Exact Tests for Contemporaneous Correlation of Disturbances in Seemingly Unrelated Regressions

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1 2000s-16 Exact Tests for Contemporaneous Correlaton of Dsturbances n Seemngly Unrelated Regressons Jean-Mare Dufour, Lynda Khalaf Sére Scentfque Scentfc Seres Montréal Ma 2000

2 CIRANO Le CIRANO est un organsme sans but lucratf consttué en vertu de la Lo des compagnes du Québec. Le fnancement de son nfrastructure et de ses actvtés de recherche provent des cotsatons de ses organsatonsmembres, d une subventon d nfrastructure du mnstère de la Recherche, de la Scence et de la Technologe, de même que des subventons et mandats obtenus par ses équpes de recherche. CIRANO s a prvate non-proft organzaton ncorporated under the Québec Companes Act. Its nfrastructure and research actvtes are funded through fees pad by member organzatons, an nfrastructure grant from the Mnstère de la Recherche, de la Scence et de la Technologe, and grants and research mandates obtaned by ts research teams. Les organsatons-partenares / The Partner Organzatons École des Hautes Études Commercales École Polytechnque Unversté Concorda Unversté de Montréal Unversté du Québec à Montréal Unversté Laval Unversté McGll MEQ MRST Alcan Alumnum Ltée Banque Natonale du Canada Banque Royale du Canada Bell Québec Développement des ressources humanes Canada (DRHC) Fédératon des casses populares Desjardns de Montréal et de l Ouest-du-Québec Hydro-Québec Imasco Industre Canada Raymond Chabot Grant Thornton Téléglobe Canada Vlle de Montréal 2000 Jean-Mare Dufour et Lynda Khalaf. Tous drots réservés. All rghts reserved. Reproducton partelle permse avec ctaton du document source, ncluant la notce. Short sectons may be quoted wthout explct permsson, provded that full credt, ncludng notce, s gven to the source. Ce document est publé dans l ntenton de rendre accessbles les résultats prélmnares de la recherche effectuée au CIRANO, afn de suscter des échanges et des suggestons. Les dées et les opnons émses sont sous l unque responsablté des auteurs, et ne représentent pas nécessarement les postons du CIRANO ou de ses partenares. Ths paper presents prelmnary research carred out at CIRANO and ams at encouragng dscusson and comment. The observatons and vewponts expressed are the sole responsblty of the authors. They do not necessarly represent postons of CIRANO or ts partners. ISSN

3 Exact Tests for Contemporaneous Correlaton of Dsturbances n Seemngly Unrelated Regressons * Jean-Mare Dufour, Lynda Khalaf Résumé / Abstract Cet artcle propose des procédures exactes pour tester la spécfcaton SURE (régressons emplées) dans le contexte des régressons lnéares multvarées,.e. s les perturbatons des dfférentes équatons sont corrélées ou non. Nous applquons la technque des tests de Monte Carlo (MC) [Dwass (1957), Barnard (1963)] pour obtenr des tests d ndépendance exacts fondés sur les crtères du quotent de vrasemblance (LR) et du multplcateur de Lagrange (LM). Nous suggérons auss un crtère du type quas-quotent de vrasemblance (QLR) dérvé sur base des mondres carrés généralsés réalsables (FGLS). Nous démontrons que ces statstques sont lbres de paramètres de nusance sous l hypothèse nulle, ce qu justfe l applcaton des tests de Monte Carlo. Par alleurs, nous généralsons le test exact proposé par Harvey et Phllps (1982) au contexte des équatons multples. En partculer, nous proposons pluseurs tests nduts basés sur des tests de type Harvey-Phllps et nous suggérons une technque basée sur des smulatons afn de résoudre le problème de combnason de tests. Nous évaluons les proprétés des tests que nous proposons dans le cadre d une étude de Monte Carlo. Nos résultats montrent que les tests asymptotques usuels présentent de séreuses dstorsons de nveau, alors que les tests de MC contrôlent parfatement le nveau et ont une bonne pussance. De plus, les tests QLR se comportent ben du pont de vue de la pussance; ce résultat est ntéressant vu que les tests (multvarés) que nous proposons sont basés sur des smulatons. La pussance des tests de MC nduts augmente sensblement par rapport aux tests fondés sur l négalté de Bonferron et, dans certans cas, dépasse la pussance des tests de MC fondés sur la vrasemblance. Nous applquons les tests sur des données utlsées par Fscher (1993) pour analyser des modèles de crossance. Ths paper proposes fnte-sample procedures for testng the SURE specfcaton n mult-equaton regresson models,.e. whether the dsturbances n * Correspondng Author: Jean-Mare Dufour, CIRANO, 2020 Unversty Street, 25 th floor, Montréal, Qc, Canada H3A 2A5 Tel.: (514) Fax: (514) emal: The authors thank Bryan Campbell, Judth Gles, Allan Wurtz and Vctora Znde-Walsh for useful comments. Ths work was supported by the Canadan Network of Centres of Excellence [program on Mathematcs of Informaton Technology and Complex Systems (MITACS)], the Canada Councl for the Arts (Kllam Fellowshp), the Natural Scences and Engneerng Research Councl of Canada, the Socal Scences and Humantes Research Councl of Canada, and the Fonds FCAR (Government of Québec). Unversté de Montréal, CRDE and CIRANO Unversté Laval and GREEN

