Liklihood Raio Basd ss for Markov Rgim Swiching Zhongjun Qu y Boson Univrsiy Fan Zhuo z Boson Univrsiy Fbruary 4, 07 Absrac Markov rgim swiching modls ar widly considrd in conomics and nanc. Alhough hr hav bn prsisn inrss s.g., Hansn, 99, Garcia, 998, and Cho and Whi, 007, h asympoic disribuions of liklihood raio basd ss hav rmaind unknown. his papr considrs such ss and sablishs hir asympoic disribuions in h conx of nonlinar modls allowing for mulipl swiching paramrs. h analysis simulanously addrsss hr di culis: i som nuisanc paramrs ar unidni d undr h null hypohsis, ii h null hypohsis yilds a local opimum, and iii h condiional rgim probabiliis follow sochasic procsss ha can only b rprsnd rcursivly. Addrssing hs issus prmis subsanial powr gains in mpirically rlvan siuaions. Bsids obaining h ss asympoic disribuions, his papr also obains four ss of rsuls ha can b of indpndn inrs: a characrizaion of condiional rgim probabiliis and hir high ordr drivaivs wih rspc o h modl s paramrs, a high ordr approximaion o h log liklihood raio prmiing mulipl swiching paramrs, a r nmn o h asympoic disribuion, and 4 a uni d algorihm for simulaing h criical valus. For modls ha ar linar undr h null hypohsis, h lmns ndd for h algorihm can all b compud analyically. h abov rsuls also shd ligh on why som boosrap procdurs can b inconsisn and why sandard informaion criria, such as h Baysian informaion cririon BIC, can b snsiiv o hypohss and modl s srucur. Whn applid o h US quarrly ral GDP growh ras, h mhods suggs fairly srong vidnc favoring h rgim swiching spci caion consisnly ovr a rang of sampl priods. Kywords: Hypohsis sing, liklihood raio, Markov swiching, nonlinariy. JEL cods: C, C, E. W hank Jams Hamilon, Chuqing Jin, Hiroaki Kaido, Frank Klibrgn, Pirr Prron, Douglas Sigrwald and sminar paricipans a Amsrdam, Brown, BU saisics, UCSD, h 06 Economric Sociy Winr Ming, h NBER-NSF im sris confrnc, 04 JSM, h rd SNDE, and h h World Congrss oh Economric Sociy for valuabl suggsions, and Carrasco, Hu and Plobrgr for making hir cod availabl. y Dparmn of Economics, Boson Univrsiy, 70 Bay Sa Rd., Boson, MA, 05 qu@bu.du. z Dparmn of Economics, Boson Univrsiy, 70 Bay Sa Rd., Boson, MA, 05 zhuo@bu.du.
Inroducion Markov rgim swiching modls ar widly considrd in conomics and nanc. Hamilon 989 is a sminal conribuion, which provids no only a framwork for dscribing conomic rcssions, bu also a gnral algorihm for lring, smoohing and maximum liklihood simaion whil building on h work of Goldfld and Quand 97 and Cossl and L 985. Survys ohis voluminous liraur can b found in Hamilon 008, 06 and Ang and immrmann 0. hr approachs hav bn considrd for dcing rgim swiching. h rs approach involvs ranslaing his issu ino sing for paramr homogniy agains hrogniy. Nyman and Sco 966 sudid h C s. Chshr 984 drivd a scor s and showd ha i is closly rlad o h informaion marix s of Whi 98. Lancasr 984 and Davidson and MacKinnon 99 ar rlad conribuions. Wason and Engl 985 dsignd a s saisic ha allows h hrogniy o follow a saionary AR procss. Carrasco, Hu and Plobrgr 04 furhr dvlopd his approach by considring gnral dynamic modls and allowing h hrogniy o follow xibl wakly dpndn procsss. hy analyzd a class oss and showd ha hy ar asympoically locally opimal agains a spci c alrnaiv characrizd in hir papr. h abov ss hav wo common faurs. Firs, hy only rquir simaing h modl undr h null hypohsis. Scond, hy ar dsignd for dcing paramr hrogniy, no paricularly Markov rgim swiching. Alhough h ss can hav powr agains a broad class of alrnaivs, hir powr can b subsanially lowr han wha is achivabl ih paramrs indd follow a ni sa Markov chain. h scond approach, du o Hamilon 996, is o conduc gnric ss oh hypohsis ha a K-rgim modl.g., K = adqualy dscribs h daa. h insigh is ha if a K-rgim spci caion is accura, hn h scor funcion should hav man zro and form a maringal diffrnc squnc. Ohrwis, h modl should b nrichd o allow for addiional faurs, in som siuaions by inroducing an addiional rgim. Hamilon 996 dmonsrad how o implmn such ss as a by-produc of calculaing h smoohd probabiliy ha a givn obsrvaion is from a paricular rgim. his maks h ss simpl and widly applicabl. Manwhil, i rmains dsirabl o hav sing procdurs ha focus on dcing Markov swiching alrnaivs. h hird approach procds undr h quasi liklihood raio principl. h quasi liklihood funcions ar consrucd assuming a singl rgim undr h null hypohsis and wo rgims undr h alrnaiv hypohsis. h analysis facs hr challngs. i Som nuisanc paramrs ar unidni d undr h null hypohsis. his givs ris o h Davis 977 problm. ii h
null hypohsis yilds a local opimum c.f. Hamilon, 990. Consqunly, a scond ordr aylor approximaion o h liklihood raio is insu cin for analyzing is asympoic propris. iii h condiional rgim probabiliy h probabiliy of bing in a paricular rgim a im givn h informaion up o im follows a sochasic procss ha can only b rprsnd rcursivly. h rs wo di culis ar also prsn whn sing for mixurs. I is h simulanous occurrnc of all hr di culis ha plagus h sudy oh liklihood raio in h currn conx. For xampl, whn analyzing high ordr xpansions oh liklihood raio, i is ncssary o sudy high ordr drivaivs oh condiional rgim probabiliy wih rspc o h modl s paramrs. So far, hir saisical propris hav rmaind lusiv. Consqunly, h asympoic disribuion of h log liklihood raio has also rmaind unknown. Svral imporan progrsss hav bn mad by Hansn 99, Garcia 998, Cho and Whi 007, and Carr and Sigrwald 0. Hansn 99 clarly documnd why h di culis i and ii caus h convnional approximaion o h liklihood raio o brak down. Furhr, h rad h liklihood funcion as a sochasic procss indxd by h ransiion probabiliis i.., h probabiliis of rmaining in h rs rgim p and rmaining in h scond rgim q and h swiching paramrs, and drivd a bound for is asympoic disribuion. His rsul provids a plaform for conducing consrvaiv infrnc. Garcia 998 suggsd an approximaion o h log liklihood raio ha would follow ih scor had a posiiv varianc a h null simas. Rsuls in h currn papr will show ha his disribuion is in gnral di rn from h acual limiing disribuion. Rcnly, Cho and Whi 007 mad a signi can progrss. hy suggsd a quasi liklihood raio QLR s agains a wo-componn mixur alrnaiv i.., a modl whr h currn rgim arrivs indpndnly of is pas valus. hr, h di culy iii is avoidd bcaus h condiional rgim probabiliy is rducd o a consan, which can furhr b rad as an addiional unknown paramr. Carr and Sigrwald 0 furhr discussd a consisncy issu rlad o QLR s. h currn papr maks us of svral imporan chniqus in Cho and Whi 007. A h sam im, i gos byond hir framwork o dircly confron Markov swiching alrnaivs. As will b sn, h powr gains from doing so can b qui subsanial. Spci cally, his papr considrs a family of liklihood raio basd ss and sablishs hir asympoic disribuions in h conx of nonlinar modls allowing for mulipl swiching paramrs. h framwork ncompasss h imporan spcial cass osing for rgim swiching in auorgrssiv modls and in auorgrssiv disribud lags modls. hroughou h analysis, h modl has wo rgims undr h alrnaiv hypohsis. Som paramrs can rmain consan
