TESTING FOR REGIME SWITCHING

Transcription

1 TESTING FOR REGIME SWITCHING JIN SEO CHO School of Economics and Finance Vicoria Universiy of Wellingon PO Box 600, Wellingon, 6001, New Zealand and HALBERT WHITE Deparmen of Economics Universiy of California San Diego 9500 Gilman Dr., La Jolla, CA, , U.S.A. Firs version: Feb. 17, This version: Apr. 7, Absrac We analyze use of a quasi-likelihood raio (QLR) saisic for a mixure model o es he null hypohesis of one regime versus he alernaive of wo regimes in a Markov regime-swiching conex. This es explois mixure properies implied by he regime-swiching process bu ignores cerain implied serial correlaion properies. When formulaed in he naural way, he seing is non-sandard, involving nuisance parameers on he boundary of he parameer space, nuisance parameers idenified only under he alernaive, or approximaions using derivaives higher han he second order. We exploi recen advances by Andrews (2001) and conribue o he lieraure by exending he scope of mixure models, obaining asympoic null disribuions differen from hose in he lieraure. We furher provide criical values for popular models or bounds for ail probabiliies useful in consrucing conservaive criical values for regime-swiching ess. We compare he size and power of our saisics o oher useful ess for regime swiching via Mone Carlo and find relaively good performance. We apply our mehods o re-examine he classic carel sudy of Porer (1983) and reaffirm Porer s findings. Key Words Markov Regime Swiching, Mixure Model, Likelihood-Raio Saisic, Null Disribuion, Bounds for Criical Values, Carel Sabiliy. Subjec Class Primary C12, Secondary L13. Acknowledgemens The co-edior, Joel Horowiz, and hree anonymous referees provided very helpful commens for which we are mos graeful. We also have benefied from discussions wih Swarnji Arora, Roger Bowden, Rober Davies, Wouer den Haan, Seven Durlauf, Graham Ellio, Parick Fizsimmons, Clive Granger, James Hamilon, Chirok Han, Bruce Hansen, Tae-Hwan Kim, Bruce Lehmann, Mark Machina, Angelo Melino, Masao Ogaki, Peer Phillips, Ehsan Soofi, Kamil Tahmiscioglu, Allan Timmermann, Peer Thomson, and seminar paricipans a UCSD, Wisconsin-Milwaukee, SRA, he Universiy of Auckland, ESAM03, VUW, and NZESG03. We are indebed o Rober Porer for provision of his daa.

2 1. Inroducion Models of regime-swiching behavior play an imporan role in analyzing economic daa. For example, in indusrial organizaion, Porer (1983) in a classic paper uses a wo regime model o invesigae carel behavior. In macroeconomics, Hamilon (1989) in anoher classic paper uses a wo regime model o invesigae he properies of poswar U.S. real GNP. Conducing inference abou such processes is ofen a main goal. I is criical ha such inferences be properly drawn, as hese can be used o argue he innocence or guil of firms accused of anirus violaions or o inform key economic policy decisions. Neverheless, as Hamilon (1996), among ohers, has poined ou, conducing proper inference in regime-swiching models is paricularly challenging. As we discuss in deail laer, his challenge arises due o he fac ha when formulaed in he naural way, esing he null hypohesis ha here is a single regime (versus he alernaive of, say, wo regimes) can involve a nuisance parameer idenified only under he alernaive, as well as a parameer on he boundary of he parameer space. Sandard likelihood raio (LR) ess (and relaed Lagrange muliplier or Wald ess) canno be conduced in he usual manner. A main goal of his paper is herefore o develop sraighforward mehods ha researchers can use o draw large sample inferences, esing he null of one regime versus he alernaive of wo regimes in a regime-swiching model. Recen significan advances by Andrews (1999, 2001) play an imporan role in aaining his goal. A furher goal of his paper is o revisi he work of Porer (1983). This serves he dual purpose of illusraing our mehods in a classical seing and, as i urns ou, affirming Porer s original inferences. In he prior lieraure, aemps o es he number of regimes have proceeded by addressing cerain aspecs of he problem. For example, Hansen (1992) considers his problem using Markov regime-swiching models, and obains a lower bound for he limiing disribuion of a sandardized LR saisic. As will be clear laer, however, he null parameer space can be pariioned ino wo muually exclusive subses: one wih he boundary parameer problem and one wihou he boundary parameer problem. Hansen s bound considers he behavior of he LR saisic on only one of hese wo subses. Garcia (1998) reviews Hansen s problem. As we see below, however, boh subses are indeed relevan, and due o he boundary parameer problem, sandard argumens canno apply o he LR saisic. As we discuss in Secion 2, applying he LR saisic o esing one vs. wo Markov regimes is challenging, because he log-likelihood for he wo-regime alernaive does no facor in he usual way. This leads o geomeric growh of he populaion variance of he log-likelihood firs derivaive under he null, ruling ou applicaion of sandard cenral limi resuls. Moreover, he power of such a es urns ou o be weaker han in he sandard case. Insead, we proceed by applying mixure models, ignoring cerain ime series dependence properies implied by he regime-swiching process. This yields a quasi-log-likelihood ha does facor in he usual way and whose analysis is much more racable. The resuling quasi-likelihood raio (QLR) es is hus sensiive o he mixure aspec of he regime-swiching process, delivering a es wih appealing power under he alernaive. 1

3 Mixure models are widely used for idenically and independenly disribued (i.i.d.) daa. Tesing he number of mixure componens also has problems similar o hose encounered in esing Markov regime swiching, namely he boundary parameer problem and nuisance parameers presen only under he alernaive. Much of he lieraure aemps o avoid hese problems or mus confron associaed difficulies in aemping o es he number of componens. For example, Chesher (1984) and Lancaser (1984) es for unobserved heerogeneiy by esing he hypohesis of correc model specificaion, an indirec mehod of esing he mixure hypohesis. Neyman and Sco s (1966) C(α) saisic ess he mixure hypohesis, moivaed by he properies of he dispersion of he dependen variable of ineres under he null and alernaive hypoheses relevan here, as well reviewed by Lindsay (1995). On he oher hand, Harigan (1985), Ghosh and Sen (1985), Liu and Shao (2003), and references herein consider he LR saisic for esing he number of componens of a mixure model, and show ha i converges weakly o a funcional of a coninuous Gaussian process on a compac parameer space. As hese auhors show, compacness plays a paricularly imporan role in deermining he null disribuion of he LR saisic. This is rue here, oo, and we devoe paricular aenion o he crucial role played by he parameer space. As we show, mixure models can give rise o variey of ineresing behaviors. In paricular, he model considered by Porer (1983) doesn have a coninuous Gaussian process as he limi of he LR saisic. We sudy he mixure model in he Markov regime-swiching conex, and conribue o he lieraure in several ways. Firs, we conribue no only by providing a way o exploi he associaed QLR saisic in such a way ha he previously encounered difficulies can be avoided, bu by doing so in a conex ha explicily allows he observable random variables o exhibi ime-series dependence. To his end, we show how he mixure model esimaors behave when he daa are generaed by a β-mixing process ha is a Markov regime-swiching process under he alernaive. This yields he asympoic null disribuion for he QLR saisic on a compac parameer space. Nex, we exend he mixure lieraure by examining models whose null weak limis are funcionals of disconinuous Gaussian processes, as well as hose models whose limis are coninuous Gaussian processes. We carefully examine hese, providing examples and comparing criical values obained wih and wihou accoun aken of he boundary parameer problem. As we show, if boundary condiions are ignored, hen criical values can be oo conservaive. Nex, we demonsrae our mehods by using hem o revisi Porer s (1983) empirical sudy. Finally, we provide approximae null disribuions for popular models and a mehod for obaining conservaive bounds when approximaions are oherwise hard o obain. This paper is organized as follows. In Secion 2, we assume ha given daa follow a Markov regimeswiching process, and we show ha when a mixure model is applied o hese daa, we can obain an associaed QLR saisic ha can be used o es he number of regimes. Furher, we discuss how o obain criical values or heir conservaive approximaions. Secion 3 provides resuls of Mone Carlo experimens in which we compare he size and he power of our saisics wih ohers in he lieraure. We revisi Porer s (1983) analysis in Secion 4. Mahemaical proofs are colleced in he Appendix. 2

