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 Fbruary 9, 07 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 hypothss and 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, Chuqing Jin, Hiroaki Kaido, Frank Klibrgn, Pirr Prron, Douglas Stigrwald and sminar participants at Amstrdam, Brown, BU statistics, UCSD, th 06 Economtric Socity Wintr Mting, th NBER-NSF tim sris confrnc, 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, 06 and Ang and immrmann 0. 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 dsirabl to hav tsting procdurs that focus 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 hypothsis and two rgims undr th altrnativ hypothsis. h analysis facs thr challngs. i Som nuisanc paramtrs ar unidnti d undr th null hypothsis. his givs ris to th Davis 977 problm. ii h

3 null hypothsis yilds a local optimum c.f. Hamilton, 990. Consquntly, a scond ordr aylor approximation to th liklihood ratio is insu cint for analyzing its asymptotic proprtis. iii h conditional rgim probability th probability of bing in a particular rgim at tim t givn th information up to 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, thir statistical proprtis hav rmaind lusiv. Consquntly, th asymptotic distribution of th log liklihood ratio has also rmaind unknown. 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 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

4 across th two rgims. h analysis is structurd into v stps: Stp charactrizs th conditional rgim probability and its high ordr drivativs with rspct to th modl s paramtrs. Whn valuatd undr th null hypothsis, th probability rducs to a constant whil th drivativs can all b rprsntd as linar rst ordr di rnc quations with 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. Stp drivs a fourth ordr aylor approximation to th liklihood ratio for xd p and q. his stp builds on Cho and Whit 007, but gos byond it to account for th ct oh tim variation in th conditional rgim probability on th liklihood ratio. h rsults ar informativ about why substantial powr gains rlativ to th QLR tst ar possibl. Stp 3 obtains an approximation to th liklihood ratio as an mpirical procss indxd by p and q. 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 04. Stp 4 provids a nit sampl r nmnt. his is motivatd by th obsrvation that, whil th limiting distribution in Stp 3 is adquat for a broad class of modls, it can lad to ovrrjctions whn a furthr singularity spci d latr is prsnt. his problm is addrssd by analyzing 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 safguard against thir cts. his lads to a r nd distribution that dlivrs rliabl approximations throughout our xprimntations. Stp 5 outlins an algorithm for simulating th r nd asymptotic distribution. For linar modls, th lmnts ndd for this algorithm can all b computd analytically. h asymptotic distribution shows som uncommon faturs. First, nuisanc paramtrs, though constraind to b constant across th rgims, can a ct th limiting distribution. Scondly, proprtis oh rgrssors i.., whthr thy ar strictly or wakly xognous also a ct th distribution. hirdly, th distribution dpnds on which paramtr i.., th intrcpt, th slop, or th varianc oh rrors 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 3

5 to th hypothsis and th modl s structur. h abov implications ar 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. Empirical studis hav stimatd rgim switching modls on a wid rang oim sris, including xchang rats, output growth, intrst rats, dbt-output ratio, bond prics, quity rturns, and consumption and dividnd procsss Hamilton, 008. Rgim switching has also bn incorporatd into DSGE modls; s Schorfhid 005, Liu, Waggonr and Zha 0, Bianchi 03, and Lindé, Smts and Woutrs 06. Doing so allows th transmission mchanism oh conomy to b occasionally fundamntally di rnt, a fatur that is byond th scop of constant-paramtr linar modls. Howvr, du to th lack of mthods with good powr proprtis, th prsnc of rgim switching is rarly formally tstd from a frquntist prspctiv. h procdur in this papr can potntially hlp to narrow this gap. 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 03 studid distributions osts for momnt rstrictions whn th associatd Jacobian matrix is dgnrat at th tru paramtr valu. his papr is th rst that simultanously tackls 4

6 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 procds 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 xd p and q. Sction 5 prsnts four sts of rsults: a th wak convrgnc oh scond ordr drivativ oh concntratd log liklihood, b th limiting distribution oh tst statistic, c a nit sampl r nmnt, and d 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 notations ar 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 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 th rgim at tim t. D n th information st at tim t as t = - ld :::; x 0 t ; y t ; x 0 t; y t : Lt fj t ; ; dnot th conditional dnsity of y t, satisfying y t j t ; s t v fjt ; ; ; if s t = ; fj t ; ; ; if s t = ; t = ; ::::; : 5

