Testing for Cross-Sectional Dependence in a Random Effects Model

Transcription

1 Opn Journl of Sttstcs,,, Publshd Onln Jnury ( stng for Cross-Sctonl pndnc n Rndom Effcts Modl Afs Slsu, Sm Olofn, Eugn Kouss Cntr for Economtrcs nd Alld Rsrch (CEAR), prtmnt of Economcs, Unrsty of bdn, bdn, gr prtmnt of Economcs, Unrsty of Cocody, Abdjn, Cot d or Eml: slsu@crorgng Rcd cmbr 4, rsd cmbr 8, ccptd Jnury, ABSRAC hs ppr xtnds nd gnrlzs th works of [,] to llow for cross-sctonl dpndnc n th contxt of two-wy rror componnts modl nd consquntly dlops LM tst h cross-sctonl dpndnc follows th frst ordr sptl utorgrss rror (SAE) procss nd s mposd on th rmndr dsturbncs t s mportnt to not tht ths ppr dos not consdr ltrnt forms of sptl lg dpndnc othr thn SAE t lso dos not llow for ndognty of th rgrssors nd rqurs th normlty ssumpton to dr th LM tst Kywords: Cross-Sctonl pndnc Error Componnts Modl Lgrngn Multplr (LM) sts ntroducton h stndrd rror componnts modl ssums, mong othrs, sptl ndpndnc cross cross-sctonl unts Howr, ths rstrct ssumpton my not hold for lot of pnl dt pplctons Whn on bgns to look t cross scton of rgons, stts, countrs, tc, ths ggrgt unts my xhbt cross-sctonl corrlton tht hs to b dlt wth (s [3]) gnorng cross-sctonl dpndnc whn n fct t xsts, rsults n bsd, nconsstnt nd nffcnt stmts of rgrsson coffcnts (s [,4,5]) n th ltrtur, srl tst sttstcs h bn dlopd for sptl conomtrcs howr n th contxt of thr cross sctonl frmwork or on-wy rror componnts modl h spcfcton of cross-sctonl dpndnc n lnr rgrsson modls by most of ths works follows thr sptl utorgrss (SAR) procss oftn dfnd s sptl lg dpndnc (s [6-]) sptl mong rg procsss (SMA) oftn clld sptl rror dpndnc (s []) sptl utorgrss rror procss (SAE) (s [6,,3]) SARMA ( combnton of SAR nd SMA) (s [,4,4]) combnton of SAR nd SARE (s [5]) drct rprsntton form of cross-sctonl dpndnc (s [6,7]) or sptl rror componnt procss (SEC) suggstd by [8] Consquntly, rous tsts s wll s stmtors wr drd gnst ths dffrnt spcfcton forms usng thr th Mxmum Lklhood (ML) pproch (s [,9] A rw of scor tst sttstcs for ltrnt spcfctons n sptl conomtrcs n th contxt of cross sctonl dt nd on-wy rror componnts modl cn b found n [,3] rspctly for sury of th ltrtur) or nstrumntl Vrbls (V) nd Gnrlzd Mthod of Momnts (GMM) (s [9,,]) h prsnt study dlops LM tst for cross-sctonl dpndnc n th contxt of pnl dt frmwork h lttr s two-wy rndom ffcts modl whr th cross-sctonl dpndnc follows th SAE nd s mposd on th rmndr dsturbncs Promnnt pprs tht h doptd th SAE nclud [,4] n th contxt of cross-sctonl frmwork, nd [,,3] n th contxt of on-wy rror componnts frmwork hus, th mn objct of ths work s to xtnd nd gnrlz th works of [,] to llow for cross-sctonl dpndnc n th contxt of two-wy rror componnts modl h pnl dt modl consdrd hr s th rstrctd twowy rndom ffcts modl ssumng no cross-sctonl dpndnc n th rmndr dsturbncs hus, th LM tst wll b smlr to th on dlopd by [] f w furthr modfy th hypothss to tst for cross-sctonl dpndnc ssumng th prsnc of rndom nddul ffct only (whl gnorng th prsnc of tm ffcts) n th sm n, th LM tst wll b smlr to [,4] f th hypothss s rconstructd to tst for cross-sctonl dpndnc gnorng th prsnc of both th rndom country nd tm ffcts n Scton, th structur of th two-wy rror rndom ffcts modl s dscrbd n th contxt of crosssctonl dpndnc n th rmndr dsturbnc trm Anlyss of th LM tst r prodd n Scton 3 nd Scton 4 concluds th ppr Copyrght ScRs

