Likelihood Ratio Based Tests for Markov Regime Switching

Transcription

1 Liklihood Ratio Basd sts for Markov Rgim Switching Zhongjun Qu y Boston Univrsity Fan Zhuo z Boston Univrsity April, 06 Abstract Markov rgim switching modls ar widly considrd in conomics and nanc. Although thr hav bn prsistnt intrsts s.g., Hansn, 99, Garcia, 998, and Cho and Whit, 007, th asymptotic distributions of liklihood ratio basd tsts hav rmaind unknown. his papr considrs such tsts and stablishs thir asymptotic distributions in th contxt of nonlinar modls allowing for multipl switching paramtrs. h analysis simultanously addrsss thr di cultis: i som nuisanc paramtrs ar unidnti d undr th null hypothsis, ii th null hypothsis yilds a local optimum, and iii th conditional rgim probabilitis follow stochastic procsss that can only b rprsntd rcursivly. Addrssing ths issus prmits substantial powr gains in mpirically rlvant situations. Bsids obtaining th tsts asymptotic distributions, this papr also obtains four sts of rsults that can b of indpndnt intrst: a charactrization of conditional rgim probabilitis and thir high ordr drivativs with rspct to th modl s paramtrs, a high ordr approximation to th log liklihood ratio prmitting multipl switching paramtrs, 3 a r nmnt to th asymptotic distribution, and 4 a uni d algorithm for simulating th critical valus. For modls that ar linar undr th null hypothsis, th lmnts ndd for th algorithm can all b computd analytically. h abov rsults also shd light on why som bootstrap procdurs can b inconsistnt and why standard information critria, such as th Baysian information critrion BIC, can b snsitiv to th hypothsis and th modl s structur. Whn applid to th US quartrly ral GDP growth rats, th mthods suggst fairly strong vidnc favoring th rgim switching spci cation consistntly ovr a rang of sampl priods. Kywords: Hypothsis tsting, liklihood ratio, Markov switching, nonlinarity. JEL cods: C, C, E3. W thank Jams Hamilton, Hiroaki Kaido, Frank Klibrgn, Pirr Prron, Douglas Stigrwald and sminar participants at Amstrdam, Brown, BU statistics, UCSD, th 06 Economtric Socity Wintr Mting, 04 JSM, th 3rd SNDE, and th th World Congrss oh Economtric Socity for valuabl suggstions, and Carrasco, Hu and Plobrgr for making thir cod availabl. y Dpartmnt of Economics, Boston Univrsity, 70 Bay Stat Rd., Boston, MA, 05 qu@bu.du. z Dpartmnt of Economics, Boston Univrsity, 70 Bay Stat Rd., Boston, MA, 05 zhuo@bu.du.

2 Introduction Markov rgim switching modls ar widly considrd in conomics and nanc. Hamilton 989 is a sminal contribution, which provids not only a framwork for dscribing conomic rcssions, but also a gnral algorithm for ltring, smoothing and maximum liklihood stimation whil building on th work of Goldfld and Quandt 973 and Cossltt and L 985. Survys ohis voluminous litratur can b found in Hamilton 008, 04. hr approachs hav bn considrd for dtcting rgim switching. h rst approach involvs translating this issu into tsting for paramtr homognity against htrognity. Nyman and Scott 966 studid th C tst. Chshr 984 drivd a scor tst and showd that it is closly rlatd to th information matrix tst of Whit 98. Lancastr 984 and Davidson and MacKinnon 99 ar rlatd contributions. Watson and Engl 985 dsignd a tst statistic that allows th htrognity to follow a stationary AR procss. Carrasco, Hu and Plobrgr 04 furthr dvlopd this approach by considring gnral dynamic modls and allowing th htrognity to follow xibl wakly dpndnt procsss. hy analyzd a class osts and showd that thy ar asymptotically locally optimal against a spci c altrnativ charactrizd in thir papr. h abov tsts hav two common faturs. First, thy only rquir stimating th modl undr th null hypothsis. Scond, thy ar dsignd for dtcting paramtr htrognity, not particularly Markov rgim switching. Although th tsts can hav powr against a broad class of altrnativs, thir powr can b substantially lowr than what is achivabl ih paramtrs indd follow a nit stat Markov chain. h scond approach, du to Hamilton 996, is to conduct gnric tsts oh hypothsis that a K-rgim modl.g., K = adquatly dscribs th data. h insight is that if a K-rgim spci cation is accurat, thn th scor function should hav man zro and form a martingal diffrnc squnc. Othrwis, th modl should b nrichd to allow for additional faturs, in som situations by introducing an additional rgim. Hamilton 996 dmonstratd how to implmnt such tsts as a by-product of calculating th smoothd probability that a givn obsrvation is from a particular rgim. his maks th tsts simpl and widly applicabl. Manwhil, it rmains important and usful to hav tsting procdurs that focus spci cally on dtcting Markov switching altrnativs. h third approach procds undr th quasi liklihood ratio principl. h quasi liklihood functions ar constructd assuming a singl rgim undr th null and two rgims undr th altrnativ hypothsis. h analysis facs thr challngs. i Som nuisanc paramtrs ar

3 unidnti d undr th null hypothsis. As a rsult, th log liklihood ratio is locally non-quadratic, causing th Chi-squar approximation to its distribution to brak down. his givs ris to th Davis 977 problm. ii h null hypothsis yilds a local optimum c.f. Hamilton, 990, making th scor function idntically zro whn valuatd at th null paramtr stimats. Consquntly, a scond ordr aylor approximation to th liklihood ratio is insu cint for analyzing its asymptotic proprtis. iii h conditional rgim probability, i.., th probability of bing in a particular rgim at tim t givn th information up to th tim t, follows a stochastic procss that can only b rprsntd rcursivly. h rst two di cultis ar also prsnt whn tsting for mixturs. It is th simultanous occurrnc of all thr di cultis that plagus th study oh liklihood ratio in th currnt contxt. For xampl, whn analyzing high ordr xpansions oh liklihood ratio, it is ncssary to study high ordr drivativs oh conditional rgim probability with rspct to th modl s paramtrs. So far, th statistical proprtis oh lattr hav rmaind lusiv. Consquntly, th asymptotic distribution oh log liklihood ratio has also rmaind unknown. Manwhil, svral important progrsss hav bn mad by Hansn 99, Garcia 998, Cho and Whit 007, and Cartr and Stigrwald 0. Hansn 99 clarly documntd why th di cultis i and ii caus th convntional approximation to th liklihood ratio to brak down. Furthr, h tratd th liklihood function as a stochastic procss indxd by th transition probabilitis i.., th probabilitis of rmaining in th rst rgim p and rmaining in th scond rgim q and th switching paramtrs, and drivd a bound for its asymptotic distribution. His rsult provids a platform for conducting consrvativ infrnc. Garcia 998 suggstd an approximation to th log liklihood ratio that would follow ih scor had a positiv varianc at th null stimats. Rsults in th currnt papr will show that this distribution is in gnral di rnt from th actual limiting distribution. Rcntly, Cho and Whit 007 mad a signi cant progrss. hy suggstd a quasi liklihood ratio QLR tst against a two-componnt mixtur altrnativ i.., a modl whr th currnt rgim arrivs indpndntly of its past valus. hr, th di culty iii is avoidd bcaus th conditional rgim probability is rducd to a constant, which can furthr b tratd as an additional unknown paramtr. Cartr and Stigrwald 0 furthr discussd a consistncy issu rlatd to QLR tst. h currnt papr maks us of svral important tchniqus in Cho and Whit 007. At th sam tim, it gos byond thir framwork to dirctly confront Markov switching altrnativs. As will b sn, th powr gains from doing so can b quit substantial. Spci cally, this papr considrs a family of liklihood ratio basd tsts and stablishs thir