7 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: ps t = j t ; s t = ; s t ; ::: = ps t = js t = = p; ps t = j t ; s t = ; s t ; ::: = ps t = js t = = q: h 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; ; ; ; 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, a di rnt initial valu dos not a ct th asymptotic rsults. Whn = =, th log liklihood rducs to L N ; = log ; : 8 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. Assumption i h random vctor x 0 t; y t is strictly stationary, rgodic and -mixing with 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. 6

8 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 ; ; E[L 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: i [L N ; EL N ; ] = o p holds uniformly ovr ; with P r 0 ; 0 0 log ; r 0 ; 0 log ; bing positiv dfinit 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 ; ;. Assumptions 3 rquirs 8 and 4 to satisfy uniform laws of larg numbrs. It allows 4 to hav multipl local maximizrs. Undr Assumptions and 3, th maximizrs of 8 and 4 for 0 < p; q < convrg in probability to ; and ; ; undr th null hypothsis. Assumptions to 3 ar similar to thos usd in Cho and Whit 007, with two important di rncs. First, th liklihood function 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 th altrnativ hypothss can b statd as H 0 : = = for som unknown ; H : ; = ; for som unknown 6= and p; q 0; 0; : 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, th modl rmains in th rst rgim with probability, any statmnt about th scond rgim is irrlvant. h rvrs holds for H Blow, w introduc a modl that will b usd throughout th papr to illustrat th main componnts oh thory. 7

9 An illustrativ modl. Gaussian rrors: An important application of rgim switching is to linar modls with y t = z 0 t + w 0 t fsg + w 0 t fst=g + u t ; 9 whr ; and ar unknown nit dimnsional paramtrs and u t ar indpndntly normally distributd with man zro. h variabls z t and w t can includ laggd valus of y t. hrfor, th spci cation ncompasss nit ordr AR modls and ADL modls as spcial cass. In trms of and, 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 hypothss can b tstd dpnding on which paramtrs ar allowd to switch: a Only th varianc of u t can switch. Lt and dnot its variancs undr th two rgims. hn, in rlation to, = ; = and 0 = 0 ; 0 with = =. b Only th rgrssion co cints can switch. Lt dnot th varianc of u t. hn, =, = and 0 = 0 ;. c Both componnts can switch. hn, 0 = 0 ;, 0 = 0 ; and =. h rsults in this papr covr all thr situations. In th most gnral cas c, th dnsitis in ar givn by 3 n o 3 4 ; p yt zt xp 0 w0 t 5 6 = 4 n o 7 ; p yt zt xp 0 w0 t 5 : h 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. For Assumption, bcaus oh linarity, th -mixing of x 0 t; y t is implid by that of x t. his 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, as rviwd in Chn 03, includ thos gnratd by thrshold autorgrssiv modls, functional co cint autorgrssiv modls, and GARCH and stochastic volatilitis modls. For Assumption, its part i is satis d if Ex t x 0 t has full rank. Its part ii rquirs that, ih data ar gnratd by 6= with 0 < p; q <, thn th conditional distribution of y t should xhibit faturs that ar not capturd by th singl rgim linar spci cation. hat is, th rsulting Kullback-Liblr divrgnc should b positiv. Finally, in Assumption 3, th rank rquirmnt ssntially rquirs P x tx 0 t to b positiv d nit in larg sampls. h rst ohis assumption rquirs uniform laws of larg numbrs to hold. Bcaus tjt p; q; ; ; is boundd btwn 0 and, thy hold undr Assumption and mild conditions on th momnts of y t and x t. 8

10 3 h tst statistic his sction studis two issus. First, it considrs a family ost statistics basd on th log liklihood ratio. Scond, it xamins mpirically rlvant valus for th transition probabilitis p and q. h scond issu is important not only for making th tsts practically rlvant, but also for th tchnical analysis 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. 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 rgim. hs stimats ar rprsntativ of applications in conomics and nanc and thy strongly suggst two faturs. First, non oh valus corrspond to mixturs. 9

