Likelihood Inference for Dynamic Linear Models with Markov Switching Parameters: On the Efficiency of the Kim Filter

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1 Likelihood Inference for Dynamic Linear Models wih Markov Swiching Parameers: On he Efficiency of he Kim Filer Kyu Ho Kang Young Min Kim February 2017 Absrac The Kim filer (KF) approximaion is commonly used for he likelihood calculaion of dynamic linear models wih Markov regime-swiching parameers. Despie is populariy, is approximaion error has no been examined in a rigorous way ye. This sudy invesigaes he reliabiliy of he KF approximaion. To measure he approximaion error, we compare he oucomes of he KF mehod wih hose of he auxiliary paricle filer (APF). The APF is a numerical mehod requiring a longer compuing ime, bu is error can be sufficienly minimized by increasing simulaion size. We conduc exensive simulaion and empirical sudies for comparison purposes. According o our experimens, he likelihood values obained from he KF approximaion are pracically idenical o hose of he APF, and he KF mehod becomes more reliable when regimes are more persisen. Consequenly, he KF approximaion is a fas and efficien approach for maximum likelihood esimaion of regime-swiching dynamic linear models. This aricle conribues o he lieraure by providing evidence o jusify he use of he KF mehod. (JEL classificaion: C22, C63) Keywords: Sae space model, auxiliary paricle filer, Kalman filer, maximum likelihood esimaion Deparmen of Economics, Korea Universiy, Seoul, Souh Korea, kyuho@korea.ac.kr, Tel: Corresponding auhor, Deparmen of Economics, Korea Universiy, Seoul, Souh Korea, eccenricecon@korea.ac.kr, Tel:

2 1 Inroducion Dynamic linear models wih Markov regime-swiching parameers are widely used in empirical macroeconomics and finance because of heir flexibiliy. This flexibiliy is aribued o wo ypes of unobserved sae variables in he model: coninuous laen variables following an auoregressive process and discree laen variables governed by a firs-order Markov process. This class of models includes he regime-swiching dynamic common facor, ime-varying parameer, and unobserved componen models. These models are paricularly useful for deecing drasic regime changes in business cycles. Business cycles can be empirically characerized by a dynamic common facor among various coinciden macroeconomic variables. The asymmeric naure of expansions and conracions or volailiy reducion in he business cycles can be modeled by allowing he common facor dynamics o be subjec o regime shifs according o a Markov process. For saisical inference, maximum likelihood esimaion (MLE) is ypically employed in a classical approach. However, he exac likelihood calculaion of he Markov regimeswiching dynamic linear models is challenging. The source of he problem is ha he condiional densiy of observaions is no expressed in a closed form because he condiional densiy of coninuous laen variables depends on he enire hisory of he discree sae variables. To resolve his problem, he Kim filer (KF) approximaion was proposed in he work of Kim (1994). For convenience, we refer o coninuous and discree sae variables as facors and regimes, respecively. The key idea behind he KF mehod is o approximae he condiional densiy of he facors, so ha i depends only on he disribuion of he mos recen regimes, and no on heir enire hisory. The KF mehod makes he likelihood calculaion feasible, and i is easy o implemen pracically. Therefore, a number of sudies including Kahn and Rich (2007), Timmermann (2001), Morley and Piger (2012), Kang, Kim, and Morley (2009), and Chauve (1998) have employed his approximaion mehod for MLE of regime-swiching dynamic linear models. This work is moivaed by he fac ha he approximaion error has no been invesigaed rigorously despie he populariy of he KF mehod. The primary goal of 2

3 his sudy is o es wheher he KF mehod is efficien for he likelihood compuaion. To his end, we measure is approximaion error by comparing oucomes from he KF mehod wih hose from he auxiliary paricle filer (APF) mehod. The APF, inroduced by Pi and Shephard (1999), relies on a numerical inegraion, and is numerical error can be sufficienly minimized by increasing simulaion size. 1 As he difference beween he likelihood values from he KF and APF mehods is smaller, he KF mehod is regarded as more efficien. I should be noed ha using he APF is no desirable for MLE because his numerical mehod requires a much longer compuing ime han he KF mehod. This idea of comparing he KF and APF mehods is based on he work of Herbs and Schorfheide (2015), in which he Kalman filer and paricle filering for dynamic linear models wihou regime shifs are compared. We conduc exensive simulaion and empirical sudies for he comparison. In he simulaion sudy, he likelihood inference of he mehods of dynamic common facor, ime-varying parameer, and unobserved componen models wih Markov swiching parameers is evaluaed in erms of accuracy and compuing ime given he simulaed daa. The models considered in he empirical applicaion are Friedman s plucking model and a business cycle urning poins model. According o our experimens, he likelihood values obained from he KF approximaion are almos idenical o hose of he APF mehod, which is robus o model specificaion and sample size. Because he KF algorihm is simpler, and easier o implemen in pracice han he APF mehod, he KF approximaion is a fas and efficien approach for MLE. Our work conribues o he lieraure on regime-swiching dynamic linear models by providing evidence o jusify he use of he KF mehod. However, i should be noed ha he performance of he KF mehod ends o be less reliable as regime shifs are more frequen. By a Mone Carlo simulaion, we show ha he KF approximaion error has a larger variance when regimes are less persisen. The res of his paper is organized as follows. In Secion 2, we presen a general 1 Many sudies on a Bayesian framework have applied he APF for he likelihood compuaion, given he model parameers. For insance, Chib, Nardari, and Shephard (2002) developed he APF for a sochasic volailiy model ha can be expressed in a sae-space model wih exogenous and independen swiching parameers. Recenly, Kang (2014) exended his APF algorihm for dynamic linear models wih endogenous Markov regime-swiching parameers. 3

