On the testing of heterogeneity effects in dynamic unbalanced panel data models

Transcription

1 Economcs Letters 58 (1998) On the testng of heterogenety effects n dynamc unbalanced panel data models Serg Jmenez-Martn* Unversdad Carlos III de Madrd epartment of Economcs, Av. Madrd, 16, 8903 Getafe, Madrd, Span Receved 4 June 1997; accepted 1 October 1997 Abstract Ths artcle analyses the behavour of the Holtz-Ean test for the presence of ndvdual heterogenety effects n dynamc small-t unbalanced panel data models n the presence of endogenous but predetermned regressors. The test behaves correctly for a moderate autoregressve coeffcent. However, when ths coeffcent approaches unty, the presence of an addtonal regressor sharply affects both the power and the sze of test Elsever Scence S.A. Keywords: Monte Carlo smulaton; Panel data; Sargan testng JEL classfcaton: C3 1. Introducton Several authors (see, e.g. Maddala, 1993, and references theren) have consdered the estmaton of lnear dynamc small-t panel data models wth predetermned regressors n the presence of ndvdual heterogenety effects that, f present, are correlated wth the lagged endogenous and, often, wth most 1 of the predetermned varables. Usually, those effects are elmnated by frst dfferencng the model. However, ths method mples that we gve up an mportant share of the sample for estmaton. In partcular, those unts wth T 5 do not contrbute to the dentfcaton of the parameters of the model. Holtz-Ean (1988) (HE) proposed a very smple Sargan-dfference test for the purely frst-order autoregressve model. He exploted the fact that under the null hypothess of no effects, both the levels and the frst dfferences equatons are vald for dentfyng the parameters. Under the alternatve of sgnfcant effects, only the estmators based upon the frst dfferences equatons reman consstent. The test evaluates the addtonal orthogonalty restrctons mpled by the null hypothess relatve to *Tel.: ; fax: ; e-mal: jmenez@eco.uc3m.es 1 Naturally, there are many other possbltes (see Matyas and Sevestre, 1993, for a revew). Note that the use of devatons from means s napproprate n dynamc models (see Ncell, 1981). The re-nterpretaton of the Holtz-Ean test as a Sargan (1988) dfference test s due to Arellano (1993). Among other testng possbltes, we menton the Hausman-type test of Keane and Runle (199) / 98/ $ Elsever Scence S.A. All rghts reserved. PII S (97)0078-4

2 158 S. Jmenez-Martn / Economcs Letters 58 (1998) the alternatve hypothess by computng the correspondng Sargan overdentfyng restrctons test (Arellano, 1993), whch has an asymptotc ch-square dstrbuton. Our am s to evaluate the behavour of the HE test, usng Monte Carlo methods, when, n addton to the lagged dependent varable, there are both endogenous and tme-nvarant regressors. Ths case, whch often appears n emprcal wor, has usually been consdered as a straghtforward extenson of the autoregressve model. However, the presence of endogenous regressors may have strong consequences. We carry out the smulatons on the test n an unbalanced panel structure because unbalanced panels exhbt, for a gven number of cross-secton unts, worse testng performance than balanced ones. We mae use of a flexble estmaton procedure that can be helpful for practtoners. The remander of ths artcle s organzed as follows. In Secton the model and the test are presented. A Monte Carlo smulaton s dscussed n Secton 3. Fnally, Secton 4 concludes.. The model and testng procedure Consder the followng dynamc model wth both endogenous but predetermned and tme-nvarant regressors: y 5 ay 1 b9x 1 t9r 1h 1 e, 5 1,...,N;t5,...,T;T #T, (1) t t1 t t where yt s the model varable; xt s a K 3 1 vector of explanatory varables contemporary correlated wth the error term, e ; R s a vector of tme-nvarant regressors; d 5 (abt) are the vectors of t coeffcents; and h s an ndvdual heterogenety effect. Summarsng, the basc assumpton of our model s: E(etuy t1,...,y 1,x t1,...,x 1,h,R) 50. We want to test H 0: var(h ) 5 0 aganst H a: var(h ) ± 0. In the latter case, we have that he(h y ) ± 0; ;tj and we may have that he(h x ) ± 0; t1 t ;t, for some j. To obtan consstent estmates under H, we frst dfference (1) to elmnate the a heterogenety effects: yt 5wtd* 1u t, 51,...,N;t53,...,T;T #T, () where wt 5 (yt1x t), d *9 5 (ab ) and ut 5e t. We estmate d * usng a generalsed method of moments based on the moment condtons he(uty s)50, E(utx s)50; t. s 1 1, t 5 3,...,T,;j. Under the null, the levels equatons also produce consstent estmates of the parameters n (1). In such a case, we can combne both the levels and frst dfferences equatons and mae use of the correspondng orthogonalty condtons to dentfy the parameters. Ths procedure, whch s nown as 3 dflev, has two advantages: there s an mplct effcency gan (as more nformaton s used) and t allows for straghtforward testng procedures. Thus, consder the followng system of staced equatons: Y5 Wd 1 U, 5 1,...,N, (3) where Y5 (y9 y 9)9, W5((w 0)9w 9)9 and U5 (u9u)9. Then, a consstent estmate of d s obtaned by GMM-IV, usng the followng matrx of nstruments: 3 The name s taen from the 1991 verson of the ynamc Panel ata programme by Arellano and Bond (1988).