across h wo rgims. h analysis is srucurd ino v sps: Sp characrizs h condiional rgim probabiliy and is high ordr drivaivs wih rspc o h modl s paramrs. Whn valuad undr h null hypohsis, h probabiliy rducs o a consan whil h drivaivs can all b rprsnd as linar rs ordr di rnc quaions wih laggd co cins qual o p + q. Bcaus 0 < p; q <, hs quaions ar all sabl and amnabl o h applicaions of uniform laws of larg numbrs and funcional cnral limi horms. his novl characrizaion is a criical sp ha maks h subsqun analysis fasibl. Sp drivs a fourh ordr aylor approximaion o h liklihood raio for xd p and q. his sp builds on Cho and Whi 007, bu gos byond i o accoun for h c oh im variaion in h condiional rgim probabiliy on h liklihood raio. h rsuls ar informaiv abou why subsanial powr gains rlaiv o h QLR s ar possibl. Sp obains an approximaion o h liklihood raio as an mpirical procss indxd by p and q. h valus of p and q ar rquird o b sricly bwn 0 and saisfying p + q + wih bing som arbirarily small posiiv consan. hs rquirmns ar compaibl wih applicaions in macroconomics and nanc; s h discussion in Scion. h mpirical procss prspciv undrakn hr follows a rich array of sudis, including Hansn 99, Garcia 998, Cho and Whi 007, and Carrasco, Hu and Plobrgr 04. Sp 4 provids a ni sampl r nmn. his is moivad by h obsrvaion ha, whil h limiing disribuion in Sp is adqua for a broad class of modls, i can lad o ovrrjcions whn a furhr singulariy spci d lar is prsn. his problm is addrssd by analyzing a sixh ordr xpansion oh liklihood raio along h lin p + q = and an ighh ordr xpansion a p = q = =. h lading rms ar hn incorporad ino h asympoic disribuion o safguard agains hir cs. his lads o a r nd disribuion ha dlivrs rliabl approximaions hroughou our xprimnaions. Sp 5 oulins an algorihm for simulaing h r nd asympoic disribuion. For linar modls, h lmns ndd for his algorihm can all b compud analyically. h asympoic disribuion shows som uncommon faurs. Firs, nuisanc paramrs, hough consraind o b consan across h rgims, can a c h limiing disribuion. Scondly, propris oh rgrssors i.., whhr hy ar sricly or wakly xognous also a c h disribuion. hirdly, h disribuion dpnds on which paramr i.., h inrcp, h slop, or h varianc oh rrors is allowd o swich. hs faurs imply ha som boosrap procdurs can b inconsisn and ha sandard informaion criria, such as BIC, can b snsiiv
o h hypohsis and h modl s srucur. h abov implicaions ar discussd in Scion 6. W conduc simulaions using a daa gnraing procss DGP considrd in Cho and Whi 007. h rsuls show ha h powr di rnc can b larg whn h rgims ar prsisn, a siuaion ha is common in pracic. W also apply h sing procdur o h US quarrly ral GDP growh ras, ovr h priod 960:I-04:IV and a rang of subsampls. h rsuls consisnly favor h rgim swiching spci caion. In addiion, h smoohd rgim probabiliis closly mirror NBER s rcssion daing. o our knowldg, his is h rs im such consisn vidnc for rgim swiching in h man oupu growh is documnd hrough hypohsis sing. Empirical sudis hav simad rgim swiching modls on a wid rang oim sris, including xchang ras, oupu growh, inrs ras, db-oupu raio, bond prics, quiy rurns, and consumpion and dividnd procsss Hamilon, 008. Rgim swiching has also bn incorporad ino DSGE modls; s Schorfhid 005, Liu, Waggonr and Zha 0, Bianchi 0, and Lindé, Sms and Wours 06. Doing so allows h ransmission mchanism oh conomy o b occasionally fundamnally di rn, a faur ha is byond h scop of consan-paramr linar modls. Howvr, du o h lack of mhods wih good powr propris, h prsnc of rgim swiching is rarly formally sd from a frqunis prspciv. h procdur in his papr can ponially hlp o narrow his gap. From a mhodological prspciv, his papr conribus o h liraur ha sudis hypohsis sing whn som rgulariy condiions fail o hold. Bsids h works mniond abov, closly rlad sudis includ h following. Davis 987, King and Shivly 99, Andrws and Plobrgr 994, 995, and Hansn 996 considrd ss whn a nuisanc paramr is unidni- d undr h null hypohsis. Andrws 00 sudid ss whn, in addiion o h abov faur, som paramrs li on h boundary oh mainaind hypohsis. Harigan 985, Ghosh and Sn 985, Lindsay 995, Liu and Shao 00, Chn and Li 009, and Gu, Konkr and Volgushv 0 ackld h issus of zro scor and/or unidni d nuisanc paramrs in h conx of mixur modls. Chn, Ponomarva and amr 04 considrd uniform infrnc on h mixing probabiliy in mixur modls whn nuisanc paramrs ar prsn. Ronizky, Cox, Boai and Robins 000 dvlopd a hory for driving h asympoic disribuion oh liklihood raio saisic whn h informaion marix has rank on lss han full; also s h discussions in hir papr pag 44 for ohr sudis on h sam issu in various conxs. Dovonon and Rnaul 0 sudid disribuions oss for momn rsricions whn h associad Jacobian marix is dgnra a h ru paramr valu. his papr is h rs ha simulanously ackls 4
h di culis i o iii in h hypohsis sing liraur. W conjcur ha h chniqus dvlopd can hav implicaions for hypohsis sing in ohr rlad conxs ha involv modls wih hiddn Markov srucurs. h papr procds as follows. Scion prsns h modl and h hypohss. Scion inroducs a family os saisics. Scion 4 sudis h asympoic propris oh log liklihood raio for xd p and q. Scion 5 prsns four ss of rsuls: a h wak convrgnc oh scond ordr drivaiv oh concnrad log liklihood, b h limiing disribuion oh s saisic, c a ni sampl r nmn, and d an algorihm for obaining h rlvan criical valus. Scion 6 discusss som implicaions oh hory for boosrapping and informaion criria. Scion 7 xamins h s s ni sampl propris. Scion 8 considrs an applicaion o h US ral GDP growh ras. Scion 9 concluds. All proofs ar in h appndix. h following noaions ar usd. jjxjj is h Euclidan norm of a vcor x. jjjj is h vcor inducd norm of a marix. x k and k dno h k-fold Kronckr produc of x and, rspcivly. h xprssion vca sands for h vcorizaion of a k dimnsional array A. For xampl, for a hr dimnsional array A wih n lmns along ach dimnsion, vca rurns a n -vcor whos i + j n + k n -h lmn quals Ai; j; k. fg is h indicaor funcion. For a scalar valud funcion f of R p, r f 0 dnos a p-by- vcor of parial drivaivs valuad a 0, r 0f 0 quals h ranspos of r f 0, and r j f 0 dnos is j-h lmn. In addiion, r j r j r jk f 0 dnos h k-h ordr parial drivaiv of f akn squnially wih rspc o h j ; j ; :::; j k -h lmn of valuad a 0. h symbols,! d and! p dno wak convrgnc undr h Skorohod opology, convrgnc in disribuion and in probabiliy, and O p and o p is h usual noaion for h ordrs of sochasic magniud. Modl and hypohss h modl is as follows. L fy ; x 0 g b a squnc of random vcors wih y bing a scalar and x a ni dimnsional vcor. L s b an unobsrvd binary variabl, whos valu drmins h rgim a im. D n h informaion s a im as = - ld :::; x 0 ; y ; x 0 ; y : L fj ; ; dno h condiional dnsiy of y, saisfying y j ; s v fj ; ; ; if s = ; fj ; ; ; if s = ; = ; ::::; : 5
