Short T Dynamic Panel Data Models with Individual and Interactive Time E ects

Transcription

1 Short T Dynamc Panel Data Models wth Indvdual and Interactve Tme E ects Kazuhko Hayakawa Hroshma Unversty M. Hashem Pesaran Unversty of Southern Calforna, USA, and Trnty College, Cambrdge L. Vanessa Smth Unversty of York February 8, 8 Abstract Ths paper proposes a quas maxmum lkelhood estmator for short T dynamc xed e ects panel data models allowng for nteractve tme e ects through a mult-factor error structure. The proposed estmator s robust to the heterogenety of the ntal values and common unobserved e ects, whlst at the same tme allowng for standard xed and tme e ects. It s applcable to both statonary and unt root cases. Order condtons for dent caton of the number of nteractve e ects are establshed, and condtons are derved under whch the parameters are almost surely locally dent ed. It s shown that global dent caton s possble only when the model does not contan lagged dependent varables. The QML estmators are shown to be consstent and asymptotcally normally dstrbuted. A sequental multple testng lkelhood rato procedure s also proposed for estmaton of the number of factors whch s shown to be consstent. Fnte sample results obtaned from Monte Carlo smulatons show that the proposed procedure for determnng the number of factors performs very well and the quas ML estmator has small bas and RMSE, and correct emprcal sze even when the number of factors s estmated. An emprcal applcaton, revstng the growth convergence lterature s also provded. JEL Class catons: C, C3, C3 Keywords: short T dynamc panels, unobserved common factors, quas maxmum lkelhood, nteractve tme e ects, multple testng, sequental lkelhood rato tests, output and growth convergence. The authors would lke to thank Vasls Sara ds for helpful comments on a prelmnary verson of the paper. Part of ths paper was wrtten whlst Hayakawa was vstng the Unversty of Cambrdge as a JSPS Postdoctoral Fellow for Research Abroad. He acknowledges nancal support from the JSPS Fellowshp and the Grant-n-Ad for Scent c Research (KAKEHI 7378, 57853) provded by the JSPS. Pesaran and Smth acknowledge nancal support from the ESRC Grant o. ES/I366/.

2 Introducton There now exsts an extensve lterature on the estmaton of lnear dynamc panel data models where the tme dmenson (T ) s short and xed relatve to the cross secton dmenson (), whch s large. Such panels are usually referred to as mcro panels, and often arse n mcroeconometrc applcatons. For example, many emprcal applcatons based on survey data such as the Brtsh Household Panel Surveys (BHPS) and the Panel Study n Income Dynamcs (PSID) are charactersed by data coverng relatvely short tme perods. Short T panels also arse n the cross country emprcal growth lterature where data s typcally averaged over ve to seven years to elmnate the busness cycle e ects. It s now qute common to nclude dynamcs n such studes n addton to ndvdual and tme xed e ects, the former beng partcularly mportant to capture ndvdual characterstcs, and the latter to control for common shocks and the n uence of aggregate trends. Emprcal applcatons of dynamc panel data models wth both ndvdual and tme e ects usng survey data nclude, for example, the studes of Guargla and Ross () and Pror (). In the context of growth emprcs these nclude Islam (995), Casell et al. (996), and Agnger and Falk (5) among others. Although such studes feature ndvdual and tme e ects along wth dynamcs, t s rare to nd studes that allow for error cross secton dependence as well. In many emprcal applcatons tme dummes are used to deal wth cross secton dependence, whch s vald only f the tme e ect s homogeneous over the cross secton unts. Both generalzed method of moments (GMM) and lkelhood approaches have been advanced to estmate such panel data models. See, for example, Anderson and Hsao (98), Arellano and Bond (99), Arellano and Bover (995), Blundell and Bond (998), Hsao et al. (), Bnder et al. (5) and Moral-Bento (3). However, ths lterature assumes that the errors are cross sectonally ndependent, whch mght not hold n many applcatons where cross secton unts are subject to common unobserved e ects, or possbly spatal or network spllover e ects. Ignorng cross secton dependence can have mportant consequences for conventonal estmators of dynamc panels. Phllps and Sul (7) study the mpact of cross secton dependence modelled as a factor structure on the nconsstency of the pooled least squares estmate of a short dynamc panel regresson. Sara ds and Robertson (9) nvestgate the propertes of a number of standard wdely used GMM estmators under cross secton dependence and show that such estmators are nconsstent. In applcatons where the spatal patterns are mportant and can be charactersed by known spatal weght matrces, error cross secton dependence s typcally modelled as spatal autoregressons and estmated jontly wth the other parameters of the dynamc panel data model. See Lee and Yu () for a revew. Such models wth short T are consdered, for example, by Elhorst (5) and Su and Yang (5) for random e ects as well as xed e ects spec catons. In the latter case they apply the rstd erencng operator to elmnate the xed e ects and then use the transformed lkelhood approach of Hsao et al. () to deal wth the ntal value problem. The treatment of the ntal values n spatal dynamc panel data models poses addtonal d cultes and requres further nvestgaton. Jacobs et al. (9) dscuss GMM estmaton of dynamc xed e ect panel data models featurng spatally correlated errors and endogenous nteracton. In addton to the spatal e ects t s also lkely that the error cross secton dependence could be a result of omtted unobserved common factor(s). Ths class of models has been the subject of ntensve research over the recent years and robust estmaton procedures have been advanced n the case of panels where and T are both large. See, for example, Pesaran (6), Ba (9), Pesaran and Tosett (), Chudk et al. (), and Kapetanos et al. (). In contrast, less work has been done so far on the estmaton of short T dynamc panels where error cross secton dependence s due to unobserved common factors, also known as nteractve e ects. An early contrbuton by MaCurdy (98) features panel models wth an error structure that combnes factor schemes wth autoregressve-movng average models estmated by maxmum lkelhood and used to analyse the error process assocated wth

