Markov switching in stochastic variance. Bayesian comparison of two simple models

Vol. X LIX -L (2008-2009) PL ISSN 0071-674X

MARKOV SWITCHING IN STOCHASTIC VARIANCE.

BAYESIAN COMPARISON OF TWO SIMPLE MODELS

ŁUKASZ KWIATKOWSKI1

Katedra Metod Statystycznych Krakowska Akademia im. Andrzeja Frycza Modrzewskiego

PL 30-705 Kraków, ul. Gustawa Herlinga-Grudzińskiego 1

e-mail: kiuintkoiuski.lukas@gmnil.com

Praca przedstaw iona na posiedzeniu K om isji N auk E konom icznych i Statystyki O d działu PAN w K rakow ie 13 stycznia 2009 roku przez autora.

ABSTRACT

Ł. Kwiatkowski. Markov Switching in Stochastic Variance. Bayesian Comparison o f Two Simple Models.

Folia Oeconomica Cracoviensia 2008-2009, 49-50: 109-143.

In the paper tw o particular M arkov Sw itching Stochastic Volatility m od els (M SSV ) are under consideration: one w ith a sw itching intercep t in the Iog-volatility equation, and the oth er — with a regim e-dependent autoregression param eter. W hile the form er one is fairly com m on in the literature (as a tool of taking accoun t for regim es of d ifferent m ean v olatility level), the latter has not been paid alm ost any attention so far. We note the fact, that state-varyin g m ean v olatility m ay arise from sw itches in the intercept or in the autoregression param eter. Hence, w e aim to com pare these tw o m odels in respect of goodness o f fit to the data from the Polish financial m arket, em ploying B ayesian techniques of estim ation and m od el com p a­ rison. C lear evidence of structural shifts in the volatility p attern is found. Tw o different re­ gimes of the econ om y are characterized in term s of the m ean volatility level and the v arian ­ ce of volatility.

KEY WORDS — SŁOWA KLUCZOWE

M arkov sw itching, Stochastic Volatility m odel, B ayesian inference przełączanie typu M arkow a, m odel zm ienności stochastycznej (SV), w nioskow anie

b ay e so w sk ie

1 Assistant at The Andrzej Frycz Modrzewski Krakow University College, Departm ent of Sta­ tistics. The author is deeply indebted to Prof. ]. Osiewalski and Dr. A. Pajor for their invaluable help and advice.

1. INTRODUCTION

Recent years have w itnessed a rapid increase in the interest in m odelling time- varying volatility of financial tim e series. A m ong the m ost p o p u lar tools de­ vised to capture som e of its com m on features have been tw o param etric model families: the ARCH processes, intro d u ced by Engle (1982) (along w ith their num ero u s extensions starting w ith GARCH process of Bollerslev, 1986), and the stochastic volatility (SV) processes, of w hose the m ain concept has been presented by Clark (1973). A lthough in formally different ways, in both of the above conditional variance equation is defined explicitly.

The und erly ing assum ption of these constructions is hom ogeneity of the m o d elled tim e series, w hich m eans exclusion of potential structural breaks occurring in the analyzed period. It allows one to presum e that the parameters of interest rem ain constant over tim e. H ow ever, volatility clustering, a com­ m on p hen om eno n observed in stock retu rn s series, m ay question this belief. It is so d u e to som e heuristic reasoning that less volatile periods alternating w ith these of hig h er un certain ty m ay som ehow co rresp o n d w ith structural breaks occurring in the data. In view of potential heterogeneity of a certain tim e series, m odels such as GARCH or SV are of too restrictive nature (Hwang et al., 2004). N ot being able to capture discrete shifts of states of the economy m ay be the cause for these m odels to yield som ew hat m isleading results. For instance, G rang er an d H y u n g (1999) an d D iebold an d Inoue (2001) suggest th at structu ral breaks in the m ean of volatility m ay be a source of volatility persistence. It follows that a p rop er m odel should include an explicit m echa­ nism capable of accounting for possible regim e changes. One of the m ost popular in this respect is M arkov sw itching (MS) m echanism introduced by Ham ilton (1989). W hat he suggested is an autoregressive process w hose param eters are subject to changes over tim e according to a latent hom ogeneous M arkov chain. Since then m an y studies have been undertaken to em ploy the idea of MS into volatility m odels, m ainly those of the GARCH family (see Bauwens et al., 2006, am ong m any).

The aim of the p ap e r is Bayesian estim ation and com parison (in terms of goodness of fit to the data) three SV specifications: a non-sw itching basic sto­ chastic volatility (BSV) m odel and tw o M arkov Switching SV (MSSV) models (one w ith a regim e-changing intercept and the other w ith a sw itching autore­ gression p a ra m e te r in the volatility equation). The d ata se t com prises daily ob se rv a tio n s on the 1-m onth W arsaw In te rb an k O ffered Rate (WIBOR1M) interest rates over the period from A pril 17, 2000 to April 7, 2008. Incidental to the analysis of the regim e-sw itching constructions is a search for potential structural shifts occurring in the series a n d — if any are found — character­ ization of the identified states of the economy.