4 Before proceeding, we inroduce some useful mahemaical noaion. We le denoe weakly converges o, and and are he Euclidean and he uniform norms respecively. 2. Markov Regime-Swiching Processes and Mixure Models We consider a specific framework designed o faciliae analysis of key aspecs of he problem of ineres, using he following daa generaing process (DGP). A1 : (i) The observable random variables {X R d } n =1, d N, are generaed as a sequence of sricly saionary β-mixing random variables such ha for some c > 0 and ρ [0, 1), he β-mixing coefficien, β τ, is a mos cρ τ. (ii) The sequence of unobserved regime indicaors, {S {1, 2}} n =1, is generaed as a firs-order Markov process such ha P (S = j S 1 = i) = p ij wih p ii [0, 1] (i, j = 1, 2). (iii) The given {X } is a Markov regime-swiching process (hidden Markov process). Tha is, for some θ := (θ 0, θ 1, θ 2 ) Rr 0+2, F ( X 1 ; θ0 X F 1, θ 1 ), if S = 1, F ( X 1 ; θ0, θ 2 ), if S = 2, where F 1 := σ(x 1, S ) is he smalles σ-algebra generaed by (X 1, S ) := (X 1, X 1, S,, S 1 ); r 0 N; and he condiional cumulaive disribuion funcion (CDF ) of X F 1, F ( X 1 ; θ0, θ j ), has a probabiliy densiy funcion (PDF ) f( X 1 ; θ0, θ j ) (j = 1, 2). Furher, for (p 11, p 22 ) [0, 1) [0, 1) \ {(0, 0)}, θ is unique in R r 0+2. The β-mixing condiion is suiable for he Markov regime-swiching process, as discussed by Davydov (1973), Doukhan (1994), and Vidyasagar (2003). As well reviewed by Ephraim and Merhav (2002), he populariy of his DGP exends far beyond economics. In economics, Porer (1983) examines he carel sabiliy problem, assuming p 11 = p 21, as we discuss in Secion 4. Hamilon (1989) considers he case in which X F 1 is a funcion of S m,, S (m N {0}) and X 1, so ha he unobserved wo sae process, {S }, induces a DGP for X F 1 wih 2 m+1 unobserved saes. In his paper, we resric our aenion o he case m = 0 and focus sricly on esing for regime swiching. Also, noe ha we canno assume ha p 11 = p 22 = 0, because if so {S } becomes deerminisically periodic, implying ha {X } is uncondiionally heerogeneous, hus violaing he saionariy assumpion. Many models for his DGP have been proposed. We consider he following model. A2 : (i) A model for f( X 1 ; θ0, θ j ) is {f( X 1 ; θ j ) : θ j := (θ 0, θ j ) Θ}, where Θ := Θ 0 Θ R r0+1 ; and Θ 0 and Θ are convex and compac ses in R r 0 and R respecively. Furher, for each θ j Θ, f( X 1 ; θ j ) is a measurable PDF wih CDF F ( X 1 ; θ j ) (j = 1, 2). (ii) For every x R d, f(x X 1 ; ) C (2) ( Θ) (he se of wice coninuously differeniable funcions on Θ) almos surely. (a.s.) 3

5 For noaional simpliciy, we will abbreviae F (X X 1 ; θ j ) and f(x X 1 ; θ j ) as F (θ j ) and f (θ j ) respecively (j = 1, 2). Also, unless confusion will oherwise resul, we omi he funcion argumen placeholder, so ha as an example, f ( ) is also denoed as f. Suppose ha he researcher wishes o es wheher here is only a single regime. Formally, relevan hypoheses are: for an unknown θ, H 0 : p 11 = 1 and θ1 = θ ; p 22 = 1 and θ2 = θ ; or θ1 = θ2 = θ ; H 1 : (p 11, p 22) [0, 1) [0, 1) \ {(0, 0)} and θ1 θ2, where θ is defined in he following A3. A3 : (θ 0, θ ) maximizes n 1 E[ n =1 l ] uniquely in he inerior of Θ, where for each θ 1, l (θ 1 ) := log(f (θ 1 )). Given he Assumpions A1 and A2(i), he log-likelihood funcion can be represened as ( [ n ] ) L n (p 11, p 22, θ) := log PF (θ) ι, where for each and θ Θ := Θ 0 Θ Θ, F (θ) is a 2 2 diagonal marix wih j-h diagonal elemen f (θ j ); P := [p ij ] parameerizes he ransiion marix of S (i, j = 1, 2); := [, 1 ] wih := (1 p 22 )/(2 p 11 p 22 ); and ι is a 2 1 vecor of ones. Because he log-likelihood funcion canno be represened as a sum of individual log-likelihood funcions, he LR saisic urns ou o behave in unappealing ways. Specifically, if p 11 = 1 (or p 22 = 1), hen he associaed null firs-order derivaive of he log-likelihood funcion has a populaion variance growing geomerically as he sample size increases, and he sandard cenral limi heorem canno be applied. Furher, as implied in Secion 2.3, he power of he LR saisic is weaker han in he sandard n 1/2 case when θ1 = θ 2 = θ. Because of hese difficulies, we ake a differen approach. = A Quasi-LR Saisic We avoid hese difficulies by focusing insead on he quasi-likelihood funcion for a mixure model. As we show, his permis us o esimae key aspecs of he Markov regime-swiching process wihou sacrificing much. Thus, consider he mixure model quasi-log-likelihood funcion defined as follows: for each (, θ) [0, 1] Θ, le L n(, θ) := n =1 l (, θ), where l (, θ) := log(f (θ 1 ) + (1 )f (θ 2 )). This model capures he mixure aspec of he condiional PDF of X σ(x 1 ) and he uncondiional PDF of S under he alernaive. More precisely, he condiional PDF of X σ(x 1 ) is f (θ 0, θ 1 )P (S = 1 σ(x 1 )) + f (θ 0, θ 2 )P (S = 2 σ(x 1 )), and he mixure weighs, P (S = 1 σ(x 1 )) and P (S = 2 σ(x 1 )), are random variables wih uncondiional means, and 1, respecively, where := (1 p 22 )/(2 p 11 p 22 ). We replace hese wih an unknown parameer and esimae by maximizing he quasi-log-likelihood funcion. This specificaion ignores he serial correlaion in {S }, whereas he serial correlaion of X is capured by f. Thus, we work wih a model ha ignores he serial correlaion in he unobserved sae; Theorem 1 below shows his does no maer for esing he number of regimes. 4