4 asymptotic distributions in th contxt of nonlinar modls allowing for multipl switching paramtrs. h framwork ncompasss th important spcial cass osting for rgim switching in autorgrssiv modls and in autorgrssiv distributd lags modls. hroughout th analysis, th modl has two rgims undr th altrnativ hypothsis. Som paramtrs can rmain constant across th two rgims. h analysis is structurd in v stps:. W charactriz th dynamics oh conditional rgim probability i.., th probability of bing in a particular rgim at tim t givn th information up to th tim t and its high ordr drivativs with rspct to th modl s paramtrs. W show that, whn valuatd at th null paramtr stimats, th formr rducs to a constant whil th lattr can all b rprsntd as linar rst ordr di rnc quations with th laggd co cints qual to p + q. Bcaus 0 < p; q <, ths quations ar all stabl and amnabl to th applications of uniform laws of larg numbrs and functional cntral limit thorms. his novl charactrization is a critical stp that maks th subsqunt analysis fasibl.. W x p and q and driv a fourth ordr aylor approximation to th liklihood ratio. his stp builds on th analysis in Cho and Whit 007, but accounts for th ct oh tim variation in th conditional rgim probability. h rsults ar informativ about why substantial powr gains rlativ to th QLR tst ar possibl whn th data ar not gnratd by simpl mixturs. 3. W viw th liklihood ratio as an mpirical procss indxd by p and q and driv its limiting distribution. h valus of p and q ar rquird to b strictly btwn 0 and satisfying p + q + with bing som arbitrarily small positiv constant. hs rquirmnts ar compatibl with applications in macroconomics and nanc; s th discussion in Sction 3. h mpirical procss prspctiv undrtakn hr follows a rich array of studis, including Hansn 99, Garcia 998, Cho and Whit 007, and Carrasco, Hu and Plobrgr Whil th abov limiting distributions ar adquat for a broad class of modls, thy can lad to ovr-rjctions whn a furthr singularity th sourc of which is spci d latr is prsnt. o ovrcom this problm, w analyz a sixth ordr xpansion oh liklihood ratio along th lin p + q = and an ighth ordr xpansion at p = q = =. h lading trms ar thn incorporatd into th asymptotic distribution to saf guard against thir cts. his lads to a r nd distribution that dlivrs rliabl approximations throughout our xprimntations. his r nmnt is valid whthr or not this singularity is truly prsnt. 3

5 5. W provid a uni d algorithm for simulating th r nd asymptotic distribution. For modls that ar linar undr th null hypothsis, th lmnts ndd for this algorithm can all b computd analytically. his prmits dvloping a computr program, which mainly rquirs th rsarchr writing down th modl undr th null hypothsis, spcifying which paramtrs ar allowd to switch, and providing th prmissibl valus for th two transition probabilitis. h asymptotic distribution shows som uncommon faturs. First, nuisanc paramtrs, though constraind to b constant across th rgims, can a ct th limiting distribution. Scond, proprtis oh rgrssors i.., whthr thy ar strictly or wakly xognous also a ct th distribution. hird, th distribution dpnds on which paramtr i.., th intrcpt, th slop or th rsidual varianc is allowd to switch. hs faturs imply that som bootstrap procdurs can b inconsistnt and that standard information critria, such as BIC, can b snsitiv to th hypothsis and th modl s structur. h abov implications ar furthr discussd in Sction 6. W conduct simulations using a data gnrating procss DGP considrd in Cho and Whit 007. h rsults show that th powr di rnc can b larg whn th rgims ar prsistnt, a situation that is common in practic. W also apply th tsting procdur to th US quartrly ral GDP growth rats, ovr th priod 960:I-04:IV and a rang of subsampls. h rsults consistntly favor th rgim switching spci cation. In addition, th smoothd rgim probabilitis closly mirror NBER s rcssion dating. o our knowldg, this is th rst tim such consistnt vidnc for rgim switching in th man output growth is documntd through hypothsis tsting. From a mthodological prspctiv, this papr contributs to th litratur that studis hypothsis tsting whn som rgularity conditions fail to hold. Bsids th works mntiond abov, closly rlatd studis includ th following. Davis 987, King and Shivly 993, Andrws and Plobrgr 994, 995, and Hansn 996 considrd tsts whn a nuisanc paramtr is unidnti- d undr th null hypothsis. Andrws 00 studid tsts whn, in addition to th abov fatur, som paramtrs li on th boundary oh maintaind hypothsis. Hartigan 985, Ghosh and Sn 985, Lindsay 995, Liu and Shao 003, Chn and Li 009, and Gu, Konkr and Volgushv 03 tackld th issus of zro scor and/or unidnti d nuisanc paramtrs in th contxt of mixtur modls. Chn, Ponomarva and amr 04 considrd uniform infrnc on th mixing probability in mixtur modls whn nuisanc paramtrs ar prsnt. Rotnitzky, Cox, Bottai and Robins 000 dvlopd a thory for driving th asymptotic distribution oh liklihood ratio statistic whn th information matrix has rank on lss than full; also s th discussions in thir papr pag 44 for othr studis on th sam issu in various contxts. Dovonon and Rnault 4