2 A SALSU E AL 89 h Modl mtrx, s k ctor nd u s ctor h mtrx X s ssumd to b of full column rnk nd ts W consdr th followng pnl dt rgrsson modl: lmnts r ssumd to b symptotclly boundd n yt xt ut,, t,, () bsolut lu Gn Equton (6), Equton (3) cn b whr th ndx dnots rgonl unts nd th ndx t r-wrttn s: rfrs to th obsrtons of ch rgon h subscrpt, u t W t t (9) thrfor, dnots th cross-sctonl dmnson W cn wrt Equton (8) n ctor from s: whrs t dnots tm-srs dmnson h totl numbr of obsrtons s yt s th obsrton on th u B () th rgon or th tth tm prod x t s th tth obsrton on k xplntory rbls nd u t s th h rnc-cornc (VCV) mtrx () of Equton () (tht s, th unrstrctd modl) cn b x- rgrsson dsturbnc trm h rror trm u follows t two-wy rndom ffcts wth both rgonl spcfc nd prssd s: tmporl ffcts tht s, BB J ut t t () () whr dnots rgonl spcfc ffcts, dnots t tmporl ffcts nd t rprsnts th rmndr dstur- whr J nd t s mtrx of ons of dmnson bnc trm Stckng th obsrtons of ch tm- o obtn th spctrl dcomposton of Equton (), prod t, Equton () my b wrttn s: w us th [4] mthod Essntlly, w rplc J by ut t t (3) J nd by E J whr E J nd J J nd consquntly, w obtn 3 : whr u u,, t ut, ut,, t, s ctor of ons of dmnson, t t, t,, t nd E,,, BB () Assumpton : Both nd t r ssumd ndpndnt J BB nd normlly dstrbutd ccordng to, Also, usng th [5] mthod of nrson, Equton () ~, t (4) cn b xprssd s: h rmndr dsturbnc trm ( t ) s ssumd to follow th frst ordr sptl rror corrlton (s [,3]), tht s: W (5) t t t,,,, whr t t t nd t t t h trm s th sclr sptl utorgrss coffcnt wth h mtrx W s n sptl wght mtrx whch rprsnts th dgr of potntl ntrcton btwn nghborng loctons whos dgonl lmnts r zro nd off-dgonl lmnts r non-zro Equton (5) cn b furthr smplfd s: W (6) t t Gn Equton (6), th wght mtrx W lso stsfs th condton tht W s nonsngulr for ll t s lso ssumd to b ndpndnt nd normlly dstrbutd s: ~, (7) t t h t procss s lso ndpndnt of th nd trms h modl () cn b r-wrttn n mtrx notton s: y X u (8) whr y s of dmnson ctor, X s n k E BB whr J BB (3) E A J A A BB A BB 3 rton of th LM st nd n ths scton, w dr th LM tst for tstng for no cross-sctonl dpndnc n two-wy rndom ffcts modl W mploy th Mxmum Lklhood (ML) pproch nd consquntly, th log-lklhood functon h LM tst drd s bsd on th d tht th scor of th lklhood functon lutd undr th null s qul to zro whn th null hypothss s tru, so tht tst bsd on th squr of th scor ddd by th pproprt lmnt of th nformton mtrx (snc ths s th rnc of th scor) cn b constructd h us of th norml lklhood functon rqurs th ssumpton of S th ppndx for th drton 3 ot tht E nd J r symmtrc dmpotnt mtrcs Copyrght ScRs