6 In paricular, our QLR es is poenially powerful agains any regime-swiching process, even if S is no a Markov process. Raher han esing for he specific serial correlaion implied by regime swiching, we es for he mixure properies of X σ(x 1 ) generaed by S. Here we leave aside esing for his serial correlaion. (See Carrasco, Hu, and Ploberger (2004), who recenly propose a es saisic of his sor.) Despie he exensive analysis of mixure models for he case of i.i.d. daa (see Harigan (1985), Ghosh and Sen (1985) Liu and Shao (2003) and references herein), mixure models have no ofen been used for esing for regime swiching. Our analysis hus no only conribues o he mixure model lieraure by showing is uiliy for esing he number of regimes, bu also conribues o boh he Markov swiching and mixure model lieraure by demonsraing he uiliy of mixure models in esing he number of Markov regimes in a β-mixing conex. We also exend he scope of he mixure models by considering oher popular mixures yielding regime-swiching es saisics whose null limiing disribuions are differen from hose in he lieraure. We furher conribue by providing formulae and algorihms ha can be used o calculae criical values and/or upper bounds for our es saisics ha are useful in applicaions. The almos sure limis of he esimaors maximizing L n can be represened in erms of he coefficiens of he DGP boh under he null and he alernaive. For his, we assume he following regulariy condiions. A4 : For all (, θ) [0, 1] Θ, n 1 E[ n =1 l (, θ)] exiss and is finie. A5 : (i) There exiss a sequence of posiive, sricly saionary, and ergodic random variables, {M }, such ha (a) E[M ] < < ; (b) sup (,θ) [0,1] Θ (,θ) l (, θ) M. These assumpions are mild and enable us o apply he srong uniform law of large numbers (SULLN) o he mixure model quasi-log-likelihood. Theorem 1: (a) Given A1, A2 (i, ii), A3, A4, A5 (i), and H 0, (ˆ n, q ˆθ q 0,n, ˆθ q 1,n, ˆθ q 2,n ) {(, θ 0, θ 1, θ 2 ) [0, 1] Θ : = 1 and θ 1 = θ ; or θ 1 = θ 2 = θ ; or = 0 and θ 2 = θ } a.s., and (ˆθ 0,n n, ˆθ 1,n n ) (θ 0, θ ) a.s., where (ˆ n, q ˆθ q 0,n, ˆθ q 1,n, ˆθ q 2,n ) is he ( unresriced ) quasi-mle (QMLE) of he mixure model, and (ˆθ 0,n n, ˆθ 1,n n ) is he ( resriced ) QMLE imposing H 0. 1 (b) Given A1, A2 (i, ii), A3, A4, A5 (i), and H 1, (ˆ n, q ˆθ q 0,n, ˆθ q 1,n, ˆθ q 2,n ) (, θ0, θ 1, θ 2 ) a.s. Under he null, he QMLE is he MLE, and i converges o a se in Theorem 1(a). The conclusion of Theorem 1(b) is crucial o he goal of his paper. As poined ou by Levine (1983), a correc model specificaion for he condiional mean is imporan for consisen esimaion of he condiional mean, bu correc specificaion of DGP dynamics is no necessary. Theorem 1 assumes ha X F 1 is correcly specified, and ignores he dynamics induced by {S }. Levine s (1983) poin applies o he curren conex, and from his, i follows ha (ˆθ q 0,n, ˆθ q 1,n, ˆθ q 2,n ) (θ 0, θ 1, θ 2 ) a.s. We show addiionally ha he esimaor for he parameer replacing he random weighs, P (S = 1 σ(x 1 )), is consisen for he uncondiional mean of he random weighs. Tha is, ˆ q n converges o a.s. Thus, he mixure model is correcly specified 1 The superscrips q and n are used o denoe quasi-mle and null-imposing MLE respecively. 5

7 for boh X σ(x 1 ) and he uncondiional mean of {S }, bu misspecified in erms of he dynamics of {S }. Regime-swiching ess can be based on he limis of he esimaors. Noe ha = 1 if and only if p 11 = 1 (and = 0 if and only if p 22 = 1), so ha here are wo regimes if and only if (0, 1). We exploi his fac and es hese hypoheses using he QLR saisic defined as he log-likelihood raio compued from he QMLEs under he null and he alernaive. This modifies our prior hypoheses as follows: for an unknown θ, H 0 : = 1 and θ1 = θ ; θ1 = θ2 = θ ; or = 0 and θ2 = θ ; versus H 1 : (0, 1) and θ1 θ2. The null H 0 can be furher pariioned: for an unknown θ, H 01 : = 1 and θ 1 = θ ; H 02 : θ 1 = θ 2 = θ ; or H 03 : = 0 and θ 2 = θ. In his conex, several sandard assumpions are violaed, summarized as follows. Firs, if = 1 (resp. = 0), hen θ2 (resp. θ 1 ) is no idenified, so ha he Davies problem (1977, 1987) occurs: a nuisance parameer is presen only under he alernaive. A he same ime, is on he boundary of [0, 1], which also violaes he sandard condiion yielding he chi-square limiing disribuion for he QLR saisic. Second, if θ 1 = θ 2, hen is no idenified. Hence, he nuisance parameer problem again occurs, bu he boundary parameer problem does no appear. The resuls of Andrews (2001) hus play a key role in analyzing H 01 and H 03, bu no H 02. To analyze H 02, i urns ou ha he sandard second-order derivaive-based approximaion o he QLR saisic has o be improved o approximaions using higher-order derivaives. We resolve hese challenges under each hypohesis and combine he ensuing resuls o derive he null limiing disribuion of he QLR saisic Null Disribuion of he QLR Saisic under H 01 and H 03 We firs examine he QLR behavior under H 01 and H 03 wih suiable regulariy condiions. A5 : (ii) There exiss a sequence of posiive, sricly saionary, and ergodic random variables, {M }, such ha (a) for some δ > 0, E[M 1+δ ] < < ; (b) sup (,θ) [0,1] Θ (,θ) l (, θ) (,θ) l (, θ) M ; and (c) sup (,θ) [0,1] Θ 2 (,θ) l (, θ) M. A6 : (i) For each (, θ 0, θ 1, θ 2 ) wih θ 1 θ 2, λ min(b(, θ 0, θ 1, θ 2 )) 0 such ha (a) if λ min(b(, θ 0, θ 1, θ 2 )) > 0, hen λ max(b(, θ 0, θ 1, θ 2 )) < ; or (b) if λ min(b(, θ 0, θ 1, θ 2 )) = 0, hen = 1 or 0, and for each θ 2 θ and θ 2 θ, λ min (C (θ) (θ 2, θ 2 )) > 0 and λ max(c (θ) (θ 2, θ 2 )) <, where for each (, θ), B(, θ) = E[ (,θ) l (, θ) (,θ) l (, θ) ]; for each (θ 2, θ 2 ), C (θ) (θ 2, θ 2) := C(θ) 11 (θ 2, θ 2 ) C(θ) 12 (θ 2 ) C (θ) 21 (θ := E[r (θ 2 )r (θ 2 )] 1 E[r (θ 2 )r(1) (θ )], 2) C (θ) 22 E[r (θ 2 )r (1) (θ )] E[r (1) (θ )r (1) (θ ) ] r (θ 2 ) := f (θ0, θ 2)/f (θ0, θ ), and r (1) (θ 2 ) := θ 1f (θ0, θ 2)/f (θ0, θ ); and λ max ( ) and λ min ( ) denoe he maximum and he minimum eigenvalues of a given marix, respecively. 6

