Testing for Regime Switching in State Space Models

Transcript

1 Tesing for Regime Swiching in Sae Space Models Fan Zhuo Boson Universiy November 10, 2015 Absrac This paper develops a modified likelihood raio MLR es for deecing regime swiching in sae space models. I apply he filering algorihm inroduced in Gordon and Smih 1988 o consruc a modified likelihood funcion under he alernaive hypohesis of wo regimes and exend he analysis in Qu and Zhuo 2015 o esablish he asympoic disribuion of he MLR saisic under he null hypohesis. I also presen a pracical applicaion of he es using U.S. unemploymen raes. This paper is he firs o develop a es for deecing regime swiching in sae space models ha is based on he likelihood raio principle. Keywords: Hypohesis esing, likelihood raio, sae space model, Markov swiching. I am deeply indebed o my advisor Zhongjun Qu for his invaluable advice and suppor for my research. I also wish o hank Pierre Perron, Hiroaki Kaido, Ivan Fernandez-Val, and seminar paricipans a Boson Universiy for valuable suggesions. Deparmen of Economics, Boson Universiy, 270 Bay Sae Rd., Boson, MA, zhuo@bu.edu. 1

2 1 Inroducion Economiss have long recognized he possibiliy ha model parameers may no be consan hrough ime, and ha insead here can be variaions in model srucure. If hese variaions are emporary and recurren, hen he Markov regime swiching model can offer a naural modeling choice. Hamilon 1989 makes a seminal conribuion ha no only inroduces a framework wih Markov regime swiching for describing economic growh, bu also provides a general algorihm for filering, smoohing, and maximum likelihood esimaion. A survey of he lieraure on regime swiching models can be found in Hamilon Meanwhile, sae space models are widely used in economics and finance o sudy ime series wih laen sae variables. Harvey 1981 and Meinhold and Singpurwalla 1983 inroduced economiss o he use of he Kalman 1960 filer for consrucing likelihood funcions hrough he predicion error decomposiion. Laen sae variables and regime swiching can arise a he same ime, which poses a challenge for modeling hem joinly. However, as Hamilon 1990 poined ou, conducing formal ess for he presence of Markov swiching is challenging. There are generally hree approaches for deecing regime swiching. The firs approach ess for parameer homogeneiy versus heerogeneiy. Early conribuions include Neyman and Sco 1996, Chesher 1984, Lancaser 1984 and Davidson and MacKinnon Recenly, Carrasco, Hu and Ploberger 2014 furher developed his approach and proposed a class of opimal ess for he consancy of parameers in random coefficiens models where he parameers are weakly dependen under he alernaive hypohesis. The second approach, from Hamilon 1996, offers a series of specificaion ess of regime swiching in ime series models. These ess only need researchers o esimae he model under he null hypohesis and have powers agains a wide range of alernaive models. However, heir powers can be lower han wha is achievable if he parameers indeed follow a finie sae Markov chain. The hird approach is based on he quasi likelihood raio principle. Several imporan advances have been made by Hansen 1992, Garcia 1998, Cho and Whie 2007, and Carer and Seigerwald Qu and Zhuo 2015 is a recen developmen, which analyzes likelihood raio based ess for Markov regime swiching allowing for muliple swiching parameers. The purpose of he presen paper is o deec he regime swiching in sae space models. The likelihood funcion for a sae space model wih regime swiching is hard o consruc, as discussed in Kim and Nelson Differen approximaions o he likelihood funcion have been considered in he lieraure, such as in Gordon and Smih 1988 and Highfield This paper 2

3 uses he approximaion applied in Gordon and Smih Based on his approximaion, I develop a modified likelihood raio MLR es. I exend he echniques developed in Qu and Zhuo 2015 o handle he nonsandard feaures associaed wih he MLR es. These nonsandard feaures include he following: 1 Some nuisance parameers are unidenified under he null hypohesis, which violaes he sandard condiions ha yield he chi-squared asympoic disribuion for he es saisic. This gives rise o he Davies 1977 problem. 2 The null hypohesis yields a local opimum c.f. Hamilon, 1990, making he score funcion idenically zero a he null parameer esimaes. Consequenly, a second order Taylor approximaion of he likelihood raio is insufficien o sudy is asympoic properies. 3 Condiional regime probabiliies follow sochasic processes ha can only be consruced recursively. Moreover, his paper ackles an addiional difficuly inroduced by he laen sae variables when expanding he MLR. The asympoic disribuion of he MLR es saisic is analyzed in five seps. 1. I describe he algorihm used o consruc he modified likelihood funcion for Markov swiching sae space models inroduced in Gordon and Smih I characerize he condiional regime probabiliy, he filered laen sae, he mean squared error of he filered laen sae, and heir high order derivaives wih respec o he model parameers. 3. I firs fix p and q and derive a fourh order Taylor approximaion o he MLR. Then, I view he MLR as an empirical process indexed by p and q, and derive is asympoic disribuion. 4. While he above limiing disribuions are adequae for a broad class of models, hey can lead o over-rejecions in some siuaions specified laer. To resolve he issue of over-rejecion, he higher order erms in he likelihood expansion are incorporaed ino he asympoic disribuion o safe guard agains heir effecs. 5. I apply a unified algorihm proposed in Qu and Zhuo 2015 o simulae he above refined asympoic disribuion. Three Mone Carlo experimens are conduced o examine he MLR saisic. The firs experimen checks he improvemen inroduced by he refined asympoic disribuion. The second and hird experimens check he size and power of he MLR saisic. I also apply my mehod o sudy changes in U.S. unemploymen raes and find srong evidence favoring he regime swiching specificaion. 3

4 This paper is he firs o develop a likelihood raio based es for deecing regime swiching in general sae space models and conribues o he lieraure in several ways. Firs, I demonsrae he consrucion of he modified likelihood funcion under a wo regimes specificaion for general sae space models. Nex, I sudy he Taylor expansion of he MLR when some regulariy condiions fail o hold. Finally, I apply my mehod o an empirical example and find he comovemen beween he U.S. business cycle and changes in monhly U.S. unemploymen raes. The paper is srucured as follows. In Secion 2, I provide he general model, he basic filer, and he hypoheses. Secion 3 inroduces he es saisic. Secion 4 sudies asympoic properies of he MLR for prespecified p and q. Secion 5 provides he limiing disribuion of he MLR es saisic and inroduces a finie sample refinemen. Secion 6 examines he finie sample properies of he es saisic. Secion 7 considers an empirical applicaion o he U.S. unemploymen rae. Secion 8 concludes. All proofs are colleced in he appendix. The following noaion is used. x is he Euclidean norm of a vecor x. X is he vecor induced norm for a marix X. x k and X k denoe he k-fold Kronecker produc of x and X, respecively. The expression veca sands for he vecorizaion of a k dimensional array A. For example, for a hree dimensional array A wih n elemens along each dimension, veca reurns a n 3 -vecor whose i+j 1n+k 1n 2 -h elemen equals Ai, j, k. 1 } is he indicaor funcion. For a scalar valued funcion fθ, le θ R p, θ f θ 0 denoes a p 1 vecor of parial derivaives wih respec o θ and evaluaed a θ 0. θ fθ 0 equals he ranspose of θ f θ 0 and θj fθ 0 denoes is j-h elemen. For a marix funcion P θ, θj P θ denoes he derivaive of P θ wih respec o he j-h elemen in θ. The symbols, d and p denoe weak convergence under he Skorohod opology, convergence in disribuion and in probabiliy, respecively. O p and o p are he usual noaions for he orders of sochasic magniude. 2 Model and hypoheses This secion presens he model and hypoheses. The discussion consiss of he following: he model, he log likelihood funcion under he null hypohesis i.e., one regime, he modified log likelihood funcion under he alernaive hypohesis i.e., wo regimes, and some assumpions relaed o hese hree aspecs. 4