3 the earnngs of prme age males. Further recent related lterature wll be consdered n the next secton. A recent survey of panel data models wth error cross secton dependence and short T can be found n Sara ds and Wansbeek (). Motvated by the practces and requrements of the emprcal lterature, n ths paper we explctly consder ndvdual and tme e ects wthn a dynamc panel data model wth short T; allowng n addton for nteractve e ects. In the analyss of output and growth convergence for example, accountng for nteractve e ects allows to capture the dea that all economes have access, possbly wth d erent degrees, to the same pool of technologcal knowledge (Pesaran, 7). Buldng on the work of Hsao et al. (), we propose an alternatve quas maxmum lkelhood (QML) approach appled to the panel data model after rst-d erencng. In ths way, we account for heterogenety of the ntal values and the common factors n an ntegrated framework. We establsh order condtons for dent caton of the number of nteractve e ects, and derve condtons under whch the parameters are almost surely locally dent ed. Global dent caton s possble only when the model does not contan lagged dependent varables. The QML estmators are shown to be consstent and asymptotcally normally dstrbuted both for statonary and unt root cases. Most mportantly, for the practcal mplementaton of our approach we propose a sequental multple testng lkelhood rato (MTLR) procedure to estmate the number of nteractve e ects, whch delvers a consstent estmator of the true number of factors. The proposed method can be readly extended to a panel VAR framework as n Bnder et al. (5). Monte Carlo smulatons are carred out to nvestgate the nte sample performance of the QML estmator and the MTLR procedure, followed by an applcaton of the approach to growth convergence. The rest of ths paper s organsed as follows. Secton revews the recent related lterature. Secton 3 sets out the dynamc panel data model and ts assumptons. Secton 4 develops the quas lkelhood approach and derves a soluton usng an egenvalue approach. Ident caton of the number of factors and the parameters of the model are dscussed n Secton 5. Secton 6 establshes the consstency of the QML estmator and derves ts asymptotc dstrbuton. Secton 7 presents the sequental MTLR procedure for estmatng the number of factors. Secton 8 descrbes the Monte Carlo experments and provdes nte sample results on the performance of the sequental MTLR estmator for the number of factors, and the proposed QML estmator. An emprcal applcaton to growth convergence s provded n Secton 9. Some concludng remarks are provded n the nal secton. Techncal dervatons are gven n the Appendx. Detals of alternatve GMM estmators used n the Monte Carlo experments together wth addtonal Monte Carlo results are provded n an onlne supplement. otatons: Let w = (w ; w ; :::; w n ) and A = (a j ) be an n vector and an n n matrx, respectvely. Denote the largest egenvalue of A by % max (A), the Eucldean norm of w and the Frobenus norm of A by kwk = n = = w and kak = [T r(a A)] =, respectvely. T s a T vector of ones, T = (; ; :::; ). If fy n g n= s any real sequence and fx ng n= s a sequence of postve real numbers, then y n = O(x n ) f there exsts a postve nte constant C such that jy n j =x n C for all n. y n = o(x n ) f f n =g n! as n!. If fy n g n= and fx ng n= are both postve sequences of real numbers, then y n = (x n ) f there exsts and postve nte constants K and K such that nf n (y n =x n ) K and sup n (y n =x n ) K. Postve, possbly large, xed constants wll be denoted by K (and f needed by K ; K and so on) that could take d erent values n d erent equatons. Small postve constants wll be denoted by. E (:) denotes expectatons taken under the null dstrbuton.! p and a:s:! denote convergence n probablty and almost sure (a.s.) convergence, respectvely.! d denotes convergence n dstrbuton for a xed t and as!. Related Lterature There s a substantal lterature on estmaton of short T dynamc panels. Such models are typcally estmated usng the generalzed method of moments (GMM) appled to the rst-d erenced verson of panel data models. The GMM approach s qute general and has been appled to a varety of dynamc panels. See, for example, Anderson and Hsao (98 and 98), Holtz-Eakn et al. (988),

4 Arellano and Bond (99), Ahn and Schmdt (995), Arellano and Bover (995), and Blundell and Bond (998). However, these papers prmarly focus on models wth ndvdual e ects and when they consder tme e ects ths s done assumng they are homogeneous across the ndvdual unts. Short T dynamc panels wth heterogeneous tme e ects modelled as mult-factor error processes are consdered by Ahn, Lee and Schmdt (,3), and more recently by Ba (3). Ahn et al. () consder a sngle factor error structure and propose a quas-d erencng approach to elmnate the factor, and then apply GMM to consstently estmate the parameters. The quas-d erencng transformaton was orgnally proposed by Chamberlan (984). Holtz-Eakn et al. (988) mplement t n the context of a bvarate panel autoregresson. auges and Thomas (3) follow the same approach n addton to pror rst-d erencng to elmnate the xed e ects, whch they consder separately from the sngle factor error structure they assume for the errors. Ahn et al. (3) extend ther quas-d erencng approach to a multfactor error structure. More recently, Hayakawa () proposes a GMM estmator based on the projecton method whle Robertson and Sara ds (5) propose an nstrumental varable estmaton procedure that ntroduces new parameters to represent the unobserved covarances between the nstruments and the unobserved factors. They show that the resultng estmator s asymptotcally more e cent than the GMM estmator based on quas-d erencng as t explots extra restrctons assumed. See also comments on ths approach by Ahn (5) and Hayakawa (6). As an alternatve to GMM, Ba (3) proposes a quas-maxmum lkelhood approach appled to the orgnal dynamc panel data model wthout d erencng, treatng tme e ects as free parameters, and wthout explctly allowng for ndvdual e ects. To deal wth possble correlatons between the factor loadngs and the regressors Ba follows Mundlak (978) and Chamberlan (98) and spec es lnear relatonshps between the factor loadngs and the regressors to be estmated along wth the other parameters. However, he contnues to assume that all factor loadngs (ncludng the ones assocated wth the ndvdual e ects) are uncorrelated wth the errors. We also use a lkelhood framework, but unlke Ba (3) we allow for unrestrcted ndvdual e ects possbly correlated wth the errors. Our procedure also d ers from the one suggested by Ba (3) snce we apply the maxmum lkelhood estmaton to rst-d erences wth ndvdual e ects elmnated. Our proposed estmaton method can be vewed as a generalzaton of the transformed lkelhood approach of Hsao et al. () where we now allow the errors to have a mult-factor error structure to deal wt the error cross sectonal dependence. In ths way we also deal wth the dependence of the ntal values on the model parameters. Fnally, we propose a sequental multple testng lkelhood procedure to consstently estmate the number of factors whch s not consdered by Ba (3). 3 A dynamc panel data model wth nteractve error components We begn wth the followng standard dynamc panel data model wth tme and xed e ects: y t = y ;t + x t + + t + t ; for t = ; ; :::; T; and = ; ; :::; ; () where x t s a k vector of regressors that vary both across and t, jj, s a k vector of unknown coe cents, wth kk < K, and K denotes a nte postve constant. and t denote a unt spec c xed e ect and, a tme e ect, respectvely. Ths s the standard short T dynamc panel used extensvely n the emprcal lterature assumng that the errors, t, are ndependently dstrbuted across and t (IID error case). In ths paper we extend ths standard model by allowng the errors to have the followng mult-factor structure t = f t + u t ; () where f t s an nteractve e ect wth f t an m vector of unobserved common factors, an m vector of factor loadngs, and u t denotes the remanng dosyncratc error term. The above spec caton Ba (3) refers to models wth mult-factor error structures as panels wth nteractve e ects. In the present paper where T s xed, one could also consder explosve values of, so long as jj < K. 3