11 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 spcifying 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. 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. his sction bgins with studying this probability and its drivativs with rspct to ; and. his will nabl us to dvlop xpansions oh concntratd log liklihood undr th null hypothsis. h rsults rportd in this sction hold uniformly ovr p; q [; ] [; ] with bing an arbitrary constant satisfying 0 < < =. 4. h conditional rgim probability h following two obsrvations ar important. a h xprssions 6 and 7 can b combind to rprsnt t+jt p; q; ; ; rcursivly as t = ; ; :::: t+jt p; q; ; ; 4 ; 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 implis that th drivativs of t+jt p; q; ; ; with rspct to ; ; ar also rst ordr di rnc quations. b Although ths di rnc quations ar nonlinar at gnral 0

12 valus of and, thy simplify substantially if =. Bcaus th asymptotic xpansions considrd latr ar around th null paramtr stimats, considring = will b su cint. h nxt lmma charactrizs t+jt p; q; ; ; and its drivativs valuatd at = =, whr is 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 : Lt t+jt and f t dnot t+jt p; q; ; ; and ; 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 = =. Not that th 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. Lt I a ; I b dnot j I a and k I b ; a; b = 0; ;. hn, E jk;t is givn by: 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 :

13 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 any sampl siz. It shows that 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. It is worthwhil to tak a closr look at th four rsults in th lmma. Lmma. is intuitiv. Bcaus th two rgims ar idntical whn =, obsrving th data provids no furthr information about th rgim probability. Lmma. quanti s th rst ordr ct of changing a paramtr s valu on th rgim probability. hr, changing has no ct; t+jt p; q; ; ; rmains qual to. Changing th valus of and has xactly opposit cts, i.., r j t+jt = r j+n t+jt for any j I. Lmma.3 quanti s th scond ordr cts. h rst cas, I 0 ; I 0, shows that changing still has no ct. h nxt two cass show that changing and aftr a chang in still hav qual opposit cts. h rmaining thr cass ar mor complx, but thy all show that E jk;t only dpnd on r j tjt j I [ I and quantitis rlatd to th dnsity functions. Lmma.4 consist on di rnt cass with di rnt combinations of j; k and l. For th analysis latr, th xact xprssions of E jkl;t is unimportant. What is important is that thy dpnd only on lowr ordr drivativs of tjt and quantitis rlatd to th dnsity functions. h rcursiv structur within th rsults highr ordr drivativs dpnd succssivly on th lowr ordrs with th rst ordr dpnding only on r j log f t and r j log f t suggsts a stratgy for analyzing thir statistical proprtis. W start with th rst ordr drivativs, which ar simpl to analyz. hn, w us th rsults cumulativly to study th scond ordr followd by th third ordr drivativs. his 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, Lmma. bcoms t+jt = q + tjt. h othr rsults also continu to hold with and r rplacd by tjt and tjt tjt rspctivly. Bcaus jj <, tjt convrgs at an xponntial rat to as t incrass. Consquntly, th rgim probability and its drivativs all convrg to thir countrparts in th lmma at an xponntial rat. his rat of convrgnc implis that using a gnric nit initial valu will not altr th asymptotic rsults prsntd latr.

14 h illustrativ modl cont d. Considr th gnral cas whr th rgrssion co cints and th varianc oh rrors 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 follow stationary AR procsss with man zro. hir variancs ar nit and satisfy E r j t+jt = r Ew jt ; Er t+jt r = ; 4 whr dnots th tru valu of and r j dnots th rst ordr drivativ w.r.t. th j-th lmnt of. h procsss spci d by Lmma.3-.4 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. 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 rstrictd MLE ; ;. his is infasibl hr bcaus 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 follow thir insightful stratgy. his allows us to brak th analysis into two stps. h rst stp quanti s 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. h scond stp xpands th concntratd liklihood around = s Lmma 3 blow and obtains an approximation to LRp; q. Bcaus th conditional rgim probability is tim varying, th task hr is mor challnging than that of Cho and Whit 007. Lt ^ and ^ b th maximizr oh log liklihood for a givn valu th dpndnc of ^ and ^ on p and q is supprssd to simplify th notation: ^ ; ^ = arg max ; L A p; q; ; ; : 6 3

15 Lt Lp; q; dnot th concntratd log liklihood: 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 : 7 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 -th; :::; 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 8 L 4 jklm p; q; d j d k d l d m ; whr in th last trm is a valu btwn and. 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. his assumption is slightly strongr than Assumption A5 iii in Cho and Whit 007. hr, instad of k=k, th rspctiv valus ar 4; ; and for k = ; ; 3 and 4. h assumption prmits th application of laws of larg numbrs and cntral limit thorms to th trms in 8. 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 arbitrarily small constant satisfying 0 < < =. 4