6 03 studid distributions osts for momnt rstrictions whn th associatd Jacobian matrix is dgnrat at th tru paramtr valu. h currnt work is th rst that simultanously tackls th di cultis i to iii in th hypothsis tsting litratur. W conjctur that th tchniqus dvlopd can hav implications for hypothsis tsting in othr rlatd contxts that involv modls with hiddn Markov structurs. h papr is structurd as follows. Sction prsnts th modl and th hypothss. Sction 3 introducs a family ost statistics. Sction 4 studis th asymptotic proprtis oh log liklihood ratio for prspci d p and q. Sction 5 prsnts four sts of rsults. It stablishs th wak convrgnc oh scond ordr drivativ oh concntratd log liklihood. It provids th limiting distribution oh tst statistic. It introducs a nit sampl r nmnt. Finally, it outlins an algorithm for obtaining th rlvant critical valus. Sction 6 discusss som implications oh thory for bootstrapping and information critria. Sction 7 xamins th tst s nit sampl proprtis. Sction 8 considrs an application to th US ral GDP growth rats. Sction 9 concluds. All proofs ar in th appndix. h following notation is usd. jjxjj is th Euclidan norm of a vctor x. jjjj is th vctor inducd norm of a matrix. x k and k dnot th k-fold Kronckr product of x and, rspctivly. h xprssion vca stands for th vctorization of a k dimnsional array A. For xampl, for a thr dimnsional array A with n lmnts along ach dimnsion, vca rturns a n 3 -vctor whos i + j n + k n -th lmnt quals Ai; j; k. fg is th indicator function. For a scalar valud function f of R p, r f 0 dnots a p-by- vctor of partial drivativs valuatd at 0, r 0f 0 quals th transpos of r f 0, and r j f 0 dnots its j-th lmnt. In addition, r j r j r jk f 0 dnots th k-th ordr partial drivativ of f takn squntially with rspct to th j ; j ; :::; j k -th lmnt of valuatd at 0. h symbols,! d and! p dnot wak convrgnc undr th Skorohod topology, convrgnc in distribution and in probability, and O p and o p is th usual notation for th ordrs of stochastic magnitud. Modl and hypothss W squntially discuss th following issus: th modl, th log liklihood functions undr th null i.., on rgim and th altrnativ i.., two rgims hypothsis, and som assumptions rlatd to ths two aspcts. h modl is as follows. Lt fy t ; x 0 tg b a squnc of random vctors with y t bing a scalar and x t a nit dimnsional vctor. Lt s t b an unobsrvd binary variabl, whos valu dtrmins 5

7 th rgim at tim t. D n th information st at tim t as Lt fj t t = - ld :::; x 0 t ; y t ; x 0 t; y t : ; ; dnot th conditional dnsity of y t, satisfying fjt ; ; ; if s t = ; y t j t ; s t v t = ; ::::; : fj t ; ; ; if s t = ; his spci cation allows th vctor to switch btwn and, whil rstricting th vctor to rmain constant across th rgims. Hncforth, w abbrviat th two dnsitis on th right hand sid of as ; and ;, rspctivly. h rgims ar Markovian, i.., ps t = j t ; s t = ; s t ; ::: = ps t = js t = = p and ps t = j t ; s t = ; s t ; ::: = ps t = js t = = q. h rsulting stationary or invariant probability for s t = is givn by p; q = q p q : 3 Evaluatd at 0 < p; q <, th log liklihood function associatd with is = L A p; q; ; ; 4 n o log ; tjt p; q; ; ; + ; tjt p; q; ; ; ; t= whr tjt dnots th probability of s t = givn t, i.., tjt p; q; ; ; = ps t = j t ; p; q; ; ; t = ; :::; ; 5 which satis s th following rcursiv rlationship tjt p; q; ; ; = ; tjt p; q; ; ; ; tjt p; q; ; ; + ; tjt p; q; ; ; ; 6 t+jt p; q; ; ; = p tjt p; q; ; ; + q tjt p; q; ; ; : 7 hroughout th papr, w st th initial valu j0 =. As shown latr, using a di rnt initial valu dos not a ct th asymptotic rsults. Whn = =, th log liklihood rducs to L N ; = log ; : 8 t= his papr studis tsts basd on 8 and 4 for th singl rgim spci cation against th two rgims spci cation givn in. o procd, w impos th following rstrictions on th DGP and th paramtr spac. Lt n and n dnot th dimnsions of and. 6

8 Assumption i h random vctor x 0 t; y t is strict stationary, rgodic and -mixing with th mixing co cint satisfying c for som c > 0 and [0;. ii Undr th null hypothsis, y t is gnratd by fj t ; ;, whr and ar intrior points of R n and R n with and bing compact. Part i is th sam as Assumption A.i in Cho and Whit 007. As discussd thr, th -mixing condition is commonly usd whn analyzing Markov procsss. It allows x t to b a ctd by rgim switching undr th null hypothsis. Part ii spci s th tru paramtr valus. h intrior point rquirmnt nsurs that th asymptotic xpansions considrd latr ar wll d nd. Assumption Undr th null hypothsis: i ; uniquly solvs max ; E[L N ; ]; ii for any 0 < p; q <, ; ; uniquly solvs max ; ; EL A p; q; ; ;. Part i implis that ; is globally idnti d at ; undr th null hypothsis. Part ii implis that thr dos not xist a two-rgim spci cation i.., with 6= that is obsrvationally quivalnt to th singl-rgim spci cation i.., with = =. h nxt assumption rlats th idnti cation proprtis in Assumption to som asymptotic proprtis oh stimators. Assumption 3 Undr th null hypothsis, w hav: i [L N ; EL N ; ] = o p holds uniformly ovr ; with P t= r 0 ; 0 0 log ; r 0 ; 0 log ; bing positiv d nit in an opn nighborhood of ; for su cintly larg ; ii for any 0 < p; q <, [L A p; q; ; ; EL A p; q; ; ; ] = o p holds uniformly ovr ; ;. h abov assumption stats that 8 and 4 both satisfy uniform laws of larg numbrs. Along with Assumption, it implis that, undr th null hypothsis, th maximizrs of 8 and 4 for 0 < p; q < convrg in probability to ; and ; ; rspctivly. his assumption allows 4 to hav multipl local maximizrs. h lattr fatur will b accountd for whn analyzing th liklihood xpansions. Assumptions to 3 ar similar to thos usd in Cho and Whit 007, with two important di rncs. First, th liklihood 4 corrsponds to a Markov switching modl, not a mixtur modl. Scond, multipl paramtrs ar allowd to b a ctd by th rgim switching. Using th abov notation, th null and altrnativ hypothss can b mor formally statd as: H 0 : = = for som unknown ; H : ; = ; for som unknown 6= and p; q 0; 0; : 7