There are several reasons b eh in d o u r research. Firstly, em ploying n o n ­ sw itching m odels in view of potential structural breaks in the tim e series m ay

lead to a m odel misspecification error. In this regard, sw itching specifications, like the MSSV processes, m ay be of value as they account for discrete shifts in the param eters. Secondly, w e note that ab rup t changes in the m ean volatility level, which is the reason w idely cited in the literature for em ploying the MSSV structures, m ay be attributed not only to the sw itching intercept but, alterna­ tively, to the regim e-changing autoregression param eter in the volatility eq u a­ tion. Finally, neither the issue above nor the MSSV m odels w ith a sw itching elasticity of volatility2 are tackled in the literature kno w n to the a u th o r.3

As reg a rd s the cu rre n t state of the litera tu re on the MSSV m o d els, in a predom inant p art of the studies only two-state specifications w ith a sw itch­ ing intercept are concerned (Smith, 2000; Kalimipali an d Susmel, 2001; Casarin, 2003; Shibata an d W atanabe, 2005; C arv alh o a n d Lopes, 2006). T hree-state m odels are analyzed in So et al. (1998) and H w ang et al. (2003, 2004). In term s of the estim ation tools the Bayesian approach prevails, w ith use of either stan­ d a rd MCMC procedures (the Gibbs sam pler; So et al., 1998; K alim ipali an d Susmel, 2001, Shibata and Watanabe, 2005) or m ore recent (auxiliary) particle filters (Casarin, 2003; Carvalho an d Lopes, 2006). Some of the m odels feature additional elem ents such as term structure (Smith, 2000; Kalim ipali an d Sus­ mel, 2001) and heavy-tailed distributions of the noise term in the observable process (Casarin, 2003).

We conduct the analysis in the Bayesian setting, w hich allows fully p ro b ­ abilistic inference on all the unknow n quantities of the m odel as w ell as well- founded m odel com parison. As o pposed to the 'classical' (i.e. non-Bayesian) tools, Bayesian m eth o d o lo g y in the context of sw itch in g m o d els (or, m o re generally, m ixture m odels) is found even m ore app ealin g . The latter stem s from the possibility of inference on the latent regim es unconditionally upo n the p aram eter estim ates (see G artner, 2007).

The rem ainder of the p aper is organized as follows. In Section 2 w e present the m odels un d er consideration and selected regim e characteristics, of w hich use is m ade in the further parts. Bayesian estim ation of the m odels and their comparison are briefly discussed in Section 3, followed by an em pirical illus­ tration of the presented m ethodology in Section 4. Finally, Section 5 concludes.

2 'Elasticity of volatility' is the term used by Smith (2000) with reference to the autoregression parameter, <p, in the log-volatility equation of a SV model given as: ln/i, = n + tplnh,^ + at],. Assu­ ming 7], ~ » 0 ( 0 ,1 ) (i.e. each T], is an independent and identically distributed random variable with zero mean and unit variance) the third parameter, a, is a standard deviation of the innovation term

arh, and hence referred to as 'volatility of volatility'.

3 The only works in which the autoregression parameter is allowed to switch over the regimes are of Hwang et al. (2003, 2004). However, not only the estimation approach employed in these studies (Quasi-ML), but also the specification of the log-volatility equation is different than in our work.

2. SELECTED MARKOV SWITCHING SV (MSSV) MODELS

In this p a rt the basics of selected MSSV processes are presented. We start with the follow ing definition of a general M arkov Sw itching SV process.

A stochastic p ro cess4 [yt, t e N u{0)} follow s a tw o -state M arkov Switching Stochastic Volatility (MSSV) process if an d only if for each t e N u {0} the fol­ low ing assum p tion s hold:

{Sf/ t e N u{0}} — a hom ogenous, ergodic and irreducible two-state

Mar-The observable variable, y(, is defined as a p ro d u ct of a G aussian white noise and conditional standard deviation.5 Equation 2 defines the log-volatil- ity w hich evolves over tim e according to a sim ple sw itching autoregressive process. Since all of the param eters in the latter feature regim e-changing pro p ­ erty, the definition m ay be view ed quite general, although further extensions are possible (a heavy-tailed distribution for et can be considered, for instance, as in C asarin, 2003). The sw itching m echanism , represented by the family of discrete ran d o m variables S('s, is assum ed to follow a sim ple two-state Markov chain, in accordance w ith the idea p roposed by H am ilton (1989). For the sake of ou r study, ergodicity and irreducibility of the chain are assum ed by restrict­ ing the transition probabilities, p-, to lay strictly w ithin the un it interval.

O ne should note, th at a basic stochastic volatility process (BSV), w ith the log-volatility equation defined as:

4 By N we denote the set of positive integers.

5 It is straightforward to show that — conditionally upon a a-algebra with respect to which h, is measurable — the latter constitutes conditional variance of the process {y(), i.e: Var(yt IF , //(, S() =

ht, where FM is the past information about the process (/i( , f e N u (0(( up to the moment f-1. D e f i n i t i o n 1 V t = £ t y f i t ) \nh t = n Si +(pSi \nht_ i + a Sirjt i

(

1

)

(

2

)

ln/i( - ju + p]nht_i + crrjt,

m ay be view ed as a particular case of the general MSSV process once fdx - //2,

(px = (p2 and CTj = <J2 hold. However, the transition probabilities, p-, rem ain then

unidentified.

In ou r w ork tw o special cases of the general MSSV process are of particu­ lar interest: the one with a switching intercept and the other — w ith a regim e- d ep e n d en t autoregression p aram eter. A concise discussion of b o th follow s.6

2.1. M S S V m o d e l w i t h a s w i t c h i n g i n t e r c e p t , M S S V ( ^ )

In this case Equation 2 collapses to:

, , , , \V--i + + cm, <=> St = 1

\nh, = jus +(p]nh,_1 +077, = < . (3)

f HS‘ * ‘ 1 " \ v 2 +(p\nht_1 + oilt « St = 2 w For the sake of identifiability of the m odel w e reparam etrize the sw itching param eter as (see So et al., 1998):

t*Sl = n + r j i s t = 2)'

w here yx e R, y2 c 0 and I(.) denotes the indicator function w hich takes one if the condition in the parentheses is satisfied and zero otherwise. Such a rep re­ sentation of the sw itching intercept results in inequality /j.y > /ur It m ay be show n that the latter is equivalent to predeterm ining states V an d '2' as ones of high and low m ean log-volatility level, respectively, th at is:

/ /, > f i 2 <=> E(ln/tf I St = 1) > E(ln/tf I S( = 2) .