his spci caion allows h vcor o swich bwn and, whil rsricing h vcor o rmain consan across h rgims. Hncforh, w abbrvia h wo dnsiis on h righ hand sid of as ; and ;, rspcivly. h rgims ar Markovian: ps = j ; s = ; s ; ::: = ps = js = = p; ps = j ; s = ; s ; ::: = ps = js = = q: h saionary or invarian probabiliy for s = is givn by p; q = q p q : Evaluad a 0 < p; q <, h log liklihood funcion associad wih is = L A p; q; ; ; 4 n o log ; j p; q; ; ; + ; j p; q; ; ; ; whr j dnos h probabiliy of s = givn, i.., j p; q; ; ; = ps = j ; p; q; ; ; = ; :::; ; 5 which sais s h following rcursiv rlaionship j p; q; ; ; = ; j p; q; ; ; ; j p; q; ; ; + ; j p; q; ; ; ; 6 +j p; q; ; ; = p j p; q; ; ; + q j p; q; ; ; : 7 hroughou h papr, w s h iniial valu j0 =. As shown lar, a di rn iniial valu dos no a c h asympoic rsuls. Whn = =, h log liklihood rducs o L N ; = log ; : 8 his papr sudis ss basd on 8 and 4 for h singl rgim spci caion agains h wo rgims spci caion givn in. o procd, w impos h following rsricions on h DGP and h paramr spac. L n and n dno h dimnsions of and. Assumpion i h random vcor x 0 ; y is sricly saionary, rgodic and -mixing wih mixing co cin saisfying c for som c > 0 and [0;. ii Undr h null hypohsis, y is gnrad by fj ; ;, whr and ar inrior poins of R n and R n wih and bing compac. 6
Par i is h sam as Assumpion A.i in Cho and Whi 007. As discussd hr, h -mixing condiion is commonly usd whn analyzing Markov procsss. I allows x o b a cd by rgim swiching undr h null hypohsis. Par ii spci s h ru paramr valus. h inrior poin rquirmn nsurs ha h asympoic xpansions considrd lar ar wll-d nd. Assumpion Undr h null hypohsis: i ; uniquly solvs max ; E[L N ; ]; ii for any 0 < p; q <, ; ; uniquly solvs max ; ; E[L A p; q; ; ; ]. Par i implis ha ; is globally idni d a ; undr h null hypohsis. Par ii implis ha hr dos no xis a wo-rgim spci caion i.., wih 6= ha is obsrvaionally quivaln o h singl-rgim spci caion i.., wih = =. h nx assumpion rlas h idni caion propris in Assumpion o som asympoic propris oh simaors. Assumpion Undr h null hypohsis: i [L N ; EL N ; ] = o p holds uniformly ovr ; wih P r 0 ; 0 0 log ; r 0 ; 0 log ; bing posiiv dfini in an opn nighborhood of ; for su cinly larg ; ii for any 0 < p; q <, [L A p; q; ; ; EL A p; q; ; ; ] = o p holds uniformly ovr ; ;. Assumpions rquirs 8 and 4 o saisfy uniform laws of larg numbrs. I allows 4 o hav mulipl local maximizrs. Undr Assumpions and, h maximizrs of 8 and 4 for 0 < p; q < convrg in probabiliy o ; and ; ; undr h null hypohsis. Assumpions o ar similar o hos usd in Cho and Whi 007, wih wo imporan di rncs. Firs, h liklihood funcion 4 corrsponds o a Markov swiching modl, no a mixur modl. Scond, mulipl paramrs ar allowd o b a cd by h rgim swiching. Using h abov noaion, h null and h alrnaiv hypohss can b sad as H 0 : = = for som unknown ; H : ; = ; for som unknown 6= and p; q 0; 0; : chnically, as discussd in Cho and Whi 007, h null hypohsis can also b formulad as: H0 0 : p = and = or H0 00 : q = and =. In H0 0, h modl rmains in h rs rgim wih probabiliy, any samn abou h scond rgim is irrlvan. h rvrs holds for H 00 0. Blow, w inroduc a modl ha will b usd hroughou h papr o illusra h main componns oh hory. 7
An illusraiv modl. Gaussian rrors: An imporan applicaion of rgim swiching is o linar modls wih y = z 0 + w 0 fsg + w 0 fs=g + u ; 9 whr ; and ar unknown ni dimnsional paramrs and u ar indpndnly normally disribud wih man zro. h variabls z and w can includ laggd valus of y. hrfor, h spci caion ncompasss ni ordr AR modls and ADL modls as spcial cass. In rms of and, = - ld :::; z 0 ; w0 ; y ; z 0 ; w 0 ; y and x 0 = z 0 ; w 0. hr hypohss can b sd dpnding on which paramrs ar allowd o swich: a Only h varianc of u can swich. L and dno is variancs undr h wo rgims. hn, in rlaion o, = ; = and 0 = 0 ; 0 wih = =. b Only h rgrssion co cins can swich. L dno h varianc of u. hn, =, = and 0 = 0 ;. c Boh componns can swich. hn, 0 = 0 ;, 0 = 0 ; and =. h rsuls in his papr covr all hr siuaions. In h mos gnral cas c, h dnsiis in ar givn by n o 4 ; p y z xp 0 w0 5 6 = 4 n o 7 ; p y z xp 0 w0 5 : h normaliy assumpion in his modl can b rplacd by ohr disribuional assumpions, providd ha ; and ; ar rplacd by h appropria dnsiis. W now illusra Assumpions - using his modl. For Assumpion, bcaus oh linariy, h -mixing of x 0 ; y is implid by ha of x. his is sais d if x follows a saionary VARMAP,Q procss P P j=0 B jx j = P Q j=0 A j" j wih " bing man zro i.i.d. random vcors whos dnsiy is absoluly coninuous wih rspc o Lbsgu masur on R dim" ; s Mokkadm 988. Ohr procsss ha ar -mixing wih a gomric ra of dcay, as rviwd in Chn 0, includ hos gnrad by hrshold auorgrssiv modls, funcional co cin auorgrssiv modls, and GARCH and sochasic volailiis modls. For Assumpion, is par i is sais d if Ex x 0 has full rank. Is par ii rquirs ha, ih daa ar gnrad by 6= wih 0 < p; q <, hn h condiional disribuion of y should xhibi faurs ha ar no capurd by h singl rgim linar spci caion. ha is, h rsuling Kullback-Liblr divrgnc should b posiiv. Finally, in Assumpion, h rank rquirmn ssnially rquirs P x x 0 o b posiiv d ni in larg sampls. h rs ohis assumpion rquirs uniform laws of larg numbrs o hold. Bcaus j p; q; ; ; is boundd bwn 0 and, hy hold undr Assumpion and mild condiions on h momns of y and x. 8