4 dfferent equatons are contemporaneously uncorrelated or not. We apply the technque of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] to obtan exact tests based on standard LR and LM zero correlaton tests. We also suggest a MC quas-lr (QLR) test based on feasble generalzed least squares (FGLS). We show that the latter statstcs are pvotal under the null, whch provdes the justfcaton for applyng MC tests. Furthermore, we extend the exact ndependence test proposed by Harvey and Phllps (1982) to the mult-equaton framework. Specfcally, we ntroduce several nduced tests based on a set of smultaneous Harvey/Phllps-type tests and suggest a smulaton-based soluton to the assocated combnaton problem. The propertes of the proposed tests are studed n a Monte Carlo experment whch shows that standard asymptotc tests exhbt mportant sze dstortons, whle MC tests acheve complete sze control and dsplay good power. Moreover, MC-QLR tests performed best n terms of power, a result of nterest from the pont of vew of smulaton-based tests. The power of the MC nduced tests mproves apprecably n comparson to standard Bonferron tests and n certan cases outperform the lkelhood-based MC tests. The tests are appled to data used by Fscher (1993) to analyze the macroeconomc determnants of growth. Mots Clés : Keywords: Régressons emplées, système SURE, test d ndépendance, régresson lnéare multvarée, corrélaton contemporane, test exact, test à dstance fne, test de Monte Carlo, bootstrap, test ndut, test LM, quotent de vrasemblance, test de spécfcaton, macroéconome, crossance. Seemngly unrelated regressons, SURE system, multvarate lnear regresson, contemporaneous correlaton, exact test, fnte-sample test, Monte Carlo test, bootstrap, nduced test, LM test, lkelhood rato test, specfcaton test, macroeconomcs, growth JEL: C3, C12, C15, C30, C33, C52

5 Contents Lst of Defntons, Propostons and Theorems Lst of Tables v v 1. Introducton 1 2. Framework 2 3. Test statstcs for cross-equaton dsturbance correlaton Lkelhood-based tests Induced Harvey-Phllps tests Fnte-sample theory 9 5. Smulaton experments Applcaton to growth equatons Concluson 24 References 25

6 Lst of Defntons, Propostons and Theorems 4.1 Proposton : Standardzed representaton of LM and Harvey-Phllps statstcs Proposton : Standardzed representaton of the LR statstc Proposton : Standardzed representaton of QLR statstcs Proposton : Pvotal property of tests for cross-equaton correlaton Lst of Tables 1 Covarance matrces used n the Monte Carlo experments Emprcal szes of LM and quas-lr ndependence tests Emprcal rejectons of varous ndependence tests GDP growth SURE systems: ndependence tests Captal growth SURE systems: ndependence tests Productvty growth SURE systems: ndependence tests Labor force growth SURE systems: ndependence tests v

7 1. Introducton Mult-equaton models whch use both cross-secton and tme seres data are common n econometrc studes. These nclude, n partcular, the seemngly unrelated regressons (SURE) model ntroduced by Zellner (1962). The SURE specfcaton s expressed as a set of lnear regressons where the dsturbances n the dfferent equatons are correlated. The non-dagonalty of the error covarance matrx usually entals that ndvdual equaton estmates are sub-optmal; hence, generalzed least squares (GLS) estmaton whch explots the correlatons across equatons may mprove nference. However, the mplementaton of GLS requres estmatng the error covarance from the data. Further the cross-equaton dependence must be taken nto account when testng cross-equaton parameter restrctons. As t s well known, the feasble generalzed least squares (FGLS) estmators need not be more effcent than ordnary least squares (OLS); see Srvastava and Gles (1987, Chapter 2). Indeed, the closer the error covarance comes to beng sphercal, the more lkely t s that OLS estmates wll be superor. Ths has extensvely been dscussed n the SURE lterature; see, for example, Zellner (1962, 1963), Mehta and Swamy (1976), Kmenta and Glbert (1968), Revankar (1974, 1976), Kuntomo (1977), Karya (1981a), and Srvastava and Dwved (1979). In ths sense, choosng between GLS and OLS estmaton n the SURE model corresponds to the problem of testng for sphercty of the error covarance matrx. Ths paper studes and proposes fnte-sample tests for ndependence aganst contemporaneous correlaton of dsturbances n a SURE model. Independence tests n multvarate models have been dscussed n both the econometrc and statstcal lteratures. In partcular, Breusch and Pagan (1980) derved a Lagrange multpler (LM) test for the dagonalty of the error covarance matrx. Karya (1981c) derved locally best nvarant tests n a two-equaton framework. Shba and Tsurum (1988) proposed Wald, lkelhood rato (LR), LM and Bayesan tests for the hypothess that the error covarance s block-dagonal. Related results are also avalable n Karya (1981b), Karya, Fujkosh, and Krshnaah (1984) and Cameron and Trved (1993). Except for one specal case, these test procedures are only justfed by asymptotc arguments. The excepton s Harvey and Phllps (1982, Secton 3) who proposed exact ndependence tests between the errors of an equaton and those of the other equatons of the system. These tests (whch we wll denote EFT) nvolve conventonal F statstcs for testng whether the (estmated) resduals added to each equaton have zero coeffcents. EFT tests may be appled n the context of general dagonalty tests; for example, one may assess n turn whether the dsturbances n each equaton are ndependent of the dsturbances n all other equatons. Such a sequence of tests however rases the problem of takng nto account the dependence between multple tests, a problem not solved by Harvey and Phllps (1982). A major problem n the SURE context comes from the fact that relevant null dstrbutons are ether dffcult to derve or too complcated for practcal use. Ths s true even n the case of dentcal regressor matrces. Hence the applcable procedures rely heavly on asymptotc approxmatons whose accuracy can be qute poor. Ths s evdent from the Monte Carlo results reported n Harvey and Phllps (1982) and Shba and Tsurum (1988), among others. In any case, t s wdely acknowledged by now that standard multvarate LR-based asymptotc tests are unrelable n fnte samples, n the sense that test szes devate from the nomnal sgnfcance levels; see Dufour and Khalaf (1998) for related smulaton evdence. 1