8 These assumpions enable us o apply he CLT on he se of unidenified parameers. In paricular, we impose A6(i) o approximae he quasi-log-likelihood funcions by quadraic funcions. More precisely, alhough n 1 E[ l ] is no uniquely maximized under H 01 and H 03 and herefore canno be usefully approximaed by a quadraic funcion, A6(i) neverheless ensures ha for each θ 2 ( θ ), n 1 E[ l (1,,, θ 2 )] (or, for each θ 1 ( θ ), n 1 E[ l (0,, θ 1, )]) can be locally approximaed by quadraic funcions under H 01 (or H 03 ), providing he necessary degree of idenificaion hrough C(θ) (θ 2, θ 2 ). By A6(i), he null model is idenified for each θ 2. The QLR scores have he following properies. Lemma 1: (a) For each elemen in {(1, θ0, θ, θ 2 ) [0, 1] Θ : θ 2 Θ \ {θ }}, define [ ] 1 [ ] n n S 1,n (θ 2 ) := n 1 (, θ 1 )l (, θ) (, θ 1 )l (, θ) n 1/2 (, θ 1 )l (, θ). =1 Given A1, A2 (i, ii), A3, A4, A5 (ii), A6 (i), and H 01, S 1,n S 1 over Θ (ɛ) := {θ 2 Θ : θ 2 θ > ɛ} for each ɛ > 0 such ha for each θ 2 Θ (ɛ), S 1 (θ 2 ) N(0, C (θ) (θ 2, θ 2 ) 1 ), and for each θ 2, θ 2 Θ (ɛ), E[S 1 (θ 2 )S 1 (θ 2 )] = C(θ) (θ 2, θ 2 ) 1 C (θ) (θ 2, θ 2 )C(θ) (θ 2, θ 2 ) 1. (b) Given he same assumpions as in Lemma 1 (a), G : Θ (ɛ) R is differeniable in he mean, where for each θ 2, θ 2, G(θ 2) := Ω (θ) (θ 2, θ 2 ) 1/2 S [1:1] 1 (θ 2 ) and Ω (θ) (θ 2, θ 2 ) := C(θ) 11 (θ 2, θ 2 ) C(θ) 12 (θ 2)[C (θ) 22 ] 1 C (θ) 21 (θ 2 ). Furher, A [i:j] is a sub-vecor conaining he i-h hrough j-h elemens of he vecor A. 2 =1 We omi explici analysis for H 03, as he same score as S 1,n is obained by symmery. The proof of Lemma 1 involves considering he join behavior of a coninuum of random variables. For each θ 2, we can use S 1,n (θ 2 ) o approximae he quasi-log-likelihood funcion by a quadraic funcion. Le he associaed quasi-log-likelihood funcion be defined as QLR 1,n (θ 2 ) := 2(L n(ˆ q n(θ 2 ), ˆθ q 0,n (θ 2), ˆθ q 1,n (θ 2), θ 2 ) L n(1, θ 0, θ, θ 2 )), where (ˆ q n(θ 2 ), ˆθ q 0,n (θ 2), ˆθ q 1,n (θ 2)) := arg max (,θ 1 ) [0,1] Θ L n(, θ). Then, for given θ 2, QLR 1,n (θ 2 ) = S 1,n (θ 2 ) C (θ) (θ 2, θ 2 )S 1,n (θ 2 ) + o p (1) under H 01, if we ignore he boundary parameer for he momen. Theorem 1(a) shows ha ˆθ q 2,n does no converge o any paricular value in Θ (ɛ). Thus, he limi of S 1,n needs o be derived insead. For his, he finie dimensional disribuions of S 1,n are firs shown o converge weakly o hose of S 1, and we show furher ha his disribuion is igh. The desired resuls of Lemma 1(a) hen follow by heorem 7.1 of Billingsley (1999), and his yields he specified Gaussian process as he limiing process of S 1,n. Tighness is proved by relying on Doukhan, Massar and Rio (1995) and Hansen (1996, 2004), who provide sufficien condiions for ighness in he β-mixing conex. Nex, we show ha he covariance funcion of G has a generalized second-order derivaive. Differeniabiliy in he mean follows by Grenander (1981, heorem 1, ch. 2-2). Laer, his yields a conservaive rejecion region. There are several ineresing aspecs o Lemma 1. Firs, because he model is correcly specified under he null, each score is a maringale difference sequence, and he informaion marix equaliy holds. This ensures ha we can represen he covariance of S 1 by C (θ) (θ, θ ). Noe also ha if θ 2 = θ, hen S 1,n (θ 2 ) isn defined, as L n (, θ ) 0, so ha [ n 1 2 (, θ 1 ) L n(, θ)] 1 isn necessarily defined uniformly 2 We call a sochasic process, {G : Θ R k, k N} L 2 (Θ), differeniable in he mean on Θ, if here is a sochasic process, {G : Θ R k } L 2 (Θ), such ha for all θ Θ, lim h 0 E[[(G(θ + h) G(θ))/ h G (θ)] 2 ] = 0. 7

9 on {(1, θ 0, θ, θ 2 ) [0, 1] Θ : θ 2 Θ \ {θ }} or in n. I could even be negaive definie near θ for some n, so ha he usual approximaion using he Hessian may behave quie badly. To preven his, we replace he convenional score wih S 1,n, exploiing he informaion marix equaliy. Second, for each θ 2, S 1 (θ 2 ) C (θ) (θ 2, θ 2 )S 1 (θ 2 ) can be decomposed ino G 1 (θ 2 ) 2 and oher erms. As shown below, G 1 (θ 2 ) 2 forms he weak limi of he QLR saisic under H 01, bu he boundary parameer condiion needs o be adjused. Third, he disribuion of G may be saionary or non-saionary. Tha is, E[G(θ 2 )G(θ 2 + τ)] can be a funcion of τ only or of boh τ and θ 2. This propery depends on he DGP as well as he model. Finally, as θ 2 ends o θ, Ω (θ) (θ 2, θ 2 ) ends o zero, because C11 θ (θ, θ) and Cθ 12 (θ) converge o zero. This raises a quesion abou he exisence of plim θ2 θ G(θ 2 ). We invesigae his afer he QLR saisic is examined under H 02. Also, noe ha for given θ 2 and θ 2, Ω(θ) (θ 2, θ 2 ) is he asympoic covariance beween n 1/2 (1 f (ˆθ n 0,n, θ 2)/f (ˆθ n 0,n, ˆθ n 1,n )) and n 1/2 (1 f (ˆθ n 0,n, θ 2 )/f (ˆθ n 0,n, ˆθ n 1,n )). The asympoic disribuion of he QLR saisic under H 01 can be derived by using Lemma 1. Theorem 2: Given A1, A2 (i, ii), A3, A4, A5 (ii), A6 (i), and H 01, for each ɛ > 0, QLR n(ɛ) := max θ2 Θ (ɛ)(qlr 1,n (θ 2 ) QLR 2,n ) H(ɛ) := sup θ2 Θ (ɛ)(min[0, G(θ 2 )]) 2, where QLR 2,n := 2(L n(1, ˆθ n 0,n, ˆθ n 1,n, θ 2) L n(1, θ 0, θ, θ 2 )), and θ 2 in L n(1, ˆθ n 0,n, ˆθ n 1,n, θ 2) is a placeholder whose value is irrelevan. An advanage of he QLR saisic under H 01 is ha is weak convergence limi exiss under mild condiions. Theorem 2 exends Ghosh and Sen s (1985) resul for i.i.d. daa o he β-mixing ime series conex. To inerpre he QLR saisic, we noe ha he firs piece, max θ2 Θ (ɛ) QLR 1,n (θ 2 ), ess a join hypohesis: here is a single regime and X F 1 F ( X 1 ; (θ 0, θ )), whereas QLR 2,n ess he single hypohesis X F 1 F ( X 1 ; (θ 0, θ )). Thus, he QLR saisic ess only he number of regimes, as desired. The boundary parameer condiion involves only he negaive par of he score under H 01. The boundary parameer associaed wih S 1,n splis G ino posiive and negaive pieces and discards he posiive piece, resuling in he appearance of he min operaor in he conclusion of Theorem 2. Technical consideraions relevan o he boundary parameer problem race from Chernoff (1954), Self and Liang (1987) and Andrews (1999). As hese resuls resolve he boundary parameer problem only for idenified models, we can apply heir conclusions only o QLR 1,n (θ 2 ) for given θ 2. Andrews (2001) provides furher relevan heory for unidenified models wih boundary parameer problems. We uilize his advances o obain Theorem 2. In paricular, from he given approximaions and he boundary parameer condiion, max θ2 Θ (ɛ) QLR 1,n (θ 2 ) H(ɛ)+Z Z under H 01, where Z Z is idenically he probabiliy limi of QLR 2,n, following he chi-square disribuion. Thus, QLR n (ɛ) weakly converges o H(ɛ) under H 01. We emphasize ha he convergence limi of max θ2 Θ (ɛ) QLR 1,n (θ 2 ) separaes ino wo pieces, H(ɛ) and Z Z (depending on wheher he boundary parameer problem arises or no), and he probabiliy limi of QLR 2,n, which is he idenical random variable Z Z. As poined ou by one of he referees, he esimaion error for parameers no on he boundary has he same limi as he esimaion error obainable when he boundary parameers are known in advance. 8