9 chnically, as discussd in Cho and Whit 007, th null hypothsis can also b formulatd as: H0 0 : p = and = or H0 00 : q = and =. In H0 0, bcaus th modl rmains in th rst rgim with probability, any statmnt about th scond rgim bcoms irrlvant. h rvrsd holds for H Blow, w introduc a modl that will b usd throughout th papr to illustrat th main componnts oh thory. An illustrativ modl. Gaussian rrors: An important application of rgim switching is to linar modls with y t = z 0 t + w 0 t fst=g + w 0 t fst=g + u t ; 9 whr ; and ar unknown nit dimnsional paramtr vctors and u t ar i.i.d. Normally distributd whos unknown varianc can also potntially switch. h variabls z t and w t can includ laggd valus of y t. hrfor, th spci cation ncompasss nit ordr autorgrssiv modls and autorgrssiv distributd lags modls as spcial cass. In rlation to and, w hav t = - ld :::; z 0 t ; w0 t ; y t ; z 0 t; w 0 t; y t and x 0 t = z 0 t; w 0 t. hr situations can aris dpnding on which paramtrs ar allowd to b a ctd by rgim switching: a Only th varianc of u t is a ctd. Lt and dnot its variancs undr th two rgims. hn, in rlation to, w hav = ; = and 0 = 0 ; 0 with = =. b Only th rgrssion co cints ar a ctd. Lt dnot th varianc of u t. hn, w hav =, = and 0 = 0 ;. c Both componnts ar a ctd. W hav 0 = 0 ;, 0 = 0 ; and =. h rsults in this papr will ncompass all thr situations. In th most gnral situation c, th dnsitis corrsponding to ar givn by 3 4 ; p xp 5 6 = 4 ; p xp n yt z 0 t w0 t n yt zt 0 w0 t Not that th normality assumption in this modl can b rplacd by othr distributional assumptions, providd that ; and ; ar rplacd by th appropriat dnsitis. W now illustrat Assumptions -3 using this modl. Rgarding Assumption, bcaus oh linarity, th -mixing rquirmnt of x 0 t; y t rducs to that of x t. h lattr is satis d if x t follows a stationary VARMAP,Q procss P P j=0 B jx t j = P Q j=0 A j" t j with " t bing man zro i.i.d. random vctors whos dnsity is absolutly continuous with rspct to Lbsgu masur on R dim"t ; s Mokkadm 988. Othr procsss that ar -mixing with a gomtric rat of dcay, o o : 8

10 as rviwd in Chn 03, includ thos gnratd by thrshold autorgrssiv modls, functional co cint autorgrssiv modls, and GARCH and stochastic volatilitis modls. Rgarding Assumption, part i is satis d if Ex t x 0 t has full rank. Part ii rquirs that, ih data ar gnratd by 6= with 0 < p; q <, th conditional distribution of y t will xhibit faturs that ar not capturd by th singl rgim linar spci cation. hat is, th rsulting Kullback-Liblr divrgnc will b positiv. Finally, in Assumption 3, th rank rquirmnt ssntially rquirs P t= x tx 0 t to b positiv d nit in larg sampls. h rst oh assumption rquirs uniform laws of larg numbrs to hold. Bcaus tjt p; q; ; ; ar boundd btwn 0 and, th lattr holds undr Assumption and mild conditions on th momnts of y t and x t. 3 h tst statistic his sction studis thr issus. First, it considrs a family ost statistics basd on th log liklihood ratio. Scond, it prviws th di cultis involvd in driving th limiting distribution and outlins th stratgis for addrssing thm. hird, it xamins mpirically rlvant valus for th transition probabilitis p and q. h lattr is important not only for making th tsts practically rlvant, but also for th tchnical analysis ndd latr in th papr. Lt and dnot th maximizr oh null log liklihood: ; = arg max ; LN ; : 0 h log liklihood ratio valuatd at som 0 < p; q < thn quals LRp; q = max L A p; q; ; ; ; ; L N ;. his lads to th following tst statistic: SupLR = Sup LR p; q ; p;q whr is a compact st spci d blow and th suprmum is takn to obtain th strongst vidnc against th null hypothsis. Oprators othr than th 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 function that assigns wights on p and q. Such considrations lad to a family ost statistics basd on LRp; q. his papr focuss on SupLR ; th rsults xtnd immdiatly to ExpLR. 9

11 h tst statistic SupLR is not nw. For xampl, it has bn studid by Hansn 99 and Garcia 998. h contribution ohis papr is in obtaining an adquat approximation to its nit distribution and in providing an algorithm for simulating it. W now discuss nonstandard faturs associatd with this statistic and highlight our stratgy for tackling thm. First, th tim varying rgim probability tjt p; q; ; ; prsnt challngs. On th on hand, it can only b xprssd rcursivly; s 6 and 7. On th othr hand, whn obtaining asymptotic xpansions, it is ssntial to study its high ordr drivativs with rspct to ; and. So far, its ct on th log liklihood ratio has rmaind unknown, vn for th simplst situation whr p and q ar prspci d. In an important contribution, Cho and Whit 007, p. 675 suggstd to avoid this di culty by rplacing th liklihood 4 with that for a mixtur i.., a modl whr th rgim arrivs indpndntly oh past with tjt p; q; ; ; = throughout th sampl: P t= log f; + ; g. Howvr, this quasi log liklihood function bhavs di rntly from th actual liklihood whn p+q is di rnt from zro. his can translat into larg powr di rncs as sn latr in this papr. his is troubling bcaus in conomic and nancial applications th rgims ar typically substantially srially dpndnt. In this papr, w mak progrsss by obsrving that tjt p; q; ; ; and its drivativs can all b charactrizd as rst ordr di rnc quations, whos proprtis furthr simplify drastically onc w valuat thm at ;. his is a critical stp that maks th subsqunt analysis fasibl. Scond, th log liklihood ratio has thr nonstandard faturs as in mixtur modls: i h valus of p and q ar unidnti d undr th null hypothsis. Consquntly, thr ar in nit dirctions to approach any on distribution in th null hypothsis i.., th scor spac is in nit dimnsional. his complicats mattrs bcaus a ky stp in stablishing th asymptotic proprty oh liklihood ratio lis in dtrmining what happns to th scor function as w approach th null hypothsis. o addrss this, w trat th log liklihood ratio as an mpirical procss indxd by p and q, such that onc thy ar xd, th scor spac bcoms nit dimnsional. Such an mpirical procss prspctiv follows from a rich array of studis, with th most closly rlatd bing Hansn 99, Garcia 998, Cho and Whit 007, and Carrasco, Hu and Plobrgr 04. ii h scor of 4 is idntically zro whn valuatd at th null paramtr stimats. Consquntly, a scond ordr aylor xpansion is insu cint for analyzing th liklihood ratio. o addrss this, w obtain liklihood xpansions oh fourth ordr, and in som spci cations, oh ighth ordr. h obstacls for driving such xpansions ar substantial, spcially givn that w allow for multipl switching paramtrs. iii h valus p = and q = fall on th boundary of paramtr spac. 0