8 In ths paper, we reemphasze ths fact and propose to use the technque of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] n order to obtan provably exact procedures. We apply the MC test technque to: () the standard lkelhood rato (LR) and Lagrange multpler (LM) crtera, and () OLS and FGLS-based quas-lr (QLR) statstcs. We also ntroduce several nduced tests based on a set of smultaneous Harvey/Phllps-type tests and suggest a smulaton-based soluton to the assocated combnaton problem. The crtcal regons of conventonal nduced tests are usually computed usng probablty nequaltes (e.g., the well know Boole-Bonferron nequalty) whch yelds conservatve p-values whenever non ndependent tests are combned [see, for example, Savn (1984), Folks (1984), Dufour (1990) and Dufour and Torrès (1998)]. Here, we propose to construct the nduced tests such that sze-correct p-values can be readly obtaned by smulaton. The frst step towards an exact test procedure nvolves dervng nusance-parameter-free null dstrbutons. In the context of standard ndependence tests, nvarance results are known gven two unvarate or multvarate regresson equatons [Karya (1981c), Karya (1981b), Karya, Fujkosh, and Krshnaah (1984)]. The problem of nusance parameters s yet unresolved n models nvolvng more than two regresson equatons. Here, we show that the LR, LM and QLR ndependence test statstcs are pvotal under the null, for mult-equaton SURE systems. Though the proof of ths result s not complex, t does not appear to be known n the lterature. Of course, exstng work n ths area has typcally focused on dervng p-values analytcally. By contrast, the approach taken n ths artcle does not requre extractng exact dstrbutons; the technque of MC tests allows one to obtan provably exact randomzed tests n fnte samples usng very small numbers of MC replcatons of the orgnal test statstc under the null hypothess. In the present context, ths technque can easly be appled whenever the dstrbuton of the errors s contnuous and specfed up to an unknown covarance matrx (or lnear transformaton). Note ths dstrbuton does not have to be Gaussan. For further references regardng MC tests, see Dufour (1995), Dufour and Kvet (1996, 1998), Kvet and Dufour (1997), Dufour, Farhat, Gardol, and Khalaf (1998), and Dufour and Khalaf (2000). We nvestgate the sze and power of suggested tests n a Monte Carlo study. The results show that, whle the asymptotc LR and LM tests serously overreject, the MC versons of these tests acheve perfect sze control and have good power. The power of the MC nduced tests mproves apprecably n comparson to the standard Bonferron tests and n several cases outperform the correspondng MC-LR and LM tests. The outlne of ths study s as follows. In Secton 2, we present the model and the estmators used, whle the test statstcs are descrbed n Secton 3. In Secton 4, we show that the proposed test statstcs have nusance-parameter free dstrbutons under the null hypothess and descrbe how exact MC tests can be mplemented. In Secton 5, we report the smulaton results. In Secton 6, we apply the tests to data used by Fscher (1993) to analyze the macroeconomc determnants of growth. We conclude n Secton Framework Consder the seemngly unrelated regresson model Y = X β + u, =1,..., p, (2.1) 2