For the m odel in question w e shall also assum e covariance stationarity of the log-volatility process following Equation 3, for w hich it is necessary an d sufficient7 to guarantee that I cp I < 1.

2.2. M S S V m o d e l w i t h a s w i t c h i n g a u t o r e g r e s s i o n p a r a m e t e r , M S S V ( ^ )

In our stu d y w e note that discrete shifts in the m ean volatility level m ay result from not only a switching intercept, b u t — alternatively — a regim e-changing autoregression param eter. Hence, w e consider a MSSV m odel w ith Equation

2 assum ing the form:

In ht = n + (ps \nht._! + OTit, (4)

6 For a comprehensive work on non-switching SV models we refer to Pajor (2003).

7 General results on second-order and strict stationarity of switching vector autoregression processes may be found in Francq and Zakolan (2001).

log-volatility

Fig. 1. Simulated path of a MSSV(^) process (// = -2 .5 ; tp} = 0.2; <p2 = 0.5; a 2 = 0.6132; pu = 0.98;

p22 = 0.95) and the corresponding log-volatility and regim e-switching processes

For the identifiability reasons w e shall im pose the inequality (pA < <pT Once form ulas for conditional expectations E(ln/z( I Sf = 1) (i - 1, 2) are available (see the follow ing subsection), it is easily sh o w n that:

H < 0 => [<pi < q>2 <=> £(lnht I St = 1) > E(\nht I St = 2)] an d

/1 > 0 => [(p\ < ( p 2 <=> E ( \ n h t I S t = 1) < E ( \ n h t I S t = 2 ) ] ,

w hich indicates different m ean volatility levels in each of the regimes. F urther, w e assu m e covariance statio n a rity of the log-volatility process following Equation 4. The relevant (necessary an d sufficient) condition is giv­ en b y the set of inequalities:8

p i <1

\ R 2 < 2, w here:

K l = Pll<Pl + P 2292 + ( 1 - P l l - P22)<Pl<P2 /

R2 = pn (pi + P22P2 •

8 The condition is also valid for the general case, in which all the three volatility parameters are allowed regime shifts (see Francq and Zakoian, 2001). One should note that the condition is some­ w hat contrary to an initial 'intuition' according to which it should be 'enough' to assume that I (ft I < 1 and I (fa I < 1 . The latter constitutes neither a necessary nor a sufficient condition for second-order stationarity of a two-state switching first-order autoregression (ibid.).

Figure 1 depicts a sim ulated p a th of a certain MSSV($>) process a n d the corresponding regim e-sw itching an d (stationary) log-volatility processes. The latter displays evid en t shifts in the m ean level (according to the sw itch in g m echanism ), w hich m anifest them selves in the form of volatility clustering.

2.3. S e l e c t e d r e g i m e c h a r a c t e r i s t i c s

While allowing different states of the economy, it is n atural to characterize the regimes in som e system atic way. In our w ork w e do so by calculating selected regime-specific characteristics both of the log-volatility process a n d the sw itching

m echanism as well, including: ,

— state-conditional m ean log-volatility level:9

E1 3 E(\nht I Sf = 1) = --- ^ l l - _(p2p22) + fi2(p]( l - p n )---^ I - M 2 ~ V2P220-~ <Pl) ~ flPuO--<P2) ’

E2 s E (ln ^ I S t = 2 ) = - ■ ^ ( 1 - W l l ) + ^1<P2( 1 - P 2 2 ) .

1 - (p{(p2 ~ <P2V220- - 9 l ) - № l l ( l - 9 2 ) — state-conditional variance of the log-volatility process:

Vi s Var(\nht I St = i) = E(ln2 ht I St = i) - E? for i = 1, 2, !

w here: and E (ln 2 h t I S t = l ) = d 1Q .-(p % p 2 2 ) + d 2< p f ( l - p n ) ^ 1 - < P l P l l - 9 2 .V 2 2 + ( P l f 2 ( ~ ^ J r P n + P22) , p fln2 h K = ?V = ^2(1-<Pi2P i i ) + d l q > l Q . - p 22) \ t t / - 2 2 2 2/ i \ ' 1- W l l - ^ 2P22+ W 2 ( - 1 + P l l + P22) di = n f + 2 fi i(piE { \n h t_ 1 I S t = i) + o f f o r i = 1 , 2 , E(Infif_1 I St = i) = i ?)-,-E(ln/!, I S, = j) . /=1

w ith py; = = i I Sj = / ) being the inverse transition probabilities;10

9 First- and second-order state-conditional moments of the log-volatility process have been obtained for the general case (that is the one in which all three parameters are regim e-changing) under assumption of covariance stationarity of that process and based on the results of Nielsen

and Olesen (2000). , •

10 In the case o f a two-state M arkov chain the inverse transition probabilities defined as p* = Pr(SM = i \ S l = j ) are easily shown to equal the ordinary transition probabilities, = Pr^S, =

.11

— ergodic probabilities:

7C\ -Pr(S{ =1) = -- 1~ ^ 22 ,

2 ~ P n~ P 2 2 n 2 - Pr(Sf = 2) = 1 - K \ )

— expected duratio n12 of state T (once the system has sw itched to that state; see H am ilton, 1989):

Dun = — -— for i = 1, 2. 1 - Pa

3. BAYESIAN ESTIMATION AN D COMPARISON

OF THE MSSV MODELS13

Estim ation o f the MSSV m odels is n o t trivial. H andling the m axim um likeli­ h o o d pro ced u re is rid d le d w ith serious num erical obstacles d u e to the p res­ ence of (as m u ch as) tw o latent processes underlying the observable process: the conditional volatilities, ht's, a n d the states, St's. In o u r w ork w e resort to Bayesian m ethodology, w hich prevails in the MSSV literature.14 A lthough new m eth o d s — based on the (auxiliary) particle filters — have been developed of recent,15 w e em ploy the already 'classical' MCMC procedures: the Gibbs sam ­ pler a n d the M etropolis-H astings algorithm , to sim ulate from the joint poste­ rior distrib u tio n of all the u n k n o w n quantities of the m odel.