3 9 A SALSU E AL normlty of th rror trm Essntlly, th drton of th LM tst nols th followng stps: Stp : r th VCV mtrx for th unrstrctd modl Stp : r th VCV mtrx for th rstrctd modl Stp 3: r th spctrl dcomposton for th mtrcs obtnd n stps nd Stp 4: r th nrs of th mtrcs obtnd n stps nd usng th rsults from stp 3 Stp 5: r th gnrl log-lklhood functon Stp 6: Us th nformton n stps - 5 to dr th scor functons of th lklhood lutd from th rstrctd ML undr H Stp 7: r th nformton mtrx nd ts nrs Stp 8: Us th rsults obtnd n stps 6 nd 7 to dlop th LM tst h log lklhood functon, L undr normlty of dsturbncs s gn s: og u L, c l u (4) whr u y X nd th ctor of prmtrs s d- notd s,,,,, whr,,, Snc our tst sttstc rqurs nformton only on th ctor of prmtrs, consquntly, nformton du to s gnord Followng [6], th grdnt of th log lklhood wth rspct to cn b xprssd s: L tr u L E j u By furthr smplfcton, t s sy to show tht: tr j (5) (6) For, j,,3,4 Equtons (5) nd (6) rprsnt th scor functon nd th nformton mtrx rspctly h nformton mtrx- s block dgonl h LM sttstc cn, thrfor, b wrttn gnrlly s: LM (7) whr nd r th scor functon nd nformton mtrx rspctly lutd t th null hypothss h LM tst sttstc xprssd n (7) s dstrbutd s ( ch-squr dstrbutd) wth k dgrs of k frdom, k bng th numbr of prmtrs n th ctor Bsd on Equton (7), thrfor, th followng hypothss cn b tstd n rlton to cross-sctonl dpndnc: H : (8) hs s tst of no cross-sctonl dpndnc ssumng th prsnc of rndom nddul nd tm ffcts hs s th null hypothss ths study sts out to tst H (9) b : hs hypothss tsts for cross-sctonl dpndnc ssumng th prsnc of rndom nddul ffct only (whl gnorng th prsnc of tm ffcts) hs tst s smlr to [] LM tst for sptl rror corrlton s wll s rndom country ffcts H () c : hs hypothss tsts for cross-sctonl dpndnc gnorng th prsnc of both th rndom country nd tm ffcts hs s smlr to th LM tst by [,4] W dr blow th scor functon for th null hypothss xprssd n (8) bo whch s th focus of ths ppr tht s: H : Undr th null hypothss n (8), th VCV mtrx rducs to: 4 E () J Gn tht thn t t nd, thrfor, Vr t Vr t h Equton () s th VCV mtrx for th rstrctd modl Usng [5] Lmm, th nrs of Equton () cn b xprssd s: E J () E A J A whr A nd A h Equton () s th rducd form of Equton (3) nd s lso th VCV mtrx for th fmlr two-wy rndom ffcts rror componnts modl n ddton, t s prncpl componnt rqurd n th log-lklhood functon to dr th LM tst n prtculr, both Equtons () nd () r rqurd to dr th prtl drts nd nformton mtrx for th LM tst 4 S th ppndx for th drton Copyrght ScRs

4 A SALSU E AL 9 Usng th gnrl formuls on log lklhood dffrntton, w dr ts grdnts lutd t th rstrctd ML undr H s follows: Rcll Equton (5): L tr u u Assumpton : Lt M BB, thn M BB WB BW Rcll, B W nd snc undr H, thn B nd M W W Assumpton 3: f E, ndj r dmpotnt nd symmtrc mtrcs, w cn wrt tht E J whr E J hn, E nd J r orthogonl (s [3]) Proposton : Bsd on ssumptons nd 3, w cn wrt th drts for th prmtrs,,, nd Proof: (A), rspctly, s: M J E J BB E BB J BB BB BB E BB J (4) Bsd on ssumpton, t s sy to stblsh from Equton (4) tht: E J E J M M 5 W lso rplc by whr E J (5) 5 B E J E J E J E J C E J E J J J J E J E J E J E J E J (6) (7) Copyrght ScRs

