LM-type tests for slope homogenety n panel data models Jörg Bretung Unversty of Cologne Chrstoph Rolng Deutsche Bundesbank azar Salsh BGSE, Unversty of Bonn. July 4, 206 Abstract Ths paper employs the Lagrange Multpler LM prncple to test parameter homogenety across cross-secton unts n panel data models. The test can be seen as a generalzaton of the Breusch-Pagan test aganst random ndvdual effects to all regresson coeffcents. Whle the orgnal test procedure assumes a lkelhood framework under normalty, several useful varants of the LM test are presented to allow for non-normalty, heteroskedastcty and serally correlated errors. Moreover, the tests can be convenently computed va smple artfcal regressons. We derve the lmtng dstrbuton of the LM test and show that f the errors are not normally dstrbuted, the orgnal LM test s asymptotcally vald f the number of tme perods tends to nfnty. A smple modfcaton of the score statstc yelds an LM test that s robust to non-normalty f the number of tme perods s fxed. Further adjustments provde versons of the LM test that are robust to heteroskedastcty and seral correlaton. We compare the local power of our tests and the statstc proposed by Pesaran and Yamagata 2008. The results of the Monte Carlo experments suggest that the LM-type test can be substantally more powerful, n partcular, when the number of tme perods s small. JEL classfcaton: C2; C33 Keywords: panel data model; random coeffcents; LM test; heterogenous coeffcents We are grateful to the edtor Mchael Jansson and anonymous referee for helpful and constructve comments on an earler verson of ths paper. Correspondng author: Unversty of Cologne, Insttute of Econometrcs, 50923 Cologne, Germany. Emal: bretung@statstk.un-koeln.de, chrstoph.rolng@bundesbank.de, salsh@un-bonn.de. The vews expressed n ths paper do not necessarly reflect the vews of Deutsche Bundesbank or ts staff.
Introducton In classcal panel data analyss t s assumed that unobserved heterogenety s captured by ndvdual-specfc constants, whether they are assumed to be fxed or random. In many applcatons, however, t cannot be ruled out that slope coeffcents are also ndvdualspecfc. For nstance, heterogenous preferences among ndvduals may result n ndvdualspecfc prce or ncome elastctes. Ignorng ths form of heterogenety may result n based estmaton and nference. Therefore, t s mportant to test the assumpton of slope homogenety before applyng standard panel data technques such as the least-squares dummy-varable LSDV estmator for the fxed effect panel data model. If there s evdence for ndvdual-specfc slope parameters, economsts are nterested n estmatng a populaton average lke the mean of the ndvdual-specfc coeffcents. Pesaran and Smth 995 advocate mean group estmaton, where n a frst step the model s estmated separately for each cross-secton unt. In a second step, the untspecfc estmates are averaged to obtan an estmator for the populaton mean of the parameters. Alternatvely, Swamy 970 proposes a generalzed least squares GLS estmator for the random coeffcents model, whch assumes that the ndvdual regresson coeffcents are randomly dstrbuted around a common mean. In ths paper we derve a test for slope homogenety by employng the LM prncple wthn a random coeffcents framework, whch allows us to formulate the null hypothess of slope homogenety n terms of K restrctons on the varance parameters. Hence, the LM approach substantally reduces the number of restrctons to be tested compared to the set of K lnear restrctons on the coeffcents mpled by the test proposed by Pesaran and Yamagata 2008, henceforth referred to as PY. Ths does not mean, however, that our test s confned to detect random devaton n the coeffcents. In fact our test s optmal aganst the alternatve of random coeffcents but t s also powerful aganst any systematc varatons of the regresson coeffcents. Our approach s related but not dentcal to the condtonal LM test recently suggested by Juhl and Lugovskyy 204 whch s referred to as the JL test. The man dfference s that the latter test s derved for a more restrctve alternatve, where t s assumed that the ndvdual-specfc slope coeffcents attached to the K regressors have dentcal varances. In contrast, our test focuses on the alternatve that the coeffcents have dfferent varances whch allows us to test for heterogenety n a subset of the regresson coeffcents. Furthermore, the dervaton of our test follows the orgnal LM prncple nvolvng the nformaton matrx, whereas the JL test employs the outer product of the scores as an estmator of the nformaton matrx. Our smulaton study suggest that both non-standard features of the latter test may result n sze dstortons n small samples and a szable loss n power. An mportant advantage of the JL test s however that t s robust aganst non-gaussan and heteroskedastc errors. We therefore propose varants of the
orgnal LM test that share the robustness aganst non-gaussan and heteroskedastc errors. Furthermore, we also suggest a modfed LM test that s robust to serally correlated errors. Another contrbuton of the paper s the analyss of the local power of the test that allows us to compare the power propertes of the LM and PY tests. Specfcally, we fnd that the locaton parameter of the LM test depends on the cross-secton dsperson of the regresson varances, whereas the locaton parameter of the PY test only depends on the mean of the regressor varances. Thus, f the regressor varances dffer across the panel groups, the gan n power from usng the LM test may be substantal. The outlne of the paper s as follows. In Secton 2 we compare two tests for slope heterogenety recently proposed n the lterature. We ntroduce the random coeffcents model n 3 and lay out the standard assumptons for analyzng the large-sample propertes. In Secton 4 we derve the LM statstc and establsh ts asymptotc dstrbuton. Secton 5 dscusses several varants of the proposed test. Frst, we relax the normalty assumpton and extend the result of the prevous secton to ths more general settng. Second, we propose a regresson-based verson of the LM test. Secton 6 nvestgates the local asymptotc power of the LM test. Secton 7 descrbes the desgn of our Monte Carlo experments and dscusses the results. Secton 8 concludes. 