9 where Y s a vector of n observatons on a dependent varable, X a full-column rank n k matrx of regressors, β a vector of k unknown coeffcents, and u =(u 1,u 2,...,u n ) a n 1 vector of random dsturbances. When X = X j,,j=1,..., p,we have a multvarate lnear regresson (MLR) model; see Anderson (1984, chapters 8 and 13), Berndt and Savn (1977), and Karya (1985). The system (2.1) may be rewrtten n the stacked form y = Xβ + u (2.2) where y = Y 1 Y 2. Y p,x= X X X p,u= u 1 u 2. u p,β= β 1 β 2. β p, (2.3) so that X s a (np) k matrx, yand u each have dmenson (np) 1 and β has dmenson k 1, wth k = p =1 k.set U= [ ] u 1 u 2 u p = where U t =(u t1,u t2,...,u tp ) s the dsturbance vector for the t-th observaton. In the sequel, we shall also use, when requested, some or all of the followng assumptons and notatons: where J s a fxed lower trangular p p matrx such that where we set σ σ 1/2,=1,...,p; U 1 U 2. U n (2.4) U t = JW t, t =1,..., n, (2.5) Σ JJ = [ σ j s nonsngular, (2.6) ],j=1,...,p W 1,...,W n are p 1 random vectors whose jont dstrbuton s completely specfed; (2.7) u s ndependent of X. (2.8) Assumpton (2.8) s a strct exogenety assumpton, whch clearly holds when X s fxed. The assumptons (2.5) - (2.7) mean that the dsturbance dstrbuton s completely specfed up an unknown lnear transformaton that can modfy the scalng and dependence propertes of the dsturbances n dfferent equatons. Note (2.5) - (2.7) do not necessarly ental that Σ s the covarance matrx of 3

10 U t. However, f we make the addtonal assumpton that or, more restrctvely, W 1,...,W n are uncorrelated wth E(W t )=0, E(W t W t)=i p, t =1,...,n, (2.9) W 1,..., W n..d. N[0, I p ], (2.10) we have: E ( U t ) =0, E ( Ut U t ) =Σ, t =1,..., n, (2.11) E(u) =0,E(u u j )=σ ji n,, j =1,..., p, (2.12) and E(uu )=Σ I p. (2.13) The coeffcents of the regresson equatons can be estmated by several methods among whch the most well known are: () ordnary least squares (OLS) appled to each equaton, () two-step feasble generalzed least squares (FGLS), () teratve FGLS (IFGLS), and (v) maxmum lkelhood (ML) assumng u follows a multnormal dstrbuton. The OLS estmator of β s ˆβ OLS =(ˆβ 1,..., ˆβ p), ˆβ =(X X ) 1 X Y, =1,..., p. (2.14) An assocated estmate ˆΣ for Σ can be obtaned from the OLS resduals: û = Y X ˆβ = M(X )u, M(X )=I n X (X X ) 1 X, =1,..., p. (2.15) The two-step FGLS estmate based on any consstent estmate S of Σ, s gven by β F GLS = [ X (S 1 I n )X ] 1 X (S 1 I n )y. (2.16) If the dsturbances are normally dstrbuted, we have the log-lkelhood functon L = np 2 ln(2π) n 2 ln( Σ ) 1 2 (y Xβ) (Σ 1 I n )(y Xβ). (2.17) The correspondng maxmum lkelhood (ML) estmators β and Σ of β and Σ satsfy the followng normal equatons: X ( Σ 1 I n )X β = X ( Σ 1 1 I n )y, Σ= nũ Ũ= [ r ] j (2.18),j=1,...,p 4

11 where β =( β 1,..., β p ) and Ũ =[ũ 1,..., ũ p ], ũ = Y X β, r j =ũ ũ j /[(ũ ũ )(ũ jũ j )] 1/2. (2.19) Of course, the estmators n (2.18) are well defned provded the matrx Σ has full column rank, an assumpton we shall make n the sequel. Iteratve procedures are typcally appled to obtan the ML estmates. Suppose Σ (0) s an ntal estmate of Σ. Usng (2.18), we can solve for a frst GLS estmate of β, β (0) = [ X ( Σ (0) I n ) 1 X ] 1 X ( Σ (0) I n ) 1 y, (2.20) from whch a new estmate of u may be obtaned: ũ (1) = y X β (0). (2.21) Ths resdual leads to further estmators Σ (1) and β (1) of Σ and β. Pursung ths teratve process, we see that the estmators at the h-th teraton take the form: h =1,2,...,where β (h) = [ X ( Σ (h) I n ) 1 X ] 1 X ( Σ (h) I n ) 1 y, (2.22) Σ (h) = 1 nũ(h) Ũ (h) (h)] = [ r j,j=1,...,p, (2.23) Ũ (h) =[ũ (h) 1,..., ũ p (h) ], ũ(h) = Y X β(h 1), r (h) j =ũ (h) ũ (h) j /n. (2.24) Under standard assumptons, teratng ths procedure to convergence yelds the ML estmates [see Oberhofer and Kmenta (1974)]. For a more general dscusson of the propertes of such partally terated estmators, the reader may consult Robnson (1988). 3. Test statstcs for cross-equaton dsturbance correlaton 3.1. Lkelhood-based tests Gven the setup descrbed above, we consder the problem of testng the hypothess H 0 that Σ s dagonal. For any vector (d 1,..., d N ), let us denote D N (d ) the dagonal matrx whose dagonal elements are d 1,..., d N : D N (d )=dag(d 1,..., d N ). (3.1) D N (d ) represents a dagonal matrx of dmenson N, wth d =(d 1,..., d N ) along the dagonal. Then H 0 may be expressed as H 0 :Σ=D p (σ 2 ). (3.2) 5