3 F G L Z 0 0 Z 5 L*, 0 Z R S. Jmenez-Martn / Economcs Letters 58 (1998) where the (T )x( 1 1)(T )(T 1)/ matrx Z contans the restrctons vald for (), the (T 1)x( 1 1) matrx Z L* contans the non-redundant restrctons mpled by the levels equatons he(uty t1)50, E(utx t1)50; t 5,...,T, ;j and, 0 denotes matrces of zeros of approprate ˆ L dmenson. The effcent dflev estmator, d, s gven by: S ˆ d 5 OW9Z A OZ 9W OW9Z A OZ 9Y, (4) L L L L 1 L L L N N L 1 L L 1 where A 5 (N o Z 9(Y W 9 d )(Y W 9 N d )9Z ) s a frst-stage estmate of the optmal weghtng matrx based on a consstent but neffcent frst-stage estmate of d. The followng Sargan test of overdentfyng restrctons evaluates the adequacy of the proposal (Arellano, 1993): S S S 5 OUˆ 9Z A OZ 9U ˆ x, (5) L L L L N q where Uˆ s the two-stage vector of resduals for a gven ndvdual, and q s the number of overdentfyng restrctons. Ths statstc must be compared wth the correspondng statstc under the alternatve hypothess, whch s derved from estmates of (). However, the computaton of those estmates s cumbersome because not all the parameters are dentfed under the alternatve hypothess. The easest way to avod ths problem s the followng. Use the frst-stage resduals for frst dfferences equatons n (3) to construct the weghtng matrx, B N. Usng ths matrx, compute the second-step estmates and the correspondng resduals, u ˆ. Havng both, obtan the test of overdentfyng restrctons, S, whch follows an asymptotc ch-square dstrbuton wth r degrees of freedom. Fnally, construct the followng Sargan-dfference test (or pseudo lelhood rato test): L S 5S S x qr, (6) where q r s the column dmenson of Z L*. 3. The Monte Carlo experment and results Consder the extended verson of Eq. (1), whch descrbes the experment: K yt 5 ayt1 1O51bxt 1 wt 1 l j 1h 1 e t, ut 5h 1 e t, 5 xt 5mxt 1gxhh1gxeet1n t, 51,...,5, () where w s a tme effect and l s a tme-nvarant ndustry effect (not dentfed under the H estmaton method). The dstrbutonal assumptons and the parameter values are: t j a Effects: h N(0,s ), w,l N(0,1), h t j Errors: e N(0,s ), n N(0,), 5 1,...,00, t 5 1,...,15, j 5 1,...,5, () t e t

4 160 S. Jmenez-Martn / Economcs Letters 58 (1998) Parameters: b 5 1 ;, g 5 0.0, g xh xe The ntal condtons of the model are y0 5 0 and x 5 0. Note that, for example, the mplct 1 correlaton of x wth ether e or h s 0.19 and 0.38, respectvely. In general, the correlaton between x and y ncreases wth m, gxh and gxe and decreases wth. Four groups of 50 unts each are consdered. For each unt, 15 tme seres observatons are drawn. However, the ntal nne observatons are dscarded. In formng an unbalanced panel, r 1 addtonal observatons are dscarded for groups r 5 1,...,4. The sample used has the followng structure: r,n,t l;(1,50,6l,,50,5l,3,50,4l,4,50,3l). Consequently, the ntal sample s formed by 900 r r 4 observatons. We have developed a program (avalable on request) devoted to solvng the more complcated parts of the test. Ths program uses the 1991 verson of the P program of Arellano and Bond (1988). Four cases are reported. Two sze experments: S1 (s 5 0, s 5 1) and S (s 5 0, s 5 ), and h e h e two power experments: P1 (s 5 1, s 5 1) and P (s 5 1, s 5 ). For each experment, three h e h e parameters have been vared: a (0.5 and 0.75), m (0 and 0.5) and, fnally, (from0to5). Table 1 summarses the emprcal rejecton frequency (ERF) at nomnal sze of Fg. 1 plots the trade-off between the sze and the power for each combnaton of a and m ( 5 0, 1 and 5). 1 Table 1 Emprcal rejecton frequency Sze a at the 5% nomnal sze. 500 replcatons Power g ± 0, s 5 1 g ± 0, s 5 g ± 0, g xh± 0 g ± 0, g xh± 0 sh 50, s 5 1 sh 5 0, s 5 sh 5 1, s 5 1 sh 5 1, s 5 S1 S P1 P m a 50.5 a a 50.5 a a 50.5 a a 50.5 a a The emprcal rejecton frequency (ERF) s defned as: 500 l ERF5O 1( pr(x q.s ),0.05)/500, r l51 l where S s the statstc of the test and 1(?) s an ndcator functon. The number of degrees of freedom n any partcular cell s q r 5 5*( 1 1), where s the number of predetermned varables, apart from the lagged dependent. 4 When a dynamc model s estmated n levels, T years of data yeld T 1 observatons enterng the estmaton process. Lewse, when the model s estmated n frst dfferences, T years of data yeld T observatons.