5 2.1 The model Consider he following sae space represenaion of a dynamic linear model wih swiching in boh ransiion and measuremen equaions: x = G s + F s x 1 + u, 2.1 y = H s x + A s z, 2.2 u N 0, Q s. 2.3 The ransiion equaion 2.1 describes he dynamics of he unobserved sae vecor x as a funcion of a J 1 vecor of shocks u and x 1. The measuremen equaion 2.2 describes he evoluion of an observed scalar ime series as a funcion of x and a K 1 vecor of weakly exogenous variables z. The measuremen error, normally included in 2.2, is reaed as a laen variable in x. F s is of dimension J J, G s is of dimension J 1, H s is of dimension J 1, and A s is of dimension K 1. Q s is a posiive semidefinie symmeric marix of dimension J J. The subscrips in F s, G s, H s, A s, and Q s imply ha some of he parameers in hese marices are dependen on an unobserved binary variable s whose value deermines he regime a ime. The regimes are Markovian, i.e., ps = 1 s 1 = 1 = p and ps = 2 s 1 = 2 = q. The resuling saionary or invarian probabiliy for s = 1 is given by p, q = 1 q 2 p q. 2.4 In he subsequen analysis, p, q is abbreviaed as. Because his paper seeks o es regime swiching in sae space models based on he likelihood raio principle, subsecions will focus on consrucing he likelihood funcion under he wo regimes specificaion, and he likelihood funcion under one regime specificaion is a by-produc of he sandard Kalman filer. 2.2 Modified Kalman filer When consrucing he likelihood funcion for a general sae space model wih regime swiching, each ieraion of he Kalman filer produces a wo-fold increase in he number of cases o consider under a wo regimes specificaion, as noed by Gordon and Smih 1988 and Harrison and Sevens This means here can be more han 1000 componens in he likelihood funcion for a sample of size 5

6 T = 10. This makes sudying his likelihood funcion and is expansion infeasible. Therefore, an approximaion is considered here o collapse he filered saes when s = 1 and s = 2 o a single filered sae a each, as in Gordon and Smih Define he informaion se a ime 1 as Ω 1 = σ-field..., z 1, y 2, z, y 1 }. Suppose he model parameers are known. The modified Kalman filer algorihm, condiional on s = i, is given by: x i := G i + F i x 1 1, 2.5 P i := F ip 1 1 F i + Q i, 2.6 µ i := y H ix i A iz, 2.7 C i := H ip i H i, 2.8 x i P i := xi + P i H ic i 1 µ i, 2.9 i := I P H ic i 1 H ip i, 2.10 where x 1 1 is an esimae of x 1 based on informaion up o ime 1; x i is an esimae of x based on informaion up o ime 1 given s = i; P i is an esimae of he mean squared error of x i ; µi esimaes he condiional forecas error of y based on informaion up o ime 1 given s = i; and C i esimaes he condiional variance of he forecas error µi. Le ξ be an esimae of P rs = 1 Ω. The collapse sep combines he wo filered saes x 1 and x 2 ino a single esimae of x based on Ω by x := ξ x ξ x Then, he mean squared error of x can be compued as: P := ξ P ξ P 2 + ξ 1 ξ x 1 x2 x 1 x A he end of each ieraion, equaions 2.11 and 2.12 are employed o collapse he wo filered saes ino one filered sae x and calculae he mean squared error of x, i.e. P. 6

7 2.3 Modified Markov swiching filer To complee he modified Kalman filer, we need o calculae ξ for = 1, 2,..., T. The calculaion of ξ is based on he Markov regime swiching filer inroduced in Hamilon 1989 and conduced in hree seps. 1. A he beginning of he -h ieraion, given ξ 1 1, we have ξ := pξ q1 ξ An esimae of he densiy of y is obained by fy Ω 1 := ξ fy s = 1, Ω ξ fy s = 2, Ω 1, where he condiional densiy saisfies fy s = i, Ω 1 := 2πC i 1/2 exp µ i 2 2C i, i = 1, where µ i and Ci are given in 2.7 and Once y is observed, we can updae he modified condiional regime probabiliy ξ : = pξ q1 ξ 1 1 fy s = 1, Ω 1 fy s = 2, Ω 1 + pξ q1 ξ 1 1 fy s = 1, Ω 1 fy s = 2, Ω Figure 1 presens a flowchar for he filer described in subsecions The modified log likelihood funcion under he wo regimes specificaion, i.e. T log fy Ω 1, is given as a by-produc of he filer. The iniial values ξ 0 0, x 0 0 and P 0 0 will be discussed in secion Hypoheses Le δ represen parameers ha are affeced by regime swiching, aking a value of δ 1 in regime 1 and δ 2 in regime 2. Le β represen parameers ha remain consan across he regimes. Then, for any 7

8 Figure 1: Flowchar for he filer: sae space models wih regime swiching 8

9 prespecified 0 < p, q < 1, he modified log likelihood funcion is given by = L A p, q, β, δ 1, δ 2 where } log f 1 p, q, β, δ 1, δ 2 ξ p, q, β, δ 1, δ 2 + f 2 p, q, β, δ 1, δ 2 1 ξ p, q, β, δ 1, δ 2, 2.16 f i p, q, β, δ 1, δ 2 = fy s = i, Ω 1, which is defined in When δ 1 = δ 2 = δ, he modified log likelihood funcion reduces o L N β, δ = log f 1 p, q, β, δ, δ 2.17 := log f β, δ, which can be compued using he sandard Kalman filer. This paper sudies a es saisic based on 2.17 and 2.16 for he one regime specificaion versus he wo regimes specificaion. To sar, I impose he following resricions on he DGP and he parameer space, following Assumpion 1-3 in Qu and Zhuo Assumpion 1. i The random vecor z, y is sric saionary, ergodic and β-mixing wih he mixing coefficien β τ saisfying β τ cρ τ for some c > 0 and ρ 0, 1. ii Under he null hypohesis, y is generaed by f Ω 1 ; β, δ where β and δ are inerior poins of Θ R n β and R n δ wih Θ and being compac. Assumpion 2. Under he null hypohesis: i β, δ uniquely solves max β,δ Θ E L N β, δ ; ii for any 0 < p, q < 1, β, δ, δ uniquely solves max β,δ1,δ 2 Θ E L A β, δ 1, δ 2. Assumpion 3. Under he null hypohesis, we have: i T 1 L N β, δ EL N β, δ = o p 1 holds uniformly over β, δ Θ, wih T 1 T β,δ log f β, δ β,δ log f β, δ being posiive definie over an open neighborhood of β, δ for sufficienly large T ; ii for any 0 < p, q < 1, T 1 L A β, δ 1, δ 2 EL A β, δ 1, δ 2 = o p 1 holds uniformly over β, δ 1, δ 2 Θ. 9

10 Using he above noaion, he null and alernaive hypoheses can be more formally saed as: H 0 : δ 1 = δ 2 = δ for some unknown δ, H 1 : δ 1, δ 2 = δ 1, δ 2 for some unknown δ 1 δ 2 and p, q 0, 1 0, 1. For he remainder of his paper, I use an ARMAK, L model o illusrae he main resuls. The illusraive model. Le us consider a general ARM AK, L model: K L m = φ k,s m k + ε + θ l,s ε l, ε i.i.d. N0, σs 2, 2.18 k=1 l=1 y = α s + m, 2.19 where only y is observable bu no m or s. In his illusraive model, some or all of he model parameers can be affeced by s. This ARMA model can be wrien ino he seing in as follows: Define n r = maxk, L + 1}. Inerpre φ j,s = 0 for j > K and θ j,s = 0 for j > L. Le F s = φ 1,s φ 2,s φ nr1,s φ nr,s , G s = 0, u = ε 0. 0 N 0, Q s, Q s = σ 2 s , H s = 1 θ 1,s. θ nr1,s, A s = α s, and z = 1. 10