12 Snce J s lower trangular, t s easy to see that: Σ=D p (σ 2 )f and only J = D p(σ ). Thus, under H 0,u t = σ W t,=1,..., p,where W t =(W 1t,W 2t,...,W pt ). If (2.9) holds, H 0 s equvalent to the absence of contemporaneous correlaton between the components of U t.ifthe components of W t are ndependent, H 0 s equvalent to the ndependence between the components of u t ; when W 1,...,W n are ndependent, the latter condton entals that the dsturbance vectors u 1,..., u p are ndependent. In the sequel, we wll frequently refer to the standardzed dsturbances w = ( w 1,..., w p), where w =(1/σ )u, =1,..., p. (3.3) Under the assumptons (2.5) - (2.7), the vector w has a completely specfed dstrbuton f H 0 holds. Let us now consder the case where, n addton to (2.5) - (2.7), we make the normalty assumpton (2.10). Then the dsturbance vectors U t = JW t, t =1,...,n,are..d. N[0, Σ] where Σ=JJ and we have the log-lkelhood functon (2.17). In ths case, the LR and LM statstcs for testng H 0 take relatvely smple forms. The LR statstc s ξ LR = n ln( Λ) where whle the LM crteron s Λ = D p (ˆσ 2 ) / Σ, (3.4) ξ LM = n p 1 rj 2 (3.5) =2 j=1 where r j =û ûj/[(û û)(û jûj)] 1/2. Under standard regularty condtons, both ξ LR and ξ LM follow a χ 2( p(p 1)/2 ) dstrbuton asymptotcally under H 0 [see Breusch and Pagan (1980)]. In the sequel, we shall also consder quas-lr statstcs ξ (h) LR = n ln( Λ (h) ) where Σ (h) s used nstead of the unrestrcted ML estmator Σ : Λ (h) = D p (ˆσ 2 ) / Σ (h). (3.6) Snce unrestrcted ML estmators of the SURE model parameters are usually obtaned through teratve numercal methods, such QLR statstcs are easer to compute than the fully-terated LR statstc Induced Harvey-Phllps tests A fnte-sample exact ndependence test was developed by Harvey and Phllps (1980). Ther procedure s applcable under the assumptons (2.5) - (2.10) to test a null hypothess of the form [ ] σ 2 H 01 :Σ= 1 0 (3.7) 0 Σ 11 6

13 where Σ 11 s a (p 1) (p 1) matrx. Specfcally, they propose the followng statstc: EFT = û ˆV 1 1 (ˆV 1 M 1 ˆV ) 1 1 ˆV 1û1/(p 1) ( û 1[ I ˆV1 ˆV 1 M 1 ˆV ) 1 1 ˆV 1 ]û1 /(n k 1 p +1), (3.8) whch follows an F dstrbuton wth (p 1, n k 1 p+1)degrees of freedom under H 01.The EFT statstc can be obtaned as the usual F -statstc for testng whether the coeffcents on ˆV 1 are zero n the regresson of Y 1 on X 1 and ˆV 1. More generally, we can consder any partcular dsturbance vector u (or equaton) from the p regressons n (2.1) and test n ths way whether u s ndependent of V K() [ u j, where ]j K () K () s some non-empty subset of {j :1 j p, j }. Ths can be done by estmatng an extended regresson of the form Y = X β + û j γ j + u (3.9) j K () and testng the hypothess H 0 [K () ] : γ j = 0 for j K (). Under the null hypothess H 0 of ndependence [see (3.2)], the correspondng F -statstc F [K () ]= (û û SS(K () ) ) /p SS(K () )/(n k p ) (3.10) follows an F (p,n k p ),where p s the number of elements n K () and SS(K () ) s the unrestrcted resdual sum of squares from regresson (3.9). As thngs stand, the latter procedures only test the ndependence of one dsturbance vector u wth respect to the other dsturbance vectors. It s straghtforward to see that the test of H 0 based on F [K () ] can only detect correlatons between u and the other dsturbances. In order to test H 0 aganst all possble covarance matrces Σ, we need a dfferent procedure. A smple way to do ths, whch stll explots the Harvey-Phllps procedure, conssts n usng nduced tests that combne several tests of the form F [K () ]. Here we shall consder two methods for combnng tests. Denote G F [x ν 1,ν 2 ] the survval functon of the Fsher dstrbuton wth (ν 1,ν 2 ) degrees of freedom;.e., f F s a random varable that follows an F (ν 1,ν 2 ) dstrbuton, we have G F [x ν 1,ν 2 ]=P[F x].we consder the test statstcs EFT F [K ], where K {j:1 j n, j },=1,...,p, (3.11) each of whch tests whether u s ndependent of all the other dsturbance vectors. The p-value assocated wth EFT s: pv [K ]=G F [EFT p 1, n k p+1] (3.12) whch follows a unform dstrbuton on the nterval [0, 1]. The level-α F-test based on EFT s equvalent (wth probablty 1) to rejectng the null hypothess when pv [K ] α, or equvalently 7

