for 3 then [ ] * * E Cov E GM Cov as requred.
7 Fnally t s worth notng that the and the Swamy's estmators are n fact algebracally equvalent for suffcently large namely. lm 6 o prove that from 6 and when we get lm lm K ε ε 7 substtutng 7 nto we get [ ] [ ] GM as requred. It s worth notng that * when to prove that we substtutng 7 nto 3 we get * * * 8 as requred.
3. Classcal Poolng Estmator When coeffcents are equal for all ndvduals. We are assumng that the ndvduals n our database are drawn from a populaton wth a common regresson parameter vector. In ths case the observatons for each ndvdual can be pooled and a sngle regresson performed to obtan a more effcent estmator of. he equaton system s now wrtten as Y Z ε 9 y y Y y Z K ε ε ε ε and s a K vector of coeffcents to be estmated. If the error varance can be assumed equal for each ndvdual s estmated effcently and wthout bas by E ε ε I then Z Z Z Y. 3 hs estmator has been termed the Classcal Poolng estmator. But f the error has dfferent varances for each ndvdual then the estmator under ths assumpton would be Z Ω Z Z Ω Y 3 I Ω I I. 3 he unnown parameters can be consstently estmated by S K t t ε for 33 ε t are the resduals obtaned from applyng OLS to equaton number. 8
4. Desgn of the Smulaton We wll use the Mote Carlo smulaton for mang comparsons between the behavor of and estmators n three models fxed and Mxed models. he settngs of the model and results of the smulaton study are dscussed below. he values of the ndependent varable x t were generated as ndependent normally dstrbuted random varates wth mean µ and standard devaton. he values of x t were allowed to dffer for each cross-sectonal unt. However once generated for all crosssectonal unts the values were held fxed over all Monte Carlo trals. he value of µ was set equal to zero and the value of was set equal to. he dsturbances ε t were generated as ndependent normally dstrbuted random varates ndependent of the x t values wth mean zero and standard devaton ε. he dsturbances were allowed to dffer for each crosssectonal unt on a gven Monte Carlo tral and were allowed to dffer between trals. he standard devaton of the dsturbances was set equal to ether or and held fxed for each cross-sectonal unt. he values of and were chosen to be and to represent small medum and large samples for the number of ndvduals and the tme dmenson. he values were chosen to represent small samples and the values were to represent medum samples whle the values were to represent large samples. he parameters and were set at several dfferent values to allow study of the estmators under condtons the model was both properly and mproperly specfed. he fve dfferent combnatons of and used are detaled n able by gvng the means and varances of the coeffcents. ote that a varance of zero smply means that the coeffcent s fxed and equal over all cross-sectonal unts. able Values of Coeffcent Means and Varances Used In the Smulaton Model Var Var 3 3 3 4 3 3 For each of the expermental settngs Monte Carlo trals were used and results were recorded n ables through 6 wth each table consstng of two panels numbered I and II for the dfferent samples sze and. And each panel from ths panels correspondng to two settngs of the dsturbance standard devaton and. Each of the tables provdes the results for a partcular scheme of generaton of the regresson coeffcents. 9
. Monte Carlo Results In tables results several estmators and test statstcs are of nterest. ables through 6 are set up to show the followng nformaton: he estmators for the coeffcent mean are computed as n equaton 9. estmators for the coeffcent mean are computed as n equaton 3. Whle estmators for the coeffcent mean are computed as n equaton. able Results of Dfferent Estmaton Methods When ~ 3 and ~ 3 he ε Estmaton Method I. II. Bas -.6 -.9.8. -.3. MSE 3. 3.7...3.3 % Rejectons H : 7.8 73.3 99. 99... % Coeffcents Contaned n 9. 99. 9.3 98.9 94.8 98.9 Bas.6 -.7.3.4 -..4 MSE...... % Rejectons H : 86.8 99. 97.4... % Coeffcents Contaned n 3.8..6.9.8.3 Bas -.6 -.9.8. -.3. MSE 3. 3.7...3.3 % Rejectons H : 7.8 73.3 99. 99... % Coeffcents Contaned n 9. 99. 9.3 98.9 94.8 98.9 Bas -.84 -.4 -.. -. -.8 MSE 4.33 3.7.368.98.3.3 % Rejectons H : 6.6 73. 98.3 99... % Coeffcents Contaned n 93.3 98.9 9.3 99. 9. 99. Bas -.4 -..7.9. -.8 MSE.78.8.47... % Rejectons H : 7. 98.8 9.3... % Coeffcents Contaned n 34.4.4 4.9 6.8 9. 3. Bas. -. -.. -. -.8 MSE 4.4 3.8.37.98.3.3 % Rejectons H : 6. 73. 98.4 99... % Coeffcents Contaned n 94.6 98.9 9.3 99. 9. 99.