10 The naure of he parameer space Θ fundamenally affecs he probabiliy law of H(ɛ). As poined ou by Harigan (1985) and Lindsay (1995), if we allow Θ o be unbounded, hen even he exisence of H(ɛ) is in quesion. Theorem 3 formally underscores he imporance of assuming compac Θ. Theorem 3: Given A1, A2 (i, ii), A3, A4, A5 (ii), A6 (i), and H 01, for all ɛ > 0, P (sup θ 2 Θ (ɛ) G(θ 2 ) < ) = 1. We prove Theorem 3 using he fac in Lifshis (1995) ha a Gaussian process is bounded wih probabiliy one if and only if i has a finie oscillaion a.s. when he given parameer space, Θ (ɛ), can be covered by a finie number of open balls wih radius measured by he semi-meric E[(G(θ 2 ) G(θ 2 ))2 ] for θ 2 θ 2 Θ (ɛ). If Θ is unbounded and he covariance beween G(θ 2 ) and G(θ 2 ) converges o 0 as θ 2 θ 2 ges large, hen for given η (0, 1) and K R +, we can choose a se of parameers, say {θ 2i } n(η,k) i=1, such ha P (sup θ2 {θ 2i } n(η,k) G(θ 2 ) > K) > 1 η. This implies ha by leing K grow we can ensure ha H(ɛ) i=1 evenually diverges o infiniy in probabiliy. We avoid his by requiring Θ o be bounded. In erms of Harigan (1985) and Lindsay (1995), herefore, our focus here should be undersood as invesigaing how o exploi he QLR saisic for a given compac parameer space. Our boundedness requiremen is furher underscored by he warnings raised by Azaïs, Gassia, and Mercadier (2006) agains using an unbounded parameer space. The Mone Carlo experimens for mixures of exponenials in Mosler and Seidel (2001) show he lack of convergence in disribuion in his conex. Anoher ineresing aspec of Theorem 3 is ha he QLR saisic has a model-dependen null disribuion. As menioned following Lemma 1, he null disribuion of G depends on boh he DGP and he model. This siuaion has been recognized in he goodness-of-fi es lieraure by Darling (1955) and Durbin (1973). Their insighs apply here. Furher, he parameer space, Θ, is anoher source of model dependence for our QLR saisic. For example, if here are wo parameer spaces, say Θ (1) and Θ (2), such ha Θ (1) Θ (2), hen P (sup θ2 Θ (1) (ɛ) (min[0, G(θ 2)]) 2 > K) P (sup θ2 Θ (2) (ɛ) (min[0, G(θ 2)]) 2 > K) and P (sup θ2 Θ (1) (ɛ) (min[0, G(θ 2)]) 2 = 0) P (sup θ2 Θ (2) (ɛ) (min[0, G(θ 2)]) 2 = 0). Thus, for differen parameer spaces, differen criical values will apply, and he poin mass given o 0, he effec due o he boundary parameer problem, can evenually disappear. We invesigae his in he model exercises of Secion Null Disribuion of he QLR Saisic under H Non-zero Second-Order Derivaive Case. To examine he QLR saisic under H 02, we firs inroduce relevan noaion for our analysis. For given (, θ 2 ) (0, 1) Θ, le ( θ q 0,n (θ 2), θ q 1,n (θ 2)) := arg max θ 1 Θ L n(, θ), which saisfies he firs-order condiions (FOCs), so ha for each (, θ 2 ), θ0 L n(, θ q 0,n (θ 2), θ q 1,n (θ 2), θ 2 ) = f (1,0) ( θ q 0,n (θ 2), θ q 1,n (θ 2)) + (1 )f (1,0) ( θ q 0,n (θ 2), θ 2 ) f ( θ q 0,n (θ 2), θ q 1,n (θ 2)) + (1 )f ( θ q 0,n (θ 0, 2), θ 2 ) θ1 L n(, θ q 0,n (θ 2), θ q 1,n (θ 2), θ 2 ) = (1 )f (0,1) ( θ q 0,n (θ 2), θ q 1,n (θ 2)) f ( θ q 0,n (θ 2), θ q 1,n (θ 2)) + (1 )f ( θ q 0,n (θ 2), θ 2 ) 0, 9

11 where f (i,j) := i θ 0 j θ 1 f. Noe ha ( θ q 0,n, θ q 1,n ) should have been represened as a funcion of, oo, bu we omi his, as will be aken as given under H 02. Each componen in he expressions above can be appropriaely exploied for our furher analysis. For each (, θ 2 ), we hus simplify by leing h (θ 2 ) := f (0,1) ( θ q 0,n (θ 2), θ 2 ), k (θ 2 ) := f (0,1) ( θ q 0,n (θ 2), θ q 1,n (θ 2)), m (, θ 2 ) := f (1,0) ( θ q 0,n (θ 2), θ q 1,n (θ 2)) + (1 )f (1,0) ( θ q 0,n (θ 2), θ 2 ), g (, θ 2 ) := 1/(f ( θ q 0,n (θ 2), θ q 1,n (θ 2)) + (1 )f ( θ q 0,n (θ 2), θ 2 )), and L n (, θ 2 ) := L n(, θ q 0,n (θ 2), θ q 1,n (θ 2), θ 2 ). Then we can wrie: M (1) n (, θ 2 ) := θ0 L n (, θ q 0,n (θ 2), θ q 1,n (θ 2), θ 2 ) = m (, θ 2 )g (, θ 2 ) 0, K (1) n (, θ 2 ) := θ1 L n (, θ q 0,n (θ 2), θ q 1,n (θ 2), θ 2 ) = (1 ) k (θ 2 )g (, θ 2 ) 0. In he expressions immediaely above, he superscrip (1) denoes he firs-order derivaive wih respec o θ 2. Laer, we use he superscrip (2) o denoe he second-order derivaive wih respec o θ 2, ec. Thus, implying ha and ˆr (1) L (1) n (, θ 2 ) := θ2 Ln (, θ 2 ) = (1 )f (0,1) ( θ q 0,n (θ 2), θ 2 ) f ( θ q 0,n (θ 2), θ q 1,n (θ 2)) + (1 )f ( θ q 0,n (θ 2), θ 2 ), (1) L n (, θ 2 ) = (1 ) h (θ 2 )g (, θ 2 ). We furher le ˆr (i,j) := f (i,j) (ˆθ 0,n n, ˆθ 1,n n )/f (ˆθ 0,n n, ˆθ 1,n n ) := θ 1f (ˆθ n 0,n, ˆθ n 1,n )/f (ˆθ n 0,n, ˆθ n 1,n ); and where no confusion arises, for a given funcion of (, θ 2), say q, q (, ˆθ n 1,n ) is abbreviaed as ˆq. For example, g (, ˆθ n 1,n ) and h (ˆθ n 1,n ) are denoed as ĝ and ĥ. The QLR saisic can be represened using his noaion under H 02. Noe ha for each, we have QLR n () := 2(L n(, ˆθ q 0,n (), ˆθ q 1,n (), ˆθ q 2,n ()) L n(1, ˆθ 0,n n, ˆθ 1,n n, θ 2)), where (ˆθ q 0,n (), ˆθ q 1,n (), ˆθ q 2,n ()) := max θ L n(, θ); his is idenical o 2( L n (, ˆθ q 2,n ()) L n (, ˆθ 1,n n )), because ( θ q 0,n (ˆθ q 2,n ()), θ q 1,n (ˆθ q 2,n ()), ˆθ q 2,n ()) and ( θ q 0,n (ˆθ 1,n n ), θ q 1,n (ˆθ 1,n n )) saisfy he FOCs under he alernaive and he null model assumpions respecively, so ha ˆθ q 0,n () = θ q 0,n (ˆθ q 2,n ()), ˆθ q 1,n () = θ q 1,n (ˆθ q 2,n ()), ˆθ n 0,n = θ q 0,n (ˆθ n 1,n ) and ˆθ n 1,n = θ q 1,n (ˆθ n 1,n ). In addiion o he idenificaion problem, he QLR saisic exhibis anoher nonsandard propery under H 02. By he definiions of h and k, ĥ := h (ˆθ 1,n n ) = ˆk := k (ˆθ 1,n n ), implying ha L (1) n (, ˆθ n 1,n) = (1 ) ĥ ĝ = (1 ) ˆk ĝ = K (1) n (, ˆθ n 1,n) = 0. Tha is, he firs-order derivaive is idenically zero under H 02. The score given by he firs-order derivaive canno approximae he null disribuion. Neymann and Sco (1966) and Lee and Chesher (1986) consider similar problems in he conex of he C(α) saisic and recommend approximaing he log-likelihood funcion using higher-order derivaives. Lindsay (1995) elaboraes and shows his approach exends beyond he models of Neymann and Sco (1966). We follow hese insighs. A lile algebra gives ha for each (, θ 2 ), L (2) n (, θ 2 ) = (1 ) {h (1) (θ 2 )g (, θ 2 ) + h (θ 2 )g (1) (, θ 2 )}, M (2) n (, θ 2 ) = {m (1) (, θ 2 )g (, θ 2 ) + m (, θ 2 )g (1) (, θ 2 )} = 0, 10