5 contans a number of models consdered n the lterature and revewed n Secton above as specal cases. It also provdes a drect generalzaton of Hsao and Tahmscoglu (8) who consder estmaton of () wth IID errors usng the transformed MLE procedure. The model consdered by Ahn et al.(3) allows for errors to have the mult-factor error structure as n (), but does not explctly allow for tme e ects n (). We propose an extenson of the transformed MLE by treatng the unknonw factors as xed parameters to be estmated for each t, but followng Ahn, Lee and Schmdt (,3) assume the factor loadngs to be random and dstrbuted ndependently of the errors, u t and the regressors, x t. We also contrbute to the analyss of dent caton of short T dynamc models wth a multple factor error structure, and derve order condtons for dent caton of m and the parameters of nterest, and. Intally, we develop our proposed estmaton method assumng that m s known, and consder the problem of consstent estmaton of m n Secton 7.. We make the followng assumptons: Assumpton The dosyncratc errors, u t, for = ; ; :::; are dstrbuted ndependently across and over t wth zero means and constant varance,, such that < < < <, and sup ;t E ju t j 4+ < K. Assumpton The tme e ects, t, for t = ; ; :::; T, and the m vector of factors f t, vary across t, so that t 6= and g t = f t 6= at least for some t = ; :::; T; m < T; and sup t kg t k < K. Assumpton 3 The regressors, x t, for = ; ; ::::; are dstrbuted ndependently of u t and, for all ; t, and t, and ther rst-d erences, x t, follow general lnear statonary tme seres processes x t = c x + X j= j" ;t j ; for = ; ; :::; ; (3) where c x and j for j = ; ; ::: are k vector and kk matrces of xed constants such that kc x k < K, and P j= k jk <. Further " t s IID(; I k ), wth sup ;t E k" t k 4+ < K. Assumpton 4 The unt spec c xed e ects,, for = ; ; :::; are allowed to be correlated wth x jt, j, and u jt, for all ; j and t, and could be determnstc and unformly bounded, sup j j < K, or stochastc and unformly bounded, sup E j j < K. Assumpton 5 The unobserved m factor loadngs,, for = ; ; ::::; are dstrbuted ndependently of u jt, and the common factor, f t, for all, j and t; and are ndependently and dentcally dstrbuted across wth zero means, and a nte covarance matrx, namely, s IID(; ); (4) where s an m m symmetrc postve de nte matrx wth k k < K and sup E k k 4+ < K. The above assumptons are standard n the lterature on short T dynamc panels. Assumpton s nnocuous and requres tme e ects and the factors to be tme-varyng. ote that the case where t = and/or f t = f for all t s already covered by the presence of the xed-e ects,. Assumpton 3 requres the regressors to be strctly exogenous wth respect to t. Ths can be relaxed by consderng a vector autoregressve verson of () and () where z t = (y t ; x t ) s modelled jontly as n Holtz-Eakn et al. (988) and Bnder et al. (5). Whle n practce the choce of strctly exogenous varables s typcally drven by economc theory and pror knowledge, tests for strct exogenety are also avalable, see for example Su et al. (6). Regardng possble correlaton between and the regressors x ; ths can be controlled for by usng the methods of Mundlak (978) and Chamberlan (98). Furthermore, whle the composte error term, t ; n () s cross-sectonally heteroskedastc through the presence of 4

6 the nteractve e ects, allowng explctly for the same n the dosyncratc error, u t ; of () can be pursued along the lnes of Hayakawa and Pesaran (5). These authors extend the cross-sectonally ndependent homoskedastc dosyncratc errors of Hsao et al. () to the heterosketastc case. The above extensons are not consdered here as they are beyond the scope of the present focus of the paper. Assumpton 4 permts a very general spec caton of xed e ects, whch s one of the man strengths of the proposed method for emprcal applcatons where lttle s known about the ndvdual e ects. Assumpton 5 s requred for dent caton of the factors and the parameters. Combnng () and (), and elmnatng the ndvdual e ects by rst-d erencng we have y t = y ;t + x t + d t + g t + u t, for t = ; 3; ::::; T ; = ; ; :::; ; (5) where d t = t 6= and g t = f t 6= for some t, and t = g t + u t, for t = ; 3; :::; T: (6) For the spec caton of y we make the followng assumpton about the ntalzaton of (5): Assumpton 6 Suppose that for each, fy t g s started from tme t = S +, for some S >, wth the ntal rst d erences, y ; S+, as random draws from a dstrbuton such that E (y ; S+ jx ) = a S + Sx ; (7) where x = (x ; x ; :::; x T ) s the kt vector of observatons on the regressors, a S s a xed coe cent that allows for non-zero means, and S s the kt vector of coe cents, such that sup S ja S j < K, and sup S k S k < K. Furthermore, let $ = y ; S+ E (y ; S+ jx ) ; (8) and suppose that $ s IID(; $), < $ < K, and sup E j$ j 4+ < K: Ths assumpton s not that restrctve and allows the ntal values, y ; S and y ; S+ to depend on the xed e ects,. Also t s redundant f jj < and S s su cently large, and obvously does not apply f there are no regressors n (). The man restrcton s the assumed lnearty of (7). The followng proposton summarzes the result for y Proposton Under Assumptons, 3 and 6 y = d + x + ; for = ; ; :::; ; (9) where d and are unknown parameters of dmensons and kt, respectvely, and s the composte error de ned by = ~g + v : () The component v s dstrbuted ndependently of x and and sats es for some small >, and a xed K >, and v s IID(;! ); E jv j 4+ < K; () Cov (v ; u t ) = where <! mn <! <! max <, and! mn and! max are xed constants. A proof s provded n Secton A. of the Appendx. for t = for t = 3; 4; :::; T, () 5