Let y = (j/j, y z, y T)' den o te the m o d elled tim e series, vector h = (hv h2, ..., h T)' b e th e se rie s of th e la te n t c o n d itio n a l v o la tilitie s a n d v e c to r

S = (Sj, S2, ..., ST)' — the unobserved M arkov chain. We define the param eter vector as 0= (/?', <j2, pn , p22y w ith a 2, pn an d p22 being the param eters comm on to b o th MSSV m odels, an d p com prising the m odel-specific param eters:

0 _ [(Y vY i'V Y for MSSV(n)

\{li,<pi,(p2)' for MSSV(<p)'

Further, w e em ploy the data-augm entation technique introduced by Tan­ n er a n d W ong (1987), w ithin w hich all the u n know n quantities of the m odel

11 Ergodic probabilities calculated for an ergodic Markov chain tell us approximately for how long (in terms of a part of the analyzed time series) the chain remains in each of its states. In the study we assume that pj;- e (0,1) for i = 1, 2, which ensures ergodicity of the switching process.

12 Expected duration of a certain regime is calculated conditionally upon being in that state. 13 In the paper w e discuss only the estimation of the switching SV models. For a detailed description of the Bayesian estimation of simple SV constructions we refer to Pajor (2003).

14 For the relevant references see Section 1. 15 D itto.

are treated as random variables subject to estim ation an d taking values in the com m on space:

c o ' = ( 0 ' , h ' , S ' ) e n = 0 x H x Q T,

w here 0 e <9c R 6, h e H c 7?J, S e QT an d Q = {1, 2}. , The joint p osterior distribu tio n of co is factorized a s:16

p ( 9 , h , S l j ) o c p ( y \ h ) p ( h \ S , e ) p ( S \ 0 ) p ( 0 ) t ( 5 ) w hich reveals its hierarchical structure. Individual com ponents of (5) are p re ­ sented in the A ppendix. Here, w e focus on the p rio r structure of the p a ra m ­ eters, for (almost) all of w hich m u tu al independence is assum ed:

p (0 ) = | p ( j8)p(<72)p (P i1)p (P22) for MSSV(u), '

\ p ( P \ P n ’P2 2) P ( ° 2)P(Pu)P(P2 2) for MSSV(<p).

Conditioning on the transition probabilities in p(J3\pn , p ^ ) for the m odel w ith a sw itching autoregression param eter stem s from im posing p rio r restric­ tions of covariance statio n arity of th e u n d e rly in g log-volatility process (see

Section 2.2). . , ■

In the stu d y w e choose fairly diffuse priors, letting the posterior results arise m ainly from the inform ation contained in the data. M ore specifically, w e

have: .. . , - ; ■ ■

1. prior distributions for the param eters com m on to both MSSV m o d els:17

— p(cr2) = f IG(cr2 1 vv v2), v x = 1, v2 = 200; — p(Pii) = U p,s 1 av b) r ai = bi = i / for 1 2;

w here f B(Pjj I «,■, bj) denotes the density function of a Beta-distributed ra n d o m variable, pu, w ith the shape an d scale param eters equal a; an d fy, respectively; , 2. p rio r distribution for /}:■

— for the MSSV(//) model:

pifi) = (J3\J30, A Q~1)I(y2 < 0)7(1 ^1 < 1), J3Q = 0(3xl), A0 = 0,01 • 73, where I A 0_1) denotes the density function of a norm ally distributed fc-variate ran d o m variable, /3, w ith the m ean vector an d the precision m atrix equal f3Q and A 0, respectively;

16 The analysis is conducted conditionally on h0 = 1, dependence on w hich is omitted in the notation.

17 We parametrize the density of the inverse gamma distribution as:

p ( e 2) = f , c ( v 2 ^ , v 2) = |- 2y , -e x p

v 2a 2

— for the MSSV(^) m odel: M

VWVXV P2 2) = / N(3) V ) r<Ri < ° № < OX f i 0 = o0xl), A0 = 0,01 • V

As reg ard s p rio r distributions for the param eters of the BSV m odel, we follow the stru ctu re em ployed in Pajor (2003), namely:

— p ( a 2) = f lG{a 2 \vv v2), vl = 1, v2 = 200;

— P(M,<P) = / n(2)( A <P10(2xl), A 0~')I(\<p\ < 1 ), A 0 = 0,01 • I3.