14 when 1 pv [K ] 1 α. A dffculty we meet here conssts n controllng the overall level of a procedure based on several separate tests. A smple way to do ths conssts n runnng each one of the p tests F [K ] at level p α, so that α = α, and rejectng H 0 when at least one of the p separate tests rejects the null =1 hypothess; for example, we may take α = α/p, =1,...,p.By the Boole-Bonferron nequalty, ths ensures that the probablty of rejectng H 0 s not greater than α (although t could be smaller). When α = α/p, ths procedure s equvalent to rejectng H 0 when pv mn α/p, where pv mn mn{pv [K ]:=1,...,p} (3.13) s the mnmum of the p-values. Note that usng the mnmum of several p-values as a test statstc was orgnally proposed by Tppett (1931) and Wlknson (1951), n the case of ndependent test statstcs. The ndependence condton does not however hold here for the EFT statstcs, hence the necessty of takng nto account the dependence. Because t s conservatve, the Boole-Bonferron bound may lead to a power loss wth respect to a procedure that avods the use of a bound. In the next secton, we wll see that the conservatve property of the Bonferron-based pv mn procedure can be corrected by usng the technque of Monte Carlo tests. In other words, we consder the procedure that rejects H 0 when pv mn, as defned by (3.12) and (3.13), s small, and we shall show that ts sze can be controlled by usng the Monte Carlo test technque. A second farly natural way of aggregatng separate tests conssts n rejectng H 0 when the product pv = p pv [K ] (3.14) =1 s small. Such a procedure was orgnally suggested by Fsher (1932) and Pearson (1933), agan for ndependent test statstcs. As for the pv mn procedure, we wll see that the sze of such a test based on pv can be controlled by Monte Carlo technques, even f the ndvdual p-values pv [K ] are not ndependent. For convenence reasons, we shall mplement both these tests by takng the test crtera: F mn =1 pv mn, (3.15) F =1 pv, (3.16) each one of whch rejects H 0 when t s large. We also consdered a sequental approach n whch we test the sequence of hypotheses H 0 : u s ndependent of u +1,...,u p (3.17) 8

15 for =1,..., p 1, usng Harvey-Phllps tests based on regressons of the form Y = X β + p û j γ j + u, (3.18) j=+1 =1,..., p 1. Clearly H 0 s equvalent to the conjuncton of the p 1 hypotheses H 0, = 1,..., p 1, so that we should reject H 0 when at least one of these tests s sgnfcant. Ths yelds the p 1 test statstcs F [{ +1,..., p}],=1,..., p 1 for whch t s easy to see that F [{ +1,...,p}] F(p, n k p +) under H 0. The problem then conssts agan n controllng the overall level of ths combned procedure. Snce t s not clear the test statstcs are ndependent, one way to acheve ths control conssts n usng agan the Boole-Bonferron nequalty. p For ths, we test H 0 at level α, where α = α,and reject H 0 when one of the tests s sgnfcant. =1 In a sequental context, a standard way of dong ths conssts n consderng geometrcally declnng levels, such as α 1 = α/2,α 2 =α/(2 2 ),..., α p 2 = α/(2 p 2 ),α p 1 =α/(2 p 2 ); (3.19) see Anderson (1971, Chapter 4) and Lehmann (1957). Here we shall consder the bound procedure based on (3.19), as well as tests on the mnmum and the product of the p separate p-values assocated wth the test statstcs F [{ +1,..., p}]: FS mn =1 mn {pv [{ +1,..., p}]:=1,...,p 1}, (3.20) FS =1 p 1 pv [{ +1,..., p}]. (3.21) =1 Agan the levels of the two latter procedures wll be controlled through the Monte Carlo test technque. For further dscusson of multple test procedures, the reader may consult Mller (1981), Folks (1984), Savn (1984), Dufour (1989, 1990), Westfall and Young (1993), Gouréroux and Monfort (1995, Chapter 19), and Dufour and Torrès (1998, 1999). 4. Fnte-sample theory We proceed next to examne the fnte-sample dstrbutons of the above defned LM, LR and QLR test crtera. In partcular, we show that the assocated null dstrbutons are free of nusance parameters. To do ths, we wll frst demonstrate n the three followng propostons that all the statstcs consdered are functons of the standardzed dsturbances w,=1,...,p.interestngly, these propertes hold under very weak dstrbutonal assumptons on u and X. 9