able 3 Results of Dfferent Estmaton Methods When ~ and ~ he ε Estmaton Method I. II. Bas -.9 -...3 -..3 MSE..3.4.4.. % Rejectons H :.7 99.3.3..7. % Coeffcents Contaned n 9. 99. 9. 98.9 94.8 98.9 Bas.3 -..7. -.. MSE.9..... % Rejectons H : 46.. 47.7. 48.4. % Coeffcents Contaned n 6.9.7 4.4.4 3.. Bas -.9 -...3 -..3 MSE..3.4.4.. % Rejectons H :.7 99.3.3..7. % Coeffcents Contaned n 9. 99. 9. 98.9 94.8 98.9 Bas.7 -.7 -.. -. -.4 MSE.84.4.63.4.. % Rejectons H :.7 99..4..3. % Coeffcents Contaned n 88. 98.9 9. 99. 9. 99. Bas -.9 -.6 -... -. MSE.78.8.47... % Rejectons H : 3.7. 3.3. 33.. % Coeffcents Contaned n 3. 9.8 4..9 33.4 6.4 Bas. -.6 -.. -. -.4 MSE.38.4.68.4.. % Rejectons H :.9 99....4. % Coeffcents Contaned n 94.4 98.9 9.4 99. 9. 99. he bas values of the coeffcents mean estmators and are computed as bas s a vector of coeffcents mean estmators and s a true vector of coeffcents mean. he bas values shown n the frst row of each panel I and II.
able 4 Results of Dfferent Estmaton Methods When and he ε Estmaton Method I. II. Bas..3 -. -.4.. MSE.9..... % Rejectons H : 99. 99.9.... % Coeffcents Contaned n 67. 7.4 8.7 8. 94.3 98.9 Bas...... MSE.8..... % Rejectons H :...... % Coeffcents Contaned n 9. 97.9 93.9 99. 94.7 99. Bas.. -.... MSE...... % Rejectons H :...... % Coeffcents Contaned n 94.8 99. 9. 99. 9. 99.3 Bas -.3.3 -.33.698 -.. MSE.96..84.487.. % Rejectons H : 9.4 99. 99. 99.9.. % Coeffcents Contaned n 68. 7. 8. 8.4 94.4 98.7 Bas.. -.7. -.. MSE.78.8.47... % Rejectons H : 99...... % Coeffcents Contaned n 9. 98. 94. 98.9 94. 99. Bas.4. -.. -.. MSE.6..66... % Rejectons H : 99...... % Coeffcents Contaned n 9. 99. 9. 99. 9. 99. he Mean Square Error MSE of coeffcents mean estmators that are computed as ^ ^ MSE [ Var bas ] Var s the estmated varance of the coeffcent mean estmator and s computed as the th dagonal element of the varance-covarance matrx. he estmated varances of estmators are the dagonal elements n equaton. he estmated varances of estmators are the dagonal elements n equaton 3. Whle
the estmated varances of estmators are the dagonal elements n equaton 3. he MSE values shown n the row four of each panel. able Results of Dfferent Estmaton Methods When and ~ 3 he ε Estmaton Method I. II. Bas -.3..9...4 MSE..99..98..99 % Rejectons H : 99.7 7.. 99... % Coeffcents Contaned n 7.8 99. 88.6 99. 94.4 99. Bas.74.7 -.3. -.4.3 MSE.3..... % Rejectons H : 89. 99. 98.7... % Coeffcents Contaned n 4...6.7.6.4 Bas..6....4 MSE..99..98..99 % Rejectons H :. 7.. 99... % Coeffcents Contaned n 94. 99. 9.3 99. 9. 99. Bas. -.8 -.9 -. -..3 MSE.73 3.4.74.9..3 % Rejectons H : 93.6 7.7 99.7 99.3.. % Coeffcents Contaned n 7.9 98.8 88. 99. 94. 98.9 Bas -.8.7 -.9 -...3 MSE.79.8.48... % Rejectons H : 78. 98.8 97.... % Coeffcents Contaned n 39.. 8. 6.7 8.3 3. Bas -.3.3 -. -. -..3 MSE.43 3..66.9..3 % Rejectons H : 99. 7.7. 99.3.. % Coeffcents Contaned n 94.9 99. 9. 99. 9. 98.9 he thrd row shows the percentage of rejectons of the null hypothess H o : for and. he test uses the t-statstc computed as t / se se s the square root of the th dagonal element of the varance-covarance matrx. A nomnal % level of 3
sgnfcance was used so the expected percentage of rejectons whenever the null hypothess s true s %. able 6 Results of Dfferent Estmaton Methods When ~ 3 and he ε Estmaton Method I. II. Bas. -.. -..4. MSE 3.7....3. % Rejectons H : 7.3. 99.... % Coeffcents Contaned n 94.7 78.3 9. 89.4 94.9 98.7 Bas.6....3. MSE.8..... % Rejectons H : 99...... % Coeffcents Contaned n 8..4.6 38..8 36.3 Bas.....4. MSE 3.8....3. % Rejectons H : 7.4. 99.... % Coeffcents Contaned n 9. 98.9 9. 99. 94.9 99. Bas -.3 -. -.78.43.3. MSE 4.8.4.384.3.3. % Rejectons H : 6. 99.4 98. 99.9.. % Coeffcents Contaned n 89.6 77.3 93.7 88.6 9. 98.8 Bas -.8. -.... MSE.78.8.47... % Rejectons H : 9...... % Coeffcents Contaned n 6.9 97. 47.6 97.8 8. 97.6 Bas -.. -...3. MSE 4.33..36..3. % Rejectons H : 6.. 98.6... % Coeffcents Contaned n 94.9 98.9 94.8 99. 9. 99. he percentage of tme a 9% confdence nterval estmate of contaned the true value of the coeffcent s reported n row four. he confdence nterval s computed as ± t se. 4