The prior structure presented above provides very convenient (in terms of the sam pling m ethod) conditional posterior distributions of the m odel param ­ eters.18 The latter are em ployed to construct a hybrid chain w ithin the MCMC procedure, thro u g h w hich a N-sized sam ple from the joint posterior distribu­ tion is obtained, {<D(<7)} _^+i , w here q denotes the num ber of the cycle of the sam pling algorithm , of w hich the first M cycles are discarded, an d a № signi­ fies the outcom e on co from the q-th step. Once the algorithm is complete, it is straig h tfo rw ard to obtain also a sam ple of any m easurable function of co, such as regim e characteristics, in particular. '

In o rd er to allow Bayesian m odel com parison, the m arginal likelihood for each of the estim ated m odels needs to be evaluated. In our w ork w e resort to the procedure intro d uced by N ew ton an d Raftery (1994), in w hich the q uan­ tity of interest is estim ated as:

' '' n-1 '

1 M .+N 1

I

N q = M + 1 p ( y I , M j )

u(i)

w here *(y I M ;) is the estim ator of the m arginal likelihood in the z-th m odel, M ;. D espite its lam entable num erical instability, the m eth o d p roved satisfactory in o u r applications. Finally, to com pare the m odels pair-w ise use is m ad e of Bayes factors, Btj, d efin ed as:

E p ( y \ M j )

p ( y I M j )

18 Full details on the posterior structure of all the estimated quantities are found in the Ap­ pendix.

4. EMPIRICAL STUDY 4.1. D a t a d e s c r i p t i o n

The m eth odology p rese n ted above is' illu strate d w ith an em pirical s tu d y in w hich data from the Polish financial m arket is analyzed. M ore specifically, w e consider a series of daily quotations of the 1-m onth W arsaw Interbank Offered Rate (WIBOR1M) interest rates over the p e rio d from A pril 17, 2000 to A pril 7, 2008 (which m akes the total of 2002 observations). The series is p lo tted in Figure 2.

W IBO R1M interest rates (2000.04.17-2008.04.07) ^ . 20 15 10 s. 3 X CO ft £ o No o 8 CM CM o ino 03p 8 CM § 8CM 8CM 8CM co r— If) _{a &} T" ^ h" T“ O O p T -iO si in lO (M (O O ) M O O O T­ CO CO CO CO CM CM CM CO CO CO 8 S 8 8 S 8 8 8 S S 8 S 8 _ _______ ___ _____________ *■ “ • CM CM CM CM CM CM CM CM CM CM CM CM • CM ■ CM ' CM • CM CM CM CM s N COCM COo N- o CM N I**» 8 CM 8CM 8CM 8CM

Fig. 2. The series o f W IBOR1M interest rates, wt

We calculate the daily lo g -retu m s, rt's, on the WIBOR1M in terest rates, d efined as:

rt = 100 ]n(wt/zvtA),

w here wt denotes the price of th e in stru m e n t at tim e t. The series of r ’s is presented in Figure 3.

Log-retums on W ILBO R1M (2000.04.18-2008.04.07)

Further, w e a p p ly a sim ple linear filter — a first-order autoregressive model — to the d a ta as to account for possibly non-zero conditional m ean of the data g enerating p ro cess.19 H enceforth, the analysis is co n d u cted for the resulting series,of. A R (l)-residuals2P (see Fig. 4), d e n o te d as y t w ith t = 1, 2, ..., T - 2000. The latter display features com m only found in financial data, including vola­ tility, clustering, high value of the em pirical kurtosis coefficient (see Tab. 1), no

; AR(1)-residuals for W ILBO R1M (2000.04.19-2008.04.07)

Fig. 4. The A R(l)-residuals for the daily log-retum s on the W IBOR1M Descriptive statistics for A R(l)-residuals for W IBOR1M

Table 1

M in M ax M ean Stand, deviation Asymmetry Kurtosis ARCH(2) effect -7.2629 6.5844 0.0000. ’ 0.8394 -0.4339 18.5818 TR2= 177.1391 (p-value = 0.0000) 1.4 1.2 1:0 0.8 0.6 0.4 0.2 0 . ' ... ...frfffff ■ ■ ■■tlTbl-ri mn... l ololo lolo lol olo lolo N S C M S N . N S N S N T-; C q C D C O ^ T— C O C Q C O t - ; ...o r c\j c*j i/i

Fig. 5. Empirical distribution o f the series\ y t, t = 1, 2, ..., T] with fitted normal density (left) and the autocorrelation function of the series and its square (right)

19 Estimation of the AR(1) m odel for the series of the log-retum s, r 's , yields the results:

r = -0.0464 + 0.1537 r . + u t

(0 .0 1 8 8 ). (0.0221) '

20 A n alternative approach is to sim ulate the param eters of the conditional mean modelled with an AR(1) process from their conditional posterior distributions. However, we expect that au­ tocorrelations in the log-retum s have little impact on the volatility and, hence, adopt the method used by So et alv 1998. , , ;

significant autocorrelations in the original series, yet strong au to -d e p en d e n ­ cies in the sq u a re d series (see Fig. 5). A d dition ally , left a sy m m e try in the em pirical distribution of the residuals is found (see Tab. 1).

4.2. R e s u l t s f o r t h e B a s i c S V m o d e l

For the estim ation of the BSV m odel w e em ployed the Gibbs sam p ler com ­ bined w ith the M etropolis-H astings step for sam pling the laten t conditional volatilities, ht's, as done in Pajor (2003). The first M - 500,000 b u m t-in itera­ tions are discarded and the subsequent N = 1,500,000 iterations are reg ard ed as a sim ulated sam ple from the joint posterior density.

Table 2 contains posterio r m eans an d sta n d a rd d eviations of the m o d el param eters. One notes the posterior m ean of the autoregression param eter, <p, being fairly close to unit. It is a com m on finding in the SV literature (see Pajor, 2003, am ong m any), indicating evident persistence in the conditional volatility process. Despite prior independency betw een the param eters w e observe strong posterior correlations (see Tab. 3). The latter m ay arise as a result of 'stabili­ z ation' of the unconditional characteristics of the volatility process, su ch as m ean an d variance.