4 specificaion of our dynamic linear models wih Markov swiching parameers considered in his paper. Secion 3 describes he KF and APF mehods for likelihood inference. Secions 4 and 5 provide he resuls of KF approximaion error evaluaions based on various simulaion and empirical applicaions. The conclusion is presened in Secion 6. 2 Model Framework We consider a join sochasic process, (y, β, s ) where y is a q 1 vecor of observable dependen variables and β is a k 1 vecor of coninuous sae variables a ime. s is a discree laen regime-indicaor aking one of he values {1, 2,..., N} where N is he number of regimes. The sae variables, (β, s ), are unobserved in general. s = j indicaes ha he dependen and sae variables are drawn from he jh regime a ime. Assume ha F 1 denoes he hisory of he observaion sequence up o ime 1 and x is a h 1 vecor of he predeermined or sricly exogenous variables. We hen assume ha a each ime, s, β, and y are sequenially, no simulaneously, generaed from he following dynamic linear model wih regime-swiching parameers: s P Markov(s 0, P), (1) β β 1, Θ, s N (µ s + G s β 1, Q s ), (2) y F 1, Θ, β, s N (H s β + F s x, R s ) (3) where s 0 is he iniial regime a ime 0, N (, ) denoes he mulivariae normal disribuion, and Θ = {H s, F s, R s, µ s, G s, Q s } N s =1 is he se of model parameers in he ransiion equaion (2) and measuremen equaion (3). For generaliy, we allow all parameers o change over ime depending on he regime. The regime s is governed by a firs-order finie-sae Markov process wih a ransiion probabiliy marix, P = p 11 p 12 p 1N p 21 p 22 p 2N......, (4) p N1 p N2 p NN 4

5 where p ij = Pr[s = j s 1 = i] is he ransiion probabiliy from regime s 1 = i o regime s = j, and N j=1 p ij = 1 for all i = 1, 2,.., N. Wihou loss of generaliy, we assume ha he ransiion probabiliies are consan over ime. Once regime s is generaed given s 1 independenly of he hisory of he facors and endogenous variables, {β i, y i } 1 =0, β follows a firs-order vecor-auoregressive (VAR(1)) process in equaion (2). The regime-dependen inercep, VAR coefficiens, and facor shock variance-covariance are denoed by µ s : k 1, G s : k k, and Q s : k k, respecively. Condiioned on he facors and regimes a ime, he dependen variable y is normally disribued as in equaion (3). The marginal impac of he facors on he dependen variable is capured by H s : q k. F s is a q h coefficien marix of x, and R s is a q q variance-covariance marix of he measuremen errors. Our model specificaion is compleed by iniializing he sae variable and regime, (β 0, s 0 ) a ime 0. Firs, given he ransiion probabiliy marix P, he iniial regime is assumed o be generaed from is uncondiional disribuion if he Markov-regime process is ergodic. Tha is, s 0 = j is drawn wih he seady-sae probabiliy, Pr [s 0 = 1] Pr [s 0 = 2] ] where. Pr [s 0 = N] = ( P P ) 1 P 1 N [ 0N 1 1 [ ] IN P P = : (N + 1) N. For N = 2, he uncondiional probabiliies are simply obained as : N 1 (5) 1 p 22 Pr[s 0 = 1] =, (6) 2 p 11 p 22 Pr[s 0 = 2] = 1 Pr[s 0 = 1]. If all regimes are ransien as in a changepoin process, hen s 0 is fixed a one. Finally, given he iniial regime s 0, he iniial facors are generaed differenly, depending on wheher he facor process is saionary or nonsaionary. When i is saion- 5

6 ary, β 0 is assumed o be disribued as is uncondiional disribuion of β condiioned on he iniial sae s 0 = j as β 0 Θ,s 0 = j N ( β j 0, P j 0 ) (7) where β j 0 is a k 1 uncondiional mean vecor and P j 0 is a k k uncondiional variancecovariance marix of β. Suppose ha I k denoes he k k ideniy marix, is he Kronecker produc, and vec( ) is he vecorizaion of a marix. ( β 0, j P j 0 ) can hen be derived as β j 0 = (I k G s0 =j) 1 µ s0 =j, P j 0 = (I k G s0 =j G s0 =j) 1 vec(q s0 =j), respecively. On he oher hand, if β is nonsaionary, β 0 = β j 0 is reaed as an addiional regime-independen parameer o be esimaed. As β 0 is no longer a random variable, is variance-covariance marix, P0 = P j 0 should be fixed a a k k zero marix. 3 Likelihood Inference If he regimes are non-sochasic and observable, he sae-space model presened in equaions (2) and (3) is linear and Gaussian, and he Kalman filer is applicable for he exac likelihood calculaion. The unobservabiliy of regimes, however, makes he ransiion equaion nonlinear and he likelihood compuaion analyically infeasible via he Kalman filer. To elaborae on he source of infeasibiliy, le Y = {y } T =1 denoe a sequence of observaions. Condiioned on (F 1, Θ, P), by inegraing ou he curren coninuous and discree sae variables (s, β ), he log likelihood log f(y Θ, P) can be obained by he sum of he log likelihood densiies over ime: f(y F 1, Θ, P) = f (y s, β, F 1, Θ, P) p (s, β F 1, Θ, P) d (s, β ), (8) and log f(y Θ, P) = T log f(y F 1, Θ, P). (9) =1 6