7 Remark In the case where jj < and S! we have where s de ned by (), wth v now gven by where = y = d + x + ; v = X j u ; j + ; j= X X j x ; j j x ; j= j= j jx A : Snce x t,, and u t are ndependently dstrbuted for all, t and t, t then follows that v s dstrbuted ndependently of and x, wth E (v ) =, and X V ar (v ) = V j u ; j A + V ar ( ) j= = + + V ar ( ) > : In the case of pure AR() panels, we have the further parametrc restrcton, V ar (v ) = +, whch f mposed can ncrease estmaton e cency. Wrtng (5) and (9) n matrx notaton we now have W = y = W ' + ; = G +r ; (3) where y = (y ; y ; :::; y T ), W s the T (T k + + k + T ) matrx gven by : : : x : : : x y C.. : : :. : : :. x T. y ;T C A ; (4) ' = d ; ; ; wth d = (d ; d ; :::; d T ) ; G = (~g ; g ; :::; g T ), r = (v ; u ; :::; u T ) ; and = ~ ; ; ; T ; and recall that ~ = ~g + v, and t = g t + u t ; for t = ; 3; :::; T. Proposton Consder the composte random varable; t, for t = de ned by (), and for t = ; 3; :::; T de ned by (6). Then under Assumptons,, 3, 5, and 6, the followng moment condtons hold: sup E j t j 4+ < K, for t = ; ; :::; T, (5) and A proof s provded n Secton A. of the Appendx. sup E kx t k 4+ < K. (6) ;t 6

8 4 Quas Maxmum Lkelhood Estmaton Consder the panel data model gven by (3) and note that under Assumpton, and usng () and (), we have E(r r ) = ; (7) where E(r r ) = and = (!). Snce jj = + T (! ) ;! needs to satsfy! > T de nte. Also, snce and r are ndependently dstrbuted we have = (8) C A to ensure that s postve V ar( ) = E( ) = + G G = + QQ = ( ) (9) where Q = (=)G =, rank (Q) = m; and =!; ; vec(q). Wth ths normalsaton, the quas-log-lkelhood of the transformed model (3) s gven by where ` () = ` ('; ) = T ln () ln j ( )j = T ln () T ln( ) ln + QQ X (') ( ) (') () = X (') + QQ (');() = (') = y W '; () and t s assumed that ' does not depend on. For xed m and T, the above log-lkelhood functon depends on a xed number of unknown parameters collected n the [T (m + k + ) + k + 3] vector = ' ;. To obtan the QML estmator, snce s a postve de nte matrx and QQ s rank de cent (recall that by assumpton m < T ), we rst note that + QQ = jj I m +Q Q ; and usng the Woodbury matrx dentty + QQ = Q(I m + Q Q) Q (3) = QA Q ; where A s a non-sngular matrx de ned by A = I m + Q Q: (4) Usng the above results n (), and after some smpl caton the quas-log-lkelhood functon can be wrtten as ` () / T ln( ) ln jj ln jaj T r B T r B QA Q ; (5) 7

9 where jj = + T (! ), and B (') = X = (') ('): (6) If and u t are normally dstrbuted, maxmsng () gves the maxmum lkelhood estmator of. If they are nstead IID wth mean zero and u t has nte fourth moments, maxmsng () gves the QMLE of (Whte 98). Detaled regularty condtons can be found n Secton 6. For analytcal convenence and dent caton purposes, whch wll become clearer below, we further de ne P = = QA =. ote that snce A and are non-sngular matrces, then rank (P) = m, as well. Further, t s easly seen that I m P P= I m A = Q QA = ; and usng Q Q = A I m from (4), we have Smlarly, where A = I m P P: (7) T r B QA Q = T r P C () P ; C () = = B (') = (8) and = (' ;!; ) : Usng the above results, the quas-log-lkelhood functon gven by (5) can now be wrtten as ` (; P) / T ln( ) ln [ + T (! )]+ ln I m P P T r [C ()] T r P C () P : (9) Whle as mentoned earler the transformaton from Q to P s carred out for analytcal convenence, P s stll not dent ed. It s easly seen that the value of ` (; P) s nvarant to the orthonormal transformaton of P. To see ths consder the transformaton ~P = P, where s an mm orthonormal matrx such that = I m. Then t s readly ver ed that ` (; P) = ` ; ~P. Hence, P (or ~P) s dent ed only up to an mm orthonormal rotaton matrx. Let P = (p ; p ; :::; p m ), where p t s the t th column of P, and p t s a T vector of unknown parameters. Snce rank (P) = m, then P P can be dagonalsed by an orthonormal transformaton, and wthout loss of generalty we can mpose the followng m(m )= orthogonalty condtons p tp s =, for all s 6= t = ; ; :::; m: (3) Under these restrctons the quas-log-lkelhood functon, (9), smpl es to ` (; P) / T ln( ) ln [ + T (! )]+ mx t= ln p tp t + mx t= p tc () p t T r [C ()] : (3) Takng rst dervatves wth respect to p t and settng these dervatves to zero now yelds C () ^p t ^p t^p ^p t =, for t = ; ; :::; m; (3) t where ^p t s the quas-maxmum lkelhood estmator of p t (n terms of ). Therefore, ^p t s the egenvector of C () assocated wth the rst m largest non-zero egenvalues of C (), whch we denote by 8

10 () > () > :::: > m () >. ote that C () s a symmetrc postve de nte matrx wth all real egenvalues t () > ; for t = ; ; :::; T. We also have t () = ^p t^p ; and ^p tc () ^p t = t () : t Hence, the concentrated quas-log-lkelhood functon n terms of can be wrtten as ` (;m) / T ln( ) ln [ + T (! )] mx ln [ t ()] + mx TX [ t () ] t () ; t= t= t= (33) where t () s the t th egenvalue of C (), gven by (8). Ths concentrated quas-log-lkelhood functon can now be maxmsed wth respect to = (' ;!; ). The QML estmators, ^ t (), can then be computed usng the QML estmator of and ther correspondng varance covarance matrx can be computed usng the delta method. Wth regard to the computaton of ^p t t s mportant to bear n mnd that standard egenvector routnes provde egenvectors that are typcally orthonormalsed. Whlst n the above analyss, ^p ; ^p ; ::::; ^p m are orthogonal to each other but ther length s not unty and s gven by 5 Ident caton condtons ^p t^p t = t () : (34) We shall rst derve order condtons on m and T under whch the parameters can be dent ed, and then subject to these order condtons derve addtonal condtons under whch the parameters are locally dent ed. We also show why n general t s not possble to establsh global dent caton. Frst we consder the order condton for dent caton n the case of the panel AR() model. ote that usng (5) and (9), the panel AR() model can be wrtten as y t = d t + ~g t + v t ; for t = ; y t y ;t = d t + g t + u t, for t = ; 3; ::::; T; and = ; ; :::;. To nvestgate dent caton t s more convenent to wrte the above model as B () y = d + G +r = d + where y ; d = (d ; :::; d T ) ; and are as de ned above, and B () = C A : (35) ote that, jb ()j =, and and hence B () = T y = a + B () ; C A ; (36) 9