16 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. In 8, 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 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 ; ; = ; ;. 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 ; ; = ; ;. 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 = r kft + r jft r k tjt I t ; ; 9 whr 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 xprssd 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 dtrmin L 4 jklm p; q;. Lmma Lt th null hypothsis and Assumptions -4 hold. For all k; l; m f; :::; n g:. 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 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 : 5

17 Lmma gnralizs Lmma Ba-d in Cho and Whit 007 to Markov switching modls. h rsults quantify how and nd to chang in ordr to maximiz th liklihood whn is movd away from. hy provid th ncssary inputs for th chain rul whn computing th drivativs L k i ;:::;i k p; q; k = ; ; 3; 4 in 8. his lads to th following lmma. 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 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; quals zro. his implis that th MLE of convrgs at at a slowr rat 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 U jk;t convrgs to a multivariat normal distribution, whos proprty dpnds 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 hr for obtaining 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. Its lading trm provids a consistnt stimator oh asymptotic varianc of = P U jk;t : 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. componnts ar nw and ar du to th Markov switching structur. h rmaining two 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 rspctivly. Among th thr componnts of U jk;t, th rst picks up ovrdisprsion and th othr two pick up srial dpndnc causd by th Markov rgims. Furthrmor, th magnituds oh last two componnts bcom mor pronouncd rlativ to th rst as approachs. his is bcaus th rst componnt is indpndnt of aftr division by = whil th last two componnts involv wights s. his 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 numrical rsults rportd latr. 6

18 h illustrativ modl cont d. In th linar modl 9, th lading trms of = L jk p; q; and L 4 jklm p; q; in Lmma 3 hav simpl structurs. Suppos only th rgrssion co cints ar allowd to switch. hn, U jk;t and D jk;t ar givn by 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 0 w 0 t u t i 0 Ujk;t ; whr u t dnot th rsiduals undr th null hypothsis and = P u t. h two xprssions show that U jk;t and D jk;t dpnd only on th rgrssors and th rsiduals undr th null hypothsis. As a rsult, th covarianc function of = L jk p; q; is consistntly stimabl. his fatur is valuabl for computing critical valus oh tst. 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. 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;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 9 but valuatd at p r ; q r ; ; instad of p r ; q r ; ;. Proposition Lt th null hypothsis and Assumptions -5 hold. hn, 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. 7

19 In th appndix, th rsult is provd by rst vrifying th nit-dimnsional convrgnc and thn th stochastic quicontinuity. h covarianc function! jklm p r ; q r ; p s ; q s in gnral is a ctd by th following factors: i th modl s dynamic proprtis.g., whthr th rgrssors ar strictly or wakly xognous, ii which paramtrs ar allowd to switch.g., rgrssions co cints or th varianc oh rrors, and iii whthr nuisanc paramtrs ar prsnt. h following illustration maks this clar. h illustrativ modl cont d. W considr a simplr vrsion of 9 for which th covarianc function! jklm p r ; q r ; p s ; q s can b computd analytically: y t = w t fsg + w t fst=g + u t ; whr u t i.i.d.n0; and w t is a scalar rgrssor that is ithr strictly xognous or qual to y t. Lt r = p r + q r and s = p s + q s. W continu to us subscript * to dnot th tru paramtr valu. First, considr th situation whr only is allowd to switch and is unknown. hn, whn th rgrssor is strictly xognous, th covarianc function quals p r p s V arwt + P k= r s k Ewt wt k q r q s 4 : 3 Whn th rgrssor is th laggd dpndnt variabl thrfor only wakly xognous, 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 : 4 hs two functions ar di rnt vn whn w t i.i.d.n0; and = 0. his is bcaus r tjt is indpndnt of r whn w t is strictly xognous, but not ncssarily whn it is prdtrmind. his shows that th covarianc function is a ctd by th dynamic proprtis oh modl. Now, considr th sam situation as abov but with th valu of bing known. hn, whn th rgrssor is strictly xognous, th covarianc function quals p r p s Ewt 4 + P k= r s k Ewt wt k q r q s 4 : 5 8