7 The condiional densiy of (s, β ) is hen compued by inegraing over he enire hisory of he sae variables {(s i, β i )} 1 i=0 because he facor process is regime-dependen and p (s, β F 1, Θ, P) = p (s, β F 1, Θ, P,s 1, β 1 ) p (s 1, β 1 F 1, Θ, P) d (s 1, β 1 ) = p (β F 1, Θ, P,s, s 1, β 1 ) p (s P,s 1 ) p (s 1, β 1 F 1, Θ, P) d (s 1, β 1 ), p (s 1, β 1 F 1, Θ, P) = p (β 1 F 1, Θ, P,s 1, s 2, β 2 ) f(y F 2, Θ, P) p (s 1 P,s 2 ) p (s 2, β 2 F 2, Θ, P) d (s 2, β 2 ), p (s 2, β 2 F 2, Θ, P) = p (β 2 F 2, Θ, P,s 2, s 3, β 2 ) f(y 2 F 3, Θ, P) p (s 2 P,s 3 ) p (s 3, β 3 F 3, Θ, P) d (s 3, β 3 ),. As a resul, he condiional densiy of he observaions y, f(y F 1, Θ, P) should be obained by inegraing over he enire hisory of he sae variables up o ime. Because of his high dimensionaliy, he exac likelihood calculaion is no feasible in general. To overcome he compuaional difficuly, we can consider wo approaches ha rely on differen approximaion mehods for he likelihood inference. One approach is he Kim filer, which is he approximae Kalman filer algorihm. The key idea of his mehod is o collapse N N possible regimes ino N possible regimes a each filering recursion by approximaion. Consequenly, he condiional densiy of he facor β depends on (s, s 1, β 1 ), no ({s i } i=0, {β i } 1 i=0 ) a each ieraion of he recursions. The oher approach is a sequenial Mone Carlo mehod, which is widely used for he likelihood inference of nonlinear sae space models. As one of he sequenial Mone Carlo algorihms, he APF is implemened recursively o updae he disribuion of (s, β ) condiional on he curren informaion a ime using a se of paricles wih appropriae 7

8 weighs. The following subsecions provide a deailed discussion of he KF and APF mehods. 3.1 Kim Filer The KF algorihm consiss of hree seps: he condiional Kalman filer, Hamilon filer, and approximaion. Given (Y, Θ, P), he log likelihood ln f (Y Θ, P) is iniialized a zero. A ime = 0, he iniial unobserved sae variables disribuion, p (s 0, β 0 Θ, P), is given a he uncondiional probabiliy discussed in Secion 2. For = 1, 2,..., T, he deails of he KF algorihm seps for calculaing he likelihood funcion are presened as follows. Algorihm 1: Likelihood Inference from he Kim Filer Sep 1: (Condiional Kalman Filer) The objecive of he condiional Kalman filer sep is o obain he inference of β condiional on (s 1 = i, s = j) and informaion up o ime. By running he Kalman filers for each pair of (s, s 1 ), he necessary quaniies are recursively compued as E[β F 1, Θ,s = j, s 1 = i] = β (i,j) 1 = µ j + G j β i 1 1, (10) V ar[β F 1, Θ,s = j, s 1 = i] = P (i,j) 1 = G jp i 1 1G j + Q j, (11) E[y F 1, Θ,s = j, s 1 = i] = y (i,j) 1 = y H j β (i,j) 1 F jx, (12) V ar[y F 1, Θ,s = j, s 1 = i] = f (i,j) 1 = H jp (i,j) 1 H j + R j, (13) E[β F, Θ,s = j, s 1 = i] = β (i,j) V ar[β F, Θ,s = j, s 1 = i] = P (i,j) = β (i,j) 1 + K(i,j) = P (i,j) 1 K(i,j) (y y (i,j) ), (14) 1 H jp (i,j) 1 (15) where β i 1 1 = E[β F 1, Θ,s 1 = i] and P i 1 1 = V ar[β F 1, Θ,s 1 = i] are he condiional expecaion and variance-covariance marix of β given (F 1, Θ, s 1 = i), respecively. K (i,j) s 1 = i for i, j = 1, 2,..., N, which is given by is he Kalman gain condiioned on s = j and K (i,j) = P (i,j) 1 H j(f (i,j) 1 ) 1. 8

9 Sep 2: (Hamilon Filer) In his sep, we apply he Hamilon filer (Hamilon (1989)) o obain he likelihood densiy (i.e., f(y F 1, Θ, P)) and he filered probabiliies (i.e., f(s, s 1 F, Θ, P) and f(s F, Θ, P)). Given f(s 1 F 1, Θ, P), he following quaniies can be sequenially compued: f(s, s 1 F 1, Θ, P) = Pr[s s 1 ]f(s 1 F 1, Θ, P), (16) f(y F 1, Θ, P) = f(y F 1, Θ,s, s 1 )f(s, s 1 F 1, Θ, P), s s 1 (17) f(s, s 1 F, Θ, P) = f(y F 1, Θ,s, s 1 )f(s, s 1 F 1, Θ, P), f(y F 1, Θ, P) (18) f(s F, Θ, P) = s 1 f(s, s 1 F, Θ, P), (19) where he condiional densiy of y f(y F 1, Θ,s = j, s 1 = i) = N ( ) y y (i,j) 1, f (i,j) 1 can be compued using equaions (12) and (13). Equaion (16) is he predicive probabiliy of he regimes and equaions (18) and (19) are he filered probabiliies. Sep 3: (Numerical Error) The filered probabiliies are used in he nex sep when inegraing ou he lagged regimes o obain he filered facors condiioned on he curren regime only. If he filered probabiliies are oo close o zero, hese values could be aken o be zero and he KF does no work because he filered probabiliies are used as denominaors in he nex sep. To avoid his numerical problem, we replace he filered probabiliies by 10 4 whenever hey are less han his value. Sep 4: (Approximaion) In his sep we derive he filered disribuion of he facor β condiioned on he curren regime s, β F, Θ, P,s. Given he normaliy assumpion and he filered probabiliies, he KF algorihm provides he condiional expecaion and variance-covariance of he facors as fol- 9