11 where d d a = B. () d =.. d C B C A = d + d B A : T d T T d + T d + :::: + d T + d T (37) Snce d s a T unrestrcted parameter vector, then a s also unrestrcted, namely knowng a does not help dentfy. Therefore, can only be dent ed from the T (T + )= dstnct elements of V ar(y ) whch s gven by V ar(y ) = B() V ar( )B () = B() + QQ B () = (%; Q) : where % = ;!; : But snce Q enters (%; Q) as A = QQ we need to consder the unknown elements of the symmetrc matrx A under d erent rank condtons. Frst t s clear that f A has full rank, namely f rank(a) = T, then % s not dent ed. Hence for dent caton of %; we must have rank (A) = rank (Q) = m < T. When rank (Q) = m, t s dent ed only up to an m m non-sngular transformaton. However, the number of non-redundant parameters of Q s gven by mt m(m )= (see p. 57 of Hayash et al. (7)). Hence, the order condton for dent caton of % and the non-redundant elements of Q s gven by T (T + )= 3 + T m m(m )=: (38) Ths order condton s sats ed f T 3; for m = ; ; ; ::; m max where m max s the largest value of m that sats es (38), that s m max = T. It s easly seen that the above condton s not sats ed f m = T. The maxmzed log-lkelhood values for the rank de cent cases, m = ; ; :::; m max can be computed usng (33). Consder the more general case where the panel AR() model also contans exogenous regressors, and note that the system of equatons (3) can be wrtten equvalently as y = a + ~Z () + B () ; (39) where a; B () and are as de ned above, = ( ; ), ~Z () = B () Z, and Z s the T (T k+k) matrx of observatons on the exogenous regressors de ned by x x Z = B A : (4) x T It s clear from (39) that a and, and hence d and, are unquely dent ed for a gven value of. But t s already establshed that s dent ed from the covarance of B (), gven by (%; Q) = B() ( + QQ ) B (), f the order condton (38) s met. ote that (%; Q) does not depend on d and, and hence knowng d and wll not help dent caton of. As a result, the order condton (38) contnues to be su cent for dent caton of the parameters of the panel ARX() model. To nvestgate necessary and su cent condtons for dent caton of the parameters we consder the average log-lkelhood functon de ned by () whch we reproduce here for convenence, ` () = ` ('; ) = T ln () ln j ( )j X (') ( ) ('); (4) =

12 where = ' ;, ' = ; ; ; d ; =!; ; q, and q refers to the [mt m(m )=] vector contanng the non-reduntant elements of Q. We denote the true values of ' and by ' and, respectvely, and make the followng addtonal assumpton: Assumpton 7 () = ' wth =! q ; where s a compact subset of R n wth n = T (k + ) + k T m m(m )=; and = (' ; ) les n the nteror of ; () ( ) = ( + QQ ) s postve de nte for all values of ; and () A ( ) = X = W ( ) W a:s:! A ( ) ; as! ; (4) where A ( ) s postve de nte for all values of. The rst part of ths assumpton s standard and rules out parameter values on the boundary of the parameter space. It also allows for the order condton, (38), n the settng of n. The second part of the assumpton also holds when the order condton s met and! > T. Recall that under the latter s a postve de nte matrx and Q s rank de cent, and under Assumpton < < < <. For we need to dstngush between the case where S s xed (namely ntalzaton s from a nte past) and when S!. Under the former t s only requred that jj < K and ncludes the unt root case (jj = ). Under the latter (when S! ), we must have jj <. The almost sure convergence condton of (4) holds f sup E W ( ) W < ( ) sup E kw k 4. ote that under condton () of Assumpton 7, ( ) < K, and sup E kw k 4 < K s establshed by Lemma. Fnally, the condton that A ( ) s a postve de nte matrx s needed for dent caton of '. Denotng ` ('; ) by ` ('; ) and usng (4) t s shown n Lemma 3 that (see (A.4) and (A.5)) ` (' ; ) ` ('; ) a:s:! lm E ` (' ; ) ` ('; ) ; (43)! and lm! E ` (' ; ) ` ('; ) = ( ; )+(' ' ) A ( ) (' ' )+ ( ) ( ; ); (44) where h ( ; ) = T r ( ) ( ) log (j ( )j = j ( )j) T; wth A ( ) and ( ; ) de ned by (4) and (A.7), respectvely. Denote the egenvalues of ( ) and ( ) by t and t (t = ; ; :::; T ); respectvely (note that t > and t > ) and wrte ( ; ) as ( ; ) = TX [( t = t ) log ( t = t ) ] : t= Also note that ( t = t ) log ( t = t ) wth the equalty holdng f and only f t = t, for all t, or equvalently f and only f =. 3 Therefore, ( ; ), wth equalty holdng f and only f =. Furthermore, snce A ( ) s a postve de nte matrx, then (' ' ) A ( ) (' ' ) mn [A ( )] (' ' ) (' ' ) ; where mn [A ( )] >. It s clear that the rst two terms of (44) can not be negatve, but the same s not true of the thrd term, ( ) ( ; ), and therefore, global dent caton of can not be guaranteed. To nvestgate the possblty of local dent caton we ntroduce the followng de nton: 3 ote that for any x >, log (x) x. Here x = t= t >.

13 De nton Consder the set ( ) n the open neghbourhood of de ned by ( ) = f ; j j < g ; for some small >, and de ne the subset = \ ( ): Consder now the almost sure probablty lmt of ` (' ; ) ` ('; ) on the set ; for some. We now establsh that there exsts > for whch ths lmt can be zero f and only f =. To see ths consder the rst and the thrd terms of (44) together, and note that ( ; )+ ( ) ( ; ) = f =. In such a case lm E ` (' ; ) ` ('; )! mn [A ( )] (' ' ) (' ' ) ; and ` (' ; ) ` ('; ) a:s:!, f and only f mn [A ( )] (' ' ) (' ' ) =, whch mples ' = ', as requred snce by assumpton mn [A ( )] >. Consder now the case where 6=, and note that ( ; ) >, and j( ; )j >, and therefore on we have j( ) ( ; )j j( )j j( ; )j < j( ; )j : Also note that under Assumptons, and 5, k ( )k < K for all ; and t s readly seen that j( ; )j < K. Hence, on there must exst >, such that ( ; ) + ( ) ( ; ), and hence lm E ` (' ; ) ` ('; )! mn [A ( )] (' ' ) (' ' ) : Once agan snce by assumpton mn [A ( )] > for all values of, then on there exsts > such that ' = ', and hence =, almost surely. Ths result s summarsed n the followng proposton. Proposton 3 Consder the model gven by () and (), wth the assocated log-lkelhood functon for rst-d erences gven by (). Suppose that Assumptons -7 and the order condton (38) hold. Then s almost surely locally dent ed for values of su cently close to, as formalzed by de nton. Remark It readly follows from the above analyss that n the absence of lagged dependent varables n (), s almost surely globally dent ed. 6 Consstency and asymptotc normalty The analyss of consstency and asymptotc normalty of the QML estmator, ^ = arg max ` (), now follows by applcaton of standard results from the statstcs lterature. Almost sure local consstency of ^ follows, for example, from a straghtforward adaptaton of Theorem 9.3. of Davdson () to our problem where the parameters are only locally dent ed. 4 Under Assumptons -7, t follows that (), beng a sub-set of, s compact, () s an nteror pont of and () C () = ` () a:s:! C (), and (see (43) and (44)) C () = ( ; ) + (' ' ) A ( ) (' ' ) + ( ) ( ; ) + C ( ) ; where the term C ( ) does not depend on, and (v) s the unque mnmum of C () on. The last result follows drectly from the analyss of dent caton n the prevous secton. Therefore, all a:s: condtons of Theorem 9.3. of Davdson are sats ed and ^! on the set. 4 See, also Amemya (985).