11 3 The es saisic This secion proposes a es saisic based on he MLR. Le β and δ denoe he maximizer of he log likelihood funcion under null hypohesis: β, δ = arg max β,δ LN β, δ. 3.1 The MLR evaluaed a some 0 < p, q < 1 hen equals MLRp, q = 2 max L A p, q, β, δ 1, δ 2 L N β, δ. 3.2 β,δ 1,δ 2 I is naural o consider he following es saisic: SupMLRΛ ɛ = sup MLRp, q, p,q Λ ɛ where Λ ɛ is a compac se o be specified laer. Similar es saisics have been sudied by Hansen 1992, Garcia 1998 and Qu and Zhuo MLR under prespecified p and q This secion sudies he MLR under a given p, q Λ ɛ. The choice of Λ ɛ will be discussed in he nex secion. 4.1 Condiional regime probabiliy Le us firs sudy he condiional regime probabiliy ξ +1 p, q, β, δ 1, δ 2 as well as is derivaives wih respec o β, δ 1 and δ 2 because he resuls will be needed o develop he expansion of he modified log likelihood funcion. Combining equaions 2.13 and 2.15 gives a recursive formula o calculae ξ +1 p, q, β, δ 1, δ 2 : ξ +1 p, q, β, δ 1, δ 2 f 2 p, q, β, δ 1, δ 2 ξ p, q, β, δ 1, δ 2 1 =p + p + q 1 f 1 p, q, β, δ 1, δ 2 ξ p, q, β, δ 1, δ 2 + f 2 p, q, β, δ 1, δ 2 1 ξ p, q, β, δ 1, δ 2,

12 where f i p, q, β, δ 1, δ 2 = fy s = i, Ω as in This recursive formula implies ha he derivaives of ξ +1 wih respec o he model parameers mus also follow firs order difference equaions. Because he asympoic expansions are considered around he esimaes under he null hypohesis, i is sufficien o analyze ξ +1 p, q, β, δ 1, δ 2 and is derivaives a δ 1 = δ 2 = δ for an arbirary value of δ in. Le θ = β, δ 1, δ 2 be an augmened parameer vecor. We hen hree ses of inegers hey index he elemens in β, δ 1 and δ 2, respecively: I 0 = 1,..., n β }, I 1 = n β + 1,..., n β + n δ }, I 2 = n β + n δ + 1,..., n β + 2n δ }. Le ḡ denoe ha gβ, δ 1, δ 2 is evaluaed a some β and δ 1 = δ 2 = δ, i.e., ξ +1 and denoe ha ξ +1 p, q, β, δ 1, δ 2 and f 1 p, q, β, δ 1, δ 2 or f 2 p, q, β, δ 1, δ 2 are evaluaed a some β and δ 1 = δ 2 = δ. Le θj1... θjk ξ, θj1... θjk f1 and θj1... θjk f2 denoe he k-h order derivaives of ξ p, q, β, δ 1, δ 2, f 1 p, q, β, δ 1, δ 2 and f 2 p, q, β, δ 1, δ 2 wih respec o he j 1,..., j k -h elemens of θ, evaluaed a some β and δ 1 = δ 2 = δ. Also le F denoe he marix F i i = 1 or 2 evaluaed a some β and δ 1 = δ 2 = δ and θj1... θjk Fi denoe he k-h order derivaives of he parameer marix F i evaluaed a some β and δ 1 = δ 2 = δ. By definiion, he following relaionships hold: θj1... θjk f1 = θj1... θjk f2 if j 1,..., j k all belong o I 0. The nex lemma is parallel o Lemma 1 in Qu and Zhuo 2015, which characerizes he properies of ξ +1 p, q, β, δ 1, δ 2 and is derivaives when δ 1 = δ 2 = δ. Lemma 1. Le ξ 0 0 =, ρ = p + q 1 and r = ρ 1 wih defined in 2.4. Then, for 1, we have: 1. ξ +1 =. 2. θj ξ+1 = ρ θj ξ + Ēj,, where Ē j, = r θj f1 θ j f2, wih j I 0, I 1, I 2 }. 3. θj θk ξ+1 = ρ θj θk ξ + Ējk,, where Ējk, are given by Le I a,i b denoe he siuaion wih j I a and k I b ; a, b = 0, 1, 2,: 12

13 I 0, I 0 : 0 I 0, I 1 or I 0, I 2 : r θj θk f1 I 1, I 1 or I 1, I 2 or I 2, I 2 : r θj θk f1 2r θj f1 θ j θk f2 + θ j f2 θk f2 θ j f2 θk f1 θ j θk f2 θj f1 + ρ1 2 θ j f2 θk f1 θk ξ + θk f1 1 θ j f2 θk f2 + r2 1 θj f1 θk f2 + θ j f2 θ k f2 θj ξ θk f1. 4. θj θk θk ξ+1 = ρ θj θk θk ξ + Ējkl,, where Ējkl, are given in he appendix wih j, k, l I a, I b, I c } and a, b, c = 0, 1, 2. I now discuss he firs order derivaives of f i appearing in he lemma. By 2.5, 2.6, 2.7, 2.8, 2.14, and 4.2, we have: f i = 1/2 2πH i F i P 1 1 F i + Q i H i exp y H i G i + F i x 1 1 A i z 2H i F i P 1 1 F i + Q i H i 2. The firs order derivaive of f i wih respec o he j-h componen in θ is as follows: y H i G i + F i x 1 1 A i θj f i = f z i H i F i P 1 1 F i + Q 4.3 i H i } θj H i G i + F i x θj A iz + H i θj G i + θj F i x F i θj x y H i G i + F i x 1 1 A i + f i z 2H i F i P 1 1 F i + Q i H i H i F i P 1 1 F i + Q 1 i H i θj H i F i P 1 1 F i + Q i H i + H i F i P 1 1 F i + Q i θj H i } +H i θj F i P 1 1 F i + F i θj P 1 1 F i + F i P 1 1 θj F i + θj Q i H i. The exac expressions for he second order derivaives of f i are included in he appendix. The properies of x and P and heir derivaives will be sudied in he nex subsecion see Lemma 3 13

14 below. Noe ha, for some β and δ 1 = δ 2, θj f1 θj f2 = f y H Ḡ + F x 1 1 Ā z H F P1 1 F Q + H θj H 1 θj H 2 Ḡ + F x1 1 + θj Ā 1 θj Ā 2 z + H } θj Ḡ 1 θj Ḡ 2 + θj F1 θj F2 x f 1 y H Ḡ + F 2 x H F P1 1 F Q Ā z + H H F P1 1 F Q + H 1 θj H 1 θj H 2 F P1 1 F Q + H + H F P1 1 F + Q θj H1 θj H2 + H θj F1 θj F2 P1 1 F + F P 1 1 θj F 1 θj F 2 + θj Q1 θj Q2 H }, in which he θj x 1 1 and θj P1 1 erms are canceled ou. Consequenly, hese wo quaniies are no needed for calculaing θj ξ+1. Similarly, θj θk x 1 1 and θj θk P1 1 are no needed for calculaing θj θk f1 θj θk f2. This is also rue for higher order derivaives of f 1 f 2 when evaluaed a δ 1 = δ 2. I now use an example o illusrae Lemma 1. The illusraive model con d. Consider he illusraive example and assume ha boh α s and σ 2 s are affeced by regime swiching. Lemma 1 implies ha α1 ξ+1 = ρ α1 ξ + r 1 σ 2 y ᾱ H F x1 1, y ᾱ H 2 F x1 1 σ 2 1 ξ+1 = ρ σ 2 1 ξ + r 1 2 σ 2 σ 2 1. Because he filer described in subsecions reduces o he sandard Kalman filer when δ 1 = δ 2, α1 ξ+1 and σ 2 1 ξ+1 boh reduce o saionary AR1 processes wih mean zero when evaluaed a he rue parameer values under he null hypohesis. Their variances are finie and saisfy E α1 ξ+1 2 = r 2 1 ρ 2 σ 2 and E σ 2 1 ξ+1 2 = r 2 21 ρ 2 σ 4, where σ 2 denoes he rue value of σ 2 s under he null hypohesis. 14