12 Cho and Whit 007 addrssd this issu by considring th surfacs spci d by H 0 ; H0 0 and H00 0 and thn combind th rsults to obtain th null limiting distribution. Hr, such an approach is no longr fasibl bcaus oh additional challng introducd by tjt p; q; ; ;. W pursu a di rnt rout. hat is, whn d ning th tst statistic, w rstrict th supports of p and q to b closd substs of 0;. his approach has also bn usd whn tsting for structural changs.g., Hawkins, 987, Andrws, 993, Andrws and Plobrgr, 994, and Bai and Prron, 998 and thrshold cts.g., Hansn, 996. It is also usd in Hansn 99 and Garcia 998. W now xamin mpirically rlvant valus for th transition probabilitis p and q. Hamilton 008, th rst paragraph in p. rviwd articls that applid rgim switching modls in a wid rang of contxts. Among thm, 0 articls considrd two-rgim spci cations with constant transition probabilitis. hs studis ar rlatd to: xchang rats Jann and Masson, 000, output growth Hamilton, 989 and Chauvt and Hamilton, 006, intrst rats Hamilton, 988, 005, Ang and Bkart, 00b, dbt-output ratio Davig, 004, bond prics Dai, Singlton and Yang, 007, quity rturns Ang and Bkart, 00a, and consumption and dividnd procsss Garcia, Lugr and Rnault, 003. Eightn sts of stimats ar rportd. h valus oh transition probabilitis ar btwn 0:855 and 0:998 for th mor prsistnt rgim and 0:740 and 0:997 for th othr. hs stimats ar rprsntativ of applications in conomics and nanc and thy strongly suggst two faturs. First, non oh valus corrspond to mixturs. hat is, th valus of p + q ar all substantially abov.0. Scond, at last on rgim is fairly prsistnt. hat is, th valu of p and q can b fairly clos to :0. Motivatd by th abov obsrvations, w suggst to spcify as follows = fp; q : p + q + and p; q with > 0g : his st can b gnralizd to allow for di rnt trimming proportions.g., rplacing p + q + and p; q with p + q + and p; q 3 with ; ; 3 > 0. h st can also b narrowd if additional information about p and q is availabl. For xampl, ihir valus ar both xpctd to b highr than 0:5, thn w can considr fp; q : 0:5 + p; q with > 0g : 3 h spci cation 3 is in fact consistnt with all th 0 studis mntiond in th prvious paragraph. In this papr, w focus on ; th rsults continu to hold for th lattr two spci cations, providd that th st in th limiting distribution is changd accordingly.

13 As will b sn in th nxt sction, for crtain modls and hypothss, th asymptotic distributions of LR p; q at p + q = and p + q > can b di rnt. his ariss whn r i r i ; = 0 i i r 0 ; 0 0 ; 4 holds for som i ; i f; :::; n +n g, whr i i is a known vctor of constants. Bcaus p+q = falls out oh st, such a chang in th distribution dos not intrfr with th rst ordr asymptotic approximation to th liklihood ratio. Howvr, th issu of approximation adquacy whn is small will aris and w shall account for it using a high ordr r nmnt as follows. First, w driv an asymptotic approximation to th liklihood ratio that is valid ovr whthr or not 4 holds. hn, w study th adquacy ohis approximation whn 4 holds. h analysis will show that th approximation bcoms lss adquat whn p + q is clos to. Nxt, w driv an highr ordr xpansion oh liklihood ratio undr p + q =. Finally, th additional trms in this xpansion ar incorporatd into th original asymptotic distribution to obtain a r nd approximation. Not that whthr or not 4 holds, as wll as th valus of i i, will b known onc th modl and th hypothss ar spci d. 4 h log liklihood ratio undr prspci d p and q h conditional rgim probability t+jt p; q; ; ; rprsnts th ky di rnc btwn Markov switching and mixtur modls. W thrfor bgin by studying this quantity as wll as its drivativs with rspct to ; and. h rsults will furthr nabl us to dvlop xpansions oh concntratd log liklihood undr th null hypothsis. h rsults rportd in this sction all hold uniformly ovr p; q [; ] [; ] with bing an arbitrary constant satisfying 0 < < =. 4. h conditional rgim probability W rst mak th following two obsrvations. a h xprssions 6 and 7 can b combind to rprsnt t+jt p; q; ; ; rcursivly as t = ; ; :::: t+jt p; q; ; ; 5 ; tjt p; q; ; ; = p + p + q ; tjt p; q; ; ; + ; tjt p; q; ; ; : his is a rst ordr di rnc quation that rlats t+jt p; q; ; ; to tjt p; q; ; ;. Immdiatly, this rlationship implis that th drivativs of t+jt p; q; ; ; with rspct to ; ; must also follow rst ordr di rnc quations. b Although ths di rnc quations

14 ar nonlinar whn valuatd at gnral valus of and, thy simplify substantially onc w lt =. Bcaus th asymptotic xpansions considrd latr ar around th null paramtr stimats, analyzing th lattr cas will b su cint. h nxt lmma contains th dtails on t+jt p; q; ; ; and its drivativs valuatd at = =, whr rprsnts an arbitrary valu in. D n an augmntd paramtr vctor = 0 ; 0 ; and thr sts of intgrs thy indx th lmnts in ; and, rspctivly I 0 = f; :::; n g ; I = fn + ; :::; n + n g ; I = fn + n + ; :::; n + n g : o as th notation, lt t+jt and f t dnot t+jt p; q; ; ; and ; or ; valuatd at som and = =. Also, lt r j :::r jk tjt, r j :::r jk ft and r j :::r jk ft dnot th k-th ordr partial drivativs of tjt p; q; ; ;, ; and ; with rspct to th j -th,:::; j k -th lmnts of valuatd at som and = =. h following rlationships hold: r j :::r jk ft = r j :::r jk ft if j ; :::; j k all blong to I 0, r j :::r jk ft = 0 if any of j ; :::; j k blongs to I, and r j :::r jk ft = 0 if any of j ; :::; j k blongs to I. Lmma Lt = p + q and r = with d nd in 3. hn, for t, w hav, undr = = :. t+jt =.. r j t+jt = r j tjt + E j;t, whr 8 >< E j;t = >: 0 rr j log rr j log f t if j I 0 if j I : if j I 3. r j r k t+jt = r j r k tjt + E jk;t, whr E jk;t is givn by with I a ; I b dnoting a cas with j I a and k I b ; a; b = 0; ; : I 0 ; I 0 : 0 I 0 ; I : rr j ft I 0 ; I : rr j ft r k ft r k ft + rr j r k ft rr j r k ft I ; I : r tjt j r k ft + r tjt k r j ft + rr j r k ft I ; I : r j tjt r k ft r k tjt r j ft + r r j ft I ; I : r j tjt r k ft + r k tjt r j ft r r j ft r k ft r k ft rr j r k ft r r j ft r k ft : 3