2 Exstng tests To prepare the theoretcal dscusson n the followng sectons, we brefly revew the random coeffcents model and exstng tests. Followng Swamy 970, consder a lnear panel data model y t = x tβ + ɛ t, for =, 2,...,, and t =, 2,..., T, where y t s the dependent varable for unt at tme perod t, x t s a K vector of explanatory varables and ɛ t s an dosyncratc error wth zero mean and varance E ɛ 2 t = σ 2. For the slope coeffcent β we assume β = β + v, where β s a fxed K vector and v s a..d. random vector wth zero mean and K K covarance matrx Σ v. For more detals and extensons of the basc random coeffcent model see Hsao and Pesaran 2008. As ponted out by a referee, ths specfcaton may be replaced by some systematc varaton of the coeffcents that depends on observed varables. For example, we may specfy the devatons as β β = Γz +η, where z s some vector of observed varables possbly correlated wth x t. The correspondng varant of the LM test whch s dfferent from our LM test based assumng that v and x t are ndependent wll be optmal aganst ths partcular form of systematc varaton. In general, our test assumng ndependent varaton wth Γ = 0 wll also have power aganst systematc varatons but admttedly our test s not 2
The null hypothess of slope homogenety s β = β 2 = = β = β, 2 whch s equvalent to testng Σ v = 0. To test hypothess 2, Swamy suggests the statstc Ŝ = = β β X X WLS β s β WLS, 2 wth X = x,..., x T and β = X X X y s the ordnary least squares OLS estmator of for panel unt, and t =,..., T. The common slope parameter β s estmated by the weghted least-squares estmator β WLS = = X X s 2 where s 2 denotes the standard OLS estmator of σ 2. = X y s 2 Intutvely, f the regresson coeffcents are dentcal, the dfferences between the ndvdual estmators and the pooled estmator should be small. Therefore, Swamy s test rejects the null hypothess of homogenous slopes for large values of ths statstc, whch possesses a lmtng χ 2 dstrbuton wth K degrees of freedom as s fxed and T. Pesaran and Yamagata 2008 emphasze that n many emprcal applcatons s large relatve to T and the approxmaton by a χ 2 dstrbuton s unrelable. PY adapt the test to a settng n whch and T jontly tend to nfnty. In partcular, they assume ndvdual-specfc ntercepts and derve a test for the hypothess β = = β = β n, y t = α + x tβ + ɛ t. 3 The analogue of the pooled weghted least squares estmator above elmnates the unobserved fxed effects, β WFE = = X M ι X σ 2 = X M ι y σ 2, where M ι = I T ι T ι T /T, and ι T s a T vector of ones. A natural estmator for σ 2 y X β Mι y X β s σ 2 = T K, optmal aganst alternatve wth systematcally varyng coeffcents. 3
where β = X M ι X X M ι y and the test statstc becomes Ŝ = = β β X M ι X WFE β σ β WFE. 2 Employng a jont lmt theory for and T, PY obtan the lmtng dstrbuton as = Ŝ K 2K d 0,, 4 provded that, T and /T 0. Thus, by approprately centerng and standardzng the test statstc, nference can be carred out by resortng to the standard normal dstrbuton, provded the tme dmenson s suffcently large relatve to the crosssecton dmenson. PY propose several modfed versons of ths test, whch for brevty we shall refer to as the tests or statstcs. In partcular, to mprove the small sample propertes of the test, PY suggest the adjusted statstc under normally dstrbuted errors see Remark 2 n PY, adj = S K T +, 5 2K T K where S s computed as Ŝ but replacng σ2 σ 2 = by the varance estmator y X βfe Mι y X βfe T, 6 where β FE = X M ι X X M ι y s the standard fxed effects wthn-group = = estmator. ote that ths asymptotc framework does not seem to be well suted for typcal panel data applcatons where s large relatve to T. Therefore, t wll be of nterest to derve a test statstc that s vald when T s small say T = 0 and s very large say = 000, whch, for nstance, s encountered n mcroeconomc panels. scores The test statstc proposed by Juhl and Lugovskyy 204 s based on the ndvdual S = û M ι X X M ι û σ 2 trx M ι X, where û = y X βfe and tr A denotes the trace of the matrx A. The condtonal LM statstc results as CLM = S = S S S. 7 = = 4
It s nterestng to compare ths test statstc to the PY test whch s based on the sum Ŝ = = Ŝ wth Ŝ = β β X M ι X WFE β σ β WFE 2 = σ 2 u M ι X X M ι X X M ι u + o p f and T tend to nfnty. ote that lm EŜ = K. The man dfference between the JL and the PY statstcs s that the statstc S neglects the addtonal nverse σ 2 X M ι X n the statstc Ŝ. Thus, although these two test statstcs are derved from dfferent statstcal prncples, the fnal test statstcs are essentally testng the ndependence of u and M ι X or Eu M ι X W X M ι u = σ 2 Etr [M ι X W X M ι ] wth W = I K for the JL test and W = σ 2 X M ι X for the PY test. 3 Model and Assumptons Consder a lnear panel data model wth random coeffcents, y = X β + ɛ, 8 β = β + v, 9 for =, 2,...,, where y s a s a T vector of observatons on the dependent varable for cross-secton unt, and X s a T K matrx of possbly stochastc regressors. To smplfy the exposton we assume a balanced panel wth the same number of observaton n each panel unt see also Remark of Lemma. The vector of random coeffcents s decomposed nto a common non-stochastc vector β and a vector of ndvdual-specfc dsturbances v. Let X = [X, X 2,..., X ]. In order to construct the LM test statstc for slope homogenety we start wth model 8-9 under stylzed assumptons. However, n Secton 5 these assumptons wll be relaxed to accommodate more general and emprcally relevant setups. assumptons are mposed on the errors and the regressor matrx: The followng Assumpton The error vectors are dstrbuted as ɛ X d 0, σ 2 I T and v X d 0, Σ v, where Σ v = dag σ 2 v,,..., σ 2 v,k. The errors ɛ and v j are ndependent from each other for all and j. Assumpton 2 For the regressors we assume E x t,k 4+δ < C < for some δ > 0, for all =, 2...,, t =, 2,..., T and k =, 2..., K. The lmtng matrx lm E X X exsts and s postve defnte for all and T. 5