20 Whn th rgrssor is th laggd dpndnt variabl, it 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 two functions ar di rnt from both 3 and 4. his shows that th prsnc of nuisanc paramtrs can also a ct th covarianc function. Nxt, considr th situation whr only is allowd to switch and is unknown. Undr both strict and wak xognity: Cov G p r ; q r ; G p s ; q s = p r p s q r q s r s r s : 7 his function is di rnt from both 3 and 4. hrfor, th covarianc function can di r dpnding on which paramtr is allowd to switch. W rport som simulation rsults to complmnt th analysis abov. h paramtr valus ar = 0:5 and =. Whn th rgrssor is strictly xognous, w t is gnratd indpndntly of u s at all lads and lags as w t = 0:5w t + " t with " t i:i:d:n0;. his nsurs that th rgrssors follow th sam DGP in both cass. Furthr, 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 3-7 Hr, corrlations instad of covariancs ar plottd to facilitat comparisons. h solid lins starting from th top corrspond to 7, 5, 3, 6, and 4, rspctivly. hs functions show 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 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: SupLR sup sup p;q R n 9 W p; q; ; 8

21 whr is givn by and W p; q; = 0 vc G p; q 4 0 p; q : h quantity plays th rol of =4 in 8. 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. If n >, th optimization will nd to b carrid out numrically. Bcaus W p; q; is a quadratic function of, th optimization is rlativly standard. h illustrativ modl cont d. W illustrat th limiting distribution 8 and also xamin its adquacy in nit sampls. Considr th following spcial cas of 9: whr u t i:i:d: N0; and ; and ar unknown. y t = + y t + u t ; 9 As shown blow, th distribution of SupLR, as wll as th adquacy oh asymptotic approximation, can di r substantially dpnding on whthr th intrcpt or th slop paramtr is allowd to switch. Figur rports nit sampl th solid lins and asymptotic distributions th long dashd lins of SupLR for tsting rgim switching in only or only. h paramtr valus ar = 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. h gur shows two faturs. First, consistntly with Proposition and th illustration in Sction 5., th distributions in panl a ar signi cantly di rnt from thos in panl b. Scondly, th asymptotic distribution provids an adquat approximation in panl a, but not in panl b. For th lattr, using th asymptotic distribution will lad to ovr rjction oh null hypothsis. h scond fatur r cts th structur of = P U jk;t. Whn tsting for rgim switching in in panl b, = P U jk;t quals n = P u t + = P Pt s= s u t s whr u t dnot th rsiduals undr th null hypothsis. o u t, 30 Bcaus = P u t, th rst summation is in fact always zro. Furthrmor, th magnitud oh scond summation dcrass as approachs 0, i.., as p+q approachs. his suggsts that, in nit sampls, th magnitud of = P U jk;t may b too small to dominat th highr ordr trms in th liklihood xpansion. 0

22 As a rsult, th asymptotic distribution that rlis on = P U jk;t can b inadquat. h situation is di rnt whn tsting for switching in, whr = P U jk;t quals n = P u t y t + = P Pt s= s y t s u t s y t o u t : h rst trm in th curly brackts now convrgs to a normal distribution indpndnt of p and q. hrfor, th complication in 30 dos not aris. Figur 3 furthr compars th nit sampl and asymptotic distributions of LRp; q for tsting rgim switching in at som slctd valus of p; q. Consistntly with th discussion abov, a gap btwn th nit sampl distribution th solid lin and th asymptotic distribution th long dashd lin appars and grows widr as p + q approachs. Simulations also show that, whn tsting for rgim switching in, ths two distributions rmain clos to ach othr in all thr cass. h dtails ar omittd. h illustration suggsts that th asymptotic approximation in Proposition nds to b improvd ih hypothss imply that L p; q; quals zro whn p + q =. his is carrid out in th nxt subsction. 5.3 A r nmnt his sction obtains a sixth ordr xpansion oh liklihood ratio along p + q = and an ighth ordr xpansion at p = q = =. h rason for why th lattr is ndd is xplaind blow. h lading trms ar thn incorporatd into th limiting distribution in Proposition to dlivr a r nd approximation. hs xpansions ar basd on th following assumption. Assumption 6 h following linar rlationship holds for all t and all i ; i f; :::; n g : whr i i and i i r i r i ft = 0 i i r ft + 0 i i r ft ; 3 ar n - and n -dimnsional known vctors of constants. his assumption can b chckd onc th modl and th hypothss ar spci d. For xampl, whn tsting for rgim switching in th intrcpt in th AR modl 9, w hav r r ft = r. h nxt assumption strngthns Assumption 4. It is similar to A.5iv in Cho and Whit 007. h subsqunt analysis maks havy us ohir rsults dvlopd in Sction.3.. Assumption 7 hr xists an opn nighborhood of ;, B ;, and a squnc of positiv, strictly stationary and rgodic random variabls g satisfying E +c t < for som c > 0,