10 lows: 2 β j = E[β F, Θ, P,s = j] M i=1 = f(s = j, s 1 = i F, Θ, P)β (i,j), (20) f(s = j F, Θ, P) P j = V ar[β F, Θ, P,s = j] M i=1 = f(s = j, s 1 = i F, Θ, P)[P (i,j) + (β j β(i,j) )(β j β(i,j) ) ]. (21) f(s = j F, Θ, P) Noe ha he filered densiy p(β F, Θ, P,s = j) is a mixure of N = N N 1 saes where N 1 is he oal number of regime sequences up o ime 1. This sep is referred o as he approximaion sage because (β j, P j ) are calculaed by inegraing over s 1, no {s i } 1 i=0. These filered values are necessary when running he condiional kalman filer sep he following ime. Sep 5: If < T, se = + 1 and go o Sep 1; oherwise go o Sep 6. Sep 6: (Log likelihood) Reurn he log likelihood ln f (Y Θ, P) = T ln f(y F 1, Θ, P) (22) =1 where he condiional densiy f(y F 1, Θ, P) is calculaed in equaion (17). 3.2 Auxiliary Paricle Filer The essenial sep in he likelihood inference is updaing he condiional disribuion of he regimes and facors given he curren informaion a ime. The APF is a class of simulaion-based mehods ha recursively updae he informaion conained in he curren observaions y o he predicive disribuion, (s, β ) F 1, Θ, P. This filering procedure uses he sampling imporance resampling mehod, which consiss of wo sages. The firs sage is o propose candidae values of he sae variables. The second sage is o reweigh hem o produce draws from he arge disribuion, p (s, β F, Θ, P). Per he Bayes heorem, he arge densiy is proporional o he 2 For he deails of he derivaion, refer o Kim and Nelson (1999b). 10

11 produc of he condiional densiy of y and he predicive densiy of (s, β ).: p (s, β F, Θ, P) f (y s, β, F 1, Θ, P) p (s, β F 1, Θ, P). The key sep in he APF mehod is o inroduce an auxiliary variable, denoed by g, where g indicaes he condiional disribuion of (s, β ) given he gh samples (s (g) 1, β (g) 1) and (F 1, Θ, P). The APF algorihm enables us o sample from s, β, g F, Θ, P, and reurn he filered disribuion, s, β F, Θ, P. Supose ha he M samples {s (g) 1, β (g) 1} M g=1 drawn from s 1, β 1 F 1, Θ, P are given, and ha (ŝ (g) (g), ˆβ ) comprise he mode of he ransiion disribuion of (s, β ), s, β s (g) 1, β (g) 1, F 1, Θ, P condiioned on (s (g) 1, β (g) 1, F 1, Θ, P) for each g = 1, 2,.., M. The firs sage is o propose R values for g generaed by a proposal disribuion whose mass funcion is given by ( q (g F, Θ, P) f y ŝ (g), In he second sage, we draw {s (r) ) (g) ˆβ, F 1, Θ, P., β (r) } R r=1 from he predicive disribuion, s, β F 1, Θ, P. Then, for each r = 1, 2,.., R, we calculae he second imporance weighs, ( ) f y s (r), β (r), F 1, Θ, P w r ( ) (23) f y ŝ (gr) (gr), ˆβ, F 1, Θ, P where {g r } R r=1 is a se of indices sampled from he M draws of g. Finally, we resample M draws for (s, β ) wih he second imporance weighs from {s (r), β (r) } R r=1. The re- 11

12 sampled M draws are aken as he samples from he filered disribuion, s, β F, Θ, P. Given (Y, Θ, P) and (M, R), he iniial paricles {s (g) 0, β (g) 0 } M g=1 are drawn from s 0, β 0 Θ, P, which is he uncondiional probabiliy as we discussed in Secion 2. For each ime poin, he seps of he APF algorihm can be summarized as follows. Algorihm 2: Auxiliary Paricle Filer Sep 1: Obain he mode of he proposal disribuion ( ŝ, ˆβ ), compue he firs sage weighs, and sample auxiliary variables. Sep 1-(a): Given P, sample {s 1, (g) β 1} (g) M g=1 from s 1, β 1 F 1, Θ, P, and {û (g) } M g=1 Unif(0, 1). Then, deermine ŝ (g) ŝ (g) ŝ (g) as = 1 if û (g) < p 1, (24) = 2 if û (g) p 1 where p 1 = p 11 if s (g) 1 = 1, and p 1 = p 21 if s (g) 1 = 2. Sep 1-(b) Given Θ, {ŝ (g), β 1} (g) M g=1, and (y, x ), calculae ˆβ (g) [ = E β β (g) 1, ŝ (g) ], Θ = µŝ(g) ˆβ (g) as + Gŝ(g) β 1, (g) (25) Sep 1-(c) Compue he firs sage weighs, w g N q (y Hŝ(g) ˆβ (g) + Fŝ(g) x, Rŝ(g) ) (26) Sep 1-(d): Draw {g r } R r=1 proporional o from he indices g = 1,..., M wih he probabiliy mass w g = w g M g=1 w, (27) g and associae sampled index wih {s (gr) 1, β (gr) 1 } R r=1 and {ŝ (gr), (gr) ˆβ } R r=1. Sep 2: Generae s, β from he ransiion densiies, compue he second sage weighs, and resample ( s, β ). 12