16 Proposton 4.1 STANDARDIZED REPRESENTATION OF LM AND HARVEY-PHILLIPS STATIS- TICS. Under the assumptons and notatons (2.1) to (2.6), the LM statstc defned n (3.5) can be wrtten n the form ξ LM = n p 1 r 2 j (4.1) =2 j=1 where r j = ŵ ŵj/[(ŵ ŵ)(ŵ jŵj)] 1/2, ŵ = û /σ = M(X )w and w =(1/σ )u, whle each statstc F [K () ] defned n (3.10) s dentcal to the F -statstc F [K () ] for testng H 0 : γ j =0for j K () n the regresson where Y =(1/σ )y. Y = X β + PROOF. The result for the LM statstc follows on observng that j K () ŵ j γ j + w (4.2) r j = û ûj [(û û)(û jûj)] = ŵŵj 1/2 [(ŵ ŵ)(ŵ jŵj)] 1/2 = r j. For F [K () ], we note that û û = u M(X )u = σ 2 w M(X )w = σ 2 ŵ ŵ,ss[k () ]=σ 2 SS where ŵ ŵj and SS regresson are the restrcted and unrestrcted resdual sum of squares from the lnear Y = X β + j K () ŵ j γ j + w. We then see that F [K () ] = = (û û SS[K () ] ) ( /p σ 2 SS[K () ]/(n k p ) = ŵ ŵ σ 2 ) SS /p σ 2 SS /(n k p ) (ŵ ŵ SS ) /p SS /(n k p ) = F [K () ]. Proposton 4.2 STANDARDIZED REPRESENTATION OF THE LR STATISTIC. Under the assumptons and notatons of Proposton 4.1, suppose the matrx Σ defned n (2.18) has full column rank. 10

17 Then the LR-based statstc Λ defned n (3.4) can be wrtten n the form Λ = p w M(X )w =1 Σ (4.3) where Σ s the ML estmator of Σ obtaned by maxmzng the Gaussan log-lkelhood L = np 2 ln(2π) n 2 ln( Σ ) 1 2 (w Xβ) (Σ 1 I n )(w Xβ) (4.4) where w =(w 1,w 2,..., w p). PROOF. From (3.4) we can wrte p Λ = D p(σ 1 ) D p (ˆσ 2 ) D p(σ 1 D p (σ 1 ) Σ D p (σ 1 ) ) = =1 D p (σ 1 ˆσ 2 /σ 2 ) ΣD p (σ 1 ) (4.5) where ˆσ 2 /σ2 = ŵ ŵ = w M(X )w. Further, t s easy to see that the Gaussan log-lkelhood (2.17) s nvarant under data transformatons of the form y = vec [ Y 1 Y 2 Y p ]wth Y = c (Y + X δ ),=1,..., p, (4.6) where c s an arbtrary non-zero constant and δ an arbtrary k 1 vector ( =1,...,p).In other words, f the log-lkelhood functon of y s gven by (2.17), the lkelhood of y has the same form wth β replaced by β = c (β + δ ) and Σ replaced by Σ = D p (c )ΣD p (c ). In partcular, f we take δ = β and c =1/σ, we get Y =(1/σ )u = w wth L as the correspondng log-lkelhood functon. Consequently, by the equvarance of maxmum lkelhood estmators [see Dagenas and Dufour (1991)], we have Σ = D p (σ 1 ) ΣD p (σ 1 ), from whch (4.3) follows. Proposton 4.3 STANDARDIZED REPRESENTATION OF QLR STATISTICS. Under the assumptons and notatons of Proposton 4.1, let Σ (0) be an ntal postve defnte estmator of Σ, and suppose the matrces Σ (h),h=1,..., H, defned n (2.23) have full column rank. Then, the approxmate LR statstcs Λ (H) defned by (3.6) can be wrtten n the form Λ (H) = p w M(X )w =1 Σ (H) (4.7) where Σ (H) s the estmate of Σ obtaned through the formulas: β (h) = [ X ( Σ (h) I n ) 1 X ] 1 X ( Σ (h) I n ) 1 w, (4.8) 11

18 Σ (h) = D p (σ 1 ) Σ (0) D p (σ 1 = 1 nũ(h) Ũ (h) ), for h =0,, for h 1, (4.9) h =0,1,...,H,where Ũ (h),h 1,obeys the recurson Ũ (h) =[ũ (h) 1,..., ũ(h) p ], ũ (h) = w X β(h 1),=1,..., p. (4.10) PROOF. From the defnton (3.6), we can wrte, for h 0, Λ (h) = D p(σ 1 ) D p (ˆσ 2 ) D p (σ 1 ) D p (σ 1 ) Σ (h) D p (σ 1 ) = D p(ˆσ 2 /σ 2 ) Σ (h) = p w M(X )w =1 Σ (h) (4.11) where Σ (h) For h =0, the result holds trvally. For h 1, we have: Σ (h) = 1 nũ(h) Ũ (h) Ũ (h) = Ũ (h) D p (σ 1 )=[ũ (h) 1,..., ũ (h) p D p (σ 1 ) Σ (h) D p (σ 1 ). (4.12) = [ ũ (h) ũ(h) j /n],j=1,...,p, (4.13) ]D p (σ 1 )=[ũ (h) 1,..., ũ(h) p ], (4.14) ũ (h) (1/σ )ũ (h) =(1/σ )[Y X β(h 1) ], =1,...,p. (4.15) Puttng (4.15) n vector form, we see that ũ (h) vec[ũ (h) 1,..., ũ(h) p ]=(D I n )ũ (h) ( = D n y X β(h 1)) (4.16) where D D p (σ 1 ) and D n D I n. Now, for h 0, the feasble GLS estmator β (h) mnmzes the quadratc form wth respect to β. Snce S(β) =(y Xβ) ( Σ(h) I n ) 1(y Xβ) S(β) = (y Xβ) (D I n )(D 1 I n ) ( Σ(h) I n ) 1(D 1 I n )(D I n )(y Xβ) ths entals that = [(D I n )(y Xβ)] [ (D Σ (h) D) I n ] 1[(D In )(y Xβ)], β (h) = [( D n X ) ( Σ(h) I n ) 1 ( Dn X )] 1( Dn X ) ( Σ(h) I n ) 1Dn y. 12