14 To establsh asymptotc normalty of ^, by applcaton of the mean value theorem to ` (), we rst note that ` () ` ( ) = ( ) s ( ) ( ) H ( ) ( ) ; (45) where s () ` () =@, H () ` () =@@, and les between and. Frst note that by result (A.8) of Lemma 4, and combnng (43) and (44) we have: s ( ) a:s:! ; ` ( ) ` () a:s:! ( ; ) + (' ' ) A ( ) (' ' ) + ( ) ( ; ): Hence, n vew of (45) we must also have ( ) H () ( ) a:s:! ( ; ) + (' ' ) A ( ) (' ' ) + ( ) ( ; ): But t has been already establshed that on the rght hand sde of the above can be equal to zero f and only f =, and hence we must also have H ( ) a:s:! H( ), where H( ) must be a postve de nte matrx gven by H( ) = lm! ` () =@@. ow applyng the mean value theorem to s () we have = p s (^) = p s ( ) H () p (^ ) where les between ^ and. Then p hp (^ ) = H ( ) s ( ) ; and snce ^ s locally consstent on, then p hp (^ ) asy s H ( ) s ( ) ow usng result (A.9) of Lemma 4, we have p s ( )! d [; J ( )], where J ( ) s gven by (A.3). Hence p (^ )! d (; V ) ; where V has the famlar sandwch form V = H ( )J ( ) H ( ): In the specal case where (' ) s Gaussan, then as usual V reduces to H ( ). A consstent estmator of V can be obtaned by substtutng ^ for n the expressons for J ( ) and H( ). The above results are summarsed n the followng theorem: Theorem Consder the dynamc panel data model gven by () wth nteractve e ects as n (). Suppose that Assumptons to 7, and the order condton (38) hold. Denote the QML estmator of by ^ = arg max ` (), where ` () s gven by (4). Then ^ s almost surely locally consstent for, for values of su cently close to, as formalzed by de nton, and p (^ )! d ; H ( )J ( ) H ( ) ; where H( ) = lm! ` () =@@, and J ( ) s de ned by (A.3) n Lemma 4. 3

15 7 Estmatng the number of factors There are a number of studes that provde nformaton type crtera for selectng the number of factors ncludng Ba and g (), Onatsk (9), Kapetanos (), Ahn and Horensten (3), among others. However, these are not applcable to short T panel data sets, and requre both and T to be large. In the case of short T panels Ahn et al. (3) estmate the true number of factors, m ; wthn a GMM framework usng standard nformaton crtera n a sequental manner. To ensure consstency of the selected number of factors, followng Bauer et al. (988) and Cragg and Donald (997), Ahn et, al. (3) choose the sgn cance level b such that b! and ln(b )=! as!. Usng smulatons they nd that the sequental method could produce better estmates f the sgn cance level depends also on T (n addton to ), when the regressors and ndvdual e ects are not hghly correlated, but do not provde theoretcal detals on how best to allow for T as well as n ther selecton procedure. In what follows we consder a sequental lkelhood rato (LR) testng procedure, but adjust the crtcal values of the tests to take account of the multple testng nature of the procedure n terms of T, as well as adjustng the crtcal values of the tests n terms of to ensure consstency of the selected number of factors. We provde a formal theory that should be of general nterest for the analyss of short T factor models. 7. A sequental multple testng lkelhood rato procedure for estmatng the number of factors Our sequental multple testng lkelhood rato (MTLR) procedure makes use of the lkelhood rato statstc and n e ect nvolves sequentally performng a number of lkelhood rato tests of the overdentfyng restrctons on the model de ned by (3). To see ths, from (38) t follows that the degree of freedom (DF) for the test s gven by DF = T (T + )= (3 + T m m(m )=); (46) and depends on m and T. When m = m max = T, DF = and therefore the panel data model s exactly dent ed, and there are no free parameters (restrctons) to test. The LR tests nvolvng overdentfyng restrctons are de ned by tests of m = f; ; ; ::; T 3g aganst m max = T. Let ^ m be the QML estmator of, assumng m unobserved common factors, usng the concentrated log-lkelhood functon gven by (33) n terms of m and = (' ;!; ) ; whch we reproduce here for convenence, makng the dependence of on m explct: ` ( m ;m) / T ln( m) ln [ + T (! m )] mx ln [ t ( m )]+ t= mx [ t ( m ) ] j= TX t ( m ) ; t= where ( m ) > ( m ) > :::: > T ( m ) > are the egenvalues of C ( m ) = m = B (' m )m =. 5 Then the LR statstcs for testng H : m = m aganst H : m = m max, for m = f; ; ; ::; T 3g and m max = T > m ; are gven by LR (m max ; m ) = h ` ^ ;m mmax max ` ^ m ;m ; (47) where ^ m = arg max m ` ( m ;m). Under the assumpton that n (3) s Gaussan, and the panel data model s correctly spec ed wth m = m, then usng standard asymptotc results we have LR (m max ; m )! d DF, as! for a xed T, where DF s gven by (46) for the relevant choces of m = m max and m. The followng sequental testng procedure can now be adopted to estmate m : ^m =, f a test based on LR (m max = T ; m = ) s not rejected. 5 Recall that B (' m ) s de ned by (6), and hence C ( m ) s a postve de nte matrx. 4