h s saisic his scion sudis wo issus. Firs, i considrs a family os saisics basd on h log liklihood raio. Scond, i xamins mpirically rlvan valus for h ransiion probabiliis p and q. h scond issu is imporan no only for making h ss pracically rlvan, bu also for h chnical analysis lar in h papr. L and dno h maximizr oh null log liklihood: ; = arg max ; LN ; : 0 h log liklihood raio valuad a som 0 < p; q < hn quals LRp; q = max L A p; q; ; ; ; ; L N ;. his lads o h following s saisic: SupLR = Sup LR p; q ; p;q whr is a compac s spci d blow and h suprmum is akn o obain h srongs vidnc agains h null hypohsis. Opraors ohr han h suprmum can also b usd. For xampl, following Andrws and Plobrgr 994 and Carrasco, Hu and Plobrgr 04, on can considr ExpLR = R LR p; q djp; q, whr Jp; q is a funcion ha assigns wighs on p and q. Such considraions lad o a family os saisics basd on LRp; q. his papr focuss on SupLR ; h rsuls xnd immdialy o ExpLR. W now xamin mpirically rlvan valus for h ransiion probabiliis p and q. Hamilon 008, h rs paragraph in p. rviwd aricls ha applid rgim swiching modls in a wid rang of conxs. Among hm, 0 aricls considrd wo-rgim spci caions wih consan ransiion probabiliis. hs sudis ar rlad o: xchang ras Jann and Masson, 000, oupu growh Hamilon, 989 and Chauv and Hamilon, 006, inrs ras Hamilon, 988, 005, Ang and Bkar, 00b, db-oupu raio Davig, 004, bond prics Dai, Singlon and Yang, 007, quiy rurns Ang and Bkar, 00a, and consumpion and dividnd procsss Garcia, Lugr and Rnaul, 00. Eighn ss of simas ar rpord. h valus oh ransiion probabiliis ar bwn 0:855 and 0:998 for h mor prsisn rgim and 0:740 and 0:997 for h ohr rgim. hs simas ar rprsnaiv of applicaions in conomics and nanc and hy srongly suggs wo faurs. Firs, non oh valus corrspond o mixurs. 9
ha is, h valus of p + q ar all subsanially abov.0. Scond, a las on rgim is fairly prsisn. ha is, h valu of p and q can b fairly clos o :0. Moivad by h abov obsrvaions, w suggs spcifying as follows: = fp; q : p + q + and p; q wih > 0g : his s can b gnralizd o allow for di rn rimming proporions.g., rplacing p + q + and p; q wih p + q + and p; q wih ; ; > 0. h s can also b narrowd if addiional informaion abou p and q is availabl. For xampl, ihir valus ar boh xpcd o b highr han 0:5, hn w can considr fp; q : 0:5 + p; q wih > 0g : h spci caion is in fac consisn wih all h 0 sudis mniond in h prvious paragraph. In his papr, w focus on ; h rsuls coninu o hold for h lar wo spci caions, providd ha h s in h limiing disribuion is changd accordingly. 4 h log liklihood raio undr prspci d p and q h condiional rgim probabiliy +j p; q; ; ; rprsns h ky di rnc bwn Markov swiching and mixur modls. his scion bgins wih sudying his probabiliy and is drivaivs wih rspc o ; and. his will nabl us o dvlop xpansions oh concnrad log liklihood undr h null hypohsis. h rsuls rpord in his scion hold uniformly ovr p; q [; ] [; ] wih bing an arbirary consan saisfying 0 < < =. 4. h condiional rgim probabiliy h following wo obsrvaions ar imporan. a h xprssions 6 and 7 can b combind o rprsn +j p; q; ; ; rcursivly as = ; ; :::: +j p; q; ; ; 4 ; j p; q; ; ; = p + p + q ; j p; q; ; ; + ; j p; q; ; ; : his is a rs ordr di rnc quaion ha rlas +j p; q; ; ; o j p; q; ; ;. Immdialy, his implis ha h drivaivs of +j p; q; ; ; wih rspc o ; ; ar also rs ordr di rnc quaions. b Alhough hs di rnc quaions ar nonlinar a gnral 0
valus of and, hy simplify subsanially if =. Bcaus h asympoic xpansions considrd lar ar around h null paramr simas, considring = will b su cin. h nx lmma characrizs +j p; q; ; ; and is drivaivs valuad a = =, whr is an arbirary valu in. D n an augmnd paramr vcor = 0 ; 0 ; 0 0 5 and hr ss of ingrs hy indx h lmns in ; and, rspcivly I 0 = f; :::; n g ; I = fn + ; :::; n + n g ; I = fn + n + ; :::; n + n g : L +j and f dno +j p; q; ; ; and ; valuad a som and = =. Also, l r j :::r jk j, r j :::r jk f and r j :::r jk f dno h k-h ordr parial drivaivs of j p; q; ; ;, ; and ; wih rspc o h j -h,:::; j k -h lmns of valuad a som and = =. No ha h following rlaionships hold: r j :::r jk f = r j :::r jk f if j ; :::; j k all blong o I 0, r j :::r jk f = 0 if any of j ; :::; j k blongs o I, and r j :::r jk f = 0 if any of j ; :::; j k blongs o I. Lmma L = p + q and r = wih d nd in. hn, for, w hav, undr = = :. +j =.. r j +j = r j j + E j;, whr 8 >< E j; = >: 0 rr j log rr j log f if j I 0 if j I : if j I. r j r k +j = r j r k j + E jk;. L I a ; I b dno j I a and k I b ; a; b = 0; ;. hn, E jk; is givn by: I 0 ; I 0 : 0 I 0 ; I : rr j f I 0 ; I : rr j f r k f r k f + rr j r k f rr j r k f I ; I : r j j r k f + r j k r j f + rr j r k f I ; I : r j j r k f r k j r j f + r r j f I ; I : r j j r k f + r k j r j f r r j f r k f r k f rr j r k f r r j f r k f :
4. r j r k r l +j = r j r k r l j + E jkl;, whr h xprssions for E jkl; wih j; k; l fi a ; I b ; I c g and a; b; c = 0; ; ar givn in h appndix. Rmark h lmma holds for any sampl siz. I shows ha h condiional rgim probabiliy +j quals h saionary probabiliy ; whil is drivaivs up o h hird ordr all follow rs ordr linar di rnc quaions. h laggd co cins always qual = p + q. Bcaus 0 < p; q <, hs di rnc quaions ar always sabl. As sn blow, hs faurs allow us o apply propris of rs ordr linar sysms o analyz h propris oh log liklihood. hy ar h ky lmns ha mak h subsqun analysis fasibl. I is worhwhil o ak a closr look a h four rsuls in h lmma. Lmma. is inuiiv. Bcaus h wo rgims ar idnical whn =, obsrving h daa provids no furhr informaion abou h rgim probabiliy. Lmma. quani s h rs ordr c of changing a paramr s valu on h rgim probabiliy. hr, changing has no c; +j p; q; ; ; rmains qual o. Changing h valus of and has xacly opposi cs, i.., r j +j = r j+n +j for any j I. Lmma. quani s h scond ordr cs. h rs cas, I 0 ; I 0, shows ha changing sill has no c. h nx wo cass show ha changing and afr a chang in sill hav qual opposi cs. h rmaining hr cass ar mor complx, bu hy all show ha E jk; only dpnd on r j j j I [ I and quaniis rlad o h dnsiy funcions. Lmma.4 consis on di rn cass wih di rn combinaions of j; k and l. For h analysis lar, h xac xprssions of E jkl; is unimporan. Wha is imporan is ha hy dpnd only on lowr ordr drivaivs of j and quaniis rlad o h dnsiy funcions. h rcursiv srucur wihin h rsuls highr ordr drivaivs dpnd succssivly on h lowr ordrs wih h rs ordr dpnding only on r j log f and r j log f suggss a sragy for analyzing hir saisical propris. W sar wih h rs ordr drivaivs, which ar simpl o analyz. hn, w us h rsuls cumulaivly o sudy h scond ordr followd by h hird ordr drivaivs. his sragy is implmnd in Lmma A. in h appndix. Using as h iniial valu for +j p; q; ; ; is no rsriciv. Wih a gnric ni iniial valu, Lmma. bcoms +j = q + j. h ohr rsuls also coninu o hold wih and r rplacd by j and j j rspcivly. Bcaus jj <, j convrgs a an xponnial ra o a incrass. Consqunly, h rgim probabiliy and is drivaivs all convrg o hir counrpars in h lmma a an xponnial ra. his ra of convrgnc implis ha using a gnric ni iniial valu will no alr h asympoic rsuls prsnd lar.