In Assumpton, the random components of the slope parameters are allowed to have dfferent varances but we assume that there s no correlaton among the elements of v. ote that ths framework s more general than the one consdered by Juhl and Lugovskyy 204 who assume Ev v = σ 2 vi K. The latter assumpton seems less appealng f there are szable dfferences n the magntudes of the coeffcents. Furthermore, the power of the test depends on the scalng of regressors, whereas the local power of our test s nvarant to a rescalng of the regressors see Theorem 5. The alternatve hypothess can be further generalzed by allowng for a correlaton among the elements of the error vector v. However, ths would ncrease the dmenson of the null hypothess to KK+/2 restrctons and t s therefore not clear whether accountng for the covarances helps to ncrease the power of the test. Obvously, f all varances are zero, then the covarances are zero as well. 2 Let u = X v + ɛ. Stackng observatons wth respect to yelds y = Xβ + u, 0 where y = y,..., y and u = u,..., u. The T T covarance matrx of u s gven by Ω E [uu X] = X Σ v X + σ 2 I T 0... 0 X Σ v X + σ2 I T The hypothess of fxed homogeneous slope coeffcents, β = β for all, corresponds to testng aganst the alternatve H 0 : σ 2 v,k = 0, for k =,..., K, H :. K σv,k 2 > 0, k= that s, under the alternatve at least one of the varance parameters s larger than zero. 4 The LM Test for Slope Homogenety Let θ = σ 2 v,,..., σ 2 v,k, σ2. Under Assumpton the correspondng log-lkelhood functon results as l β, θ = T 2 log2π 2 log Ω θ 2 y Xβ Ω θ y Xβ. 2 2 We also conducted Monte Carlo smulatons allowng for non-zero dagonal elements n the matrx Σ v. We found that the results are qute smlar to the settng where Σ v s dagonal. 6
The restrcted ML estmator of β under the null hypothess concdes wth the pooled OLS estmator β = X X X y and the correspondng resdual vector and estmated resdual varance are denoted by ũ = y X β and σ 2. The followng lemma presents the score and the nformaton matrx derved from the log-lkelhood functon n 2. Lemma The score vector evaluated under the null hypothess s gven by S l θ = H0 2 σ 4 = = ũ X X ũ σ 2 X X ũ X K X K where X k s the k-th column of X for k =, 2,..., K.. ũ σ 2 X K The nformaton matrx evaluated under the null hypothess s 0 X K, 3 [ ] I σ 2 2 l E θ θ H0 = 2 σ 4 = = X X X 2 X 2 2 = = X X 2..... 2 X K X = = X K X K X K X K 2 X X 2 X 2 X 2 2 X K X K X X X K X K T., 4 where X k denotes the k-th column of the T K matrx X, k =, 2,..., K and =,...,. Remark It s straghtforward to extend Lemma to unbalanced panel data, where observatons are assumed to be mssng at random. Let X be a T K matrx and ũ be a conformable T vector. The score vector s gven by S = 2 σ 4 = = ũ X X ũ σ 2 X X ũ X K X K. ũ σ 2 X K 0 X K, 7
where σ 2 = T = = The nformaton matrx s computed accordngly. ũ ũ. Remark 2 If ndvdual-specfc constants α are ncluded n the regresson, then a condtonal verson of the test s avalable cf. Juhl and Lugovskyy 204. The ndvdual effects can be condtoned out by consderng the transformed regresson M ι y = M ι X β + M ι u, 5 wth M ι as defned n Secton 2. The typcal elements of the correspondng score vector result as ũ σ M 4 ι X j X j M ι û σ 2 X j M ι X j, j =,..., K, where ũ = M ι y M ι X β and β s the pooled OLS estmator of the transformed model 5, and σ 2 s the correspondng estmated resdual varance. It follows that we just have to replace the vector X j by the mean-adjusted vector M ι X j n Theorem. Remark 3 It s easy to see that under the more restrctve alternatve Ev v = σ 2 vi K of Juhl and Lugovskyy 204, where σ 2 v, = = σ 2 v,k = σ2 v, the score s smply the sum of all elements of S. Remark 4 otce also that the LM-type statstcs do not requre the restrcton K < T, whch s mportant for the PY approach. Ths s of course not an ssue for the asymptotc framework, where T, however, t can be a substantve restrcton n many emprcal applcatons when T s small. In the followng theorem t s shown that when T s fxed, the LM statstc possesses a χ 2 lmtng null dstrbuton wth K degrees of freedom as. Theorem Under Assumptons, 2 and the null hypothess LM = S I σ 2 S = s Ṽ s d χ 2 K, 6 as and T s fxed, where s s defned as the K vector wth typcal element s k = 2 σ 4 T 2 ũ t x t,k 2 σ 2 = t= = T x 2 t,k, 7 t= 8
and the k, l element of the matrx Ṽ s gven by Remark 5 Ṽ k,l = 2 σ 4 T = t= x t,k x t,l 2 T = T t= x 2 t,k = T t,l x 2. 8 If T s fxed, normalty of the regresson dsturbances s requred. If we relax the normalty assumpton, an addtonal term enters the varance of the score vector and the nformaton matrx becomes an nconsstent estmator. Theorem 2 dscusses ths ssue n more detals and derves the asymptotc dstrbuton of the LM test f the errors are not normally dstrbuted. Remark 6 It may be of nterest to restrct attenton to a subset of coeffcents. For example, n the classcal panel data model t s assumed that the constants are ndvdualspecfc and, therefore, the respectve parameters are not ncluded n the null hypothess. Another possblty s that a subset of coeffcents s assumed to be constant across all panel unts. To account for such specfcatons the model s parttoned as y t = β X a t + β 2X b t + β 3X c t + u t. t= The K vector X a t ncludes all regressors that are assumed to have ndvdual-specfc coeffcents stacked n the vector β. The K 2 vector X b t comprses all regressors that are supposed to have homogenous coeffcents. The null hypothess s that the coeffcent vector β 3 attached to the K 3 vector of regressors X c t s dentcal for all panel unts, that s, β 3 = β 3 for all, where β 3 = β 3 + v 3. The null hypothess mples Σ v3 = 0. Let X a 0 0 X b X c 0 X Z = 2 a 0 X2 b X c 2....., 0 0 X a Xb Xc where X a = [X a,..., X a T ] and the matrces X b and X c are defned accordngly. The resduals are obtaned as ũ = I ZZ Z Z y and the columns of the matrx X c are used to compute the LM statstc. Some cauton s requred f a set of ndvdualspecfc coeffcents are ncluded n the panel regresson snce n ths case the ML estmator σ 2 = T = T t= ũ2 t s nconsstent for fxed T and. Ths mples that the expectaton of the score vector 3 s dfferent from zero. Accordngly, the unbased estmator σ 2 = T K K 2 K 3 T ũ 2 t 9 = t= 9