15 4. r j r k r l t+jt = r j r k r l tjt + E jkl;t, whr th xprssions for E jkl;t with j; k; l fi a ; I b ; I c g and a; b; c = 0; ; ar givn in th appndix. Rmark h lmma holds for sampls of any siz. It shows that, whn =, th conditional rgim probability t+jt quals th stationary probability ; whil its drivativs up to th third ordr all follow rst ordr linar di rnc quations. h laggd co cints always qual = p + q. Bcaus 0 < p; q <, ths di rnc quations ar always stabl. As sn blow, ths faturs allow us to apply proprtis of rst ordr linar systms to analyz th proprtis oh log liklihood. hy ar th ky lmnts that mak th subsqunt analysis fasibl. Now w tak a closr look at th four rsults in th lmma. h rst rsult is intuitiv. Bcaus th two rgims ar idntical whn =, obsrving th data brings no information about th rgim probability. h scond to fourth rsults r ct th rst to third ordr cts of a unit chang in th paramtrs valus on th rgim probability. For th rst ordr, changing th valu of has no ct; t+jt p; q; ; ; rmains qual to. Manwhil, changing th valus of and hav xactly th opposit cts, i.., r j t+jt = r j+n t+jt for any j I. h rsults concrning th scond ordr drivativs hav a similar structur. In particular, changing only has no ct, whil changing and aftr a chang in still hav qual opposit cts, as indicatd by th cass I 0 ; I and I 0 ; I. h rmaining thr cass ar mor complx, but thy all show that E jk;t dpnd only on r j tjt j I [ I and quantitis rlatd to th dnsity functions. h third ordr cts consist on di rnt cass corrsponding to di rnt combinations of j; k and l. For th analysis latr, th xact xprssions of E jkl;t will b unimportant. What mattrs is that thy dpnd only on lowr ordr drivativs of tjt functions. and quantitis rlatd to th dnsity h rcursiv structur within th rsults, i.., th highr ordr drivativs dpnd succssivly on th lowr ordrs with th rst ordr dpnding only on r j log and r j log f t, suggsts a stratgy for analyzing thir statistical proprtis. hat is, w can start with th rst ordr drivativs, and thn us th rsults cumulativly to study th scond ordr followd by th third ordr drivativs. Such a stratgy is implmntd in Lmma A. in th appndix. Using as th initial valu for t+jt p; q; ; ; is not rstrictiv. With a gnric nit initial valu, th rst rsult in th lmma bcoms t+jt = q + tjt, whil th othr rsults also hold, but with and r rplacd by tjt and tjt tjt, rspctivly. Bcaus jj <, tjt convrgs at an xponntial rat to as t incrass. Consquntly, th rst to th fourth 4

16 ordr drivativs all convrg to thir countrparts in th lmma at an xponntial rat. his fast rat of convrgnc implis that using a di rnt nit initial valu will not a ct th asymptotic rsults prsntd latr. Blow, w furthr illustrat th rsults in th lmma using th linar modl 9. h illustrativ modl cont d. Considr th gnral cas whr th rgrssion co cints and th rror varianc ar both allowd to switch. Lmma. implis: r t+jt = 0; w.r.t. th non-switching paramtrs 9 r t+jt = r tjt + r wt y t zt 0 wt 0 ; = r t+jt = r tjt + r yt w.r.t. th paramtrs in th rst rgim z 0 t w0 t ; ; 9 r t+jt = r t+jt ; = w.r.t. th paramtrs in th scond rgim r t+jt = r t+jt : ; Whn valuatd at th tru paramtr valu, th drivativs with rspct to and all rduc to stationary AR procsss with man zro. hir variancs ar nit and satisfy with dnoting th tru valu of and r j th rst ordr drivativ w.r.t. th j-th lmnt of E r j t+jt = r Ew jt ; Er t+jt r = : 4 h procsss spci d by Lmma.3-.4, although mor complx, also hav nit mans and variancs whn valuatd at th tru paramtr valus, providd that th rlvant momnts of w t ; z t and u t xist. Such rsults for th gnral modl ar stablishd in Lmma A. in th appndix. 4. Concntratd log liklihood and its xpansion o obtain an asymptotic approximation to th log liklihood ratio, a standard approach would b to xpand L A p; q; ; ; around th null stimats ; ;. Howvr, this is infasibl hr du to th complx dpndnc btwn th stimats of and as L A p; q; ; ; can hav multipl local maxima. Cho and Whit 007 ncountrd a similar problm and procdd by working with th concntratd liklihood. W adopt thir insightful stratgy. his allows us to brak th analysis into two stps. In th rst stp, w quantify th dpndnc btwn th stimats of and using th rst ordr conditions that d n th concntratd liklihood s Lmma blow. his ctivly rmovs and from th subsqunt analysis. In th scond 5

17 stp, w xpand th concntratd liklihood around = s Lmma 3 blow and obtain an approximation to LRp; q. Spci cally, lt ^ and ^ b th maximizr oh log liklihood for a givn valu hr th dpndnc of ^ and ^ on p and q is supprssd to simplify th notation, i.., ^ ; ^ = arg max ; L A p; q; ; ; : 7 Lt Lp; q; dnot th concntratd log liklihood, i.., Lp; q; = L A p; q; ^ ; ^ ;. hn, th two trms in th liklihood ratio satisfy: max ; ; L A p; q; ; ; = max Lp; q; and L N ; = Lp; q;. Consquntly: h LRp; q = max Lp; q; Lp; q; i : 8 For k, lt L k i :::i k p; q; i ; :::; i k f; :::; n g dnot th k-th ordr drivativ of Lp; q; with rspct to th i ; :::; i k -th lmnts of. Lt d j j f; :::; n g dnot th j-th lmnt of. hn, a fourth ordr aylor xpansion of Lp; q; around is givn by Lp; q; Lp; q; = n j= + 3! + 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 3 jkl p; q; d j d k d l n L jk p; q; d j d k 9 L 4 jklm p; q; d j d k d l d m ; whr in th last trm is a valu that lis btwn and. Blow, w provid two lmmas to analyz this xpansion. h nxt assumption is ndd in ordr to apply a law of larg numbrs and a cntral limit thorm to th trms in 9. It is similar to but slightly strongr than Assumption A5 iii in Cho and Whit 007. k=k, th rspctiv valus ar 4; ; and for k = ; ; 3 and 4. hr, instad of Assumption 4 hr xists an opn nighborhood of ;, dnotd by B ;, and a squnc of positiv, strictly stationary and rgodic random variabls g satisfying E +c t som c > 0, such that sup ; B ; r i :::r ik ; ; k k < t < L < for for all i ; :::; i k f; :::; n + n g ; whr k 5; k = 6 if k = ; ; 3 and k = 5 if k = 4; 5. 6