h illusraiv modl con d. Considr h gnral cas whr h rgrssion co cins and h varianc oh rrors ar boh allowd o swich. Lmma. implis: r +j = 0; w.r.. h non-swiching paramrs 9 r +j = r j + r w y z 0 w 0 ; = r +j = r j + r y w.r.. h paramrs in h rs rgim z 0 w0 ; ; 9 r +j = r +j ; = w.r.. h paramrs in h scond rgim r +j = r +j : ; Whn valuad a h ru paramr valu, h drivaivs wih rspc o and follow saionary AR procsss wih man zro. hir variancs ar ni and saisfy E r j +j = r Ew j ; Er +j r = ; 4 whr dnos h ru valu of and r j dnos h rs ordr drivaiv w.r.. h j-h lmn of. h procsss spci d by Lmma.-.4 also hav ni mans and variancs whn valuad a h ru paramr valus, providd ha h rlvan momns of w ; z and u xis. 4. Concnrad log liklihood and is xpansion o obain an asympoic approximaion o h log liklihood raio, a sandard approach would b o xpand L A p; q; ; ; around h rsricd MLE ; ;. his is infasibl hr bcaus L A p; q; ; ; can hav mulipl local maxima. Cho and Whi 007 ncounrd a similar problm and procdd by working wih h concnrad liklihood. W follow hir insighful sragy. his allows us o brak h analysis ino wo sps. h rs sp quani s h dpndnc bwn h simas of and using h rs ordr condiions ha d n h concnrad liklihood s Lmma blow. his civly rmovs and from h subsqun analysis. h scond sp xpands h concnrad liklihood around = s Lmma blow and obains an approximaion o LRp; q. Bcaus h condiional rgim probabiliy is im varying, h ask hr is mor challnging han ha of Cho and Whi 007. L ^ and ^ b h maximizr oh log liklihood for a givn valu h dpndnc of ^ and ^ on p and q is supprssd o simplify h noaion: ^ ; ^ = arg max ; L A p; q; ; ; : 6
L Lp; q; dno h concnrad log liklihood: Lp; q; = L A p; q; ^ ; ^ ; : hn, h wo rms in h liklihood raio saisfy max ; ; L A p; q; ; ; = max Lp; q; and L N ; = Lp; q;. Consqunly: h LRp; q = max Lp; q; Lp; q; i : 7 For k, l L k i :::i k p; q; i ; :::; i k f; :::; n g dno h k-h ordr drivaiv of Lp; q; wih rspc o h i -h; :::; i k -h lmns of. L d j j f; :::; n g dno h j-h lmn of. hn, a fourh ordr aylor xpansion of Lp; q; around is givn by Lp; q; Lp; q; = n j= +! + 4! L j p; q; d j +! n n n j= k= l= n n n j= k= l= m= n n j= k= L jkl p; q; d j d k d l n L jk p; q; d j d k 8 L 4 jklm p; q; d j d k d l d m ; whr in h las rm is a valu bwn and. Assumpion 4 hr xiss an opn nighborhood of ;, dnod by B ;, and a squnc of posiiv, sricly saionary and rgodic random variabls g saisfying E +c som c > 0, such ha sup ; B ; r i :::r ik ; ; k k < < L < for for all i ; :::; i k f; :::; n + n g ; whr k 5; k = 6 if k = ; ; and k = 5 if k = 4; 5. his assumpion is slighly srongr han Assumpion A5 iii in Cho and Whi 007. hr, insad of k=k, h rspciv valus ar 4; ; and for k = ; ; and 4. h assumpion prmis h applicaion of laws of larg numbrs and cnral limi horms o h rms in 8. Assumpion 5 hr xiss > 0, such ha sup p;q[; ] sup j j< jl 5 jklmn p; q; j = O p for all j; k; l; m; n f; :::; n g, whr is an arbirarily small consan saisfying 0 < < =. 4
In a sandard problm, w would nd h scond ordr drivaiv L jk p; q; o b coninuous in.g., Ammiya, 985, p., or h hird ordr drivaiv L jkl p; q; o b O p o nsur ha a local quadraic xpansion is an adqua approximaion o h log liklihood. In 8, L 4 jklm p; q; plays h sam rol as h scond ordr drivaiv in a sandard problm. his is why h abov assumpion on h fh ordr drivaiv is ndd. h nx lmma characrizs h drivaivs of ^ and ^ wih rspc o valuad a =. o shorn h xprssions, l +j and dno +j p; q; ; ; and ; valuad a ; ; = ; ;. L r i :::r ik j and r i :::r ik f dno h k-h ordr drivaiv of +j p; q; ; ; and ; wih rspc o h i -h; :::; i k -h lmns of valuad a ; ; = ; ;. D n U jk; = f r j r kf + r j j D jk; = r 0 ; 0 f 0 Ujk; ; I = r 0;0 0 f r 0;0 f ; V jklm = U jk;ulm; ; Dlm = D lm; ; I = r kf + r jf r k j I ; ; 9 whr U jk; involvs h rs and scond ordr drivaivs wih rspc o h j-h and k-h lmns of. h rm insid h curly bracks can also b xprssd as = r j r k f = r j j r k f = r j f r k j. As will b sn, Ujk; drmins L jk p; q; whil D jk; and I drmin L 4 jklm p; q;. Lmma L h null hypohsis and Assumpions -4 hold. For all k; l; m f; :::; n g:. L k b an n -dimnsional uni vcor whos k-h lmn quals, hn 4 r k ^ 5 r k^ = 4 0 5 + O p = : k. h scond ordr drivaivs saisfy 4 r k r l ^ r k r l^ 5 = I D kl; + O p = :. h hird ordr drivaivs saisfy 4 r k r l r m ^ r k r l r m^ 5 = O p : 5
Lmma gnralizs Lmma Ba-d in Cho and Whi 007 o Markov swiching modls. h rsuls quanify how and nd o chang in ordr o maximiz h liklihood whn is movd away from. hy provid h ncssary inpus for h chain rul whn compuing h drivaivs L k i ;:::;i k p; q; k = ; ; ; 4 in 8. his lads o h following lmma. Lmma Undr h null hypohsis and Assumpions -5, for all j; k; l; m f; :::; n g, w hav. L j p; q; = 0:. = L jk p; q; = = P U jk; + o p.. =4 L jkl p; q; = O p =4 : 4. L 4 jklm p; q; = f V jklm D 0 jk I Dlm + V jmkl D 0 jm I Dkl + V jlkm D 0 jl I Dkm g+o p. h rs ordr drivaiv L j p; q; quals zro. his implis ha h MLE of convrgs a a a slowr ra han =. h scond ordr drivaiv L jk p; q; is of ordr O p = rahr han O p. As sn blow, is lading rm = P U jk; convrgs o a mulivaria normal disribuion, whos propry dpnds on h im varying condiional rgim probabiliy. h hird ordr drivaiv L jkl p; q; is also of ordr O p =. h xprssion of is lading rm is no ndd hr for obaining h limiing disribuion, bu w will furhr analyz i whn providing a ni sampl r nmn. Finally, h fourh ordr drivaiv L 4 jklm p; q; is of ordr O p. Is lading rm provids a consisn simaor oh asympoic varianc of = P U jk; : Rmark h rs componn of U jk;, = r j r k f =, is also prsn whn sing agains mixur alrnaivs; s Cho and Whi 007, Lmma a. componns ar nw and ar du o h Markov swiching srucur. h rmaining wo hy can b rwrin as = P s= s r j log f s r k log and = P s= s r k log f s r j log rspcivly. Among h hr componns of U jk;, h rs picks up ovrdisprsion and h ohr wo pick up srial dpndnc causd by h Markov rgims. Furhrmor, h magniuds oh las wo componns bcom mor pronouncd rlaiv o h rs as approachs. his is bcaus h rs componn is indpndn of afr division by = whil h las wo componns involv wighs s. his suggss ha h powr di rnc bwn sing agains Markov swiching alrnaivs and mixur alrnaivs can b subsanial whn h rgims ar prsisn, i.., whn is clos o. his is con rmd by h numrical rsuls rpord lar. 6