must be employed. As a specal case, assume that the constant s ncluded n X c, whereas all other regressors are ncluded n the matrx X b, and X a s dropped. Ths case s equvalent to the test for random ndvdual effects as suggested by Breusch and Pagan 980. The LM statstc then reduces to LM = T [ ũ I ι T ι T ũ ] 2, 2 T ũ ũ where ι T s a T vector of ones, whch s dentcal to the famlar LM statstc for random ndvdual effects. 5 Varants of the LM Test In ths secton we generalzed the LM test statstc by allowng for non-normally dstrbuted, heteroskedastc and serally dependent errors. Frst we show n Secton 5 that the proposed LM test s robust aganst non-normally dstrbuted errors once we assume, T jontly and specfc restrctons on the exstence of hgher-order moments. Moreover, the varants of the test wth non-normally dstrbuted errors are proposed for the settngs when and T s fxed. Second, n Secton 5.2 we propose a varant of the LM test that s robust to heteroskedastc errors. Fnally, Secton 5.3 dscusses how to robustfy the LM test, when the errors are serally correlated. 5. The LM statstc under non-normalty In ths secton we consder useful varants of the orgnal LM statstc under the assumpton that the errors are not normally dstrbuted. Therefore, we replace Assumptons and 2 by: Assumpton ɛ t s ndependently and dentcally dstrbuted wth Eɛ t X = 0, Eɛ 2 t X = σ 2 and E ɛ t 6 X < C < for all and t. Furthermore, ɛ t and ɛ js are ndependently dstrbuted for j and t s. Assumpton 2 For the regressors we assume E x t,k 6 < C < for some δ > 0, for all T =, 2...,, t =, 2,..., T and k =, 2..., K. Further, lm E [x t x t] tend to a postve defnte matrx Q and the lmtng matrx Q := exsts and s postve defnte. T T t= lm T,T = t= T E [x t x t] Assumpton 3 The error vector v s ndependently and dentcally dstrbuted wth Ev X = 0, Ev v X = Σ v, where Σ v = dag σv,, 2..., σv,k 2 and E vk 2+δ X < C < for some δ > 0, for all and k =,..., K. Further, v and ɛ j are ndependent from each other for all and j. 0
otce that, as n Secton 3 under the null hypothess Σ v or v = 0 for all. Hence, Assumpton 3 s not requred for the dervaton of the asymptotc null dstrbuton. To study the behavour of the LM test statstc under local alternatves, Assumpton 3 wll be used n Secton 6. Wth these modfcatons of the prevous setup, the lmtng dstrbuton of the LM statstc s gven n Theorem 2 Under Assumptons, 2 and the null hypothess, LM d χ 2 K, 20 as, T jontly. Generalzng the model to allow for non-normally dstrbuted errors ntroduces a new term nto the varance of the score: the k, l element of the covarance matrx now becomes see equaton 49 n appendx A.2 V k,l + µ 4 u 3σ 4 2σ 4 2 = T x 2 t,k T t= = T t= x 2 t,k x 2 t,l T = T t= x 2 t,l, 2 where µ 4 u denotes the fourth moment of the error dstrbuton, and V k,l s as n 8 wth σ 4 replaced by σ 4. The addtonal term depends on the excess kurtoss µ 4 u 3σ 4. Clearly, for normally dstrbuted errors, ths term dsappears, but t devates from zero n the more general setup. Under Assumptons and 2, the frst term V k,l s of order T 2, whle the new component s of order T, such that, when the approprate scalng underlyng the LM statstc s adopted, t vanshes as T. Therefore, the LM statstc as presented n the prevous secton contnues to be χ 2 K dstrbuted asymptotcally. By ncorporatng a sutable estmator of the second term n 2, however, a test statstc becomes avalable that s vald n a framework wth non-normally dstrbuted errors as, whether T s fxed or T. Therefore, denote the adjusted LM statstc by LM adj = s Ṽ adj s, where Ṽadj s as n 2 wth V k,l, σ 4 and µ 4 u defned n 8, σ 4 and µ 4 u replaced by the consstent estmators Ṽk,l = T T = t= ũ4 t for k, l =,..., K. As a consequence of Theorem 2 and the precedng dscusson, we obtan the followng result. Corollary Under Assumptons, 2 and the null hypothess LM adj d χ 2 K,
as and T s fxed. Furthermore, LM adj LM p 0, as, T jontly. As mentoned above, once the regresson dsturbances are no longer normally dstrbuted, the fourth moments of the error dstrbuton enter the varance of the score. It s nsghtful to dentfy exactly whch terms gve rse to ths new form of the covarance matrx. Accordng to Lemma, the contrbuton of the -th panel unt to the k-th element of the score vector s ũ X k X k ũ σ 2 X k X k = T x 2 ũ2 t,k t σ 2 + t= T ũ t ũ s x t,k x s,k. 22 t= s t The varance of the frst term on the rght hand sde depends on the fourth moments of the errors. Snce the contrbuton of ths term vanshes f T gets large, t can be dropped wthout any severe effect on the power whenever T s suffcently large. Hence, we consder a modfed score vector as presented n the followng theorem. Theorem 3 Under Assumptons, 2 and the null hypothess, the modfed LM statstc LM = s Ṽ s d χ 2 K, as and T fxed, where s s K vector wth contrbutons for panel unt s,k = σ 4 T t ũ t ũ s x t,k x s,k, 23 t=2 s= for =,...,, k =,..., K, and the k, l element of Ṽ s gven by for k, l =,..., K. Ṽ k,l = σ 4 = T t x t,k x t,l x s,k x s,l, 24 t=2 s= Remark 7 It s mportant to note that ths verson of the LM test s nvald f the panel regresson allows for ndvdual-specfc coeffcents cf. Remark 3. Consder for example the regresson y t = α + x tβ + u t 25 2
where α are fxed ndvdual effects and we are nterested n testng H 0 : varβ = 0. The resduals are obtaned as ũ t = y t y x t x β = ut u x t x β β. It follows that n ths case Eũ t ũ s x t,k x s,k 0 and, therefore, the modfed scores 23 result n a based test. To sdestep ths dffculty, orthogonal devatons e.g. Arellano and Bover 995 can be employed to elmnate the ndvdual-specfc constants yeldng yt = β x t + u t t = 2, 3,..., T, [ t wth yt = y t t ] y s, t t where x t and u t are defned analogously. It s well known that f u t s..d. so s u t. It follows that the modfed LM statstc can be constructed by usng the OLS resduals ũ t nstead of ũ t. Ths approach can be generalzed to arbtrary ndvdual-specfc regressors x a t. Let X a = [x a,..., x a T ] denote the ndvdual-specfc T K regressor matrx n the regresson s= y = X a β + X b β 2 + X c β 3 + u, 26 see Remark 3. Furthermore, let M a = I T X a X a X a X a, and let M a denote the T K T matrx that results from elmnatng the last K rows from M a such that M a M a s of full rank. The model 26 s transformed as y = X b β 2 + X c β 3 + u, 27 where y = Ξ a y and Ξ a = M a a M /2 a M. It s not dffcult to see that Eu u = σ 2 I T K and, thus, the modfed scores 23 can be constructed by usng the resduals of 27, where the tme seres dmenson reduces to T K. ote that orthogonal devatons result from lettng X a be a vector of ones. To revew the results of ths secton, the mportant new feature n the model wthout assumng normalty s that the fourth moments of the errors enter the varance of the score. The nformaton matrx of the orgnal LM test derved under normalty does not ncorporate hgher order moments, but the test remans applcable as T. To apply the LM test n the orgnal framework when T s fxed and errors are no longer normal we can proceed n two ways. A drect adjustment of the nformaton matrx to account for 3