15 4.2 Filered sae and is mean squared error Since boh he filered sae in 2.11 and is mean squared error in 2.12 are componens in he modified log likelihood funcion, i is imporan o sudy hese funcions and heir derivaives wih respec o β, δ 1 and δ 2. As in he previous subsecion, i is sufficien o sudy hese quaniies when δ 1 = δ 2. Under Assumpions 1-3 and when δ 1 = δ 2, x in 2.1 is saionary. The uncondiional mean of x can be employed as he iniial value, denoed by x 0 0. The uncondiional mean of x saisfies Ex = Ḡ + F Ex 1. This implies x 0 0 = I F 1 Ḡ. The nex lemma provides he iniial value for P when δ 1 = δ 2, denoed by P 0 0. Is resuls are also used laer o sudy he properies of x and P. Lemma 2. Le F have all i eigenvalues inside he uni circle. Se P 0 0 = P, where P solves P = I F P F Q + H H F P F + Q 1 H H F P F Q Then, under he null hypohesis and Assumpion 1-3, P = P, P i = P and P = F P F + Q, for all = 1,..., T. In he subsequen analysis, P is abbreviaed as P. The following assumpion, which is analogous o Proposiion 13.2 in Hamilon 1994, ensures ha P and P are unique. Assumpion 4. The eigenvalues of I P H H H P H F are all inside he uni circle. I use he ARM A model in o illusrae his assumpion. The illusraive model con d. Consider he model in Then, P and P are equal 15

16 o 0 and Q respecively. The quaniy in Assumpion 4 is given by: I P H H H P H F = θ 1 θ 2 θ nr , where θ j denoes ha he parameer θ j,s is evaluaed a some β and δ 1 = δ 2 = δ. For he ARMA1, 1 model, Assumpion 4 is equivalen o 1 < θ 1 < 1. The nex lemma conains he deails on he firs and second order derivaives of he filered sae and is mean squared error when evaluaed a δ 1 = δ 2 = δ. Lemma 3. Under he null hypohesis and Assumpions 1-4, we have: 1. For any j I a, a = 0, 1, 2, vec θj P = I P H H H P H 2 F vec θj P1 1 + vec Pj,, where P H H θj P j, = I H P H F 1 P F + F P F θj 1 + θj Q 1 I H H P H P H P H H θj + 1 I H P H F 2 P F + F P F θj 2 + θj Q 2 I H H P H P H P θj H1 H + H θj H 1 P H H θj H 1 P H + H P θj H1 H P H 2 F P F + H P H Q P θj H2 H + H θj H 2 P H H θj H 2 P H + H P θj H2 1 H P H H P H 2 F P F + Q. 2. For any j I a, a = 0, 1, 2, θj x = I P H H H P H F θj x X j,, where he expression of Xj, are given in he appendix. 3. For any j I a and k I b, a, b = 0, 1, 2, vec θj θk P = I P H H H P H 2 F vec θj θk P1 1 + vec Pjk,, 16

17 where he expression of P jk, are given in he appendix. 4. For any j I a and k I b, a, b = 0, 1, 2, θj θk x = I P H H H P H F θj θk x X jk,, where he expression of Xjk, are given in he appendix. Lemma 3 shows ha he firs and second order derivaives of he filered sae and is mean squared error all follow firs order linear difference equaions and he lagged coefficien marices for hem always include I P H H P H 1 H F. The recursive srucures implied by Lemma 3 sugges ha we apply a similar sraegy o analyze x and P, as we are sudying he properies of he higher order derivaives of ξ +1. The following example illusraes he resuls in Lemma 3 wih an ARMA1, 1 model. The illusraive model con d. Consider he ARM A1, 1 model in and assume only α s swiches. Lemma 3 implies ha P and P are equal o 0 and Q, respecively. Meanwhile, furher calculaions show ha θj P = 0 for j 1,..., n β + 2} and θj x = 0 0 ξ 1+ θ 1 0 1ξ 1+ θ 1 0 j 1,..., n β }, j = n β + 1, j = n β + 2. The second order derivaives of P and x, wih respec o α 1, saisfy 2 α 1 P = θ θ 1 θ 1 1 and 2 α 1 x = + θ φ 1 θ1 2 α 1 x α1 ξ 1 0 y ᾱ H F x1 1 φ1 σ 2 2 1, 17

18 where α1 ξ = ρ α1 ξ α1 f1 α 1 f2. In his case, when δ 1 = δ 2, 2 α 1 P is a consan marix, while 2 α 1 x depends on 2 α 1 x 1 1, α1 ξ, and he predicion error, y ᾱ H F x1 1, a ime. 4.3 Modified log likelihood funcion and is expansion Because of he muliple local maxima in L A p, q, β, δ 1, δ 2, i is difficul o direcly expand his funcion around he null esimaes β, δ, δ. Boh Cho and Whie 2007 and Qu and Zhuo 2015 sugges o work wih he concenraed likelihood funcion. To derive he concenraed likelihood funcion, β and δ 1 are reaed as funcions of δ 2 and he dependence beween β, δ 1 and δ 2 is quanified using he firs order condiions ha define he concenraed and modified log likelihood funcion see Lemma A.3 in he appendix. This effecively removes β and δ 1 from he subsequen analysis and allows us o work wih he concenraed, modified log likelihood funcion, which is only a funcion of δ 2. Therefore, we can expand he concenraed, modified log likelihood funcion around δ 2 = δ see Lemma 4 below o obain an approximaion for M LRp, q. For any δ 2, we can wrie Lp, q, δ 2 = max β,δ 1 L A p, q, β, δ 1, δ 2 and ˆβδ2, ˆδ 1 δ 2 = arg max β,δ 1 L A p, q, β, δ 1, δ 2. Then, MLRp, q = 2 maxlp, q, δ 2 Lp, q, δ. δ 2 For k 1, le L k i 1...i k p, q, δ 2 i 1,...i k 1,..., n δ } denoe he k-h order derivaive of Lp, q, δ 2 wih respec o he i 1,...i k -h elemens of δ 2. Le d j j 1,..., n δ } denoe he j-h elemen of 18

19 δ 2 δ. Then, a fourh order Taylor expansion of Lp, q, δ 2 around δ is given by n δ Lp, q, δ 2 Lp, q, δ = L 1 j p, q, δd j + 1 2! j=1 n δ n δ n δ + 1 3! + 1 4! j=1 k=1 l=1 n δ n δ n δ j=1 k=1 l=1 m=1 n δ n δ j=1 k=1 L 3 jkl p, q, δd j d k d l n δ L 4 jklm p, q, δd j d k d l d m, L 2 jk p, q, δd j d k 4.5 where in he las erm δ is a value ha lies beween δ 2 and δ. Two lemmas will be provided o analyze his expansion. Here, wo more assumpions, which are similar o Assumpions 4 and 5 in Qu and Zhuo 2015, are needed. Assumpion 5. There exiss an open neighborhood of β, δ, denoed by Bβ, δ, and a sequence of posiive, sricly saionary and ergodic random variables υ } saisfying Eυ 1+c c > 0, such ha sup β,δ 1 Bβ,δ θi1... θik f β, δ 1 f β, δ 1 αk k < υ < L < for some for all i 1,..., i k 1,..., n β + n δ }, where 1 k 5; αk = 6 if k = 1, 2, 3 and αk = 5 if k = 4, 5. Assumpion 6. There exiss η > 0, such ha sup p,q ɛ,1ɛ sup δ δ <η T 1 L 5 jklmn p, q, δ = O p 1 for all j, k, l, m, n 1,..., n δ }, where ɛ is an arbirary small consan saisfying 0 < ɛ < 1/2. The nex lemma characerizes he derivaives of ˆβδ 2 and ˆδ 1 δ 2 wih respec o δ 2 evaluaed a δ 2 = δ. To shoren he expressions, le ξ +1 and denoe ξ +1 p, q, β, δ 1, δ 2 and f β, δ 1 evaluaed a β, δ 1, δ 2 = β, δ, δ. Also, le δ1i1... δ1ik ξ and δ1i1... δ1ik f1 denoe he k-h order derivaive of ξ +1 p, q, β, δ 1, δ 2 and f β, δ 1 wih respec o he i 1,..., i k -h elemens of δ 1, evaluaed a β, δ 1, δ 2 = β, δ, δ. Finally, define 2 1 ξ δ1j δ1k f1 Ũ jk, = 1 ξ δ2j δ1k f1 + 1 ξ 2 δ1j ξ θk f1 + 1 δ 1j δ 1k + 1 δ 2j δ 1k θ k f2 θj f1 + f 2 f 2 + δ2j δ2k f1 f δ 2j δ 2k + ξ δ1j δ 2k + 1 δ 1j δ 2k θ j f2 δ1k ξ, 4.6 f 2 f 2 19