Table 2 Posterior means and standard deviations (in parentheses) of the BSV param eters

<P a 2

-0.3384 0.8269 1.1053

(0.0508) (0.0215) (0.1285)

Table 3 Posterior correlation matrix of the BSV parameters

Mi _{M V} yCT2'

A 1 0.8753 -0.6590

: <p 1 -0.7185 ‘

a 2 1

In F igure 6 the m arg in al p rio r an d , p o ste rio r d istrib u tio n s of th e BSV param eters along w ith the plots of their ergodic m eans (against the n u m b er of cycles) are depicted. The results of posterior densities being of reg u lar shapes and fast convergence of the ergodic m eans co n vergence to th e ir p o s te rio r counterparts rem ain consistent w ith Pajor (2003).

4 3 2 1 O-l pO i y. m,) ntff f e r -0.20 -0.22 -0.24 -0.26 -0.28 -0.30 -0.32 -0.34 -0.36 -0.38 -0.40 0 , 0 © lO If) lO r*» CM N. CM T- ■ o in CM T ? ? ? ' *? ? ? <?<? ? ? 0 0 CM . CO o o m o) Tt o cm 4 p{a2\y, M,) 1.30 1.25 1.20 1.15 1.10 1.05 1.00 CM ( D r I f )

Fig. 6. Left column: marginal prior (solid line) and posterior distributions of the BSV parameters; right column: ergodic m eans o f the parameters against the number of cycles

4.3. R e s u l t s f o r t h e M S S V ( / / ) m o d e l

To estimate the m odel w e em ploy the sam pling algorithm presented in the A p ­ p endix.21 The first M = 2,000,000 burn-in iterations are discarded a n d the su b ­ sequent N = 1,500,000 iterations constitute a sim ulated sam ple from the joint p o sterior density.

As it can be gathered from the posterior m eans of the transition probabil­ ities located very close to u n it (see Tab. 4), the sw itching m echanism m anifests strong persistence. Once a certain state is achieved, little is the probability of a sw itch to the alternative regim e. Furtherm ore, one notes significantly n eg ­ ative p osterior m ean of y 2, w hich p rov id es com pelling evidence of discrete shifts in the value of the intercept. As com pared w ith the results for the BSV m odel, the m ean posterior of the elasticity of volatility is m ark ed ly lower. It is the m ost com m on finding cited in the MSSV literature, w here it is arg u ed that structural shifts unaccounted for b y stan d ard SV: m odels m ay im ply sp u ­ riously high persistence in the volatility process. How ever, w e w o u ld n o t jum p to such conclusions, unless the true autocorrelation functions of the log-vola- tility process in the BSV an d MSSV m odel are su rv ey ed.22 O ne m ay p resum e, that the very sam e 'spuriously' high volatility persistence im plied b y the n o n ­ switching SV specification m ay be captured by the sw itching counterpart, yet in a different m an n er (resulting, for instance, in the close-to-unit m ean p o ste­ rior probabilities pti, i = 1, 2). The issue m erits fu rth er research.

Table 4 Posterior means and standard deviations of the parameters of m odel M 2

P n P22 ft . . <p , ,

0.9 9 6 0 0 .9 9 6 4 - 0 .2 7 5 3 - 0 .7 2 9 2 0 .6 6 5 8 1 .3 5 9 4 (0.0027) (0 .0026) (0 .0647) (0 .1 0 5 1 ) (0.0 3 6 0 ) (0 .1 4 3 7 )

According to the posterior correlation m atrix of the param eters (see Tab. 5), prior assum ption of their m u tu al independence seems to be overruled by the data. In o u r belief, the non-zero p osterio r correlation coefficients m ay arise from 'stabilization' of the regim e characteristics as well as the u n conditional m om ents of the log-volatility process.

21 We note that the minimum acceptance rate while sampling h,'s via the M -H algorithm, amounted to approximately 60%, which is found much satisfying.

22 For the purpose of comparison of volatility persistence implied by the BSV and both M SSV models, we averaged posterior empirical autocorrelation function (ACF) coefficients (lags: 1 to 15) for the sampled series of ln/i/s . The results appear not to reject the individual hypotheses of equal mean ACF coefficients across different models, therefore advocating the conjecture to follow in the main text.

Table 5 Posterior correlation matrix of the param eters of model M 2

M 2 •: ■ P ll P 22 7 i 72 " <P o 2 . - f . ; P u 0 .3 9 6 6 - 0 .0 0 6 2 0 .1 9 8 6 0 .1 2 9 2 0.0 0 3 5 P 2 2 _{. 1} 0 .1 5 2 4 0 .2 2 3 2 , 0 .1 5 2 3 0 .0 0 7 8 1 : .■ ■■ ■ / i _{1 1 0 .1 2 6 2 0 .5 2 7 3 -0 .3 9 2 1 : ; i . :} 7 2 • ’ ! ■ ' . ,' 1 : 0 .8 0 1 2 -0 .4 8 7 1 ’■ , 9 : 1 - 0 .6 6 4 7 cr

M arginal p o sterior densities of the transition probabilities pü concentrate tightly: o n the left of u n it (seeF ig. 7), w hich indicates th at the analyzed dataset is very inform ative w ith reg ard to the sw itching m echanism .