h illusraiv modl con d. In h linar modl 9, h lading rms of = L jk p; q; and L 4 jklm p; q; in Lmma hav simpl srucurs. Suppos only h rgrssion co cins ar allowd o swich. hn, U jk; and D jk; ar givn by and w j w k u + P s= s wj s u s h z 0 u u wk u + P s= s wk s u s wj u 0 w 0 u i 0 Ujk; ; whr u dno h rsiduals undr h null hypohsis and = P u. h wo xprssions show ha U jk; and D jk; dpnd only on h rgrssors and h rsiduals undr h null hypohsis. As a rsul, h covarianc funcion of = L jk p; q; is consisnly simabl. his faur is valuabl for compuing criical valus oh s. 5 Asympoic approximaions L L p; q; b a squar marix whos j; k-h lmn is givn by L jk p; q; for j; k f; ; :::; n g. his scion consiss of four ss of rsuls. I sablishs h wak convrgnc of = L p; q; ovr p; q. I obains h limiing disribuion of SupLR. I dvlops a ni sampl r nmn ha improvs h asympoic approximaion whn a singulariy is prsn. 4 I dvlops an algorihm o obain h rlvan criical valus. 5. Wak convrgnc of L p; q; For any 0 < p r ; q r ; p s ; q s < and j; k; l; m f; ; :::; n g, d n! jklm p r ; q r ; p s ; q s = V jklm p r ; q r ; p s ; q s D 0 jk p r; q r I D lm p s ; q s ; whr V jklm p r ; q r ; p s ; q s = E [U jk; p r ; q r U lm; p s ; q s ] ; D jk p r ; q r = ED jk; p r ; q r, and I = EI. Hr, U jk; p r ; q r ; D jk; p r ; q r and I ar d nd as U jk;, D jk; and I in 9 bu valuad a p r ; q r ; ; insad of p r ; q r ; ;. Proposiion L h null hypohsis and Assumpions -5 hold. hn, ovr p; q : = L p; q; G p; q ; whr h lmns of G p; q ar man zro coninuous Gaussian procsss saisfying Cov[G jk p r ; q r, G lm p s ; q s ]=! jklm p r ; q r ; p s ; q s for j,k,l,m f,,:::,n g, whr! jklm p r,q r ; p s,q s is givn by. 7
In h appndix, h rsul is provd by rs vrifying h ni-dimnsional convrgnc and hn h sochasic quiconinuiy. h covarianc funcion! jklm p r ; q r ; p s ; q s in gnral is a cd by h following facors: i h modl s dynamic propris.g., whhr h rgrssors ar sricly or wakly xognous, ii which paramrs ar allowd o swich.g., rgrssions co cins or h varianc oh rrors, and iii whhr nuisanc paramrs ar prsn. h following illusraion maks his clar. h illusraiv modl con d. W considr a simplr vrsion of 9 for which h covarianc funcion! jklm p r ; q r ; p s ; q s can b compud analyically: y = w fsg + w fs=g + u ; whr u i.i.d.n0; and w is a scalar rgrssor ha is ihr sricly xognous or qual o y. L r = p r + q r and s = p s + q s. W coninu o us subscrip * o dno h ru paramr valu. Firs, considr h siuaion whr only is allowd o swich and is unknown. hn, whn h rgrssor is sricly xognous, h covarianc funcion quals p r p s V arw + P k= r s k Ew w k q r q s 4 : Whn h rgrssor is h laggd dpndn variabl hrfor only wakly xognous, i quals p r p s 4 q r q s + 4 r s r s 6 + r s r r s + 6 r s s r s + r s 6 r s r s : 4 hs wo funcions ar di rn vn whn w i.i.d.n0; and = 0. his is bcaus r j is indpndn of r whn w is sricly xognous, bu no ncssarily whn i is prdrmind. his shows ha h covarianc funcion is a cd by h dynamic propris oh modl. Now, considr h sam siuaion as abov bu wih h valu of bing known. hn, whn h rgrssor is sricly xognous, h covarianc funcion quals p r p s Ew 4 + P k= r s k Ew w k q r q s 4 : 5 8
Whn h rgrssor is h laggd dpndn variabl, i quals p r p s 6 q r q s + 4 r s r s + r s 6 + r s r r s + 6 r s s r s 6 r s r s : 6 hs wo funcions ar di rn from boh and 4. his shows ha h prsnc of nuisanc paramrs can also a c h covarianc funcion. Nx, considr h siuaion whr only is allowd o swich and is unknown. Undr boh sric and wak xogniy: Cov G p r ; q r ; G p s ; q s = p r p s q r q s 8 + r s r s : 7 his funcion is di rn from boh and 4. hrfor, h covarianc funcion can di r dpnding on which paramr is allowd o swich. W rpor som simulaion rsuls o complmn h analysis abov. h paramr valus ar = 0:5 and =. Whn h rgrssor is sricly xognous, w is gnrad indpndnly of u s a all lads and lags as w = 0:5w + " wih " i:i:d:n0;. his nsurs ha h rgrssors follow h sam DGP in boh cass. Furhr, l p r ; q r = 0:6; 0:9 and p s ; q s = 0:6; x wih x varying bwn 0: and 0:9. Figur rpors h v corrlaions funcions givn by -7 Hr, corrlaions insad of covariancs ar plod o facilia comparisons. h solid lins saring from h op corrspond o 7, 5,, 6, and 4, rspcivly. hs funcions show clarly h dpndnc on h hr facors highlighd abov. Also includd in h gur ar corrlaions compud from simulaions i.., h dashd lins. hy ar gnrad by simulaing sampls of 50 obsrvaions using h sam paramr valu as abov, compuing = P U jk; using ach sris, and hn rpaing 0,000 ims o obain h mpirical corrlaions. h valus ar clos o hir asympoic approximaions in all v cass. 5. Limiing disribuion of SupLR L p; q b an n -dimnsional squar marix whos j + k n ; l + m n -h lmn is givn by! jklm p; q; p; q. hn, Proposiion implis E[vc G p; q vc G p; q 0 ] = p; q. h nx rsul givs h asympoic disribuion of SupLR. Proposiion Suppos h null hypohsis and Assumpions -5 hold. hn: SupLR sup sup p;q R n 9 W p; q; ; 8
whr is givn by and W p; q; = 0 vc G p; q 4 0 p; q : h quaniy plays h rol of =4 in 8. Is dimnsion is una cd by h prsnc of nuisanc paramrs. If n =, hn h opimizaion ovr can b solvd analyically, lading o SupLR max[0; sup p;q G p; q = p p; q]. h righ hand sid can qual zro wih posiiv probabiliy. If n >, h opimizaion will nd o b carrid ou numrically. Bcaus W p; q; is a quadraic funcion of, h opimizaion is rlaivly sandard. h illusraiv modl con d. W illusra h limiing disribuion 8 and also xamin is adquacy in ni sampls. Considr h following spcial cas of 9: whr u i:i:d: N0; and ; and ar unknown. y = + y + u ; 9 As shown blow, h disribuion of SupLR, as wll as h adquacy oh asympoic approximaion, can di r subsanially dpnding on whhr h inrcp or h slop paramr is allowd o swich. Figur rpors ni sampl h solid lins and asympoic disribuions h long dashd lins of SupLR for sing rgim swiching in only or only. h paramr valus ar = 0; = 0:5 and =. h s is spci d as wih = 0:05, h sampl siz is 50 and all rsuls ar basd on 5000 rplicaions. h gur shows wo faurs. Firs, consisnly wih Proposiion and h illusraion in Scion 5., h disribuions in panl a ar signi canly di rn from hos in panl b. Scondly, h asympoic disribuion provids an adqua approximaion in panl a, bu no in panl b. For h lar, using h asympoic disribuion will lad o ovr rjcion oh null hypohsis. h scond faur r cs h srucur of = P U jk;. Whn sing for rgim swiching in in panl b, = P U jk; quals n = P u + = P P s= s u s whr u dno h rsiduals undr h null hypohsis. o u, 0 Bcaus = P u, h rs summaion is in fac always zro. Furhrmor, h magniud oh scond summaion dcrass as approachs 0, i.., as p+q approachs. his suggss ha, in ni sampls, h magniud of = P U jk; may b oo small o domina h highr ordr rms in h liklihood xpansion. 0