13 Sep 2-(a): Given {s (gr) 1} R r=1, P, and {u (r) } R r=1 unif(0, 1), sample s (r) s (r) = 1 if u (r) < p 1, (28) = 2 if u (r) p 1 where p 1 = p 11 if s (gr) 1 = 1, and p 1 = p 21 if s (gr) 1 = 2. Sep 2-(b): Given {s (r), β (gr) 1 } R r=1 and Θ, simulae β (r) N k (µ s (r) Sep 2-(c): Compue he second imporance weighs, w r N q (y H (r) s β (r) N q (y Hŝ(gr) ) + G (r) s β (gr) 1, Q (r) s. (29) ˆβ (gr) + F s (r) x, R s (r) + Fŝ(gr) x, Rŝ(gr) ) ). (30) Sep 2-(d): Resample M imes from he R draws wih he probabiliy mass, w r = w r R r=1 w. (31) r The resampled draws are he filered samples {s (g), β (g) } M g=1 from s, β F, Θ, P. Given he filered samples {s (g) 1, β (g) 1} M g=1 from s 1, β 1 F 1, Θ, P, i is sraighforward o sample he predicive draws, {s (g), β (g) } M g=1 from s, β F 1, Θ, P. Once he predicive draws for (s, β ) are simulaed, we can compue he one-sep-ahead condiional densiy of y, f (y F 1, Θ, P) = f (y s, β, F 1, Θ, P) p (s, β F 1, Θ, P) d (s, β ) (32) by using he simple Mone Carlo averaging of N q (y H s β + F s x, R s ) over he predicive draws. 13

14 Given (Y, Θ, P), he log likelihood ln f (Y Θ, P) is iniialized a zero. A ime = 0, iniial paricles {s (g) 0, β (g) 0 } M g=1 are drawn from he uncondiional probabiliy as discussed in Secion 2. For = 1, 2,.., T, he APF algorihm seps are summarized as follows. Algorihm 3: Likelihood Inference using he Auxiliary Paricle Filer Sep 1: Simulae he predicive densiy of (s, β ) and calculae he likelihood densiy. Sep 1-(a): Given {s (g) drawn from 1} M g=1, {u (g) } M g=1 Unif(0, 1), and P, predicive values of s are s (g) s (g) = 1 if u (g) < p 1, (33) = 2 if u (g) p 1 where p 1 = p 11 if s (g) 1 = 1, and p 1 = p 21 if s (g) 1 = 2. Sep 1-(b): Condiioned on {s (g), β 1} (g) M g=1 and Θ, draw predicive values of β from β (g) N k (µ s (g) ) + G (g) s β 1, (g) Q (g) s (34) Sep 1-(c): Compue he one-sep-ahead predicive densiy of y by a numerical inegraion as f (y F 1, Θ, P) 1 M M g=1 N q (y H (g) s β (g) + F (g) s x, R (g) s ). (35) Sep 2: Apply Algorihm 2 and sample {s (g) = + 1 and go o Sep 1; oherwise go o Sep 3. Sep 3: Reurn he log likelihood,, β (g) } M g=1 from s, β F, Θ, P. If < T, se ln f (Y Θ, P) = T ln f (y F 1, Θ, P). (36) =1 Noe ha given he parameers we can obain he filered probabiliies of he regimes and filered facors over ime by averaging {s (g) Sep 2. 14, β (g) } M g=1 drawn from s, β F, Θ, P in

15 4 Simulaion Sudy The KF approximaion mehod is evaluaed based on simulaion sudies and empirical applicaions in comparison wih he APF mehod. We begin by measuring he approximaion error of he KF mehod using simulaed daa. We consider hree represenaive examples of dynamic linear models wih Markov swiching parameers, which can be expressed as special cases of he general model specificaion in Secion Models Dynamic Common Facor Model wih Markov Swiching Parameers The firs specificaion is a dynamic common facor model wih Markov swiching parameers. This model specificaion is widely used o empirically characerize business cycles by a dynamic facor β among muliple observed variables y. The dynamic facor is subjec o regime shifs modeled by a Markov process s. A ime, (s, β, y ) are sequenially, no simulaneously, generaed from he following generic dynamic linear model wih regime-swiching parameers: s P Markov(s 0, P), (37) β β 1, Θ, s N (G s β 1, Q s ), y F 1, Θ, β, s N (H s β, R s ). The ransiion marix for regime s is given by [ p P = 11 1 p 11 1 p 22 p 22 ]. Throughou Secions 4 and 5, we assume ha he regime s is governed by a wo-sae firs-order Markov process, in which all regimes are recurren. 3 For daa simulaion, bivariae observaions y are generaed by one common facor 3 In he esimaion of regime-swiching models, regime idenificaion is an imporan issue. However, he focus of his paper is he likelihood inference given he parameers. For his reason we do no discuss he label swiching problem. 15

16 (i.e., k = 1). Our choice of paramerizaion is given by s = 1 s = 2 G s Q s 1 3 H s ( ) ( ) R s 1 I 2 4 I 2 In his paramerizaion, he common facor process is regime-dependen. Paricularly, he facor is more persisen and volaile in regime 2. Furher, he facor loadings and measuremen shock variance differ in differen regimes Time-Varying Parameer Model wih Markov Swiching Parameers Consider he case in which here is a possibiliy of gradual changes in linear regression coefficiens over ime along wih regime shifs in he coefficien process or volailiy. This can be easily modeled by a ime-varying parameer model wih Markov swiching parameers as follows: β β 1, Θ, s N (β 1, Q s ), (38) y F 1, Θ, β, s N (H β, R s ) where H Unif(0, 2) is a covariae uncorrelaed wih he measuremen errors. We fix he measuremen error variance and parameer shock variance such ha Q s=1 = R s=1 = 1, Q s=2 = 5, and R s=2 = 3. As he variances in regime 2 are larger han in regime 1, he simulaed daa is supposed o reveal more drasic coefficien changes and higher volailiy during he period corresponding o he second regime Unobserved Componen Model wih Markov Swiching Parameers In an unobserved componen model he observed ime series daa is decomposed ino he sochasic rend and cyclical componens. The rend componen is ypically assumed o be a random walk process, and he cyclical componen follows a saionary vecor 16