12 K (2) n (, θ 2 ) = (1 ) {k (1) (θ 2 )g (, θ 2 ) + k (θ 2 )g (1) (, θ 2 )} = 0. The las wo equaions hold because M (1) n (, θ 2 ) and (1) K n (, θ 2 ) are idenically zero. Thus, L (2) n (, ˆθ 1,n) n (2) = L n (, ˆθ 1,n) n (2) K n (, ˆθ 1,n) n = (1 ) (ĥ(1) (1) ˆk )ĝ. Noe ha his involves firs compuing ˆ θ(1) i := (2) and M n, hen anoher se of ideniies is obained. We can also compue Once again, ˆ θ(2) i (i = 0, 1) appears; we ierae his process o obain L (3) n (, ˆθ 1,n) n = (1 ) {(ĥ(2) (1) θ i (ˆθ 1,n n (2) ) (i = 0, 1), and if hese are plugged back ino K n (3) L n (, ˆθ 1,n n ) in he same way. (4) L n (, ˆθ 1,n n ). Then for each, (2) ˆk )ĝ + 2(ĥ(1) (1) ˆk )ĝ (1) }, L (4) n (, ˆθ 1,n) n = (1 ) {(ĥ(3) (3) ˆk )ĝ + 3(ĥ(2) (2) ˆk )ĝ (1) + 3(ĥ(1) (1) ˆk )ĝ (2) }. (2) The asympoic behaviors of L n (, ˆθ 1,n n (4) ) o L n (, ˆθ 1,n n ) are deermined by each elemen on he righ-hand side (RHS). We now collec he regulariy condiions ha enable us o apply he law of large numbers (LLN) and he cenral limi heorem (CLT) o each elemen. A2 : (iii) For every x R d, f(x X 1 ; ) C (4) ( Θ) a.s. A5 : (iii) There exiss a sequence of posiive, sricly saionary, and ergodic random variables, {M }, such ha (a) for some δ > 0, E[M 1+δ ] < < ; (b) sup θ 1 Θ i 1 f (θ 1 )/f (θ 1 ) 4 M ; (c) sup θ 1 Θ i 1 i2 f (θ 1 ) /f (θ 1 ) 2 M ; (d) sup θ 1 Θ i 1 i2 i3 f (θ 1 )/f (θ 1 ) 2 M ; (e) sup θ 1 Θ i 1 i2 i3 i4 f (θ 1 )/f (θ 1 ) M, where i 1,, i 4 {θ 01, θ 02,, θ 0r0, θ 1 }. A6 : (ii) For each (, θ 0, θ 1, θ 2 ) wih θ 1 = θ 2 and (0, 1), λ min (C (2) ) 0 such ha if λ min (C (2) ) > 0, hen λ max (C (2) ) <, where for k = 2, 3,, r (k) (θ ) := (r (0,k) (θ ), r (1) (θ ) ) and E[r(0,k) (θ ) 2 ] E[r (0,k) (θ )r (1) C (k) := E[r (k) (θ )r (k) (θ ) ] := whenever hey exis. C(k) 11 C (k) 12 C (k) 21 C(θ) 22 := E[r (0,k) (θ )r (1) (θ )] C (θ) 22 (θ )] We impose A6(ii) in order o approximae he quasi-log-likelihood funcion by a quaric funcion. Specifically, A6(ii) provides a condiion relaing he second-order and firs-order derivaives under he alernaive and he null respecively. Using hese, we can obain he following asympoic properies. Lemma 2: Given A1, A2 (i, iii), A3, A4, A5 (ii, iii), A6 (ii), and H 02, for each, (2) (a) L n (, ˆθ 1,n n ) = (1 ) (0,2) ˆr + o p (n 1/2 ); (b) n 1/2 L(2) n (, ˆθ 1,n n ) ( 1 )G(2) 0, where G(2) 0 N(0, Ω (2) ) and Ω (2) := C (2) 11 C(2) 12 [C(θ) 22 ] 1 C (2) 21 ; (3) (c) L n (, ˆθ 1,n n ) = O p(n 1/2 ); (d) n 1 L(4) n (, ˆθ 1,n n 1 ) = 3( )2 Ω (2) + o p (1). 11

13 Lemma 2 is proved in he Appendix. The fac ha (2) L n (, ˆθ 1,n n ) = O p(n 1/2 ) implies ha he QLR saisic can be non-degenerae even wih he zero firs-order derivaive. Also, i is of ineres ha esimae he asympoic variance of n 1/2 L(2) n The asympoic null disribuion of he QLR saisic under H 02 Noe ha for each and θ 2 beween θ 2 and ˆθ n 1,n, (4) L n (, ˆθ 1,n n ) can (, ˆθ 1,n n ), as a resul of he informaion marix equaliy. can now be derived using Lemma 2. L n (, θ 2 ) = L n (, ˆθ 1,n) n + 1 (2) L n (, 2! ˆθ 1,n)(θ n 2 ˆθ 1,n) n (3) L n (, 3! ˆθ 1,n)(θ n 2 ˆθ 1,n) n (4) L n (, 4! θ 2 )(θ 2 ˆθ 1,n) n 4 by he mean value heorem. Lemma 2 and heorem 3.9 of Billingsley (1999) imply ha ( ( 1/2 n L(2) n (, ˆθ 1,n), n 3/4 n L(3) n (, ˆθ 1,n), n 1 n L(4) n (, ˆθ ) [1 ] [ ] ) 1,n) n G (2) 1 2 0, 0, 3 Ω (2) for each. Thus, given he differeniabiliy and he momen condiions, for each, [ ] sup 2( L n (, θ 2 ) L n (, ˆθ 1 1,n)) n sup G (2) 0 θ 2 ξ ξ2 1 [ ] 1 2 Ω (2) ξ 4, 4 where ξ capures he asympoic behavior of n 1/4 (θ 2 ˆθ 1,n n ). From his, we obain he following. Theorem 4: Given A1, A2 (i, iii), A3, A4, A5 (ii, iii), A6 (ii), and H 02, for each (0, 1), (a) max θ2 2( L n (, θ 2 ) L n (, ˆθ n 1,n )) max[0, G 0] 2, where G 0 N(0, 1); (b) for ɛ (0, 1/2), QLR n (ɛ) := max [ɛ,1 ɛ] max θ2 2( L n (, θ 2 ) L n (, ˆθ n 1,n )) max[0, G 0] 2. There are a number of ineresing aspecs o Theorem 4. Firs, he proof of Theorem 4 is no oo differen from he sandard argumen, bu i involves a sign consrain. Noe ha he QLR saisic is approximaed mainly by he second and fourh-order derivaives, and ξ is raised o even powers, which canno be less han zero. This gives rise o he square of he half normal disribuion as he asympoic null disribuion, even wihou he boundary parameer condiion. Second, he nuisance parameer,, is presen only in scaling L (2) n (, ˆθ n 1,n ) and L (4) n (, ˆθ 1,n n ), so ha he QLR saisic urns ou o be nuisance parameer free by he informaion marix equaliy. This also implies ha he asympoic null disribuion of 2( L n (, θ 2 ) L n (, ˆθ 1,n n )) is auomaically igh, leading o Theorem 4(b). Third, n 1/2 L(3) n (, ˆθ 1,n n ) doesn have o obey asympoic normaliy for Theorem 4. I can be degenerae. Wha is required for he resul is ha n 3/4 L(3) n (, ˆθ 1,n n ) = o p(1). Fourh, he convergence rae of he esimaor is differen from he sandard n 1/2 case. In his case, he convergence rae is n 1/4, which is he same as for he C(α) saisic of Neymann and Sco (1966) and Lindsay (1995). Thus, under H 02, he QLR es can have power comparable o ha of he C(α) es asympoically. Also, in urn, he general quadraic approximaion of Lin and Shao (2003) canno be applied o he likelihood raio under H 02. Goffine, Loisel and Lauren (1992) repor he same resuls as Lemma 4(a) in he case of a mixure of normals wih unknown means bu known common variance. Our analysis provides a general heory ha ness heirs as a special case. Fifh, he given limiing random variable, G 0, is he probabiliy limi of G(θ 2 ) as θ 2 ends o θ. This feaure will be explained in furher deail below. Sixh, he lieraure ofen repors he endency of mixures of exponenial family disribuions o yield more sable simulaion resuls han oher disribuions. This is 12