Posterior-distri-p(p„|y, M2) P i P j y . M2) 200 180 160 140 120 100 80 60 40 20 0 CO ID CO <J> OCO O 250 200 150 1 0 0 -50 P(P2il/> M2) p(p22\y' M2) 04 l l l M i 'nrrrnn iTTrrni imim im m 250 200 150 100 50 CO CO CO CO CO CO h- © , . h*» . T- Tf O T- r- T- CM CM CO _ 00 T- CM CO .0 ll fTTl'ini irrrrn O) o n r t d

Fig. 7. M arginal prior (solid line) and posterior distributions of the transition probabilities in model M 2

butions of the rem aining p aram eters are clearly u n im o d al a n d cluster (w ith slight asym m etries) a ro u n d their m eans (see Fig. 8). P rior covariance station- arity of the log-volatility process is n o t rejected b y the data, as the posterior density of the autoregression param eter, (p, clusters aw ay on the left of unit.

P (r,ly . M2) P (r2ly. M2) CD 1C

? 7

m o i 111111111111 in in m in t : ^ S i n o co in t - o> CM t- o o o 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5

0

p(<p,\y,M2) p (o2 \ y , M 2) CN h-LO ' Tfr T CO

Fig. 8. M arginal prior (solid line) and posterior distributions of the log-volatility param eters in m odel M„

A som ew hat unstable b ehav io ur of the ergodic m eans of the p aram eters (except for the transition probabilities) m ay raise concerns as regards conver­ gence of the MCMC procedure (see Fig. 9). H ow ever, one sh o u ld note rath er negligible m ag n itu d e of the visible fluctuations.

Pn 0.9985 0.9980 0.9975 0.9970 0.9965 0.9960 0.9955 0.9950 0.9945 1.000 0.999 0.998 0.997 0.996 0.995 0.994 -*■ o © © o © © o o o © T' © © o © o © o o o o o o o o o. o o o o o o o

cm“

_s'

co" co‘ o’ CM

_s'

co“ CO* o" CM

_s'

co“ co“ o" cm"

CM CO •o CO h- CM ^r CM CO m ,r" CM lO h- CO O) CM CM m £ CO CO h-CM -0,23 -0.24 -0.25 -0.26 -0.27 -0.28 -0.29 -0.30 -0.31 -0.32 CD CM CO O CO CM CO CO t- in CO CO CO Q) O CO CT> 0,69 0.68 0.67 0.66 0.65 0.64 0.63 0.62 CO CM CO O CO CM CO CO T- m CO_{CO CO o? o} CM CO 05

Fig. 9. Ergodic m eans o f the parameters of m odel M2 against the number of cycles

149

6

,0

0

4.4. R e s u l t s f o r t h e M S S V ( ^j) m o d e l

As in the previous case, the quantities of interest are sam pled w ithin the MCMC procedure presented in the A pp en d ix.23 The first M = 2,000,000 b u rn -in iter­ ations are d iscarded a n d the su b seq u en t N = 1,500,000 iteratio n s constitute a sim ulated sam ple from the joint posterior density.

As far as the sw itching m echanism is concerned, sim ilar (to the previous m odel) results are obtained. Posterior m ea n s.o f the probabilities p ti are very

Table 6 Posterior means and standard deviations of the parameters of m odel M 3

Pn Pll <Pi <Pi c?

0.9939 0.9961 -0.4511 0.5465 0.8201 1.2485

(0.0046) (0.0026) (0.0642) (0.0869) (0.0233) (0.1375)

close to unit, im plying high persistence in the latent M arkov chain (see Tab. 6). Moreover, the posterior m eans of the sw itching param eter differ substantially acrpss the tw o regim es. It follows th at sw itches b etw een tw o genuinely

dis-. Table 7

Posterior correlation m atrix of the param eters of m odel M 3

M 3 pn P22 ; , M ; <Pi .. . <h o2 P u 1 . 0.4241 0.0631 0.3608 - 0.1015 ■ 0.0225 Pn 1 0.0970 0.0765 0.0071 -0.0020 : A 1 0.4967 0.8355; : -0.6405 , • 1 ' . ■ 0.5678 -0.3328- . ; <h 1 -0.6360 £ 1

25 The minimum acceptance rate while sampling ht's via the M -H algorithm , amounted to approximately 60%.

p(pjy, M3) 140 120 100 80 P° 40 20 O-li <J> 0 5 CO CO P(P2,|y. M3) 250 200 150-)-100 50

■ItefrBrr

lTiTTTrrrrrnifrTt CO CO CO £ * O .CO CO CO CO PiPjy. W3) 140 120 100 j -80 j- 60 40 20 0 C O C O C O C O C O C O C O C O C O C O C O C O CO 0 0 CO CO CO CT) CT> C7> CO CM CM o CO CO CO CO CO CO

s s s

P(P22|y. M3] 250 200 150 100 50 O) ■ o

Fig. 10. M arginal prior (blue solid line) and posterior distributions of the transition probabilities in model M,

tinct states of th e econom y do occur in the m o delled tim e series. Again, the p o s te rio r c o rre la tio n s b e tw e e n th e p a ra m e te rs a p p e a r to reject th e ir p rio r independence, a reason for w hich is believed to be the sam e as in the m odel w ith a regim e-changing intercept.

M arginal posterior densities of the transition probabilities resem ble m uch those obtained for the MSSV(//) m odel. A p art from a strong left asym m etry of para m ete r q>x, n o other irregularities are fo u n d in the posterior distributions of the p a ra m ete rs (see Fig. 11).

g R

P(<P2 l y - M3)

t- CM

d o Tf in o o

p(cr2 ly, M )

Fig. 11. M arginal prior (solid line) and posterior distributions of the log-volatility param eters in model M,

The behaviour of the ergodic m eans seem s to raise n o concerns w ith re­ gard to the convergence of the M CM C algorithm (see Fig. 12).