17 auoregressive process. We consider a simple specificaion: β β 1, Θ, s N (µ s + G s β 1, Q s ), y F 1, Θ, β, s N (β, R s ) where y is univariae and assumed o be he sum of a firs-order auoregressive process and a whie noise componen. These unobserved componens are idenified as he difference in persisence. We se he parameers as presened below, so ha he dependen variable is more persisen and volaile under regime Likelihood Comparison s = 1 s = 2 µ s 2 1 G s Q s 1 4 R s 1 2 For each model, we simulae he ime series of dependen variables and facors given he rue parameers. We consider five differen sample sizes (T = 80, 100, 200, 400, and 800) o see wheher he size of he KF approximaion error is robus o sample size. The firs and hird 25% of samples are drawn under regime 1 and he ohers are drawn under regime 2. For insance, in he case of T = 80, he firs and hird 20 observaions and facors are simulaed by he parameers under regime 1, and he ohers under regime 2. Using he simulaed daa, we run he KF and APF algorihms once o compue and compare he likelihood values a he rue parameers. For he APF, resampling and paricle sizes are given by R = 50, 000 and M = 50, 000, respecively. The likelihood values obained from he APF wih such large simulaion sizes are accurae enough o be regarded as he rue likelihood values. Table 1 presens he resuls of he likelihood compuaions. There are hree imporan findings in his able. Firs, he log likelihoods approximaed by he KF mehod are almos idenical o hose of he APF. The average approximaion error is almos zero and he maximum absolue error is 0.47 in his experimen. The KF mehod does no seem o lead o a misleading likelihood raio es. Second, he performance of he KF mehod 17

18 Table 1: Log Likelihood (ln L) and Compuing Time (Time) Comparison Beween he KF and APF mehods DCF, TVP, and UC indicae he dynamic common facor model, ime-varying parameer model, and unobserved componen model, respecively. The compuing ime is in seconds. (a) DCF APF KF ln L Time ln L Time T = T = T = T = T = (b) TVP APF KF ln L Time ln L Time T = T = T = T = T = (c) UC APF KF ln L Time ln L Time T = T = T = T = T =

19 is robus o he model specificaions and sample sizes as shown in he able. Third, he compuing speed of he KF mehod is incomparable. The APF akes hundreds of seconds whereas he KF complees wihin one second. In addiion, Figure 1 plos he log condiional densiies of he observaions over ime, and clearly indicaes ha he log likelihood densiies calculaed by he KF mehod are indisinguishable from hose of he APF mehod. Consequenly, hose simulaion sudies srongly demonsrae ha he KF mehod is fas and efficien enough o be used for likelihood inference. The KF mehod is a more desirable approach for likelihood funcion opimizaion han he APF mehod. Anoher imporan elemen of approximaion qualiy is he accuracy of he filered saes and filered probabiliies of regimes. We repor he quaniies of he filered saes and regime probabiliies obained from a single run of he KF and APF in Figures 2 and 3. These figures depic he sequence of he filered facors and filered probabiliies of he regimes along wih he corresponding rue values in he case of T =200, as an example. There is lile difference beween he filered values produced by he KF and APF, which is a finding robus o he sample sizes. 4.3 Regime Persisence and Approximaion Error We now examine he sensiiviy of he KF approximaion error o regime persisence. I is imporan o invesigae wheher he KF mehod successfully approximaes he hisorical pah of he regimes regardless of regime shif frequency. To do his, we repea he simulaion in Secion imes wih a sample size of 80. For each simulaion, he regime process is generaed according o a Markov process wih he following ransiion marices: P H = [ ] and P L = [ P H and P L indicae high and low regime persisence, respecively. As a resul, he regime shifs generaed by P L occur more frequenly han hose generaed by P H. For each of he 500 simulaed daa ses, we run he KF and APF and sore he approximaion error, ln f (Y Θ, P) KF ln f (Y Θ, P) AP F, P {P H, P L }. ] 19

20 log-likelihood log-likelihood log-likelihood Figure 1: Log Likelihood Densiies This figure plos he log likelihood densiies over ime. DCF, TVP, and UC indicae he dynamic common facor model, ime-varying parameer model, and unobserved componen model, respecively. The condiional densiies from he KF and APF mehods are he black dashed line and blue doed line, respecively KF APF Time (a) DCF 0-1 KF APF Time (b) TVP 0 KF APF Time (c) UC 20

21 Figure 2: Filered Facors This figure plos he filered facors over ime for he DCF, TVP, and UC models. The rue facors, facors filered by he KF, and facors filered by he APF are he black dashed line, blue solid line, and red doed line, respecively. 4 3 True KF APF (a) DCF 6 True KF APF (b) TVP True KF APF (c) UC 21

22 Figure 3: Filered Regimes This figure plos he filered regimes over ime for he DCF, TVP, and UC models. The rue regimes, regime probabiliies filered by he KF, and regime probabiliies filered by he APF are he black dashed line, blue solid line, and red doed line, respecively. 1 True KF APF (a) DCF 1 True KF APF (b) TVP 1 True KF APF (c) UC 22

23 Figure 4 plos he empirical kernel densiies of he 500 approximaion errors, and Table 2 repors is sample mean and sandard deviaion. For all models, empirical densiies are cenered around zero regardless of he regime persisence. However, he KF mehod seems o be more reliable when he regime is more persisen, because he error densiy wih P H is much sharper han ha wih P L. This is aribued o he fac ha approximaing he enire regime pah by he recen regimes is difficul when he regime shifs are frequen and he se of poenial regime pahs wih a high mass is large. In shor, he performance of he KF mehod becomes uncerain when regime persisence is low, alhough i is robus o he model specificaion and sample size. Table 2: Summary Saisics for Approximaion Error This able presens he mean and sandard deviaion (s.d.) of likelihood discrepancy beween he KF and APF. P H P L mean s.d. mean s.d. DCF TVP UC Empirical Applicaions Regarding empirical sudies, we consider wo popular regime-swiching dynamic linear models for business cycle analysis: Friedman s plucking model (Kim and Nelson (1999a)) and he business cycle urning poins model (Kim and Nelson (1998)). 5.1 Friedman s Plucking Model Milon Friedman s plucking model is one of he popular business cycle models o explain he naure of recessions. The mos disinguishing feaure of his model is ha business cycles are exremely asymmeric. The ime-pah of he oupu can only move along a poenial oupu process. However, he oupu is plucked downwards a irregular inervals. The plucking shocks are emporary demand shocks, whereas he flucuaions of he poenial oupu are driven by supply shocks. From his model s poin of view, recessions are caused by uncommon negaive and emporary shocks such as financial crises 23