19 Further, on notng that w (w 1,w 2,..., w p) =(D I n )u=d n u and D n X =[D p (σ 1 ) I n ]X= σ 1 1 X σ 1 2 X σ 1 p X p =XD p(σ 1 I k )=X where D p (σ 1 I k ) s a non-sngular matrx, we see that where hence β (h) = [( X ) ( Σ(h) = β + 1[ X ( Σ(h) = β+ 1 β(h) β (h) For h 1, we then see that: = [ X ( Σ (h) I n ) 1 ( X )] 1 ( X ) ( Σ(h) I n ) 1X ] 1X ( Σ(h) I n ) 1Dn y I n ) 1Dn u I n ) 1 X ] 1 X ( Σ (h) I n ) 1 w. D n ũ (h) = D n (y X β (h 1) )=D n (Xβ + u Xβ X 1 β(h 1) = (h 1) w X β ũ (h) (1/σ )ũ (h) Ths completes the proof of the proposton. = w X β(h 1),=1,..., p. ) Propostons 4.1 and 4.2 show that the dstrbutons of the LM, Harvey-Phllps and LR statstcs only depend on the dstrbutons of X and w, rrespectve whether the null hypothess H 0 holds or not. Ths property also carres to procedures based on combnng several of these test statstcs, such as the nduced Harvey-Phllps tests proposed n Secton 3.2. In partcular, under the strct exogenety assumpton (2.8), ths means that the condtonal dstrbutons (gven X) of these test statstcs only depend on the dstrbuton of w (and the known value of X). If we further assume that the jont dstrbuton of W 1,...,W n s completely specfed [assumpton (2.7)], then under H 0 the dstrbuton of w does not nvolve any unknown parameter, and smlarly for the LM, Harvey- Phllps and LR statstcs. For the QLR statstcs, the same propertes wll hold provded we assume 13

20 (0) that Σ D p (σ 1 ) Σ (0) D p (σ 1 ) can be rewrtten as a functon of X and w. In partcular, ths wll be the case f the ntal value Σ (0) s obtaned from the least squares resduals from the p separate regressons n (2.1),.e. f Σ (0) = 1 nû Û,Û=[û 1,..., û p ], û =M(X )Y,=1,..., p. (4.17) We can thus state the followng proposton. Proposton 4.4 PIVOTAL PROPERTY OF TESTS FOR CROSS-EQUATION CORRELATION. Under the assumptons and notatons (2.1) to (2.8), the LM statstc, the LR-based statstc Λ and all the statstcs of the form F [K () ], where K () s some (non-empty) subset of {j :1 j p, j }, follow a jont dstrbuton (condtonal on X) that does not depend on any unknown parameter under the null hypothess H 0 :Σ=D p (σ 2 ).If furthermore D p (σ 1 ) Σ (0) D p (σ 1 )=H(X, w) (4.18) where H(X, w) s a known functon of X and w, the same property holds for the QLR statstcs Λ (h),h 0. It s of nterest to note here that the pvotal property for the LR statstcs Λ could also be obtaned by usng the nvarance results for generalzed regressons models gven by Breusch (1980). However ths would not smplfy our proof and would not yeld the explct representaton provded by Proposton 4.2. As we wll see below, the latter can be useful for mplementng MC tests. The fact that the LM, Harvey-Phllps, LR and QLR statstcs have nusance-parameter-free null dstrbutons entals that MC tests can be appled here to obtan a fnte-sample verson of the correspondng tests. Such tests can be mplemented as follows. Consder a test statstc T for H 0 wth a contnuous nusance-parameter-free null dstrbuton, suppose H 0 s rejected when T s large [.e., whent c(α),where P [T c(α)] = α under H0 ], and denote by G(x) =P[T x]ts survval functon under the null hypothess. Let T 0 be the test statstc computed from the observed data. Then the assocated crtcal regon of sze α may be expressed as G(T 0 ) α. By Monte Carlo methods, generate N ndependent realzatons T 1,..., T N of T under H 0. Now compute the randomzed p-value ˆp N (T 0 ), where ˆp N (x) = NĜN(x)+1 N+1, (4.19) Ĝ N (x) = 1 N N I [0, ) (T x), I A (x)= =1 { 1, f x A 0, f x/ A. 14

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