20 and β D jk, =,δ 1 f1 + 1 β,δ 2 f2 Ũ jk,, ξ Ĩ = β,δ 1 f1 + 1 β,δ 2 f2 β,δ 1 f β,δ 2 f 2, Ṽ jklm = T 1 T Ũ jk, Ũ lm,, Dlm = T 1 T D T lm,, Ĩ = T 1 Ĩ. 4.7 As will be seen, Ũjk, is he leading erm in L 2 jk p, q, δ, while D jk, and Ĩ appear when consrucing he leading erm of L 4 jklm p, q, δ. The nex lemma, analogous o Lemma 3 in Qu and Zhuo 2015, presens he properies of L k i 1...i k p, q, δ 2 when δ 2 is evaluaed a he null esimae, i.e. δ. Lemma 4. Under he null hypohesis and Assumpions 1-6, for all j, k, l, m 1,..., n δ }, we have 1. L 1 j p, q, δ = T 1/2 L 2 jk p, q, δ = T 1/2 T Ũ jk, + o p T 3/4 L 3 jkl p, q, δ = O p T 1/4. 4. T 1 L 4 jklm p, q, δ = Ṽjklm D jkĩ1 Dlm + Ṽjmkl D jmĩ1 Dkl + Ṽjlkm D jlĩ1 Dkm } + o p 1. The illusraive model con d. We use he ARM A1, 1 model o illusrae he leading erms of T 1/2 L 2 jk p, q, δ and T 1 L 4 jklm p, q, δ in Lemma 4. Suppose only α s D jk, equal, respecively, 1 ξ 1 + φ φ 1 θ1 1 1 θ 1 2 σ 2 2 φ 1 + θ 1 1 θ 1 j1 j j=1 s=1 µ 2 σ ρ s1 µs σ 2 µ 1 σ 2 s=1 + φ 1 θ1 µj σ 2 swiches. Then, Ũ jk, and ρ s µs σ 2 and y 1 α µ σ 2 µ µ 1 1 µ 2 1 σ 2 2 σ 2 σ 2 µ ξ 1 φ 1 σ 2 1+ θ 1 Ũ jk,, where µ denoes he residuals under he null hypohesis and σ 2 = T 1 T µ 2. This makes he variance funcion of T 1/2 T Ũ jk,, and herefore of T 1/2 L 2 jk p, q, δ, consisenly esimable. 20

21 5 Asympoic approximaions Le L 2 p, q, δ be a square marix wih is j, k-h elemen given by L 2 jk p, q, δ for j, k 1, 2,..., n δ }. This secion includes hree ses of resuls. 1 The weak convergence of T 1/2 L 2 p, q, δ over ɛ p, q 1 ɛ. 2 The limiing disribuion of SupMLRΛ ɛ. 3 A finie sample refinemen ha improves he asympoic approximaion. 5.1 Weak convergence of L 2 p, q, δ For 0 < p r, q r, p s, q s < 1 and j, k, l, m 1, 2,..., n δ }, define ω jklm p r, q r ; p s, q s = V jklm p r, q r ; p s, q s D jkp r, q r I 1 D lm p s, q s, 5.1 where V jklm p r, q r ; p s, q s = E U jk, p r, q r U lm, p s, q s, D jk p r, q r = ED jk, p r, q r, and I = EI. Here, U jk, p r, q r, D jk, p r, q r and I have he same definiions as Ũjk,, Djk, and Ĩ in 4.6 and 4.7 bu evaluaed a p r, q r, β, δ insead of p r, q r, β, δ. The following lemma is parallel o Lemma 4 in Qu and Zhuo Lemma 5. Under he null hypohesis and Assumpions 1-6, we have, over ɛ p, q 1 ɛ : T 1/2 L 2 p, q, δ G p, q, where he elemens of G p, q are mean zero coninuous Gaussian processes saisfying CovG jk p r, q r, G lm p s, q s = ω jklm p r, q r ; p s, q s for j,k,l,m 1,2,...,n δ }, where ω jklm p r,q r ; p s,q s is given by 5.1. In he appendix, his lemma is proved by firs showing he finie dimensional convergence and hen he sochasic equiconinuiy. 5.2 Limiing disribuion of SupMLRΛ ε Le E be a se of open balls ha includes all possible values of p, q such ha L 2 jk p, q, δ 0 for any j, k 1, 2,..., n δ }. For example, if for some specific j 1 and k 1, L 2 j 1 k 1 p 1, q 1, δ 0, hen p, q E if 21

22 p p 1 ɛ 1, p 1 + ɛ 1 and q q 1 ɛ 1, q 1 + ɛ 1 for any small ɛ 1, say ɛ 1 = Define Λ ɛ = p, q : ɛ p, q 1 ɛ, and p, q / E}. 5.2 Le Ωp, q be an n 2 δ -dimensional square marix whose j + k 1n δ, l + m 1n δ -h elemen is given by ω jklm p, q; p, q. Then, Lemma 5 implies EvecG p, q vecg p, q = Ωp, q. The nex resul, which is analogous o Proposiion 2 in Qu and Zhuo 2015, gives he asympoic disribuion of SupMLRΛ ɛ. Proposiion 1. Suppose he null hypohesis and Assumpions 1-6 hold. Then SupMLRΛ ɛ sup p,q Λ ɛ η R n δ sup W 2 p, q, η, 5.3 where Λ ɛ is given by 5.2 and W 2 p, q, η = η 2 1 vecg p, q η 2 Ωp, q η 2. 4 Some imporan feaures of ω jklm p, q; p, q have been shown in secion 5.1 of Qu and Zhuo 2015 by simple examples. In he curren conex, I also observe he similar feaures of ω jklm p, q; p, q ha his funcion depends on: 1 he model s dynamic properies e.g., wheher he regressors are sricly exogenous or predeermined, 2 which parameers are allowed o swich e.g., regressions coefficiens or he variance of he errors, and 3 wheher nuisance parameers are presen. 5.3 A refinemen Qu and Zhuo 2015 provide a refinemen o he asympoic disribuion when L 2 p, q, δ 0. In such siuaions, he magniude of T 1/2 T Ũ jk, can be oo small o dominae he higher order erms in he likelihood expansion when p + q is close o 1. This indicaes ha an asympoic disribuion ha relies enirely on T 1/2 T Ũ jk, can be inadequae. Moivaed by his observaion, I consider a refinemen o he asympoic approximaion under Markov swiching sae space models. The following assumpion is parallel o Assumpion 6 in Qu and Zhuo Assumpion 7. There exiss an open neighborhood of β, δ, Bβ, δ, and a sequence of pos- 22