18 h nxt lmma charactrizs th drivativs of ^ and ^ with rspct to valuatd at =. o shortn th xprssions, lt t+jt and dnot t+jt p; q; ; ; and ; valuatd at ; ; = ; ;. Also, lt r i :::r ik tjt and r i :::r ik ft dnot th k-th ordr drivativ of t+jt p; q; ; ; and ; with rspct to th i -th; :::; i k -th lmnts of valuatd at ; ; = ; ;. Finally, d n U jk;t = ft r j r kft + r j tjt D jk;t = r 0 ; 0 ft 0 Ujk;t ; It = r 0;0 0 ft r 0;0 f t ; V jklm = U jk;tulm;t ; Dlm = D lm;t ; I = t= t= r kft + r jft r k tjt I t : t= ; 0 Not that U jk;t involvs th rst and scond ordr drivativs with rspct to th j-th and k-th lmnts of. h trm insid th curly brackts can also b rprsntd as = r j r k ft = r j tjt r k ft = r j ft r k tjt. As will b sn, Ujk;t dtrmins L jk p; q; whil D jk;t and I t appar within L 4 jklm p; q;. Lmma Undr th null hypothsis and Assumptions -4, for all k; l; m f; :::; n g, w hav:. Lt k b an n -dimnsional unit vctor whos k-th lmnt quals, thn r k ^ 5 r k^ = O p = : k. h scond ordr drivativs satisfy 4 r k r l ^ r k r l^ 3 5 = I t= D kl;t + O p = : 3. h third ordr drivativs satisfy 4 r k r l r m ^ r k r l r m^ 3 5 = O p : h abov rsults gnraliz Lmma Ba-d in Cho and Whit 007 to Markov switching modls and will b usd as inputs to analyz L k i ;:::;i k p; q; in th xpansion 9. hy show how th paramtrs and nd to chang in ordr to maximiz th liklihood whn is movd 7

19 away from. Spci cally, considr a unit chang in th j-th lmnt of. hn, Lmma. shows that, in th rst ordr, ^ j will chang by = + O p =, whil all th othr lmnts of ^ and ^ will only chang by a factor of ordr O p =. Intrstingly, th quantity = is simply th stationary probability oh scond rgim dividd by th rst. Lmma. prtains to changs in th scond ordr. hr, th tim variation in th conditional rgim probability ntrs xplicitly. Finally, th xprssion for th third ordr drivativ is not ndd for th limiting distribution and thrfor omittd. Assumption 5 hr xists > 0, such that 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 arbitrary small constant satisfying 0 < < =. In a standard problm, w would nd th scond ordr drivativ L jk p; q; to b continuous in.g., Ammiya, 985, p., or th third ordr drivativ L 3 jkl p; q; to b O p to nsur that a local quadratic xpansion is an adquat approximation to th log liklihood. Hr, L 4 jklm p; q; plays th sam rol as th scond ordr drivativ in a standard problm. his is why th abov assumption on th fth ordr drivativ is ndd. h nxt lmma provids th lading trms of L k i ;:::;i k p; q; k = ; ; 3; 4 in th xpansion 9. Lmma 3 Undr th null hypothsis and Assumptions -5, for all j; k; l; m f; :::; n g, w hav. L j p; q; = 0:. = L jk p; q; = = P t= U jk;t + o p. 3. 3=4 L 3 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 rst ordr drivativ L j p; q; is idntically zro for any sampl siz. Consquntly, th MLE of will convrg at a rat slowr than =. h scond ordr drivativ L jk p; q; is of ordr O p =, rathr than O p. As sn blow, its lading trm = P t= U jk;t convrgs to a multivariat normal distribution, whos proprty dpnds xplicitly on th tim varying conditional rgim probability. h third ordr drivativ L 3 jkl p; q; is also of ordr O p =. h xprssion of its lading trm is not ndd to driv th limiting distribution, but w will furthr analyz it whn providing a nit sampl r nmnt. Finally, th fourth ordr drivativ L 4 jklm p; q; is of ordr O p. = P t= U jk;t : Its lading trm provids a consistnt stimator oh asymptotic varianc of 8

20 Rmark h rst componnt of U jk;t, = r j r k ft =, is also prsnt whn tsting against mixtur altrnativs; s Cho and Whit 007, Lmma a. It is somtims calld th disprsion scor; s Lindsay 995, p.7. h rmaining two componnts ar nw and ar du to th Markov switching structur. hy can b rwrittn as = P t s= s r j log f t s r k log and = P t s= s r k log f t s r j log. Intuitivly, among th thr componnts, th rst picks up ovrdisprsion, whil th rmaining two pick up srial dpndnc introducd by th Markov rgims. Furthr, th magnituds oh lattr two componnts bcom mor pronouncd rlativ to th rst as approachs. his follows bcaus th rst componnt is indpndnt of aftr division by = whil th lattr two componnts involv wights s. his fatur suggsts that th powr di rnc btwn tsting against Markov switching altrnativs and mixtur altrnativs can b substantial whn th rgims ar prsistnt, i.., whn is clos to. his is con rmd by th simulations rportd latr. h illustrativ modl cont d. W illustrat th lading trms of = L jk p; q; and L 4 jklm p; q; in Lmma 3 using 9. Suppos only is allowd to switch. hn, U jk;t and D jk;t ar, rspctivly, and w jt w kt u t + t P s= s wjt s u t s h z 0 t u t u t wkt u t + t P s= s wkt s u t s wjt u t w 0 t u t i 0 Ujk;t ; whr u t dnot th rsiduals undr th null and = P t= u t. hs two xprssions show that U jk;t and D jk;t dpnd only on th rgrssors and th stimats undr th null hypothsis. his maks th covarianc function of = P t= U jk;t, and thrfor of = L jk p; q;, consistntly stimabl. his fatur will b usd whn driving th rlvant critical valus. 5 Asymptotic approximations Lt L p; q; b a squar matrix whos j; k-th lmnt is givn by L jk p; q; for j; k f; ; :::; n g. his sction consists of four sts of rsults. It stablishs th wak convrgnc of = L p; q; ovr p; q. It obtains th limiting distribution of SupLR. 3 It dvlops a nit sampl r nmnt that improvs th asymptotic approximation whn a singularity is prsnt. 4 It dvlops an algorithm to obtain th rlvant critical valus. 9

21 5. Wak convrgnc of L p; q; For 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 Djk 0 p r; q r I D lm p s ; q s ; 3 whr V jklm p r ; q r ; p s ; q s = E [U jk;t p r ; q r U lm;t p s ; q s ] ; D jk p r ; q r = ED jk;t p r ; q r, and I = EI t. Hr, U jk;t p r ; q r ; D jk;t p r ; q r and I t ar d nd as U jk;t, D jk;t and I t in 0 but valuatd at p r ; q r ; ; instad of p r ; q r ; ;. Proposition Undr th null hypothsis and Assumptions -5, w hav, ovr p; q : = L p; q; G p; q ; whr th lmnts of G p; q ar man zro continuous Gaussian procsss satisfying 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 3. In th appndix, th rsult is provd by rst showing th nit dimnsional convrgnc and thn th stochastic quicontinuity. Blow, w illustrat som important faturs of! jklm p r ; q r ; p s ; q s. It will mrg that this function dpnds on: th modl s dynamic proprtis.g., whthr th rgrssors ar strictly xognous or prdtrmind, which paramtrs ar allowd to switch.g., rgrssions co - cints or th varianc oh rrors, and 3 whthr nuisanc paramtrs ar prsnt. Consquntly, to mak th tst asy to apply in practic, w will nd a procdur that can adapt to ths faturs to obtain critical valus without rquiring laborious drivations from th practitionr. Such a procdur is dvlopd in Sction 5.4. h illustrativ modl cont d. function! jklm p r ; q r ; p s ; q s can b computd analytically: whr u t W considr a simplr vrsion of 9 for which th covarianc y t = w t fst=g + w t fst=g + u t ; i.i.d.n0;, and w t is a scalar rgrssor that is ithr strictly xognous.g., a constant or quals y t. D n r = p r + q r and s = p s + q s. h subscript * continus to dnot th tru paramtr valu. First, w allow to switch, whil assuming is unknown but rmains constant across th rgims. hn, in th strictly xognous rgrssor cas, th covarianc function 3 quals p r p s [V arwt + P k= r s k Ewt wt k ] q r q s 4 : 4 0