5 9 A SALSU E AL (8) Proposton : Bsd on proposton nd ssumptons nd 3, w cn wrt th drtons of for th prmtrs,,, nd E, rspctly, s: AJ A M E A J J A A E A A J Proof: hs drts r qut strghtforwrd to show prtculrly usng th nformton n proposton Proposton 3: Bsd on propostons nd nd ssumptons nd 3, w cn wrt th drtons of, rspctly, s: for th prmtrs,,, nd E A MA J A MA E A J A H J A E A A J A A Proof: hs drts r strghtforwrd to show usng th nformton n proposton Proposton 4: Followng propostons - 3, w cn L sly clcult th prtl drts for,, nd, rspctly, lutd t th rstrctd MLE: L M M Ξ Ξ whr L M M M M g Ξ u E Mu Ξ u J M u u E u E A u u E u J A u ** tr A A whr uu u u ** L L tr nd ** A tr A A nd whr u E u u J u Proof: S th ppndx for furthr smplfctons nd proofs of th prtl drts Rcll tht w dfn,,,, thrfor, undr H,,,, cn b dfnd s th soluton obtnd ftr mxmzton of th frst ordr condton nd u y X MLE s th corrspondng rsdul undr H ot tht ll th prmtrs,,, wr lutd whn t th rstrctd MLE xcpt hs s bcus w r tstng whthr s sttstclly dffrnt from zro hus, th prtl drts undr H r rwrttn n ctor form s: M M Ξ Ξ Also, usng th mthod dlopd by [7], w obtn Copyrght ScRs

6 A SALSU E AL 93 th nformton mtrx undr H h nformton mtrx s gn by: L E tr j j (9) Proposton 5: Usng th formulr xprssd n Equton (9) nd nformton n proposton, w cn dr rspct lmnts n undr H for th ctor of prmtrs,,, s follows: L E tr E A J A L E tr E AJ A L E J A tr L E A tr L E tr AM tr A L E tr tr AM A M L E tr A AM tr A AM E L A A tr tr L E tr L E tr A A A Gn ths nformton undr H, th LM sttstc s gn by, 6 LM Undr, LM s dstrbutd s h sttstc xprssd n (3) s th LM tst sttstc, whch tsts for no cross-sctonl dpndnc n two-wy rndom ffcts modl (3) 6 tls of drtons of th nformton mtrx cn b prodd on rqust cson Crtr: h LM sttstc s sclr nd th lu obtnd whn th tst s prformd on th two-wy rror componnts modl s comprd wth th crtcl lu for th ch-squrd dstrbuton h ntnton s to scrtn whthr to rjct th null hypothss, H, tht thr s no cross-sctonl dpndnc problm n two-wy rndom ffcts modl Essntlly, f LM s lss thn th crtcl lu for th ch-squrd dstrbuton, thn, w do not rjct th null hypothss mplyng tht thr s no cross-sctonl dpndnc othrws, w rjct t 4 Concludng Rmrks hs ppr prods frmwork for tstng for no crosssctonl dpndnc ssumng th prsnc of rndom nddul nd tm ffcts hus, srl mportnt ssus h not bn ncorportd hs nclud tstng othr hypothss rlr spcfd, tht s b : whch tsts for crosssctonl dpndnc ssumng th prsnc of rndom nddul ffct only (whl gnorng th prsnc of tm c ffcts nd : whch tsts for cross-sctonl dpndnc gnorng th prsnc of both th rndom country nd tm ffcts Also, th mprcl pplctons scton nolng Mont Crlo xprmnts s lso not yt consdrd hs r som of th suggstons for futur rsrch REFERECES [] B H Bltg, S H Song nd W Koh, stng Pnl t Rgrsson Modls wth Sptl Error Corrlton, Journl of Economtrcs, Vol 7, o, 3, pp 3-5 do:6/s34-476(3)-9 [] L Ansln, Ro s Scor sts n Sptl Economtrcs, Journl of Sttstcl Plnnng nd nfrnc, Vol 97, o,, pp 3-39 do:6/s ()349-9 [3] B H Bltg, Economtrc Anlyss of Pnl t, 6th Edton, Wly, Chchstr, 8 [4] L Ansln, Sptl Economtrcs: Mthods nd Modl, Kluwr Acdmc Publshrs, ordrcht, 988 [5] L Ansln nd S Ry, Proprts of sts for Sptl pndnc n Lnr Rgrsson Modls, Gogrphcl Anlyss, Vol 3, o, 99, pp -3 do:/j tb8x [6] J Ord, Estmton Mthods for Modls of Sptl ntrcton, Journl of th Amrcn Sttstcl Assocton, Vol 7, o 349, 975, pp -6 do:37/85387 [7] A Brndsm nd R Ktllppr, Furthr Ednc on Altrnt Procdurs for stng of Sptl Autocorrlton mong Rgrsson sturbncs, n: C Brtls nd R Ktllppr, Eds, Explortory nd Explntory Anlyss n Sptl t, Mrtnus jhoff, Boston, 979, Copyrght ScRs