14 mainly because hey are infiniely differeniable, so ha he fourh-order differeniaion condiion holds auomaically. Simulaion resuls can be unsable if he model is differeniable only up o he second-order. Finally, many mixure models can be analyzed by he fourh-order approximaion, alhough here are many oher popular mixures ha canno be approximaed using a fourh-order Taylor expansion Zero Second-Order Derivaive Case. We ofen observe mixure models o have zero second-order derivaives under he null, because he second-order derivaive urns ou o be a linear funcion of he firs-order derivaives: for each θ 1 and for some non-zero (α β) R r 0+1, so ha ˆr (0,2) f (0,2) (θ 1 ) = α f (1,0) (θ 1 ) + βf (0,1) (θ 1 ), = 0. The empirical example of Porer (1983) belongs o his case, so he quaric approximaion of he previous secion has o be improved. As will be clear laer, he required approximaion order is of he eighh order. 3 By furher elaboraing he prior derivaives, we have for i = 3,, 8, L (i) n (, ˆθ 1,n) n = (1 ) (ĥ(i 1) ) i 2 (i 1) ˆk ĝ + i 1 ( ) ĥ (i j 1) (i j 1) ˆk j As before, each componen in he RHS conribues o he asympoic null disribuion of he QLR saisic. We provide suiable regulariy condiions as follows. j=1 ĝ (j). A2 : (iv) For every x R d, f(x X 1 ; ) C (8) ( Θ) a.s. A5 : (iv) There exiss a sequence of posiive, sricly saionary, and ergodic random variables, {M }, such ha for some δ > 0, E[M 1+δ ] < < ; sup θ 1 Θ i 1 ik f (θ 1 )/f (θ 1 ) 4 M ; sup θ 1 Θ i 1 il f (θ 1 )/f (θ 1 ) 2 M ; sup θ 1 Θ 8 θ 1 f (θ 1 )/f (θ 1 ) M ; sup θ 1 Θ j 1 7 θ 1 f (θ 1 )/f (θ 1 ) M ; sup θ 1 Θ j1 j2 6 θ 1 f (θ 1 )/f (θ 1 ) M, where k = 1, 2, 3, 4; l = 5, 6, 7; i 1,, i 7 {θ 01, θ 02,, θ 0r0, θ 1 }; and j 1, j 2 {θ 01, θ 02,, θ 0r0 }. Assumpions A5(iii and iv) are no he mos efficien possible momen condiions. For exposiional purposes, we provide simple assumpions ha are sronger han is sricly necessary. Noe ha he second, hird, and fourh-order derivaive condiions in A5(iv) are srenghened compared o A5(iii), and also ha he highes momen-order condiion is of fourh order even if he eighh-order derivaive is involved. This conrass sharply wih he previous case. Recall ha in he prior case, (ˆ θ(3) o obain (4) L n (, ˆθ 1,n n ). Here we don need o compue (ˆ θ(7) 0,n, ˆ θ(7) 1,n 0,n, ˆ θ(3) 1,n ) o compue L (8) n ) mus firs be compued (, ˆθ 1,n n ), only (ˆ θ(6) 0,n, ˆ θ(6) 1,n ). This is mainly because ˆ θ(7) 1,n appears as a coefficien of ˆr (0,2) (which is zero), and ˆ θ(7) 0,n is no required o (8) compue L n (, ˆθ 1,n n ). Consequenly, our regulariy condiions do no require finie eighh-order momens even if he eighh-order derivaives are involved. Assumpion A5(iv) and he nex assumpion enable us o apply he CLT. 3 We are indebed o Rober Davies for guidance wih his aspec of he problem. 13

15 A6 : (iii) For each (, θ 0, θ 1, θ 2 ) wih θ 1 = θ 2 and (0, 1), λ min (C (2) ) 0 such ha if λ min (C (2) ) = 0, hen λ min (C (s) ) > 0 and λ max (C (s) ) <, where C (s) := := E[s (θ ) 2 ] E[s (θ )r (3) (θ ) ], E[s (θ )r (3) (θ )] C (3) C(s) 11 C (s) 12 C (s) 21 C(3) and s (θ ) := r (0,4) (θ ) 6βr (0,3) (θ ) 6α r (1,2) (θ ) + 6α r (1,1) (θ )β + 3α r (2,0) (θ )α. We pariion C (s) 12 ino [C(s) 3, C(s) 1 ] := [E[s (θ )r (3,0) (θ )], E[s (θ )r (1) (θ )] ] for fuure reference. As given below, ŝ is asympoically equivalen o he fourh-order derivaive, affecing he asympoic disribuion of he QLR saisic. Thus we require boh C (s) and C (3) o be posiive definie. We now have Lemma 3: Given A1, A2 (i, iv), A3, A4, A5 (ii, iv), A6 (iii), and H 02, for each, (a) (b) n (c) (d) (3) L n (, ˆθ 1,n n ) = (1 )(1 2) (0,3) ˆr 2 ; 1/2 L(3) n (, ˆθ n 1,n ) (1 )(1 2) L (4) n (, ˆθ n 1,n ) = O p(n 1/2 ); L (5) n (, ˆθ n 1,n ) = O p(n 1/2 ). G (3) 2 0, where G(3) 0 N(0, Ω (3) ) and Ω (3) := C (3) Lemma 3 holds for any (0, 1). Neverheless, care is needed. If = 1/2, 11 C(3) 12 [C(θ) 22 ] 1 C (3) 21 ; (3) L n (, ˆθ 1,n n ) = 0, so ha he hird-order derivaive also urns ou o be zero, implying ha we need o differeniae one more ime when = 1/2. Goffine, Loisel and Lauren (1992) observe he same phenomenon when considering he mixure of normals wih unknown differen means and an unknown common variance. Neverheless, heir analysis is incorrec, as hey approximae he log-likelihood funcion only o he fourh order when = 1/2. We suppose firs ha 1/2. In his case, he asympoic variance of n he sixh-order derivaive. 1/2 L(3) n (, ˆθ 1,n n ) can be esimaed by Lemma 4: Given A1, A2 (i, iv), A3, A4, A5 (ii, iv), A6 (iii) and H 02, if {x (0, 1) : x 1/2}, hen (a) n 1 L(6) n (, ˆθ 1,n n (1 )(1 2) ) = 10( ) 2 Ω (3) + o 2 p (1); (b) max θ2 2(L n (, θ 2 ) L n (, ˆθ n 1,n )) G2 0. The proof of Lemma 4 is sraighforward. Using Lemmas 3 and 4, we can approximae he likelihood funcion as before. For each and θ 2 beween θ 2 and ˆθ 1,n n, he mean value heorem gives L n (, θ 2 ) = L n (, ˆθ 1,n)+ n 1 (3) L n (, 3! ˆθ 1,n)(θ n 2 ˆθ 1,n) n (4) L n (, 4! ˆθ 1,n)(θ n 2 ˆθ 1,n) n (5) L n (, 5! ˆθ 1,n)(θ n 2 ˆθ 1,n) n (6) L n (, 6! θ 2 )(θ 2 ˆθ 1,n) n 6. Applying heorem 3.9 of Billingsley (1999) and Lemma 6 ensures ha for each, max 2(L n (, θ 2 ) L n (, ˆθ 1,n)) n sup 1 [ ] (1 )(1 2) θ 2 ξ 3 2 G (3) 0 ξ3 20 [ ] (1 )(1 2) 2 6! 2 Ω (3) ξ 6. Solving for he RHS gives us [Ω (3) ] 1 [G (3) ] 2, which has he same disribuion as G 2 0. This explains why he sandard chi-square disribuion is obained here as he limiing disribuion of he QLR saisic. Noe 14