5 S. Jmenez-Martn / Economcs Letters 58 (1998) Fg. 1.

6 16 S. Jmenez-Martn / Economcs Letters 58 (1998) Several comments are n order. In S1, the ERF concdes, as expected, wth the nomnal sze of the test. By contrast, n S, partcularly when m 5 0.5, the ERF overestmates the nomnal sze (see Fg. 1(6) and Fg. 1(8)). However, we note that the problem detected s less mportant when all the coeffcents are, smultaneously, dentfed both under the null and the alternatve hypotheses. In experments on the power of the test, the ERF ncreases when ncreasng from 0 to and decreases thereafter (because the number of orthogonalty restrctons to be tested grows very rapdly). It also decreases when a or s ncrease. e Consequently, three mportant features have been detected. Frst, when the null s true and s e ncreases, we run nto a problem of poor nstruments, as far as the correlaton among the explanatory varables and ther nstruments decreases. Second, when the null s false and s e /sh ncreases, the two estmators tend to behave smlarly. Ths s because heterogenety effects tend to be relatvely less and less mportant. And thrd, when a approaches unty, the presence of a moderate number of endogenous regressors ncreases the power of the test (see Fg. 1(11) Fg. 1(1) Fg. 1(16)). Even when m 1, a power gan stll remans (e.g., n P1, when 5 1, a and m , we found an ERF above 0.75.) It s also mportant to note that, as expected, the power of the test s even hgher when the varance of the specfc effects ncreases (they are easer to detect), when the sample sze ncreases, when the data set s balanced (for a gven number of cross-secton unts) and when the vector X s strctly exogenous. (All these results are not reported but are avalable upon request.) 4. Concludng remars In ths artcle, a test for the presence of ndvdual heterogenety effects n lnear dynamc small-t panel data models wth both endogenous, but predetermned, and tme-nvarant regressors has been evaluated. The test s easy to mplement and computatonally smpler than other alternatves, such as the Hausman-type test of Keane and Runle (199). We use the dflev estmator descrbed by Holtz-Ean (1988) and Arellano (1993) to estmate the model under the null hypothess. We obtan that the test lacs power when the rato s e /sh ncreases and/ or the coeffcent of the lagged endogenous varable tends to unty. However, we fnd that the presence of addtonal regressors sharply ncreases the power of the test. Fnally, we detect that the presence of nformatve (m ± 0) endogenous regressors greatly affects the sze of the test. Acnowledgements I am very grateful to J.J. olado, J.M. Labeaga and Franco Peracch for several comments and suggestons. All remanng errors are my own responsblty. Fnancal support from GES PB95-09 and PB s acnowledged References Arellano, M., On the testng of correlated effects wth panel data. Journal of Econometrcs 59, Arellano, M., Bond, S., ynamc panel data estmaton usng P: A gude for users. IFS WP 88/15, London.

7 S. Jmenez-Martn / Economcs Letters 58 (1998) Holtz-Ean,., Testng for ndvduals effects n autoregressve models. Journal of Econometrcs 39, Keane, M.P., Runle,.E., 199. On the estmaton of panel-data models wth seral correlaton when nstruments are not strctly exogenous. Journal of Busness and Economcs Statstcs 10, 1 9. Maddala, G.S. (Ed.), The Econometrcs of Panel ata, Vols. 1 and. Cambrdge Unversty Press, Cambrdge. Matyas, L., Sevestre, P., The Econometrcs of Panel ata: Handboo of Theory and Applcatons. Kubler Academc. Ncell, S., Bases n dynamc models wth fxed effects. Econometrca 49, Sargan, J.., Testng for msspecfcaton after estmatng usng nstrumental varables In: Maasoum, E. (Ed.), Contrbutons to Econometrcs: John enns Sargan, Vol. 1. Cambrdge Unversty Press, Cambrdge.