23 such that th suprmums oh following quantitis ovr B ; ar boundd from th abov by t : 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 = ; ; 3; 4, m = 5; 6; 7; i ; :::; i 7 f; :::; n + n g and j ; j f; :::; n g. Bfor procding, w rst stablish som notation. o approximat th third and sixth ordr drivativs oh concntratd log liklihood, d n s jkl;t p; q = p p q r r j r k l q 3 and lt G 3 jkl p; q b a continuous Gaussian procss with man zro satisfying! 3 jklmnu p r; q r ; p s ; q s = CovG 3 jkl p r; q r ; G 3 mnu p s ; q s = E [s jkl;t p r ; q r s mnu;t p s ; q s ] E r 0 ; 0 r s jkl;t p r ; q r I 0 ; 0 0f t s mnu;t p s ; q s ; whr s jkl;t p; q is th sam as s jkl;t p; q but valuatd at th tru paramtr valus th othr quantitis ar also valuatd at th tru paramtr valus. o approximat th fourth and ighth ordr drivativs, d n k jklm;t p; q = p p q p + q! p 3 rj r k r l r mft + 33 q r i r i r 0 ft i i ;i ;i 3 ;i 4 f 3 i 4 r i r i r 0 ft i 3 i 4 S t + 0 i i r r 0 ft i 3 i i i r r 0 ft i 3 i 4 and lt G 4 i i i 3 i 4 p; q dnot a continuous Gaussian procss with man zro satisfying! 4 i i :::i 8 p r ; q r ; p s ; q s = Cov G 4 i i i 3 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 3 i 4 ;t p r ; q r k i5 i 6 i 7 i 8 ;t p s ; q s ] r E 0 ; 0 r k i i f i 3 i 4 ;t p r ; q r I 0 ; 0 0f t k i5 i t f 6 i 7 i 8 ;t p s ; q s ; t whr th indx st S in 33 is givn by S = fjklm; jlkm; jmkl; kljm; kmjl; lmjkg, k i i i 3 i 4 ;tp; q is quivalnt to k i i i 3 i 4 ;tp; q but valuatd at th tru paramtr valus th rmaining quantitis ar also valuatd at th tru paramtr valus.

24 h nxt lmma charactrizs th asymptotic proprtis of L k i i :::i k p; p; for i ; :::; i k f; :::; n g and k = 3; :::; 8. It gnralizs Lmma 3, 4a, 5a- in Cho and Whit 007 by allowing for multipl switching paramtrs. Lmma 4 Undr th null hypothsis and Assumptions -7:. h following rsults hold uniformly ovr fp; q : p; q ; p + q = g: = L 3 jkl p; q; = = s jkl;t p; q + o p G 3 jkl p; q; = L 4 jklm p; q; = O p ; = L 5 jklmn p; q; = O p ; L 6 jklmnr p; q; =! 3 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 rsults hold at p = q = = : = L 3 jkl p; q; = o p ; = L 4 jklm p; q; = = k jklm;t 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 th lmnts of IND ar as follows: i = j; ach triplt i ; i 3 ; i 4 corrsponds to on of th 35 outcoms of picking 3 lmnts from {k; l; m; n; r; s; ug th ordring dos not mattr; and i 5 ; i 6 ; i 7; i 8 corrspond to th rmaining lmnts. h two sts of rsults charactriz th high ordr drivativs along th lin p + q =. Whn p 6= =, th third ordr trm = P s jkl;tp; p rplacs th scond ordr trm = P U jk;t to bcom th lading trm in th liklihood xpansion. Consquntly, a sixth ordr xpansion is ndd to approximat th liklihood ratio. Whn p = =, th fourth ordr trm = P k jklm;t p; p bcoms th lading trm, and an ighth ordr xpansion is ndd. h rstriction p = q is not imposd whn rprsnting th lading trm in = L 3 jkl p; q;. his nsurs that th co cint in front of r j r k r l ft = is corrct vn whn p + q 6=. For th sam rason, p = q = = is also not imposd whn xprssing th lading trm of 3