7 94 A SALSU E AL pp -36 do:7/ _5 [8] H Bloommstn, Spcfcton nd Estmton of Sptl Economtrc Modls: A scusson of Altrnt Strtgs for Sptl Economc Modlng, Rgonl Scnc nd Urbn Economcs, Vol 3, o, 985, pp 5-7 do:6/66-46(83)96-9 [9] H Kljn nd R Pruch, A Gnrlzd Momnts Estmtor for th Autorgrss Prmtr n Sptl Modl, ntrntonl Economc Rw, Vol 4, o, 999, pp do:/ [] H H Kljn, R Pruch nd E Yusfoch, nstrumntl Vrbl Estmton of A Sptl Autorgrss Modl wth Autorgrss sturbncs: Lrg nd Smll Smpl Rsults, n: J LSg nd R K Pc, Eds, Sptl nd Sptotmporl Economtrcs (Adncs n Economtrcs), Vol 8, Elsr, w York, 4, pp do:6/s73-953(4)85-5 [] R Hnng, Sptl t Anlyss n th Socl nd Enronmntl Scncs, Cmbrdg Unrsty Prss, Cmbrdg, 988 [] L Ansln nd A K Br, Sptl pndnc n Lnr Rgrsson Modls wth n ntroducton to Sptl Economtrcs, n: A Ullh nd E A Gls, Eds, Hndbook of Appld Economc Sttstcs, Mrcl kkr, w York, 998 [3] P Burrdg, On th Clff-Ord st for Sptl Autocorrlton, Journl of th Royl Sttstcl Socty, Vol 4, 98, pp 7-8 [4] J S Hung, h Autorgrss Mong Arg Modl for Sptl Anlyss, Austrln Journl of Sttstcs, Vol 6, o, 988, pp do:/j467-84x984tb3x [5] A Cs, Sptl Pttrns n Houshold mnd, Economtrc, Vol 59, o 4, 99, pp do:37/93868 [6] K V Mrd nd R J Mrshll, Mxmum Lklhood Estmton of Modls for Rsdul Cornc n Sptl Rgrsson, Bomrk, Vol 7, o, 984, pp do:93/bomt/735 [7] K V Mrd, Mxmum Lklhood Estmton for Sptl Modls, n: A Grffth, Ed, Sptl Sttstcs: Pst, Prsnt nd Futur, nsttut of Mthmtcl Gogrphy, Ann Arbor, 99, pp 3-5 [8] H Kljn nd P Robnson, Sptl Corrlton: A Suggstd Altrnt to th Autorgrss Modl, n: L Ansln nd R Florx, Eds, w rctons n Sptl Economtrcs, Sprngr-Vrlg, Brln, 995, pp do:7/ _3 [9] L Ansln nd R Morno, Proprts of sts for Sptl Error Componnts, Rgonl Scnc nd Urbn Economcs, Vol 33, o 5, 3, pp do:6/s66-46(3)8-5 [] L Ansln, Som Robust Approchs to stng nd Estmton n Sptl Economtrcs, Rgonl Scnc nd Urbn Economcs, Vol, o, 99, pp -7 do:6/66-46(9)9-j [] H Kljn nd R Pruch, A Gnrlzd Sptl wo Stg Lst Squrs Procdur for Estmtng Sptl Autorgrss Modl wth Autorgrss sturbncs, Journl of Rl Estt Fnnc nd Economcs, Vol 7, o, 998, pp 99- do:3/a: [] B H Bltg, S H Song, B C Jung nd W Koh, stng for Srl Corrlton, Sptl Autocorrlton nd Rndom Effcts Usng Pnl t, Journl of Economtrcs, Vol 4, o, 7, pp 5-5 do:6/jjconom69 [3] B H Bltg, S H Song nd J H Kwon, stng for Htroscdstcty nd Sptl Corrlton n Rndom Effcts Pnl t Modl, Computtonl Sttstcs nd t Anlyss, Vol 53, o 8, 9, pp do:6/jcsd869 [4] J Wnsbk nd A Kptyn, A Smpl Wy to Obtn th Spctrl composton of Vrnc Componnts Modls for Blncd t, Communctons n Sttstcs, Vol, o 8, 98, pp 5- do:8/ [5] J R Mgnus, Multrt Error Componnts Anlyss of Lnr nd onlnr Rgrsson Modls by Mxmum Lklhood, Journl of Economtrcs, Vol 9, o -3, 98, pp do:6/34-476(8)95-7 [6] J R Mgnus nd H udckr, Mtrx ffrntl Clculus wth Applctons n Sttstcs nd Economtrcs, Wly Srs n Probblty nd Sttstcs, Chchstr, 988 [7] A Hrll, Mxmum Lklhood Approchs to Vrnc Componnt Estmton nd to Rltd Problms, Journl of th Amrcn Sttstcl Assocton, Vol 7, o 358, 977, pp do:37/86796 Copyrght ScRs