hgher order moments yelds a vald test. Alternatvely, we can adjust the score tself and restrct attenton to that part of the score that does not ntroduce hgher order moments nto the varance. In the next secton, we further pursue the second route of dealng wth non-normalty and thereby robustfy the test aganst heteroskedastcty and seral correlaton. 5.2 The regresson-based LM statstc In ths secton we offer a convenent way to compute the proposed LM statstc va a smple artfcal regresson. Moreover, the regresson-based form of the LM test s shown to be robust aganst heteroskedastc errors. Followng the decomposton of the score contrbuton n 22 and the dscusson thereafter, we construct the Outer Product of Gradents OPG varant of the LM test based on the second term n 22. Rewrtng the correspondng elements of the score contrbutons of panel unt as s,k = T t ũ t ũ s x t,k x s,k, 28 t=2 s= for k =,..., K. ote that we dropped the factor / σ 4 as ths factor cancels out n the fnal test statstc. Ths gves the usual LM-OPG varant LM opg = = s = s s = s, 29 where s = [ s,,..., s,k]. An asymptotcally equvalent form of the LM-OPG statstc can be formulated as a Wald-type test for the null hypothess ϕ = 0 n the auxlary regresson where ũ t = K z t,k ϕ k + e t, for =,...,, t =,..., T 30 k= t z t,k = x t,k ũ s x s,k for k =,..., K. Therefore, wth the Ecker-Whte heteroskedastcty-consstent varance s= estmator, the regresson based test statstc results as LM reg = = T ũ t z t t=2 = T t=2 ũ 2 t z t z t = T ũ t z t, 3 It follows from the arguments smlar as n Theorem 3 that M reg test statstc s asymptotcally χ 2 dstrbuted but t turns out to be robust aganst heteroskedastcty: t=2 4
Corollary 2 Under Assumpton but allowng for heteroscedastc errors such that E[ɛ 2 t X] = σ 2 t < C <, Assumpton 2 and the null hypothess LM reg d χ 2 K, 32 as and T s fxed. It s mportant to note that the LM-OPG varant cannot be appled to resduals from a fxed effect regresson, see Remark 7. Furthermore, the replacement of the resduals by orthogonal forward devaton wll not fx ths problem snce orthogonal forward devatons are no longer serally uncorrelated f the errors are heteroskedastc. Therefore, a verson of the test s requred that s robust aganst autocorrelated errors. 5.3 The LM statstc under serally dependent errors In ths secton we propose a varant of the LM test statstc that accommodates serally correlated errors, that s, we relax Assumptons as follows: Assumpton The T error vector ɛ s ndependently and dentcally dstrbuted wth Eɛ X = 0, Eɛ ɛ X = Eɛ ɛ = Σ and E [ ɛ t 4+δ X ] < C < for some δ > 0 and all and t. The T T matrx Σ s postve defnte wth typcal element σ ts for t, s =,..., T. ote that Assumpton allows for heteroscedastcty and seral dependence across tme, however, t restrcts the error vector ɛ to be d across ndvduals. Under ths assumpton the expectaton of the score vector 23 s under the null hypothess E[u t u s x t,k, x s,k ] = σ ts E[x t,k x s,k ]. We therefore suggest a modfcaton for autocorrelated errors based on the adjusted K score vector s wth typcal element s k s,k = = = s,k for k =,..., K and T t ũ t ũ s σ ts x t,k x s,k, 33 t=2 s= where σ ts = = ũtũ s. The asymptotc propertes of the LM statstc based on the modfed score vector are presented n Theorem 4 Let LM ac = s Ṽ s, 5
where Ṽ s a K K matrx wth typcal element and Ṽ k,l = = T t=2 T t t δ tsτq x t,k x s,k x τ,l x q,l 34 τ=2 s= q= δtsτq = ũ jt ũ js ũ jτ ũ jq σ ts σ τq. j= Under Assumptons, 2, the null hypothess 2 and as wth T fxed the LM ac statstc has a χ 2 K lmtng dstrbuton. ote that ths verson of the test has a good sze control rrespectve of seral dependence n errors. However, the test nvolves some power loss relatve to the orgnal test statstcs when errors are serally uncorrelated, whch s not surprsng gven a more general setup of ths varant of the test. The respectve asymptotc power results are analyzed n the next secton see Remark 0. Secton 7 elaborates n detal on the sze-power propertes of the LM ac n fnte samples. 6 Local Power The am of ths secton s twofold. Frst, we nvestgate the dstrbutons of the LM-type test under sutable sequences of local alternatves. Two cases are of nterest, wth T fxed and, T jontly, whch are presented n the respectve theorems below. Second, we adopt the results of PY to our model n order to compare the local asymptotc power of the two tests. To formulate an approprate sequence of local alternatves, we specfy the random coeffcents n 9 n a setup n whch T s fxed. The error term v s as n Assumpton wth elements of Σ v gven by σ 2 v,k = c k, 35 where c k > 0 are fxed constants for k =,..., K. The asymptotc dstrbuton of the LM statstc results as follows. Theorem 5 Under Assumptons, 2 and the sequence of local alternatves 35, LM d χ 2 K µ, as and T fxed, wth non-centralty parameter µ = c Ψc, where c = c,..., c K 6
and Ψ s a K K matrx wth k, l element Ψ k,l = 2σ plm 4 T 2 x t,k x t,l T = t= = T t= x 2 t,k T t,l x 2. = t= In order to relax the assumpton of normally dstrbuted errors we adopt Assumpton for v, where the sequence of local alternatves s now gven by σ 2 v,k = c k T, 36 for k =,..., K. ote that accordng to Theorem 2 we requre T. Theorem 6 Under Assumptons, 2, 3 and the sequence of alternatves 36, LM d χ 2 K µ, as, T, wth non-centralty parameter µ = c Ψc, where c = c,..., c K and Ψ s a K K matrx wth k, l element Remark 8 Ψ k,l = 2σ 4 plm,t T = 2 T x t,k x t,l. As n Secton 5. above, when the normalty assumpton s relaxed, local power can be studed for LM under Assumptons, 2 and 3 when T s fxed. specfcaton of local alternatves as n Theorem 5 apples. The non-centralty parameter of the lmtng non-central χ 2 dstrbuton results as µ = c Ψ c wth t= The Ψ k,l = σ 4 plm T t x t,k x t,l x s,k x s,l, = t=2 s= for k, l =,..., K. Remark 9 Gven the results for the modfed statstc LM n remark 8, and the fact that s = = s = Z ũ, we expect a smlar result for the regresson-based LM statstc LM reg to hold. Recall that LM uses Ṽ as an estmator of the varance of s see 24, whle LM reg employs T = t=2 ũ2 t z t z t. Under the null hypothess, t s not dffcult to see that these two estmators are asymptotcally equvalent. Under the alternatve, when studyng the k, l element of the varance of LM reg, we obtan see 7