25 = L 4 jklm p; q;. h lmma assums that all th scond ordr drivativs with rspct to th switching paramtrs can b writtn as linar combinations oh rst ordr drivativs. Ihis rlationship holds only for a subst of drivativs, thn w simply st i i = 0 and i i th cass that do not satisfy 3. W now incorporat th lading trms in Lmma 4 to obtain a r nd approximation. G 3 p; q b a n 3 - dimnsional vctor whos j+k n +l n = 0 for Lt -th lmnt is givn by G3 jkl p; q. Lt 3 p; q dnot an n 3 by n3 matrix whos j +k n +l n ; m+n n +r n -th lmnt is givn by! 3 jklmnr p; q; p; q. D n W 3 p; q; = = vc G 3 p; q = p; q 3 : Lt G 4 p; q b an n 4 - dimnsional vctor whos j + k n + l n + m n3 -th lmnt is givn by G 4 jklm p; q. Lt 4 p; q b an n 4 by n4 matrix whos j + k n + l n + m n 3 ; n + r n + s n + u n3 -th lmnt is givn by!4 jklmnrsu p; q; p; q. D n W 4 p; q; = = 4 0 vc G 4 p; q p; q 4 : W propos approximating th distribution oh SupLR tst using S sup whr is spci d in. sup p;q R n n o W p; q; + W 3 p; q; + W 4 p; q; ; 34 Corollary Undr Assumptions -7 and th null hypothsis, w hav, ovr : Pr SupLR s Pr S s! 0: Rmark 3 h abov rsult holds irrspctiv of whthr or not th rlationship 3 holds. his follows bcaus th additional trms W 3 p; q; and W 4 both convrg to zro as!. hs trms provid r nmnt in nit sampls, having no ct asymptotically. h illustrativ modl cont d. quantitis 3 and 33 qual First, considr tsting for rgim switching in in 9. h 3 pp q ut q 3 3 ut and p p q + p q 3 3 p q 4 ut 4 6 ut + 3 : 4

26 h r nd approximations 34 ar rportd as dottd lins in Figurs b and 3. hy show that, rlativ to th original approximation, th improvmnts ar substantial. Nxt, considr tsting for rgim switching in. h quantitis 3 and 33 qual 3 pp q uty t q 3 3 uty t and p p q + p q 3 4 uty t 4 6 uty t + 3 : h r nd approximation is rportd as th dottd lin in Figur a. rlativ to th original approximation. hr is littl chang hrfor, th r nmnt substantially improvs th approximation whn tsting for rgim switching in th intrcpt. At th sam tim, it has littl ct whn tsting for switching in th slop co cint. his is dsirabl bcaus, for th lattr cas, th original approximation in Proposition is alrady adquat. 5.4 An algorithm for obtaining critical valus his sction shows how to obtain th critical valus of S d nd in 34. h ida is to sampl from th distribution oh vctor procss [vc G p; q 0 ; vc G 3 p; q 0 ; vc G 4 p; q 0 ] and thn solv th maximization problm 34 ovr p; q and R n. Bcaus this vctor procss is Gaussian with man zro, to gnrat th dsird draws it su cs to obtain a consistnt stimator of its covarianc function ovr. his obsrvation has also bn mad by Hansn 99 and Garcia 998. Lt U t p; q b an n -dimnsional vctor whos j + k n -th lmnt is givn by U jk;t in 9. Lt U 3 t p; q b an n 3 -dimnsional vctor whos j + k n + l n -th lmnt is givn by s jkl;t p; q in 3. Lt U 4 t p; q b an n 4 -dimnsional vctor whos j + k n + l n + m n3 -th lmnt is givn by k jklm;t p; q in 33. D n G t p; q = 6 4 U t p; q U 3 t p; q U 4 t p; q Lt U t p; q; U 3 t p; q; U 4 t p; q and G t p; q b d nd as U t p; q, U 3 t p; q, U 4 t p; q and G t p; q but valuatd at th tru valus undr th null hypothsis. Bcaus th vctor procss : 5