16 ha he informaion marix equaliy holds here, and ha he limiing disribuion is nuisance parameer free. If = 1/2, we examine derivaives up o eighh order. We have he following large sample properies. Lemma 5: Given A1, A2 (i, iv), A3, A4, A5 (ii, iv), A6 (iii), and H 02, if = 1/2, hen (a) (b) n (c) (d) L (4) n (, ˆθ n 1,n ) = ŝ + o p (n 1/2 ); 1/2 L(4) n (, ˆθ 1,n n ) G(s) 0, where G(s) 0 N(0, Ω (s) ) and Ω (s) := C (s) L (6) n (, ˆθ n 1,n ) = O p(n 1/2 ); L (7) n (, ˆθ n 1,n ) = O p(n 1/2 ); (e) n 1 L(8) n (, ˆθ 1,n n ) = 35Ω(s) + o p (1); (f ) max θ2 2( L n (, θ 2 ) L n (, ˆθ n 1,n )) max[0, G ] 2, where G N(0, 1). Noe he sharp difference beween he behavior of 11 C(s) 1 [C (θ) 22 ] 1 C (s) 1 ; (6) L n (, ˆθ 1,n n ) here and ha in Lemma 4(a). The O p(n) erm in Lemma 4(a) urns ou o have a zero coefficien given = 1/2. The oher aspecs are he same as before. Applying heorem 3.9 of Billingsley (1999) and Lemmas 5(a o e) leads o Lemma 5(f) by he same argumen as before. The sign condiion applies here also, so ha he square of he half-normal disribuion is obained as he limiing disribuion of max θ2 2( L n (1/2, θ 2 ) L n (1/2, ˆθ 1,n n )). The asympoic variance of he fourh-order derivaive can be esimaed by he eighh-order derivaive, and he informaion marix equaliy follows from his. The asympoic disribuion of he QLR saisic under H 02 follows as a corollary of Lemmas 4(b) and 5(f). By he Cramér-Wold device, heorem 3.9 of Billingsley (1999), and Lemmas 3 o 5, we have max 2( L n (, θ 2 ) L n (, ˆθ 1,n)) n max[g 2 0, max[0, G ] 2 ], (,θ 2 ) where cov(g 0, G ) = Ω (3,s) /[Ω (3) Ω (s) ] 1/2 and Ω (3,s) := C (s) 3 C (s) 12 [C(θ) 22 ] 1 C (3) 1. In applying heorem 3.9 of Billingsley (1999), we exploi he fac ha C (s) is posiive definie o show he exisence of he RHS. Theorem 5: Given A1, A2 (i, iv), A3, A4, A5 (ii, iv), A6 (iii), and H 02, for each ɛ > 0, QLR n(ɛ) max[g 2 0, max[0, G ] 2 ]. Several iems are noeworhy. Firs, here are only wo limis, G 0 and G under H 02, implying ha he ighness rivially follows for he same reason as in Theorem 4. Second, as explained below, he limiing random variable, G 0, can be inferred from G by moving θ 2 o θ, bu G canno. Third, he rae of convergence of he QLR saisic is n 1/8 or n 1/6 depending on wheher = 1/2 or no, so ha he power of he QLR saisic under H 02 is much weaker han he sandard case where he rae of convergence is n1/2, and also weaker han he non-zero second-order derivaive case. This propery is also expeced even for he sandard LR saisic under he same hypohesis, because he mixure model is a special case of he HMM specificaion. Finally, he analysis of Neyman and Sco s (1966) C(α) saisic requires combining ˆr (0,3) and ŝ. For our laer Mone Carlo simulaions, we use he C(α) saisic asympoically equivalen o he LR saisic under H

17 2.4. Null Disribuion of he QLR Saisic under H 0 Given he null disribuion of he QLR saisic under H 01 or H 02, we can es regime swiching by resricing our aenion o a paricular null hypohesis. As an example, we can compue QLR 1,n (ɛ) or QLR 2,n (ɛ) for a given ɛ and apply he disribuion appropriae o each es saisic. Indeed, he LR saisic of Harigan (1985) and Ghosh and Sen (1985) focuses on H 01, and he C(α) es in Neyman and Sco (1966) focuses on H 02. I is, however, of general ineres o obain he asympoic null disribuion of he QLR saisic when he null parameer space is unresriced. We derive his asympoic null disribuion, and implemen Mone Carlo simulaions below o compare i wih oher saisics. The null asympoic disribuion can be given as he disribuion of he maximum value of he random variables given under each hypohesis. We elaborae furher, however, because hese random variables are no independen: here exiss a regular relaionship beween hem. This requires a furher regulariy condiion. A6 : (iv) For each (, θ0, θ 1, θ 2 ), λ min(b(, θ0, θ 1, θ 2 )) 0 such ha (a) if λ min(b(, θ0, θ 1, θ 2 )) > 0, hen λ max (B(, θ 0, θ 1, θ 2 )) < ; or (b) if λ min(b(, θ 0, θ 1, θ 2 )) = 0, hen for each θ 2 θ and θ 2 θ, λ min (C (u) (θ 2, θ 2 )) > 0 and λ max(c (u) (θ 2, θ 2 )) <, where for each (θ 2, θ 2 ), r(u) (θ 2 ) := [1 r (θ 2 ), r (1) (θ 2 ) ], and C (u) (θ 2, θ 2) := E[r(0,2) (θ ) 2 ] E[r (0,2) (θ )r (u) (θ 2 ) ] E[r (u) (θ 2 )r (0,2) (θ )] C (θ) (θ 2, θ 2 ). (v) For each (, θ0, θ 1, θ 2 ), λ min(b(, θ0, θ 1, θ 2 )) 0 such ha (a) if λ min(b(, θ0, θ 1, θ 2 )) > 0, hen λ max (B(, θ 0, θ 1, θ 2 )) < ; or (b) if λ min(b(, θ 0, θ 1, θ 2 )) = 0, hen for each θ 2 θ and θ 2 θ, λ min (C (v) (θ 2, θ 2 )) > 0 and λ max(c (v) (θ 2, θ 2 )) <, where for each (θ 2, θ 2 ), E[s (θ ) 2 ] E[s (θ )r (0,3) (θ )] E[s (θ )r (u) (θ 2 )] C (v) (θ 2, θ 2) := E[r (0,3) (θ )s (θ )] E[r (0,3) (θ ) 2 ] E[r (0,3) (θ )r (u) (θ 2 ) ]. E[r (u) (θ 2 )s (θ )] E[r (u) (θ 2 )r (0,3) (θ )] C (θ) (θ 2, θ 2 ) A6(iv and v) ensure ha he asympoic join disribuion of he random variables obained under H 01 and H 02 is well-defined and properly behaved. Lemma 6: (a) Given A1, A2 (i, iii), A3, A4, A5 (ii, iii), A6 (iv), and H 0, plim θ 2 θ G(θ 2 ) = G 0, where G 0 is given in Theorem 4 (a). (b) Given A1, A2 (i, iv), A3, A4, A5 (ii, iv), A6 (v), and H 0, plim θ 2 θ G(θ 2 ) = G 0 and plim θ2 θ G(θ 2 ) = G 0, where G 0 is given in Lemma 4 (b). The null parameer resricions enforced hrough ɛ in Theorems 2, 4, and 5 are eliminaed here o le θ 2 approach θ. We prove Lemma 6 by approximaing he sample scores of G around θ. Lemma 6(a) implies ha he Gaussian process, G, is coninuous a θ in probabiliy (ha is, wih probabiliy approaching one), when he second-order derivaive isn zero. Oherwise, G is disconinuous a θ in probabiliy. Thus, he 16