24 Figure 4: Approximaion Error Kernel Densiy This figure plos he kernel densiy esimaes of likelihood discrepancy based on he KF and APF. The approximaion error densiies corresponding o high and low regime persisence are he solid and dashed lines, respecively p = 0.8 p = Approximaion Error (a) DCF p = 0.8 p = Approximaion Error (b) TVP p = 0.8 p = Approximaion Error (c) UC 24

25 and oil shocks. Kim and Nelson (1999a) propose an economeric model incorporaing he asymmeric behavior of business cycles. Following Kim and Nelson (1999a), we decompose he log of real GDP y ino a ransiory componen, x, and rend componen, z : y = x + z (39) where x = δ s + φ 1 x 1 + φ 2 x 2 + u, u i.i.d.n (0, σ 2 u,s ), (40) z = g + z 1 + v, v i.i.d.n (0, σ 2 v). (41) To idenify asymmeric business cycle dynamics, we impose resricions as follows: δ s=1 < 0, δ s=2 = 0, σ 2 u,s =2 = 0. The business cycle corresponds o ransiory deviaions in he oupu process away from is permanen componen. Given ha he economy says in regime 1, he ransiory componen has a negaive uncondiional mean because δ s=1 is consrained o be negaive. The period s = 1 can be inerpreed as he sae of he plucked-down economy. Meanwhile, in regime 2, he oupu flucuaes according o he sochasic rend only, no he business cycle componen, because he ransiory shock variance, σu,s 2 =2, is equal o zero. φ 1 and φ 2 deermine he persisence of he recession componen x. g is a deerminisic drif in rend. The plucking model does no conain measuremen error, bu he measuremen equaion should include measuremen error o apply he APF. To resolve his problem, we should reexpress he model as anoher sae-space form wih measuremen error. To do his, we eliminae he rend componen by muliplying boh sides of equaion (39) by (1 L), y = g + x x 1 + y 1 + v, u i.i.d.n (0, σv) 2 (42) where L is a lag operaor and x = δ s + φ 1 x 1 + φ 2 x 2 + u, u i.i.d.n (0, σ 2 u,s ). (43) 25

26 For he likelihood inference, he resuling sae-space form given he regime is expressed by β β 1, Θ, s N (µ s + G s β 1, Q s ), y F 1, Θ, β, s N (Hβ + F x, R s ) where β = [ x x 1 ], µ s = [ δ s 0 ], G s = [ φ1 φ ], Q s = diag( σ 2 u,s 0 ), and H = [ 1 1 ], F = [ g 1 ], x = [ 1 y 1 ], R s = σ 2 v. We now esimae he model using he log of quarerly real GDP daa for he Unied Saes ranging from 1951:Q1 o 2016:Q1. 4 This model is esimaed by he MLE where he likelihoods are compued by he KF mehod and he log likelihood funcions are numerically maximized. To reduce he local-maxima problem, we follow he approach of Chib and Ramamurhy (2010) and use a simulaed annealing algorihm combined wih a Newon-Raphson mehod. This numerical opimizer is especially useful in cases where he likelihood is high-dimensional and irregular. Once he likelihood is maximized, he likelihood is compued a he ML esimaes using he KF and APF mehods. MLE resuls for he model parameers and likelihood values a he esimaes are repored in Table 3. The log likelihood from he KF mehod is and ha from he APF mehod is The approximaion error is measured a 0.28, so he likelihood values calculaed by he KF and APF mehods are almos equal. Moreover, he esimaion resuls are convincing. Figures 5 and 6 plo he esimaed sochasic rend and asymmeric business cycles of he log real GDP. These esimaes are he filered values of unobserved componens (i.e., x and z ). During he normal period, he economy is subjec mosly o permanen shocks, whereas he ransiory componen plays an imporan role in he oupu flucuaions during recession. On he oher hand, Figure 7 presens he filered 4 The daa come from he FRED daabase a he Federal Reserve Bank of S. Louis. They are in billions of chained 2009 USD wih a seasonally adjused annual rae. 26

27 Table 3: Parameer Esimaes: Friedman s Plucking Model This able presens he parameer esimaes, he log likelihoods (ln L), and he compuing ime (Time) in seconds. Parameer Esimaes value g δ φ φ σv σu, p p ln L APF KF Time APF KF 0.06 probabiliies of recession, Pr[s = 1 F ], along wih he NBER indicaor. The filered probabiliies are srongly correlaed wih he NBER business cycle daes, and hisorical recessions seem o be precisely deeced by his plucking model. 5.2 Business Cycle Turning Poins Model The business cycle urning poins model used in he work of Kim and Nelson (1998) encompasses he wo key feaures of business cycles: one common facor among macroeconomic observaions conaining informaion on he sae of he economy and he regimeswiching dynamics of he common facor. To deal wih hese characerisics, he model considers a dynamic common facor of economic variables and urning poins of he business cycle. Assume ha y i represens he firs difference of he naural log of he i-h observed coinciden indicaors a ime. The observaions are assumed o be generaed by one common facor, denoed by c, and heir idiosyncraic componens are as follows: y i = γ i c + e i, i = 1, 2, 3, 4. (44) 27