As a rsul, h asympoic disribuion ha rlis on = P U jk; can b inadqua. h siuaion is di rn whn sing for swiching in, whr = P U jk; quals n = P u y + = P P s= s y s u s y o u : h rs rm in h curly bracks now convrgs o a normal disribuion indpndn of p and q. hrfor, h complicaion in 0 dos no aris. Figur furhr compars h ni sampl and asympoic disribuions of LRp; q for sing rgim swiching in a som slcd valus of p; q. Consisnly wih h discussion abov, a gap bwn h ni sampl disribuion h solid lin and h asympoic disribuion h long dashd lin appars and grows widr as p + q approachs. Simulaions also show ha, whn sing for rgim swiching in, hs wo disribuions rmain clos o ach ohr in all hr cass. h dails ar omid. h illusraion suggss ha h asympoic approximaion in Proposiion nds o b improvd ih hypohss imply ha L p; q; quals zro whn p + q =. his is carrid ou in h nx subscion. 5. A r nmn his scion obains a sixh ordr xpansion oh liklihood raio along p + q = and an ighh ordr xpansion a p = q = =. h rason for why h lar is ndd is xplaind blow. h lading rms ar hn incorporad ino h limiing disribuion in Proposiion o dlivr a r nd approximaion. hs xpansions ar basd on h following assumpion. Assumpion 6 h following linar rlaionship holds for all and all i ; i f; :::; n g : whr i i and i i r i r i f = 0 i i r f + 0 i i r f ; ar n - and n -dimnsional known vcors of consans. his assumpion can b chckd onc h modl and h hypohss ar spci d. For xampl, whn sing for rgim swiching in h inrcp in h AR modl 9, w hav r r f = r. h nx assumpion srnghns Assumpion 4. I is similar o A.5iv in Cho and Whi 007. h subsqun analysis maks havy us ohir rsuls dvlopd in Scion... Assumpion 7 hr xiss an opn nighborhood of ;, B ;, and a squnc of posiiv, sricly saionary and rgodic random variabls g saisfying E +c < for som c > 0,
such ha h suprmums oh following quaniis ovr B ; ar boundd from h abov by : r i :::r ik ; = ; 4 ; r i :::r im ; = ; ; r i :::r i8 ; = ;, r j r i :::r i7 ; = ; ; r j r j r i :::r i6 ; = ;, whr k = ; ; ; 4, m = 5; 6; 7; i ; :::; i 7 f; :::; n + n g and j ; j f; :::; n g. Bfor procding, w rs sablish som noaion. o approxima h hird and sixh ordr drivaivs oh concnrad log liklihood, d n s jkl; p; q = p p q r r j r k l q and l G jkl p; q b a coninuous Gaussian procss wih man zro saisfying! jklmnu p r; q r ; p s ; q s = CovG jkl p r; q r ; G mnu p s ; q s = E [s jkl; p r ; q r s mnu; p s ; q s ] E r 0 ; 0 r s jkl; p r ; q r I 0 ; 0 0f s mnu; p s ; q s ; whr s jkl; p; q is h sam as s jkl; p; q bu valuad a h ru paramr valus h ohr quaniis ar also valuad a h ru paramr valus. o approxima h fourh and ighh ordr drivaivs, d n k jklm; p; q = p p q p + q! p rj r k r l r mf + q r i r i r 0 f i i ;i ;i ;i 4 f i 4 r i r i r 0 f i i 4 S + 0 i i r r 0 f i i 4 + 0 i i r r 0 f i i 4 and l G 4 i i i i 4 p; q dno a coninuous Gaussian procss wih man zro saisfying! 4 i i :::i 8 p r ; q r ; p s ; q s = Cov G 4 i i i i 4 p r ; q r ; G 4 i 5 i 6 i 7 i 8 p s ; q s = E [k i i i i 4 ; p r ; q r k i5 i 6 i 7 i 8 ; p s ; q s ] r E 0 ; 0 r k i i f i i 4 ; p r ; q r I 0 ; 0 0f k i5 i f 6 i 7 i 8 ; p s ; q s ; whr h indx s S in is givn by S = fjklm; jlkm; jmkl; kljm; kmjl; lmjkg, k i i i i 4 ;p; q is quivaln o k i i i i 4 ;p; q bu valuad a h ru paramr valus h rmaining quaniis ar also valuad a h ru paramr valus.
h nx lmma characrizs h asympoic propris of L k i i :::i k p; p; for i ; :::; i k f; :::; n g and k = ; :::; 8. I gnralizs Lmma, 4a, 5a- in Cho and Whi 007 by allowing for mulipl swiching paramrs. Lmma 4 Undr h null hypohsis and Assumpions -7:. h following rsuls hold uniformly ovr fp; q : p; q ; p + q = g: = L jkl p; q; = = s jkl; p; q + o p G jkl p; q; = L 4 jklm p; q; = O p ; = L 5 jklmn p; q; = O p ; L 6 jklmnr p; q; =! i i :::i 6 p; q; p; q + o p ; i ;i ;:::;i 6 IND whr IND={jklmnr,jkmlnr,jknlmr,jkrlmn,jlmknr,jlnkmr,jlrkmn,jmnklr,jmrkln,jnrklm}.. h following rsuls hold a p = q = = : = L jkl p; q; = o p ; = L 4 jklm p; q; = = k jklm; p; q + o p G 4 jklm p; q; = L k i i :::i k p; q; = O p, whr i ; :::; i k f; :::; n g for k=5,6 and 7, L 8 jklmnrsu p; q; =! 4 i i :::i 8 p; q; p; q + o p : i ;i ;:::;i 8 IND whr h lmns of IND ar as follows: i = j; ach ripl i ; i ; i 4 corrsponds o on of h 5 oucoms of picking lmns from {k; l; m; n; r; s; ug h ordring dos no mar; and i 5 ; i 6 ; i 7; i 8 corrspond o h rmaining lmns. h wo ss of rsuls characriz h high ordr drivaivs along h lin p + q =. Whn p 6= =, h hird ordr rm = P s jkl;p; p rplacs h scond ordr rm = P U jk; o bcom h lading rm in h liklihood xpansion. Consqunly, a sixh ordr xpansion is ndd o approxima h liklihood raio. Whn p = =, h fourh ordr rm = P k jklm; p; p bcoms h lading rm, and an ighh ordr xpansion is ndd. h rsricion p = q is no imposd whn rprsning h lading rm in = L jkl p; q;. his nsurs ha h co cin in fron of r j r k r l f = is corrc vn whn p + q 6=. For h sam rason, p = q = = is also no imposd whn xprssing h lading rm of
= L 4 jklm p; q;. h lmma assums ha all h scond ordr drivaivs wih rspc o h swiching paramrs can b wrin as linar combinaions oh rs ordr drivaivs. Ihis rlaionship holds only for a subs of drivaivs, hn w simply s i i = 0 and i i h cass ha do no saisfy. W now incorpora h lading rms in Lmma 4 o obain a r nd approximaion. G p; q b a n - dimnsional vcor whos j+k n +l n = 0 for L -h lmn is givn by G jkl p; q. L p; q dno an n by n marix whos j +k n +l n ; m+n n +r n -h lmn is givn by! jklmnr p; q; p; q. D n W p; q; = =4 0 vc G p; q = 6 0 p; q : L G 4 p; q b an n 4 - dimnsional vcor whos j + k n + l n + m n -h lmn is givn by G 4 jklm p; q. L 4 p; q b an n 4 by n4 marix whos j + k n + l n + m n ; n + r n + s n + u n -h lmn is givn by!4 jklmnrsu p; q; p; q. D n W 4 p; q; = = 4 0 vc G 4 p; q 576 4 0 4 p; q 4 : W propos approximaing h disribuion oh SupLR s using S sup whr is spci d in. sup p;q R n n o W p; q; + W p; q; + W 4 p; q; ; 4 Corollary Undr Assumpions -7 and h null hypohsis, w hav, ovr : Pr SupLR s Pr S s! 0: Rmark h abov rsul holds irrspciv of whhr or no h rlaionship holds. his follows bcaus h addiional rms W p; q; and W 4 boh convrg o zro as!. hs rms provid r nmn in ni sampls, having no c asympoically. h illusraiv modl con d. quaniis and qual Firs, considr sing for rgim swiching in in 9. h pp q u q u and p p q + p q p q 4 u 4 6 u + : 4