16 ^m =, f a test based on LR (m max = T ; m = ) s rejected, AD a test based on LR (m max = T ; m = ) s not rejected. ^m =, f a test based on LR (m max = T ; m = ) and LR (m max = T ; m = ) are both rejected AD a test based on LR (m max = T ; m = ) s not rejected. Ths sequental procedure s contnued untl m = T 3. Snce T separate tests are carred out, to control the overall sze of the sequental testng procedure we need to adjust the sze of the underlyng ndvdual tests. As the true number of factors, m ; s unknown and could be T, n what follows we assume the sequental procedure nvolves T separate tests, although n some applcatons we mght end up stoppng the sequental procedure havng carred out a fewer number of tests than T. Let the null hypotheses of nterest be H T ; ; H T ; ; :::; H T ;T 3 ; and wrte the T LR tests as Pr (LR (m max = T ; m = t ) > CV ;T ;t jh T ;t ) p ;T ;t ; for t = ; ; :::; T ; where CV ;T ;t s the crtcal value for the test of H T ;t, and p ;T ;t s the realzed p-value for H T ;t. The overall sze of the test s now gven by the famly-wse error rate (FWER) de ned by h F W ER = Pr [ T t= (LR (m max = T ; m = t ) > CV ;T ;t jh T ;t ) : Suppose that we wsh to control F W ER to le below a pre-determned value,. An exact soluton to ths problem depends on the nature of the dependence across the underlyng tests, whch s generally d cult to obtan. But one could derve bounds on F W ER usng, for example, the Bonferron (936) or Holm (979) procedures. Both of these procedures are vald for all possble degrees of dependence across the ndvdual tests, and as a result tend to be conservatve n the sense that the actual sze wll be lower than the overall target sze of. In the case of the Bonferron procedure, snce the separate LR tests are mutually exclusve usng the unon probablty rule we have h Pr [ t= T (LR (m max = T ; m = t ) > CV ;T ;t jh T ;t ) TX Pr (LR (m max = T ; m = t ) > CV ;T ;t jh T ;t ) t= T X p ;T ;t : t= Hence, to obtan F W ER, t s su cent to set p ;T ;t =(T ). The ndvdual crtcal values, CV ;T ;t are based on the asymptotc crtcal values (as! ) of the dstrbuton, namely DF [= (T )], where =(T ) s the rght-tal probablty of the ndvdual tests. The above sequental MTLR procedure ensures that lm! F W ER, but ths by tself does not guarantee that m, the true value of m, wll be estmated consstently. Ths s a well known problem n the sequental testng lterature. To acheve consstency we need to allow to declne wth at a sutable rate as wll be shown n what follows. Under non-normal errors the LR statstc, de ned by (47), need not be ch-squared dstrbuted. Ths follows from known results for the lkelhood rato statstc under msspec caton. See, for example, Foutz and Srvastava (977) who show that under msspec caton the LR statstc behaves asymptotcally as a lnear combnaton of ndependent ch-squared varates. Ths s also n lne wth results n Satorra and Bentler (994) and Yuan and Bentler (7) for standard factor models. Followng ths lterature we conjecture that under non-gaussan errors the null dstrbuton of LR (m max ; m ) can also be asymptotcally approxmated as a lnear combnaton of ndependent ch-squared varates. Smulaton results reported n the onlne supplement con rm that LR (m max ; m ) s overszed when 5

17 usng ch-square crtcal values n ths case. However, even n the case of non-normal errors, the above sequental procedure usng crtcal values of the ch-square dstrbuton can stll consstently estmate the true number of factors as shown n the followng proposton and assocated theorem. Proposton P 4 Suppose under the null hypothess H, the LR test statstc, LR s dstrbuted as k = w (), where the weghts w w ::: w k >, are nte constants, and (), for = ; ; :::; k, are ndependently dstrbuted central ch-squared varates wth degree of freedom. Further suppose that under the alternatve hypothess H ; LR s dstrbuted as P k = w (; ; ) where (; ; ), for = ; ; :::; k, are ndependently dstrbuted non-central ch-squared varates wth degree of freedom and non-centralty parameter, ;, = ; ; :::; k. Denote the non-centralty parameter of the test under H by = P k = ;. Suppose k s a nte nteger, and = O(). Denote type I and II errors of the test by and, respectvely, and the crtcal value of the test by c (k). Under Assumptons -7 f c (k)! and! as! such that c (k) =! ; then both and! : For a proof see Secton A. of the Appendx. Remark 3 The standard ch-squared test s ncluded n the above proposton as a specal case by settng w =, for all. Remark 4 Clearly, the condtons of Proposton 4 are met f = p=f(), where f() =, wth a nte non-zero constant. Further, usng (A.4) from the proof of Proposton n the Appendx we have c (k) mn ln k w k ln k p ln() = = O ; (48) and snce by assumpton = O() t follows that c (k)=! as requred. Remark 5 When s set as = p=, the parameter p ( < p < ) can be vewed as the nomnal sze of the test. Then! f ln =!, whch s sats ed n the standard case where = O(). The eyman-pearson case s obtaned f we set =. The case of > relates to the Cherno test procedure that ams at mnmzng Pr(H ) + Pr(H ), where < Pr(H ) < and < Pr(H ) < are pror probabltes of H and H, respectvely. When s nte the soluton to ths problem depends on the pror probabltes. But n the case of ch-squared tests, we have Pr(H ) + Pr(H )! as!, rrespectve of the pror probabltes Pr(H ) and Pr(H ), so long as = p= for > and p >. Remark 6 In nte samples the choce of p and can matter, though for moderate values of the choce of p s lkely to be of second order mportance. In the smulaton results that follow we set = and p = 5%. Theorem Let ^m be the number of factors obtaned usng the sequental lkelhood rato procedure based on the statstc LR (m max ; m ) gven by (47) for whch Proposton 4 holds. Then P ( ^m = m )! : For a proof see Secton A. of the Appendx. From Proposton 4 and Theorem t follows that ^m obtaned usng the sequental MTLR procedure descrbed above s a consstent estmator of the true number of factors m. In lne wth the above dscusson n the ensung Monte Carlo results when performng the sequental MTLR procedure we use = p ; where s some postve constant such that condton (48) holds approxmately. (T ) 6