8 A SALSU E AL 95 Appndx (A) rton of th VCV Mtrx for th Unrstrctd Modl Hr, nd th VCV mtrx of u cn b drd s follows Rcll Equton (), u B Usng ssumpton (), th VCV mtrx cn b xprssd s: E uu B E B E E Lt E uu n ths cs b rprsntd by, nd by furthr smplfcton, (A) bcoms: BB BB J (A) (A) (A3) whr J nd t s mtrx of ons of dmnson o obtn th spctrl dcomposton of (A3), w us th [4] mthod whch nols rplcng J by J nd by E J whr E J nd J J n (A3) hs s don s follows: E J BB J E J E BB J BB J E E BB E J J BB J J E BB J BB J (A4) Usng th [5] mthod of nrson, thrfor, th nrs of Equton (A5) cn b xprssd s: BB E J BB E A J A (A5) whr A BB A BB nd (B) rton of th VCV Mtrx for th Rstrctd Modl Hr, nd s consqunc, Vr t Vr t Gn ths ssumpton, Equton () rducs to: u (B) h n, usng ssumpton (), th VCV mtrx cn b xprss d s: E E uu E E hus, (A4) undr th unrstrctd modl rducs to: BB J (B) (B3) (B4) Just s bfor, w us th [4] mthod to obtn th spctrl dcomposton of (B4) nd followng th sm procdur s Appndx A, w h: E (B5) J J Smlrly, usng th [5] mthod of nrson, cn b xprssd s: E J J E A J A whr A ( B6) nd A J (C) rton of th Prtl rts L tr u u L Copyrght ScRs

9 96 A SALSU E AL L tr u u tr E AM J AM u E AMA J AMAu tr E AM J AM u E AMA J AMAu tr A AM ue AMA uuj AMAu (C) ot tht: tr A whr g hrfor, tr A g g g Mtr AM tr A A M tr A By som lgbrc smplfctons, w cn wrt tht: tr A A M M ot furthr tht: n whch cs, thrfor Smlrly, g u E AMA u u E A E M M E A u u E u E A u, Mu Ξ u E AMA u u E u J A MA u u J A J M J A u n whch cs, thrfor u J u J A u, u J A MA u u J M u Ξ As consqunc, w cn wrt (C) s: L H whr M M Ξ Ξ ( C) M M M M g L tr u u H tr E A J A u E A J A u tr E A J A u E A J A u (C3) Usng th nformton ldng to (C), w cn pro tht: tr A A g g g And lso wth th rprsnttons tht: u E A u u E A E A u, n whch cs, thrfor nd n th sm n, u E u E A u, u E A u uu u J A u u u Gn ths nformton thrfor, (C3) bcoms: Copyrght ScRs

10 A SALSU E AL 97 L g g g (C4) L nd tr u H tr J A u J tr A u J A A Usng th nformton tht: (C5) bcoms, L u u u J A u u u tr A g g g tr E A J A u g g g (C5) (C6) L tr u u ue A Au uj A Au tr A A u E u u J u L g g g u E u nd whr u J u (C7) Copyrght ScRs