appendx A.2 for detals = T ũ 2 t z t,k z t,l = t=2 + T t t ɛ 2 tx t,k x t,l ɛ s x s,k ɛ s x s,l = t=2 = t=2 s= s= T ɛ 2 tv B t X v + o p, 37 wth the K K matrx Bt X = x t t,k s= x t sx s,k xt,l s= x sx s,l. The frst term on the rght-hand sde n 37 has the same probablty lmt as Ṽk,l, the lmtng covarance matrx element Ψ k,l. In contrast to LM, however, the varance estmator of the regresson-based test nvolves addtonal quadratc forms such as v B X t v, contrbutng to the estmator. Snce, n a setup wth fxed T and the local alternatves σv,k 2 = c k, T ɛ 2 tv B t X v = O p /2, = t=2 the varance estmator remans consstent. In small samples, however, the addtonal term results n a bas of the varance estmator and may deterorate the power of the regresson-based test. See the appendx for detals about the above result and the Monte Carlo experments n Secton 7. Remark 0 The arguments of Remark 8 can be used to derve the local power of the LM ac statstc that accounts for seral correlaton n errors. The same specfcaton of local alternatves apples. The non-centralty parameter of the lmtng non-central χ 2 dstrbuton takes the quadratc form µ = c Ψ c wth Ψ k,l = plm T t T t u t u s u τ u q σ ts σ τq x t,k x s,k x τ,l x q,l, = t=2 s= τ=2 q= for k, l =,..., K. In the absence of seral correlaton t can be shown that the LM ac test nvolve a loss of power. To llustrate ths fact assume for smplcty that K = sngle regressor case. Further, the score vector n 33 can be equvalently wrtten as ŝ = T t ũ t ũ s σ ts x t,k x s,k = = t=2 s= T t ũ t ũ s xt,k x s,k C ts, 38 = t=2 s= where C ts = = x tx s. Thus, demeanng of ũ t ũ s s equvalent wth demeanng of x t,k x s,k. In the case of no autocorrelaton and 38 t follows that Ψ Ψ s postve sem-defnte. Therefore, the modfcaton 33 tends to reduce the power of the LM ac test when compared to LM. 8
We now proceed to examne the local power of the statstc of PY n model 8 and 9 under the sequence of local alternatves 36. In our homoskedastc setup, the dsperson statstc becomes S = β β X X β β, σ 2 = wth β as the OLS estmator n 0 as above. Usng ths expresson, the statstc s computed as n 4. The next theorem presents the asymptotc dstrbuton of the statstc under the local alternatves as specfed above. Ths result follows drectly from Secton 3.2 n PY. Theorem 7 Under Assumptons, 2, 3 and the sequence of local alternatves 36 d λ,, as, T, provded /T 0, where λ = Λ c/ 2K and Λ s a K vector wth typcal element Λ k = σ 2 plm,t T T = t= x 2 t,k, for k =,..., K. In Theorem 7, the mean of the lmtng dstrbuton of s slghtly dfferent from the result n Secton 3.2 n PY. Here, v s random and ndependently dstrbuted from the regressors and, therefore, the second term of the respectve expresson n PY s zero. Remark Consder for smplcty a scalar regressor x t that s..d. across and t wth unformly bounded fourth moments. Let E [x t ] = 0 and E [x 2 t] = σ 2,x, that s, the regressor s assumed to have a unt-specfc varaton whch s constant over tme for a gven unt. We obtan E T T t= x 2 t 2 = σ 2,x 2 + O T, mplyng µ = c 2 /2σ 4 lm = σ 2,x 2 n Theorem 6. To gan further nsght, we thnk of σ 2,x 2 as beng randomly dstrbuted n the cross-secton such that the noncentralty parameter results as [ µ = c2 σ 2σ E ] 2 2 4,x = c2 V ar [ ] [ ] σ 2 2σ 4,x + E σ 2 2,x. 39 9
Smlarly, under these assumptons, we fnd λ = c σ 2 2 E [ ] σ,x 2. 40 Comparng the mean of the normal dstrbuton of the statstc n 40 wth the noncentralty parameter of the asymptotc χ 2 dstrbuton of the LM statstc n 39, we see that the man dfference between the two tests s that the varance of σ 2,x contrbutes to the power of the LM statstc but not to the power of the test. If V ar [ σ 2,x] = 0 such that σ 2,x = σ 2 x for all, the LM test and the test have the same asymptotc power n ths example. If, however, V ar [ σ 2,x] > 0, so that there s varaton n the varance of the regressor n the cross-secton, the LM test has larger asymptotc power. To llustrate ths pont, we examne the local asymptotc power functons of the LM and the test for two cases, usng the expressons n 39 and 40. Fgure see appendx C shows the local asymptotc power of the LM sold lne and the test dashed lne as a functon of c when σ 2,x has a χ 2 dstrbuton. Fgure 2 repeats ths exercse for σ 2,x drawn from a χ 2 2 dstrbuton. In both cases, the LM test has larger asymptotc power. The power gan s substantal for the frst case, but dmnshes for the second. Ths pattern s expected, as the varance of σ 2,x contrbutes relatvely more to the non-centralty parameter n the frst specfcaton. Ths dscusson exemplfes the dfference between the LM-type tests and the statstc n terms of the local asymptotc power n a smplfed framework. The analyss suggests that the LM-type tests are partcularly powerful n an emprcally relevant settng n whch there s non-neglgble varaton n the varances of the regressors between panel unts. Havng studed the large samples propertes of the LM tests under the null and the alternatve hypothess n our model, we now evaluate the fnte-sample sze and power propertes of the LM-type tests n a Monte Carlo experment. 7 Monte Carlo Experments 7. Desgn After dervng LM-type tests n the random coeffcent model, we now turn to study the small-sample propertes of the proposed test and t varants. The am of ths secton s to evaluate the performance of the tests n terms of ther emprcal sze and power n several dfferent setups, relatng to the theoretcal dscusson of Sectons 4-6. We consder the followng test statstcs: the orgnal LM statstc presented n Theorem, the adjusted LM statstc that adjusts the nformaton matrx to account for fourth moments of the error dstrbuton see Corollary, the score-modfed LM statstcs see Theorem 3 and Theorem 4 and the regresson-based, heteroskedastcty-robust LM statstc see Secton 20
5.2. As a benchmark, we consder PY s statstc adj gven n 5. Followng the notes n Table n PY, the test usng adj s carred out as a two-sded test. In addton, the CLM test n 7 s ncluded, whch s also a two-sded test. We consder the followng data-generatng process wth normally dstrbuted errors as the standard desgn: y t = α + x tβ + ɛ t, ɛ t α d 0,, 4 d 0, 0.25, x t,k = α + ϑ x t,k, k =, 2, 3, ϑ x t,k β d 0, σ 2 x,k, d 3 ι 3, Σ v, under the null hypothess: Σ v = 0 42 0.03 0 0 under the alternatve: Σ v = 0 0.02 0, 43 0 0 0.0 where =, 2,...,, t =, 2,..., T. Hence, to smulate a model under the null the slope vector β s generated as a 3 vector of ones ι 3 for all. As dscussed n Secton 6 the varances of the regressors play an mportant role. In our benchmark specfcaton we generate the varances as σ 2 x,k = 0.25 + η,k η,k d χ 2, 44 The choce of the χ 2 dstrbuton for σx,k 2 s made analogous to the Monte Carlo experment n PY. We then consder varatons of ths specfcaton below. All results are based on 5,000 Monte Carlo replcatons. We choose {0, 20, 30, 50, 00, 200}, T {0, 20, 30}, as we would lke to study the small sample propertes of the test procedures when the tme dmenson s small. In our frst set of Monte Carlo experments the errors are normally dstrbuted; therefore we focus on the standard LM test. We also nclude ther respectve heteroskedastcty-robust regresson varants for ths exercse. 2