23 iive, sricly saionary and ergodic random variables υ } saisfying Eυ 1+c < for some c > 0, such ha he supremums of he following quaniies over Bβ, δ are bounded from above by υ : θi1... θik f β, δ 1 /f β, δ 1 4, θi1... θim f β, δ 1 /f β, δ 1 2, θi1... θi8 f β, δ 1 /f β, δ 1, θj1 θi1... θi7 f β, δ 1 /f β, δ 1, θj1 θj2 θi1... θi6 f β, δ 1 /f β, δ 1, where k = 1, 2, 3, 4, m = 5, 6, 7, i 1,..., i 8 1,..., n β + n δ } and j 1, j 2 1,..., n β }. Obaining all he leading erms in an even higher order expansion of he modified likelihood funcion is very difficul in he curren conex. This paper considers incorporaing some specific erms for he refinemen. Define s jkl, p, q = δ1j δ1k δ1l f1 ξ 2, 5.4 where x and P are reaed as consan when calculaing δ1j δ1k δ1l f1 here. For j, k, l, m, n, u 1,..., n δ }, le G 3 jkl p, q be a coninuous Gaussian process wih mean zero and saisfy ω 3 jklmnu pr, qr; ps, qs = CovG 3 jkl pr, qr, G3 mnu p s, q s = E s jkl, p r, q rs mnu,p s, q s ξ β E,δ 1 f1 + 1 ξ β,δ 2 f2 ξ s jkl, p r, q r I 1 β,δ 1 f1 + 1 ξ β,δ 2 f2 s mnu,p s, q s, where s jkl, p, q is he same as s jkl, p, q bu is evaluaed a rue parameer values. The oher quaniies on he righ hand side are also evaluaed a he rue parameer values. For he fourh and eighh order derivaives, define 1 3 ξ δ1j δ1k δ1l δ1m f1 kjklm, p, q = 1 1 +, 5.5 where x and P are reaed as consan when calculaing δ1j δ1k δ1l δ1m f1 here. For i 1,..., i 8 1,..., n δ }, le G 4 i 1 i 2 i 3 i 4 p, q denoe a coninuous Gaussian process wih mean zero and saisfy ω 4 i 1 i 2...i 8 p r, q r; p s, q s = Cov G 4 i 1 i 2 i 3 i 4 p r, q r, G 4 i 5 i 6 i 7 i 8 p s, q s = E k i1 i 2 i 3 i 4, p r, q r k i5 i 6 i 7 i 8, p s, q s ξ β E,δ 1 f1 + 1 ξ β,δ 2 f2 ξ k i1 i 2 i 3 i 4, p r, q r I 1 β,δ 1 f1 + 1 ξ β,δ 2 f2 k i5 i 6 i 7 i 8, p s, q s, 23

24 where k i1 i 2 i 3 i 4,p, q equals k i1 i 2 i 3 i 4,p, q bu evaluaed a he rue parameer values. The remaining quaniies on he righ hand side are also evaluaed a he rue parameer values. The nex lemma characerizes he asympoic properies of s jkl, p, q and k jklm, p, q when j, k, l, m 1,..., n δ }. Lemma 6. Under he null hypohesis and Assumpions 1-7, we have T 1/2 T s jkl, p, q G 3 jkl p, q and T 1/2 T kjklm, p, q G 4 jklm p, q. We now incorporae he corresponding erms o obain a refined approximaion. Le G 3 p, q be a n 3 δ - dimensional vecor whose j + k 1n δ + l 1n 2 δ -h elemen is given by G3 jkl p, q. Le Ω 3 p, q denoe an n 3 δ by n3 δ marix whose j + k 1n δ + l 1n 2 δ, m + n 1n δ + r 1n 2 δ -h elemen is given by ω 3 jklmnr p, q; p, q. Define W 3 p, q, η = T 1/4 1 3 η 3 vecg 3 p, q T 1/2 1 η 3 Ω 3 p, q η Le G 4 p, q be an n 4 δ - dimensional vecor whose j + k 1n δ + l 1n 2 δ + m 1n3 δ -h elemen is given by G 4 jklm p, q. Le Ω4 p, q be an n 4 δ by n4 δ marix whose j + k 1n δ + l 1n 2 δ + m 1n 3 δ, n + r 1n δ + s 1n 2 δ + u 1n3 δ -h elemen is given by ω4 jklmnrsu p, q; p, q. Define W 4 p, q, η = T 1/ η 4 vecg 4 p, q T 1 1 η 4 Ω 4 p, q η Then, he disribuion of he SupMLRΛ ɛ es can be approximaed by: S Λ ɛ sup sup p,q Λ ɛ η R n δ } W 2 p, q, η + W 3 p, q, η + W 4 p, q, η, 5.6 where Λ ɛ is specified in 5.2. The following corollary is analogous o Corollary 1 in Qu and Zhuo Corollary 1. Under Assumpions 1-7 and he null hypohesis, we have: Pr SupMLRΛ ɛ s Pr S Λ ɛ s 0, 24

25 over Λ ɛ in 5.2. Noe ha he above resul holds irrespecive of he model. This follows because he addiional erms W 3 p, q, η and W 4 p, q, η boh converge o zero as T. These erms provide refinemens in finie samples, having no effec asympoically. The criical values can be obained by following he simulaion procedures described in secion 5.4 of Qu and Zhuo The illusraive model con d. Consider he ARMA1, 1 model in and assume α s swiches. This model can be wrien as: m = φm 1 + e + θe 1, e i.i.d. N0, σ 2 y = α s + m. To illusrae he effecs of he refined approximaion, I simulae daa using he ARM A1, 1 model wih T = 2000, α 1 = α 2 = 0, φ = 0.9, θ = 0.70 and σ = 0.2. Then, for each simulaed sample and fixed p, q, I calculae MLRp, q, he approximaion o MLRp, q using only he second and fourh order erms in he Taylor expansion, and he approximaion o MLRp, q using he second order, fourh order and refinemen erms in he Taylor expansion. Afer simulaing 500 samples, I calculae boh he correlaion beween M LRp, q and is approximaion using only he second and fourh order erms, and he correlaion beween M LRp, q and is approximaion using he second order, fourh order and refinemen erms. Table 1: Comparison of correlaions beween M LRp, q saisic and original and refined approximaions Beween MLRp, q and p, q original approximaion refined approximaion 0.90, , , I also check hese correlaions for differen p and q. Resuls are summarized in Table 1. The resuls show ha including he refinemen erms brings he approximaion closer o he M LRp, q saisic. 25

26 6 Mone Carlo This secion examines he size and power properies of he SupMLR es saisics. The DGP is m = φm 1 + e + θe 1, e i.i.d. N0, σ 2 y = α s + m, where y is observable and α s swiches wih ps = 1 s 1 = 1 = p and ps = 2 s 1 = 2 = q. Assign φ = 0.9, θ = 0.70 and σ = 0.2. The choice of his DGP is moivaed by boh he model sudied in Perron 1993 and he empirical applicaion in he nex secion. In his secion, Λ ɛ is specified as in 5.2 wih ɛ = The criical values and rejecion frequencies are all based on 3000 replicaions. Table 2: Rejecion frequencies under he null hypohesis Level T = 200 SupMLRΛ T = 500 SupMLRΛ Table 2 repors he sizes of he SupMLRΛ ɛ es saisic a five differen nominal levels. Under he null hypohesis, we se α 1 = α 2 = 0. The rejecion frequencies overall are close o he nominal levels wih mild over-rejecions in some cases. For example, he rejecion raes a he 5% and 10% levels are 7.40% and 14.23% respecively, for ɛ = 0.01 and sample size T = 200. Similar rejecion raes are observed when T = 500. For power properies, I se α 1 = τ and α 2 = τ, wih τ = 0.05, 0.10, 0.15, 0.20 and The sample size T = 500 and he number of replicaions is Three pairs of values for p, q are considered: 0.70, 0.70, 0.70, 0.90 and 0.90, Table 3: Rejecion frequencies under he alernaive hypoheses p, q τ , 0.70 SupMLRΛ , 0.90 SupMLRΛ , 0.95 SupMLRΛ Nominal level, 5%. The rejecion frequencies a 5% nominal levels are repored in Table 3. The power of he SupM LR saisic increases consisenly as he magniude of α 1 α 2 increases. 26