28 Figure 5: The log Real GDP and Esimaes of he Permanen Componen This figure plos he log U.S. real GDP (doed, blue) along wih is sochasic rend (dashed, black) Trend log Real GDP Time The common facor c and error erm e i are assumed o follow an AR(1) process. c = δ s + φc 1 + v, v i.i.d.n (0, σ 2 v), (45) e i = ψ i e i, 1 + ɛ i, ɛ i i.i.d.n (0, σ 2 i ), i = 1, 2, 3, 4 (46) where he inercep erm in he facor process, δ s, is subjec o regime shifs, φ deermines he persisence of he common facor, σv 2 is he variance of he facor shock, ψ i capures he persisence of each shock, and σi 2 is he idiosyncraic shock variance. We also assume ha v and ɛ i are muually independen. For he facor idenificaion, he firs facor loading of he common facor is aken o be uniy(i.e., γ 1 = 1). The dynamic common facor is exraced by he only source of comovemens among he observed macroeconomic variables, so he facor can be inerpreed as he business cycle. By allowing he inercep erm δ s o swich beween he wo regimes, we can deec he urning poins of he business cycles. As in he plucking model, we muliply boh sides of equaion (44) by (1 ψ i L), and we can eliminae he serial correlaion of he idiosyncraic componen for consideraion 28

29 Figure 6: Transiory Componen of he log Real GDP This figure plos he cyclical componen of he log U.S. real GDP Cycle Componen Time of he measuremen error in he measuremen equaion, y i = ψ i y i, 1 + γ i c ψ i c 1 + ɛ i, i = 1, 2, 3, 4. (47) For he likelihood inference, he resuling sae-space represenaion is obained as β β 1, Θ, s N (µ s + G s β 1, Q s ), (48) y F 1, Θ, β, s N (Hβ + F x, R s ) where β = [ c c 1 ], µ s = [ δ s 0 ], G s = [ φ ], Q s = diag[ σ 2 v 0 ], 29

30 Figure 7: Probabiliies of Recessions: Friedman s Plucking Model This figure plos he filered probabiliies of recession (dashed, blue) along wih he NBER recession indicaor (doed, black) over ime NBER Recession Probabiliies of Recession Time and [ H = 1 γ 2 γ 3 γ 4 ψ 1 ψ 2 γ 2 ψ 3 γ 3 ψ 4 γ 4 F = diag[ ψ 1 ψ 2 ψ 3 ψ 4 ], ], x = [ y 1, 1 y 2, 1 y 3, 1 y 4, 1 ], R s = diag[ σ 2 1 σ 2 2 σ 2 3 σ 2 4 ]. The daa used for esimaion perain o U.S. monhly indusrial producion, personal income less ransfer, manufacuring and rade sales, and non-farm employee daa. The ime span is January 1967 o Augus Table 4 summarizes he esimaes of he model parameers and likelihood values from he KF and APF mehods. Figures 8 and 9 plo he filered common facor process and he filered probabiliy of recession over ime, respecively. These recession probabiliies are in close agreemen wih he NBER recession indicaor. In addiion, he asymmeric naure of business cycle phases is capured because expansions display a longer duraion on average han recessions. 30

31 Table 4: Parameer Esimaes: Business Cycle Turning Poins Model This able presens he parameer esimaes, he log likelihoods (ln L), and he compuing ime (Time) in seconds. Parameer Esimaes value δ δ φ σv ψ ψ ψ ψ γ γ γ σ σ σ σ p p ln L APF KF Time APF KF 0.37 According o Table 4, he KF approximaion error in his applicaion is around 0.37, which is very small, similar o he experimen wih he plucking model. This finding also srongly suppors he reliabiliy of he KF approximaion approach. We can conclude ha, alhough he KF approximaion does no inegrae he enire hisory of he regimes, his mehod produces an accurae likelihood value ha is close enough o he exac likelihood value. 6 Conclusion In his paper, we evaluae he KF approximaion mehod for he likelihood inference of dynamic linear models wih Markov swiching parameers. Is approximaion error is measured by he difference beween he likelihood values obained by he KF and APF 31

32 Figure 8: New Coinciden Index This figure plos he new coinciden index exraced from he esimaion of he dynamic common facor model wih Markov swiching parameers Baseline New Coinciden Index Time mehods, given ha he likelihood from he APF can be rendered sufficienly close o he exac likelihood value obained when using a large simulaion size. According o he likelihood comparison based on our exensive simulaion and empirical sudies, he KF approximaion error is sufficienly small, paricularly when he regimes are persisen, and he performance is robus o he sample size and model specificaions. Therefore, he KF mehod is a fas and efficien likelihood inference approach for dynamic linear models wih Markov-regime swiching parameers. We believe ha our work jusifies he use of he KF mehod in he exising lieraure on MLE of he models. The KF approximaion can be useful for he Bayesian poserior simulaion as well. Typically, model parameers are sampled from heir full condiional disribuions, given regimes and facors. Because of he dependency on he regimes and facors, he parameer sampling efficiency is no high, especially when he models are highly nonlinear o parameers. By using he KF mehod, we can opimize he likelihood or poserior densiy inegraing over he laen variables, and obain a proposal disribuion ha successfully approximaes he poserior disribuion. Consequenly, we may achieve a 32

33 Figure 9: Probabiliies of Recession: Business Cycle Turning Poins Model This figure plos he filered probabiliy of recession (dashed, blue) along wih he NBER recession indicaor (doed, black) over ime NBER Recession Probabiliies of Recession Time high efficiency of poserior sampling. We will consider he applicaion of he KF mehod in he Bayesian poserior sampling in fuure work. 33

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