h r nd approximaions 4 ar rpord as dod lins in Figurs b and. hy show ha, rlaiv o h original approximaion, h improvmns ar subsanial. Nx, considr sing for rgim swiching in. h quaniis and qual pp q uy q uy and p p q + p q 4 uy 4 6 uy + : h r nd approximaion is rpord as h dod lin in Figur a. rlaiv o h original approximaion. hr is lil chang hrfor, h r nmn subsanially improvs h approximaion whn sing for rgim swiching in h inrcp. A h sam im, i has lil c whn sing for swiching in h slop co cin. his is dsirabl bcaus, for h lar cas, h original approximaion in Proposiion is alrady adqua. 5.4 An algorihm for obaining criical valus his scion shows how o obain h criical valus of S d nd in 4. h ida is o sampl from h disribuion oh vcor procss [vc G p; q 0 ; vc G p; q 0 ; vc G 4 p; q 0 ] and hn solv h maximizaion problm 4 ovr p; q and R n. Bcaus his vcor procss is Gaussian wih man zro, o gnra h dsird draws i su cs o obain a consisn simaor of is covarianc funcion ovr. his obsrvaion has also bn mad by Hansn 99 and Garcia 998. L U p; q b an n -dimnsional vcor whos j + k n -h lmn is givn by U jk; in 9. L U p; q b an n -dimnsional vcor whos j + k n + l n -h lmn is givn by s jkl; p; q in. L U 4 p; q b an n 4 -dimnsional vcor whos j + k n + l n + m n -h lmn is givn by k jklm; p; q in. D n G p; q = 6 4 U p; q U p; q U 4 p; q L U p; q; U p; q; U 4 p; q and G p; q b d nd as U p; q, U p; q, U 4 p; q and G p; q bu valuad a h ru valus undr h null hypohsis. Bcaus h vcor procss 7 5 : 5
= P G p; q convrgs wakly o [vc G p; q 0 ; vc G p; q 0 ; vc G 4 p; q 0 ] ovr p; q, is covarianc funcion provids a consisn simaor for h limi. Furhr, = G p; q = = G p; q G p; q r 0 ; 0 I = r 0 ; 0 0 + o p ; whr all h quaniis on h righ hand sid ar valuad a h ru paramr valus undr h null hypohsis. h rm insid h curly bracks convrgs o a nonrandom marix. hrfor, a consisn simaor oh dsird covarianc funcion is givn by G p r ; q r G p s ; q s 0 5 G p r ; q r r 0 ; 0 I G p s ; q s r 0 0 ; 0 ; whr I is h simad informaion marix, i.., I = P [r 0 ; 0 0 = ][r 0 ; 0 = ]. Rmark 4 h simaor 5 has hr faurs. Firs, h paramr valus ar h rsricd MLE. hy ar simpl o obain. Scondly, h rlvan quaniis can all b xprssd as funcions of r j f = ; r j r k r l f = and r j r k r l r m f =. For modls ha ar linar undr h null hypohsis, hy can all b compud analyically. hirdly, nuisanc paramrs do no a c h dimnsion oh opimizaion in 4. hrfor, hy do no noicably incras h compuaional cos. h illusraiv modl con d. W show how o compu h quaniis in 5 whn sing for swiching in in h AR modl 9. In mor gnral linar modls wih mulipl swiching paramrs, h rlvan quaniis can b obaind in a similar mannr. h vcor G p r ; q r consiss ohr lmns u dnos h OLS rsidual. hy dpnd on h modl only hrough h OLS rsiduals: U p; q = p P q U pp q p; q = q U 4 p; q = p p q s= p + q s u s u ; u + p q u 6 ; p q 4 u 4 6 u + :
h vcor r 0 ; 0 = also consiss ohr lmns: r 0 ; 0 h = y u u f u i : 6 hy dpnd on h modl only hrough h OLS rsiduals and h rgrssor y. Finally, I = which follows immdialy from 6. [r 0 ; 0 0 = ][r 0 ; 0 = ]; 6 Implicaions for boosrap procdurs and informaion criria h rsuls in h prvious scion provid a plaform for valuaing h consisncy of various boosrap procdurs. Alhough a comprhnsiv sudy of such procdurs is byond h scop of h papr, i is possibl o illusra som imporan aspcs using h linar modl 9. hroughou his scion h s is compud ovr d nd by. Boosrap procdurs. W bgin wih h imporan spcial cas whr h rgrssors conain only a consan and laggd valus of y, and h rrors ar normally disribud. A sandard paramric boosrap procdur procds as follows. Esima h modl undr h null hypohsis.g., sima an auorgrssiv modl. Sampl from h normal disribuion whos man quals zro and varianc quals h sampl varianc oh rsiduals. Us h draws and h simad co cins o gnra a nw auorgrssiv sris. Compu h s using h nwly gnrad sris. 4 Rpa h sps -. h abov procdur is asympoically valid. his is bcaus all h paramrs ar simad consisnly, and h normaliy and h AR srucur ar also prsrvd. Consqunly, h covarianc funcion in h boosrap world is consisn wih wha drmins h asympoic disribuion in Proposiion. Nx, considr h mor gnral siuaion whr a scond variabl is prsn in h rgrssors;.g., an auorgrssiv disribud lags ADL modl. Bcaus h modl dos no spcify h join disribuion oh dpndn variabl and h rgrssors, h boosrap procdur dscribd abov is no longr applicabl. wo alrnaiv approachs dsrv som considraion. h rs approach involvs kping h rgrssors xd a hir original valus whn gnraing h daa, i.., using h xd rgrssor boosrap. his procdur has bn shown o b asympoically valid in h conx osing for srucural braks Hansn, 000. Howvr, i is in 7
gnral inconsisn in h currn conx. his is bcaus, in conras o h original modl, h rgrssors ar sricly bu no wakly xognous in h boosrap world. his alrs h covarianc funcion apparing in Proposiion c.f. and 4 and h accompanying discussions. W provid som simulaion rsuls o illusra h ponial svriy oh siz disorion. h daa ar gnrad using h modl 9 wih h sam spci caions. h sampl siz = 50. h solid lin in Figur 4 shows h ni sampl disribuion, whil h dashd lin corrsponds o h xd rgrssor boosrap. h di rnc is qui subsanial. his di rnc dos no dcras whn h sampl siz is incrasd o 500. h scond approach involvs spcifying h join disribuion oh daa. For xampl, if w hav an ADL modl wih normal rrors, w spcify a full modl ha corrsponds o a Gaussian vcor auorgrssion. hn, w can apply h paramric boosrap o h augmnd modl. his boosrap procdur will b consisn if i asympoically producs h sam covarianc funcion in Proposiion. A ky propry ohis procdur is ha i nails spcifying a paramric modl for h rgrssors. Invsigaing h snsiiviy o such spci caions is usful bu is byond h scop ohis papr. Informaion criria. h asympoic rsuls also imply ha h ni sampl propris of convnional informaion criria, such as BIC, can b snsiiv o h srucur oh modl and also which paramrs ar allowd o swich. his is bcaus h disribuion oh liklihood raio dpnds on which paramr is allowd o swich, whil in BIC h pnaly rm dpnds only on h dimnsion oh modl and h sampl siz. W illusra such snsiiviis using h modl 9 by conrasing h oucoms from h following wo applicaions. a W apply BIC o drmin whhr hr is rgim swiching in h inrcp. h ohr paramrs ar assumd o b consan. b h sam as a xcp ha h slop paramr is allowd o swich. In h simulad daa, no rgim swiching is prsn; = 0; = 0:5 and =. h s is spci d as wih = 0:05. h sampl siz is 50. Ou oh 5000 ralizaions, BIC falsly classi s :5% in h rs applicaion, whil only :4% in h scond applicaion. Bcaus h pnaly rms in h Akaik informaion cririon and h Hannan Quinn informaion cririon hav h sam srucur, hy ar also xpcd o xhibi h sam snsiiviy. 8