27 7 Applicaion In his secion, I apply he MLR es developed in he preceding secions o sudy he changes in monhly U.S. unemploymen raes. The daa are from he labor force saisics repored in he Curren Populaion Survey. The full sample is from January 1960 o July A simple ARMA1, 1 model as in 2.18 and 2.19 is considered, i.e. m = φm 1 + e + θe 1, e i.i.d. N0, σ 2 y = α s + m. Here, α s swiches and indicaes he mean level of change in he unemploymen rae a ime. SupMLRΛ 0.01 equals for he full sample wih he criical value being 8.83 a he 5% level. The es saisic herefore provides srong evidence favoring he regime swiching specificaion. To provide some furher evidence for he relevance of he regime swiching specificaion, I esimae he regimes implied by he model. Le s = 1 and s = 2 represen he igh and slack labor marke regimes, respecively. Here he igh and slack labor marke regimes are defined from he perspecive of he job-seekers. Esimaion resuls show ha he regime shifs closely follow he recession and expansion periods daed by he Naional Bureau of Economic Research NBER and shown in Figure 2. Comparing he smoohed regime probabiliies and he daes of he recession and expansion periods, I find ha he labor marke normally akes ime o reac a he beginning of a recession. Esimaion under wo regimes specificaion also provides deailed informaion on changes in he labor marke and he duraions of each regime. In he igh labor marke regime, he unemploymen rae increases by 0.26% per monh on average and his regime lass abou 8.8 monhs. In he slack labor marke regime, he unemploymen rae decreases by 0.03% per monh and he regime lass abou 79.4 monhs. The model assigns low probabiliies, around 20 30%, o he igh regime during he relaively shallow recessions of July 1990 o March 1991 and March 2001 o November This makes sense, because he increases in he unemploymen rae during hese wo recessions are relaively moderae when compared wih oher recessions, such as he recen Grea Recession from December 2007 o June

28 Figure 2. Smoohed probabiliies of in igh labor marke regime Noe. The shaded areas correspond o he NBER defined recessions. The solid line indicaes he smoohed probabiliies of being in he igh labor marke regime. 8 Conclusion This paper develops a modified likelihood raio MLR based es for deecing regime swiching in sae space models. The asympoic disribuion of his es saisics is also esablished. When applied o changes in U.S. monhly unemploymen raes, he es finds srong evidence favoring he regime swiching specificaion. This paper is he firs o develop a es ha is based on he likelihood raio principle for deecing regime swiching in sae space models. The echniques developed in his paper can have implicaions for hypohesis esing in more general conexs, such as esing for regime swiching in sae space models wih muliple observables. 28

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30 Hansen, B. E. 1992: The Likelihood Raio Tes under Non-Sandard Condiions: Tesing he Markov Swiching Model of GNP, Journal of Applied Economerics, 7, S61 S82. Harrison, P. J. and C. F. Sevens 1976: Bayesian Forecasing, Journal of he Royal Saisical Sociey. Series B Mehodological, 38, Harvey, A. C. 1981: Time Series Models, Oxford, UK: Philip Allan and Humaniies Press. Highfield, R. A. 1990: Bayesian Approaches o Turning Poin Predicion, in Proceedings of he Business and Economics Secion, Washingon, DC: American Saisical Associaion, Kalman, R. E. 1960: A New Approach o Linear Filering and Predicion Problems, Journal of Basic Engineering, 82, Kim, C.-J. and C. R. Nelson 1999: Sae-Space Models wih Regime Swiching: Classical and Gibbs-Sampling Approaches wih Applicaions, Cambridge, MA: The MIT Press. Lancaser, T. 1984: The Covariance Marix of he Informaion Marix Tes, Economerica, 52, Meinhold, R. J. and N. D. Singpurwalla 1983: Undersanding he Kalman Filer, The American Saisician, 37, Neyman, J. and E. Sco 1966: On he Use of Ca Opimal Tess of Composie Hypoheses, Bull. Ins. In. Sais., 41, Perron, P. 1993: Non-saionariies and Non-lineariies in Canadian Inflaion, in Economic Behaviour and Policy Choice under Price Sabiliy : Proceedings of a Conference Held a he Bank of Canada. Qu, Z. and F. Zhuo 2015: Likelihood Raio Based Tess for Markov Regime Swiching, Working Paper. 30

31 Appendix A Derivaives of he densiy funcion The derivaives of f i in 2.16 are calculaed here. By 2.5, 2.6, 2.7, 2.8, 2.14 and 4.2, we have: f i = 2πC i 1/2 exp µ i 2C i 2, where and µ i = y H i G i + F i x 1 1 A iz C i = H i F i P 1 1 F i + Q i H i. Then he firs order derivaive of µ i wih respec o he j-h componen in θ is θj µ i = θ j H i G i + F i x 1 1 H i θj G i + θj F i x F i θj x 1 1 θj A iz. and he firs order derivaive of C i wih respec o he j-h componen in θ is θj C i = θ j H i + H i F i P 1 1 F i + Q i H i + H i F i P 1 1 F i + Q i θj H i A.1 θj F i P 1 1 F i + F i θj P 1 1 F i + F i P 1 1 θj F i + θj Q i H i. A.2 Then he firs order derivaive of f i wih respec o he j-h componen in θ is θj f i = f i µi C i θj µ i + f i 1 2C i µ i 2 C i 1 θj C i. 31

32 Similarly, he derivaive of θj µ i respec o he k-h componen in θ is θj θk µ i = θ j θk H i G i + F i x 1 1 θj H i θk G i + θk F i x F i θk x 1 1 θk H i θj G i + θj F i x F i θj x 1 1 H i θj θk G i + θj θk F i x θj F i θk x F i θj θk x 1 1 θj θk A iz. A.3 The derivaive of θj C i respec o he k-h componen in θ is θj θk C i = θ j θk H i θj H i + θk H i + H i + θk H i + H i + H i + H i + H i + H i F i P 1 1 F i + Q i H i + θj H i F i P 1 1 F i + Q i θk H i θk F i P 1 1 F i + F i θk P 1 1 F i + F i P 1 1 θk F i + θk Q i H i F i P 1 1 F i + Q i θj H i + H i F i P 1 1 F i + Q i θj θk H i θk F i P 1 1 F i + F i θk P 1 1 F i + F i P 1 1 θk F i + θk Q i θj H i θj F i P 1 1 F i + F i θj P 1 1 F i + F i P 1 1 θj F i + θj Q i H i θj F i P 1 1 F i + F i θj P 1 1 F i + F i P 1 1 θj F i + θj Q i θk H i θj θk F i P 1 1 F i + θj F i θk P 1 1 F i + θj F i P 1 1 θk F i θk F i θj P 1 1 F i + F i θj θk P 1 1 F i + F i θj P 1 1 θk F i θk F i P 1 1 θj F i + F i θk P 1 1 θj F i + F i P 1 1 θj θk F i θj θk Q i H i. H i H i H i A.4 32