22 In th laggd dpndnt variabl cas, it 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 : 5 hs two functions ar di rnt vn whn w t i:i:d:n0; and = 0. his follows bcaus r tjt is indpndnt of r whn w t is strictly xognous, but not whn it is only prdtrmind. his comparison shows that th covarianc function is a ctd by th dynamic proprtis oh modl. Now, w considr th sam situation as abov but assum is known. hn, in th strictly xognous rgrssor cas, th covarianc function quals p r p s [Ewt 4 + P k= r s k Ewt wt k ] q r q s 4 ; 6 whil in th laggd dpndnt variabl cas, it quals p r p s 6 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 hs two functions ar di rnt from 4 and 5. his shows that th prsnc of nuisanc paramtrs can also altr th covarianc function. Nxt, w allow to switch but rquir th unknown rgrssion co cint to rmain constant across th rgims. hn, irrspctiv of whthr w t is strictly xognous, w hav Cov G p r ; q r ; G p s ; q s = p r p s 3 q r q s + r s : 8 r s his function is di rnt from both 4 and 5. hus, vn aftr conditioning on th modl, th covarianc function can still b di rnt dpnding on which paramtr is allowd to switch. W rport som simulation rsults to complmnt th abov analysis. h paramtr valus ar = 0:5 and =. In th strictly xognous rgrssor cas, w t is gnratd as bing indpndnt of u s at all lads and lags by w t = 0:5w t 8 : 7 + " t with " t i:i:d:n0;. his nsurs that th rgrssors follow th sam DGP in both cass. Furthr, w lt p r ; q r = 0:6; 0:9 and p s ; q s = 0:6; x with x varying btwn 0: and 0:9. Figur rports th v corrlations functions givn by 4-8 Hr, corrlations instad of covariancs ar plottd to as comparisons.

23 h solid lins starting from th top corrspond to 8, 6, 4, 7 and 5, rspctivly. hs functions dmonstrat clarly th dpndnc on th thr factors highlightd abov. Also includd in th gur ar corrlations computd from simulations i.., th dashd lins. hy ar gnratd by simulating sampls of 50 obsrvations using th sam paramtr valu as abov, computing = P t= U jk;t using ach sris and thn rpating 0,000 tims to obtain th mpirical corrlations. h valus ar clos to thir asymptotic approximations in all v cass. 5. Limiting distribution of SupLR Lt p; q b an n -dimnsional squar matrix whos j + k n ; l + m n -th lmnt is givn by! jklm p; q; p; q. hn, Proposition implis E[vc G p; q vc G p; q 0 ] = p; q. h nxt rsult givs th asymptotic distribution of SupLR. Proposition Suppos th null hypothsis and Assumptions -5 hold. hn: whr is givn by and SupLR sup sup p;q R n W p; q; ; 9 W p; q; = 0 vc G p; q 4 0 p; q : h quantity plays th rol of =4 in 9. Its dimnsion is una ctd by th prsnc of nuisanc paramtrs. If n =, thn th optimization ovr can b solvd analytically, lading to SupLR max[0; sup p;q G p; q = p p; q]. h right hand sid can qual zro with positiv probability, with th valu oh lattr dpnding on th covarianc function of G p; q = p p; q ovr p; q. If n >, th optimization will nd to b carrid out numrically. Howvr, bcaus W p; q; is a quadratic function of, th optimization rmains rlativly standard. Blow, w illustrat th abov limiting distribution and also xamin its adquacy in nit sampls. h illustration also suggsts th dsirability for a nit sampl r nmnt whn a furthr singularity is prsnt. h illustrativ modl cont d. W considr th following spcial cas of 9: y t = + y t + u t ; 30

24 whr u t i:i:d: N0; and ; and ar unknown. As shown blow, th distribution of SupLR, as wll as th adquacy oh asymptotic approximation, can di r substantially dpnding on whthr or is allowd to switch. Figur summarizs th nit sampl and asymptotic distributions of SupLR for tsting rgim switching in only or only. W considr = 0; = 0:5 and =. h st is spci d as 3 with = 0:05, h sampl siz is 50 and all rsults ar basd on 5000 rplications. wo faturs mrg. First, th nit sampl th solid lins and asymptotic th long dashd lins distributions ar both quit di rnt btwn th two cass. his is consistnt with th covarianc function of = P t= U jk;t bing paramtr dpndnt; s th illustration in Sction 5.. Scond, th asymptotic distribution provids an adquat approximation whn tsting for switching in, but not whn tsting for switching in. For th lattr cas, th asymptotic distribution falls to th lft oh nit sampl distribution. h structur of = P t= U jk;t is informativ about th scond fatur. It is givn by: For switching: For switching: n = P n = P t= u t t= u t + = P Pt t= y t + = P t= o u t, s= s u t s Pt s= s y t s u t s y t whr u t dnot th rsiduals undr th null and = P t= u t. Whn tsting for switching in, th rst trm in th curly brackts is in fact idntically zro for any sampl siz. Also, th magnitud oh scond trm dcrass as approachs 0, i.., as p+q approachs. Consquntly, in nit sampls, th magnitud of = P t= U jk;t can b too small to dominat th highr ordr trms in th liklihood xpansion. his xplains why th asymptotic distribution that rlis ntirly on = P t= U jk;t can b inadquat. In contrast, whn tsting for switching in, th rst trm in th curly brackts convrgs to a normal distribution that is indpndnt of p and q. hrfor, th issus discussd do not aris. Figur 3 provids som furthr information by comparing th nit sampl and asymptotic distributions of LRp; q whn tsting for switching at som slctd valus of p; q that qual 0:90; 0:90; 0:90; 0:75 and 0:90; 0:60. Consistnt with th discussion abov, a gap btwn th nit sampl distribution th solid lin and th asymptotic distribution th long dashd lin opns up and grows widr as p + q approachs. W hav also found in unrportd simulations, that whn tsting for switching, ths two distributions rmain clos to ach othr in all thr cass. In summary, th illustration suggsts that th asymptotic approximation in Proposition nds o u t ; 3