7.2 ormally dstrbuted errors Panel A of Table see Appendx B shows the rejecton frequences when the null hypothess s true. The adj test has rejecton frequences close to the nomnal sze of 5% for all combnatons of and T, whle the CLM test rejects the null hypothess too often, n partcular for small. Devatons from the nomnal sze for the the standard LM test and the regresson-based test are small and dsappear as ncreases, as expected from Theorem. Panel B of Table shows the correspondng rejectons frequences under the alternatve hypothess. The LM test outperforms the adj and the CLM test n general. Ths observaton holds n partcular for T = 0 where the power gan s consderable. The LM reg varant, although as powerful as the adj test for T = 0, suffers from a power loss relatve to the standard LM test. Ths power loss may be due to the small sample bas of the varance estmator, see Remark 9. Followng Remark 7 the varants of the LM tests are computed as follows. Frst, the ndvdual-specfc fxed effects α are elmnated by transformng the data usng orthogonal forward devatons see Arellano and Bover 995. The LM statstcs are then computed usng the transformed data. The results presented n Panel A of Table 2 ndcate that by employng forward orthogonalzaton all varants of the LM test have sze reasonably close to the nomnal level. By comparng panel B of Table and the rejecton rates under the alternatve n panel B of Table 2 we see that the power s very smlar n both setups confrmng usefulness of the forward orthogonalzaton procedure for the LM tests. 7.3 on-normal errors We now nvestgate the LM test when the errors are no longer normally dstrbuted, thereby buldng on the results of Secton 5.. The errors n 4 are generated from a t-dstrbuton wth 5 degrees of freedom, scaled to have unt varance. All other specfcatons of the standard desgn reman unchanged. In addton to the statstcs already consdered, we now nclude the adjusted LM statstc see corollary and the scoremodfed statstc see Theorem 3. Panel A n Table 3 reports the rejecton frequences under the null hypothess n ths case. We notce that the LM test has substantal sze dstortons when T s fxed and ncreases, whch s expected from Theorem 2. However, the adjusted LM statstc LM adj and the modfed score statstc LM are both successful n controllng the type-i error. Panel B of Table 3 shows rejecton frequences under the alternatve hypothess. The power gan of the LM test relatve to the adj test s notceable when T = 0 or T = 20. We found smlar results when the errors are χ 2 dstrbuted wth two degrees of freedom, centered and standardzed to have mean zero and varance equal to one. Gven the smlarty of the results for t and χ 2 dstrbuted errors, we do not present the latter results. 22
7.4 Serally correlated errors To study the mpact of serally correlated errors on the test statstcs we adjust the DGP as follows: y t = x tβ + ɛ t, ɛ t = ρɛ t + ρ 2 /2 et, d for =, 2,...,, t =, 2,..., T, where e t 0,. Under the null hypothess β = for all whle under the alternatve β s generated as n 43. The regressors, x t,k, k =, 2, 3 are generated as x t,k = φ,k x t,k + φ 2,k /2 ϑ x t,k, φ,k ϑ x t,k d U[0.05, 0.95], d 0, σ 2 x,k, where σ 2 x,k = 0.25 + η,k wth η,k d χ 2. Parameters φ,k and σ x,k are fxed across replcatons. Results of ths smulaton experment are reported n Table 4. Panel A and B show the rejecton frequences under the null hypothess n case of small seral dependence.e., ρ = 0.2, Panel A and moderate dependence.e., ρ = 0.5, Panel B. For all LM based test statstcs, except the LM ac test, we observe substantal sze devatons from the nomnal level. However, the LM ac test s successful n controllng the type-i error. Further, sze propertes of PY test are also sgnfcantly affected by autocorrelated errors. ote that ths fact s already documented and studed n Blomqust and Westerlund 203. Panel C of Table 4 reports power propertes of the test under no seral correlaton.e., ρ = 0, buldng on the dscusson n Remark 0. We observe that the LM ac test nvolve a 5 0% power loss compared to the LM test. Ths relatve power loss des out f T ncreases. 8 Concludng remarks In ths paper we examne the problem of testng slope homogenety n a panel data model. We develop testng procedures usng the LM prncple. Several varants are consdered that robustfy the orgnal LM test wth respect to non-normalty, heteroscedastcty and serally correlated errors. By studyng the local power we dentfy cases where the LMtype tests are partcularly powerful relatve to exstng tests. In sum, our Monte Carlo experments suggest that the LM test are powerful testng procedures to detect slope 23
homogenety n short panels n whch the tme dmenson s small relatve to the crosssecton dmenson. The LM approach suggested n ths paper may be extended n future research by allowng for dynamc specfcatons wth lagged dependent varables and cross sectonally or serally correlated errors. References Arellano, M. and O. Bover 995. Another look at the nstrumental varable estmaton of error-components models. Journal of Econometrcs 68, 29 5. Baltag, B., Q. Feng, and C. Kao 20. Testng for sphercty n a fxed effects panel data model. The Econometrcs Journal 4, 25 47. Blomqust, J. and J. Westerlund 203. Testng slope homogenety n large panels wth seral correlaton. Economcs Letters 2 3, 374 378. Breusch, T. S. and A. R. Pagan 980. The Lagrange multpler test and ts applcatons to model specfcaton n econometrcs. Revew of Economc Studes 47, 239 253. Harvlle, D. A. 977. Maxmum lkelhood approaches to varance component estmaton and to related problems. Journal of the Amercan Statstcal Assocaton 72, 320 338. Honda, Y. 985. Testng the error components model wth non-normal dsturbances. Revew of Economc Studes 52, 68 690. Hsao, C. and M. H. Pesaran 2008. Random coeffcent models. In L. Mátyás and P. Sevestre Eds., The Econometrcs of Panel Data, Chapter 6. Sprnger. Juhl, T. and O. Lugovskyy 204. A test for slope homogenety n fxed effects models. Econometrc Revews 33, 906 935. Pesaran, M. H. and R. Smth 995. Estmatng long-run relatonshps from dynamc heterogenous panels. Journal of Econometrcs 68, 79 3. Pesaran, M. H. and T. Yamagata 2008. Testng slope homogenety n large panels. Journal of Econometrcs 42, 50 93. Phllps, P. C. B. and H. R. Moon 999. Lnear regresson lmt theory for nonstatonary panel data. Econometrca 67 5, 057. Swamy, P. A. V. B. 970. Effcent nference n a random coeffcent regresson model. Econometrca 38, 3 323. Ullah, A. 2004. Fnte-sample econometrcs. Oxford Unversty Press. Wand, M. P. 2002. Vector dfferental calculus n statstcs. The Amercan Statstcan 56, 8. 24