33 Then he derivaive of θj f i respec o he k-h componen in θ is θj θk f i = θk f i µi C i θk µ i f i C i µi + θk f i 1 2C i θk C i f i 2 C i + f i 1 2C i + f i 1 2C i θj µ i θk C i f i µi C i C i 2 µ i 2 C i 1 µ i C i 2 µ i µ i 2 C i θk µ i C i 1 θj µ i θj C i θj C i µ i θj θk C i. θj θk µ i 2 θk C i C i 2 θj C i B Proofs Proof of Lemma 1. The equaion 4.1 can be wrien as ξ +1 = p + ρ A B, B.1 where A = f 2 ξ 1 and B = f 1 f 2 ξ +f 2. Le - e.g. ξ denoe ha he quaniy is evaluaed a β, δ, δ. Consider Lemma 1.1. Because f 1 = f 2 =, i follows ha Ā = ξ 1 and B =. B.2 Plugging his ino B.1, we have ξ +1 = p + ρ ξ 1. This, ogeher wih 2.13, implies ξ 2 1 = p + ρ ξ = p + ρ 1 =, where he las equaliy follows from he definiion of ρ and. This can be ieraed forward, leading o ξ +1 = for all 1. 33

34 Consider Lemma 1.2. Differeniae B.1 wih respec o he j-h componen in θ, we have θj A θj ξ +1 = ρ B A } θj B B 2, B.3 where θj A = θj f 2 ξ 1 + f 2 θj ξ and θj B = θj f 1 θj f 2 ξ + f 1 f 2 θj ξ + θj f 2. Below, we evaluae he righ hand side of B.3 a β, δ, δ for wo possible siuaions: 1. If j I 0, hen θj f1 = θj f2 and f 1 = f 2 =. Consequenly θj Ā = θj f2 1 + θj ξ, θj B = θj f2. B.4 Combining B.4 wih B.2, we have θj ξ+1 = ρ θj ξ. This implies, a = 1, we have θj ξ2 1 = ρ θj ξ1 0 = ρ θj = 0. This can be ieraed forward leading o θj ξ = If j I 1 or j I 2, hen f 1 = f 2 = and θj Ā = θj f2 1 + θj ξ, θj B = θj f1 + 1 θj f2. B.5 Combining his wih B.2, we have θj f1 θj ξ+1 = ρ θj ξ 1 } θ j f2 θj f1 = ρ θj ξ + r θ j f2, where r = ρ1. Noe ha θj ξ+1 can also be wrien as 1 θj ξ+1 = r ρ s θj f1s s=0 θ j f2s. B.6 34

35 Because θj f1 θ j f2 = θjnδ f 1 θ jnδ f2, B.7 when j I 2, we have θj ξ+1 = θjnδ ξ+1. In addiion, from 2.13 and B.6, we have 1 θj ξ = 1 ρ s θj f1s s=0 = ρ θj ξ θj f1 θ j f2s θ j f2, B.8 when j I a, a = 1, 2, and θj ξ = θjnδ ξ. Consider Lemma 1.3. Differeniaing B.3 wih respec o θ k : θj θk A θj θk ξ +1 = ρ B θ j A θk B θ k A θj B A θ j θk B B 2 B 2 B 2 } + 2 A θ j B θk B, B.9 B 3 where θj θk A = θj θk f 2 ξ 1 + θj f 2 θk ξ + θk f 2 θj ξ + f 2 θj θk ξ, θj θk B = θj θk f 1 θj θk f 2 ξ + θj f 1 θj f 2 θk ξ + θk f 1 θk f 2 θj ξ + f 1 f 2 θj θk ξ + θj θk f 2. We now evaluae he righ hand side of B.9 a δ 1 = δ 2 = δ under hree possible siuaions: 1 If j I 0 and k I 0, hen f 1 = f 2 =, θj f1 = θj f2, θk f1 = θk f2, θj θk f1 = θj θk f2 and θj ξ+1 = θk ξ+1 = 0, implying θj θk Ā = θj θk f2 ξ 1+ θj θk ξ and θj θk B = θj θk f2. Combining hem wih B.4 and B.2, θj θk ξ+1 equals θj θk f2 ξ 1+ ρ θj θk ξ ξ 1 θj f2 θk f2 2 ξ 1 θk f2 θj f2 f 2 = ρ θj θk ξ. ξ 1 θj θk f2 + 2 ξ 1 θj f2 θk f2 f 2 } 35

36 Saring a = 1 and ieraing forward, we have θj θk ξ+1 = 0 for all 1. 2 If j I 0 and k I 1, hen θj f1 = θj f2 and θj ξ+1 = 0, which imply ha θj θk Ā = θj θk f2 1 + θj f2 θk ξ + θj θk ξ and θj θk B = θj θk f1 + 1 θj θk f2. Combing hese wo equaions wih B.2, B.4 and B.5, θj θk ξ+1 equals 1 ρ θj θk f2 1 + θj f2 θk ξ + f f 1 θj θk ξ θj f2 1 2 θk f1 + 1 θk f2 1 f θj f2 θk f2 1 + f 2 θk ξ ξ 1 1 f ξ θj θk f1 + 1 θj θk f f 2 θj f2 ξ θk f1 + 1 θk f2 }. The resul follows from rearranging he erms. For he case j I 0 and k I 2, we have he same resul. 3 If j I 1 and k I 1, hen θj θk Ā = θj θk f2 1 + θj f2 θk ξ + θk f2 θj ξ + θj θk ξ and θj θk B = θj θk f1 + 1 θj θk f2 + θj f1 θj f2 θk ξ + θk f1 θk f2 θj ξ. Applying he similar derivaive above, we have θj θk ξ+1 equals ρ 1 θj θk f2 1 + θj f2 θk ξ + θk f2 θj ξ + θj θk ξ 1 f 2 θj f2 1 + θj ξ ξ θk f1 + 1 θk f2 1 f 2 ξ θj f1 + 1 θj f2 θk f2 1 + θk ξ 1 1 f ξ θj θk f1 + 1 θj θk f2 + θj f1 θj f2 θk ξ + θk f1 θk f2 θj ξ f 2 ξ θj f1 + 1 θj f2 ξ θk f1 + 1 θk f2 }. The resul follows from rearranging he erms. For he cases j I 1 and k I 2, and j I 2 and k I 2, we have he same resul. 36

37 Consider Lemma 1.4. Differeniaing B.9 wih respec o θ l : θj θk θl ξ +1 = ρ θj θk θl A B θ k θl A θj B B 2 θ l A θj θk B B θ l A θj B θk B B 3 θ j θk A θl B B 2 θ k A θj θl B B 2 A θ j θk θl B B 2 + 2A θ j θl B θk B B 3 θ j θl A θk B B θ k A θj B θl B B 3 + 2A θ j θk B θl B B 3 + 2A θ j B θk θl B B 3 θ j A θk θl B B θ j A θk B θl B B 3 } 6A θ j B θk B θl B, B 4 where θj θk θl A = θj θk θl f 2 ξ 1 + θj θl f 2 θk ξ + θk θl f 2 θj ξ + θl f 2 θj θk ξ + θj θk f 2 θl ξ + θj f 2 θk θl ξ + θk f 2 θj θl ξ + f 2 θj θk θl ξ and θj θk θl B = θj θk θl f 1 θj θk θl f 2 ξ + θj θl f 1 θj θl f 2 θk ξ + θk θl f 1 θk θl f 2 θj ξ + θl f 1 θl f 2 θj θk ξ + θj θk θl f 2 + θj θk f 1 θj θk f 2 θl ξ + θj f 1 θj f 2 θk θl ξ + θk f 1 θk f 2 θj θl ξ + f 1 f 2 θj θk θl ξ. We now evaluae he above erms a δ 1 = δ 2 = δ for 4 possible cases. We only repor he values of Ē jkl, bu omi he derivaion deails. r 1 If j I 0, k I 0 and l I 0, hen Ējkl, = 0. 2 If j I 0, k I 0 and l / I 0, hen Ējkl, equals 1 1 θj θk θl f1 θj θk θl f2 θj 2 θk f2 θl f1 θl f2 1 f θk f2 2 θj θl f1 θj θl f2 1 f 2 θj f2 θk θl f1 θk θl f f 3 θj f2 θk f2 θl f1 θl f2 }. 37