18 8 Monte Carlo desgn and results In ths secton, we nvestgate the nte sample propertes of the proposed estmator usng Monte Carlo (MC) smulatons. We begn by presentng the MC desgns that we shall be employng for the pure AR() panels and dynamc panels wth regressors. 8. Monte Carlo desgn 8.. The AR() model The observatons on y t are generated assumng m unobserved factors as y t = + t + y ;t + t ; for = ; ; :::; ; t = S + ; S + ; ::; ; ; :::; T; t = f t + u t ; wth the dosyncratc errors generated as u t IID (; ) under Gaussan errors, and u t IID p ( 6 6) under non-gaussan errors, where 6 s a ch-square varate wth sx degrees of freedom. The factor loadngs are generated as In the statonary where jj <, we start the process wth ` IID (; `); ` = ; ; :::; m: (49) y ; S = S + X j ; j ; and set S = 5 to reduce the mpact of the ntal values on the sample perod used n the analyss, whch we take to be t = ; ; :::; T. After rst-d erencng we end up wth T observatons that are used n estmaton. The unobserved common factors, f`t, are generated as q f`t = f`f`;t + f`"f`t, " f`t IID (; ), for ` = ; ; :::; m; t = S + ; :::; ; ; ; ::; T; (5) wth f` = :9, and wthout loss of generalty we set f`; S =. The resultant f`t values are re-scaled such that T P T t= f `t =, for all `. Spec cally we mpose the followng normalsatons on the common factors T P T t= f`t = ; T P T t= f `t =, and T P T t= f`tf`t =, for ` 6= `: (5) We generate the tme e ects as t = t t whch are further normalsed so that j= T P T t= t =, T P T t= t =, and T P T t= tf`t =, for all `. (5) The xed e ects,, are generated as = b u + b v ; where u = T P T t= u t, v IID (; ). b and b are xed constants to be set later. Ths set up ensures that the xed e ects are correlated wth the dosyncratc errors when b 6=. The values of the remanng parameters are set as =, ` = =m; for all `: Fnally, as shown n the Secton A.3 of the Appendx, the average t of the panel AR() model s determned by and does not depend on = V ar(u t ), and hence we set =. For the key parameter of the model,, we consder a medum and a hgh value, namely = :4 and :8; and consder the followng combnatons of sample szes, T = f5; g and = f; 3; 5g. We report smulaton results for the autoregressve parameter. Spec cally, we report the bas and root mean square error (RMSE). In addton, we present sze and power estmates. Power s presented for = f:3; :7g for the null values of = f:4; :8g. All tests are carred out at the 5% sgn cance level and all experments are replcated, tmes, unless otherwse stated. 7

19 8.. The ARX() model The observatons on y t for the panel ARX() model are generated assumng k = (one exogenous regressor) and m unobserved factors as y t = + t + y ;t + x t + t ; for = ; ; :::; ; t = S + ; S + ; ::; ; ; :::; T; t = f t + u t ; (53) wth the dosyncratc errors and the factor loadngs generated as n the prevous secton. The values of and ` are set below to ensure a certan degree of average t for the panel regresson n (53). We set y ; S = S + X SX j x ; j + j ; j ; j= wth S = 5 and dscard the rst 5 observatons, usng the observatons t = through T for estmaton, endng up wth T observatons for estmaton. The regressor, x t, s generated as j= x t = + # f t + x t ; ; x t = x x ;t + p x" t ; (54) wth x ; S = for t = S + ; :::; ; ; :::; T, where j x j < ; IID (; ), # = (# ; # ; :::; # m ) ; and " t IID (; ): We set x = :8. The factor loadngs, # ; n the x t process are generated as #` IID ( #`; #`); for ` = ; ; :::; m: (55) The tme e ects and unobserved common factors are generated as n the AR() case. The xed e ects,, are generated as = b x + b u + b v ; where x = T P T t= x t, u = T P T t= u t, v IID (; ) and b, b, b are xed constants to be de ned later. Ths set up ensures that the xed e ects are correlated both wth the regressors and the dosyncratc errors when b 6= and b 6=. We calbrate the rest of the parameters to ensure a gven average measure of t, as de ned by the average R derved n Secton A.3 of the Appendx. In ths way we ensure that the t of the underlyng model does not change wth m, the number of factors. Usng (53) and (54) we have h V ar(x t ) + Ry + P m `= P = c` + = V ar(x t ) + + P m `= P : = c` + where c` = #` + `. Also, n vew of (54) we have V ar(x t ) =. Hence + Ry + P = c + = + + P = c + ; where c = P m `= c`. But P = c = P m `= = P m `= + P m `= P P = c` = P m `= P = (#` + ` ) + P m `= P = ` = #` P = `#` ; and for su cently large and notng that #` and ` are generated ndependently, we have (see (49) and (55)) P = #`! p V ar (#` ) + [E (#` )] = #` + #`; 8

20 P = `! p V ar (` ) + [E (` )] = `; and snce ` and #` are ndependently dstrbuted and E (` ) =, we also have P = `#`! p : Hence P = c! p P m `= #` + P #` + m`= `: Usng the above results and settng = we obtan R y = + + P m `= #` + P #` + m`= ` + : We control the value of R y to be the same for all values of m. To ths end, the value of the remanng parameters are set as =, ` = #` = =m; #` = = p m, for all `; and we obtan R y = ( ) +8, from whch t follows that = R y 8 Ry : (56) For m =, = ( Ry)=5 Ry. We consder = f:4; :8g and set such that Ry = :8 for all values of m. We consder the same combnatons of T and as n the AR() case, namely T = f5; g and = f; 3; 5g and report smulaton results for the same set of statstcs, for both and, ncludng sze and power. Power s presented for = f:38, :78g and = :98 for the null values of = f:4; :8g and =. As prevously, all tests are carred out at the 5% sgn cance level and all experments are replcated, tmes, unless otherwse stated. The standard errors used for nference are based on the same formulas as those used n the AR() case wth all dervatves computed numercally. 8. Monte Carlo results We begn by reportng on the performance of the sequental multple testng LR (MTLR) procedure for selectng the true number of factors. We consder the performance of the QML estmator when the number of factors s estmated usng the MTLR procedure as well as when the number of factors s set to ts true value, m. For ths set of experments the xed e ects are allowed to be correlated wth the errors, and wth the regressors n the panel ARX case. In the above Monte Carlo desgns ths corresponds to settng b = b =, wth the addtonal b parameter set to for the ARX() model. We conclude ths secton by presentng results for the QML estmator together wth the GMM quasd erence (QD) and rst-d erence (FD) estmators of ALS, when the number of factors s assumed to be known. In ths set of experments the xed e ects are not correlated wth the errors, as ths would render the GMM estmators nconsstent. Ths corresponds to settng b = and b = ; wth the addtonal b parameter set to for the ARX() model. However, xed e ects are allowed to be correlated wth the regressors n the case of the ARX() desgn. 8.. Performance of the sequental multple testng lkelhood rato procedure Tables and provde results on the performance of the sequental MTLR procedure for the AR() and the ARX() models, respectvely. Spec cally they report the number of tmes, n percent, that the estmated number of factors, ^m; based on the sequental MTLR procedure outlned n Secton 7. s equal to the true number of factors m : The sequental MTLR procedure s mplemented usng the 9