Whte, H. 200. Asymptotc theory for econometrcans. Emerald. Wens, D. P. 992. On moments of quadratc forms n non-sphercally dstrbuted varables. Statstcs 23 3, 265 270. 25
A Appendx: Proofs nstead of full expressons and T through- To economze on notaton we use out ths appendx. and t = t= A. Prelmnary results We frst present an mportant result concernng the asymptotc effect of the estmaton error β β on the test statstcs. Defne A k = X k X k X k X k I T. T X A k u for k =,..., K. Fur- Lemma A. Let R k XAX = thermore let R k = σ 4 σ 4 X A k X and R k XAu = k k β β R 2σ 2 XAX β β 2 β β R XAu, for k =,..., K. Under Assumptons, 2 and the null hypothess the followng propertes hold f T s fxed: R k XAX = O p, R k XAu = O p /2, R k = O p, for k =,..., K. Proof. Usng the defnton of A j R k XAX = X X k X k yelds X T t x 2 t,k X X. The frst term s a K K matrx wth typcal l, m element x t,l x t,k x t,m x t,k = O p, t t as a consequence of Assumpton 2, whle t x2 t,k /T = O p and X X = O p. Recall that under the null hypothess, u = ɛ. Thus R k XAu = X X k X k u T t x 2 t,k X u. 26
The frst and the second term are O p /2 by a the central lmt theorem CLT for ndependent random varables and Assumpton 2. Combnng and together wth the fact that β β = O p yelds the result. Lemma A.2 Under Assumptons, 2 and the null hypothess the followng propertes hold for and T : R k XAX = O p T 2, R k XAu = O p /2 T 3/2, R k T = O p T, whch s defned as R k for k =,..., K. n Lemma A., Proof. Followng the proof of Lemma A. the element of the frst term of R k XAX s O p T 2, whereas the second term s O p T by Assumpton 2 whch yelds statement. otce n R k XAu has two terms as n Lemma A., where the frst one has zero mean and varance of order T 3. Therefore by Lemma n Baltag, Feng, and Kao 20 we have that X X j X j u = O p T 3/2 and by Lemma 2 n PY that X X j X j u = O p /2 T 3/2 and X u = O p /2 T /2. These results and the fact that T β β = O p mply. A.2 Proofs of the man results Proof of Lemma We use the followng rules for matrx dfferentatons: l = [ ] θ k 2 tr Ω Ω + θ k 2 [ ] l E = 2 [ Ω θ k θ tr Ω l θ k [ u Ω Ω Ω Ω θ l ] Ω u, 45 θ ] k, 46 for k, l =, 2,..., K +, see, e.g., Harvlle 977 and Wand 2002. Frst, X Σ v X = k σv,kx 2 k X k, wth X k denotng the k-th column vector of X. Hence X Σ v X 0... = k 0 X Σ v X σ 2 v,ka k, 27
wth the T T matrx, A k = X k X k 0... 0 X k Xk, for k =,..., K, and X k denotes the k-th column of the T K matrx X. Thus, Ω = k σ 2 v,ka k + σ 2 I T and { Ω A k, for k =, 2,..., K, = θ k I T, for k = K +. Under the null hypothess we have Ω = σ 2 I T. Usng 45 we obtan { l tr [A 2 σ θ k = 2 k ] + ũ A 2 σ 4 k ũ, for k =, 2,..., K H0 0, for k = K +, where σ 2 = T ũ ũ, ũ = I T X X X X y. The representaton of the score vector follows from tr [A k ] = Xt,k 2 = X k X k, t where X k denotes the k-th column of the T K matrx X. Smlarly, 46 yelds [ ] l tr [A 2σ 4 k A l ], for k, l =, 2,..., K, E = X θ k θ 2σ l H 0 k X k, for k =, 2,..., K, and l = K +, 4 T, for k = l = K +, 2σ 4 Usng the fact that A k and A l are block-dagonal, tr [A k A l ] = [ tr X k X k X l ] X l = X k X l 2, where X k denotes the -th column of X, whch yelds the form of the nformaton matrx presented n the lemma. Proof of Theorem 28
Recall that A k = X k X k T X k X k and rewrte the elements of the scores as 4 σ s k = ũ σ 4 2σ A k 4 ũ, for k =,..., K. Snce ũ = u X β β we have s k = σ 4 u σ 4 2σ A k 4 u + R k, where R k = O p from Lemma A.. Snce [ ] tr A k 0 and, therefore, lm E s = 0. The covarances are obtaned as [ ] Cov u A k u, u A l u X = 2σ 4 tr A k A l 2 = 2σ 4 X k X l X k X k X l T + T X k X k X l X l, T T X l I T, = 0 t follows that Eu A k u = T X l X l and snce u A k u s ndependent of u ja l u j for all j condtonal on X, 2 Cov 2σ 4 = 2σ 4 = V k,l. X k u A k u, 2 X l T u A l u X X k X k X l X l X k The Lapounov condton n the central lmt theorem for ndependent random varables see Whte 200, Theorem 5.0 s satsfed by Assumpton 2 and therefore /2 Ṽ d s 0, I K, where Ṽ replaces σ4 n V by σ 4. By the formula for the parttoned nverse { I σ 2 } :K,:K = Ṽ, X k 29
where { } :K,:K denotes the upper-left K K block of the matrx, t follows fnally that S I σ 2 S = s Ṽ s d χ 2 K. Proof of Theorem 2 The proof proceeds n three steps: we derve the covarance matrx of the score vector, we establsh the asymptotc normalty of the score vector and we use these results to establsh the asymptotc dstrbuton of the LM statstc. Defne the K vector s = [s,..., s K ] wth typcal element s k = u 2σ A k 4 u = s 2σ 4,k, 47 where s,k = u A k u and k K. Usng standard results for quadratc forms see e.g., Ullah 2004, appendx A.5, E [ ] [ s,k s,l X = 2σ 4 tr A k E [ ] [ s,k X = σ 2 tr ] [ A l + σ 4 tr where a k denotes the fourth moment of u t. Snce A k ] A k [ tr ] A l ] + µ 4 u 3σ 4 a k a l, s a vector consstng of the man dagonal elements of the matrx A k and µ 4 u E [ ] [ ] [ s,k X E s,l X = σ 4 tr A k ] [ tr A l ], we have Cov s,k, s,l X [ = 2σ 4 tr A k A l ] + µ 4 u 3σ 4 a k a l. 48 Due to the ndependence of u A k Cov s,k, s,l X = 2σ 4 u and u ja l u j for j, t follows that j [ tr A k A l ] + µ 4 u 3σ 4 [ Let V T denote the covarance matrx of s. Insertng the expresson for tr determne the k, l element of V T as V k,l = 2 x 2σ 4 t,k x t,l T t t µ 4 u 3σ 4 + 2σ 4 2 x 2 t,k T = V,k,l + V 2,k,l. t x 2 t,k t t,k x 2 x 2 t,k t x 2 t,l T a k a l. A k t ] A l, we To verfy that a central lmt theorem apples to s, let λ R k, λ = and Z,T = T λ s, where s s a K vector wth elements s,k for k K. Further, E [Z,T ] = 0 30 x 2 t,l 49