As a gude to nterpretng the tables let us consder able as an example. he estmators when ε and as follows: he values of bas and MSE for are -.6 and 3. respectvely. he percentages of rejectons of the null hypothess H : for and are 7.8 and 73.3. he percentages of tme a 9% confdence nterval estmate of and are 9. and 99.. As the varaton n the dsturbances ncrease from ε to ε the estmators get worst. Increasng the sample sze wll mae the estmators better. 6. Concludng Remars From ables through 6 several observatons concernng the and estmators for small medum and large samples can be made: - he estmators of the fxed coeffcent perform well when the coeffcent s fxed but ths s not true for the fxed coeffcent n the mxed models of ables and 6. - When coeffcents are random the estmators appear to be unbased for the proof see Delman 989. he problem wth usng when coeffcents are random s not bas n the estmates but n the performance of the hypothess test for sgnfcance of the coeffcents and n the performance of confdence nterval estmators. For example comparng the results for the three estmaton methods n the able 3 the and hypotheses tests for sgnfcance are obvously superor to the test. he test has rejecton rates much hgher than the % level of sgnfcance set for the test. he and rejecton rates are much closer to the nomnal % level. Also note that the enclosure rates for the 9% confdence nterval are very low when the coeffcents are random. 3- he estmator performs well when the coeffcents are random even though the samples are small. From ables and 3 the bas and MSE are dong better n small and large varaton of the parameters. In general the estmator performs best when both coeffcents are random. 4- When one of the coeffcents s fxed and the sample sze s small the estmator wll not perform as well as mght be expected. But f the samples szes are medum or large the estmator performs well. - he and methods perform well when the respectve requred assumptons are met. However both deterorate rapdly when used mproperly. hs suggests the mportance of beng able to choose the assumptons whch are approprate n each partcular stuaton. he test for randomness should prove useful n ths respect. 6- he estmators for the three models fxed and Mxed performs well even n small samples: When coeffcents are fxed able 4 the estmators for the coeffcents are better than the estmators.
7- he estmators for the fxed coeffcent n the mxed models ables and 6 perform well and better than the estmators. By usng method t s not possble to obtan negatve estmates of the coeffcents varances. So we can say that the method s the general estmaton method for fxed and Mxed models. he Monte Carlo smulaton results suggest that the estmators perform well n small samples f the coeffcents are random and but t does not n fxed or Mxed models. But f the samples szes are medum or large the estmators perform well for the three models. Whle estmators perform well n the fxed model only. But the estmators perform well f the coeffcents are random or fxed. So we can say that the method s the general estmaton method for fxed and Mxed models. hs smulaton has been lmted n scope as all smulatons must be. Hopefully t wll shed some lght on performance of several estmaton methods for the panel data models when the regresson coeffcents are random. 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. 43-6. 3. Delman. E. 989 Pooled Cross-Sectonal and me Seres Data Analyss. ew Yor: Marcel Deer. 4. Gendreau B. and Humphrey D. 98 Feedbac Effects n the Maret Regulaton of Ban Leverage: A me-seres and Cross-Secton Analyss. Revew of economc Statstcs Vol. 6 pp. 76-8.. Hsao C. 98 Benefts and Lmtatons of Panel Data. Econometrc Revew Vol. 4 pp. -74. 6. Hsao C. 3 Analyss of Panel Data. th ed. Cambrdge: Cambrdge Unversty Press. 7. Hsao C. and Pesaran M. H. 4 Random Coeffcent Panel Data Models. IEPR Worng Paper 4. Unversty of Southern Calforna. 8. 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. 39-. 9. Pesaran M.H. and R. Smth 99 Estmaton of Long-Run Relatonshps from Dynamc Heterogeneous Panels. Journal of Econometrcs Vol. 68 pp. 79-4.. Rao C. R. and Mtra 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. 3-33.. Swamy P. 97 Statstcal Inference n Random Coeffcent Regresson Models. ew Yor: Sprnger-Verlag. 3. Swamy P. 973 Crtera Constrants and Multcollnearty n Random Coeffcent Regresson Model. Annals of Economc and Socal Measurement Vol. pp. 49-4. 6
4. Swamy P. 974 Lnear Models wth Random Coeffcents. n Fronters n Econometrcs Ed. P. Zaremba. ew Yor: Academc Press Inc. pp. 43-68.. Zellner A. 96 An Effcent Method of Estmatng Seemngly Unrelated Regressons and ests of Aggregaton Bas. J.A.S.A. Vol. 7 pp. 348-368. 6. Zellner A. 963 Estmators for Seemngly Unrelated Regressons Equatons: Some Exact Fnte Sample Results. J.A.S.A. Vol. 8 pp. 977-99. 7