To analyze the validity of the p rio r constraints for second-order statio n ­ arity of the log-volatility process is a m ore d em an d in g task th an in the p rev i­ ous cases. Therefore, w e present Figure 13, p lo ttin g the values of Rx an d R2, which are required to satisfy the inequalities: Rj < 1 an d R2 <2 (see Section 2.2). We note that only the d a rk area in the figure represents the set of p airs (Rv

R2) that guarantee stationarity of the log-volatility process.24 W ithin the region

tw o-dim ensional contours of the p osterio r d en sity of ra n d o m vector (Rv R2) are plotted. D espite the location of the latter close to the stationarity border, the d ata a p p ears rath e r n o t to reject the p rio r statio n arity restrictions.

24 The stationarity region has been obtained by sim ulation and therefore displays slight inac­ curacies.

■ Pit 0.9965 0.9960 0.9955 0.9950 0.9945 0.9940 0.9935 0.9930 0.9925 0.9920 0.9915 0.9910 -0.34 -0.36 -0.38 -0.40 -0.42 -0.44 -0.46 -0.48 -0.50 -0.52 -0.54 o o o o o o o o o o O O O O O O O O O ' o 05 oo" fC CD in CO- og T-" o" CO h- v- in 05 CO N. m 05 T - C M T j - l O C D C O O ) ^ C M C O Cl CO s co in CO h- t- in Oi t- CM Tt in CD © © T** o o co' o cm" o T-o o" 00 o> J- CM CO 0.9965 0.9964 0.9963 0.9962 0.9961 0.9960 0.9959 0.9958 0.9957 © © o© ©^© ©©_ oo ©© ©^o 1 ©© o©„ 0) CO 00N CM N 5 CO m in in O) CD cSoo CO N ­ a> CM in .CM o O) CO 0.835 0.830 0.825 0.820 0.815 0.810 0.805 0.800 0.795 0.790 05 CO CO h­ r- CM m co oo 05 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 03 CO CO N. t- CM CO lO rf in CT5 CO lO to CO ’ 1 8 o© ■ © © ©© CO CM ‘in CM © ©> CO

G (G rey) — area of the lowest posterior density Pr[(R ,, R2) є G |y , M3] = 0.1344 . і 1--- 1--- 1----1--- 1--- 1--- 1 ЮЮЮЮІ ЛІ ПЮЮЮІ Л ІПЮ O h - r f T - O O l O C M O t O C O O r « * . O q r C M N ( O T t T f l f l ( D N ( 4 o o o o o o d o o d o o o o o' ' і : і _{Ю І ПЮІ ПЮІ ПЮЮ ЮЮЮЮЮЮЮ} і і і і і і і і і r C OW CM Oe nO Nt ^C O' If l- .N O) 0) 0) O T - T - C N C O ^ ^ l f i t D C O S C O C O 0.245 0.165 0.085 0.005

R,

13 0-20 20-40 a 40-60 H 60-80 В 80-100 ■ 100-120 ■ 120-140 ■ 140-160 *160-180

Fig. 13. Simulated region o f possible values of restrictions (Rt < 1, R 2 < 2) for covariance stationarity of a MSSV(^) process (dark area) along w ith contours of their bivariate posterior

distribution obtained for the data

4.5. R e g i m e c h a r a c t e r i s t i c s

Both regim e-sw itching SV specifications im p ly existence of tw o d istin g u ish ­ able states of the econom y. It is e v id e n t even m o re in Figure 14, d ep ictin g averaged posterior probabilities25 Pr(Sf = 1 \y) in each of the tw o m odels along w ith the m odelled tim e series an d the averaged posterior log-volatilities, ln/if's, extracted from m odel M2.26 Unit-close values of the 1-state m ea n probabilities clearly correspond w ith the p erio d of relatively h ig h er volatility of , the daily WIBOR1M interest rates (from about A pril, 2001 to Septem ber, 2004). M ost of the rem aining p a rt of the sam ple p erio d is definitely labelled as state '2'. There is a rather short sub-period, how ever, lasting from M arch, 2005 to September, 2005, that cannot be ascribed to any of the regim es unam biguously. It m ay the

25 Mean posterior probabilities of state '1 ' have been obtained as: Pr(S, =1|>’) = ^ X ^ , (,) =1), t = 1, 2 T.

N

ijsA! 11 .

: 26 The series of the averaged posterior ln/i/s only from m odel M 2 is presented, as it coincides quite much with the ones obtained from other specifications, i.e. M 1 and M y

■ Pr{S(f) = 1|y, M(2)> — Pr{S(f) = 1|y, M(3)} 1.2 1.0 0.8 0.6 0.4 0.2 0 “---7s*--- 71" -...V, - (—— ! VM 1-- ,--~r~- 1"1---r—1--1--1

--!--AR(1)-residuals for WINOR1M ---- aver.post. log-volat. shifted by -5 [model M(2)]

Fig. 14. M ean posterior probabilities of state '1 ' in m odels M 2 and M 3 (upper plot) along with the m odelled tim e series (lower plot) and averaged posterior lnfy's model M 2

case th a t yet another state (i.e. the one representing a m ed iu m volatility level) should be intro duced to the m odel, yielding a three-state MSSV specification.

Posterior m eans of the regim e characteristics (see Tab. 8) indicate th at the m o d els differentiate the tw o regim es in term s of eith er only the m ean log- volatility level (m odel M 2) or, ad d itio n ally , in term s of the state-d ep en d en t variances of the log-volatility process (m odel M 3), w ith the low-volatility state '2' featuring relatively increased 'variability of volatility'. O ne should note that w h a t characterizes the expected du ratio n s of each of the states is considerable d isp e rsio n (in term s of th e s ta n d a rd d ev iatio n ) in th eir p o ste rio r densities featuring very long an d heav y righ t tails (see Fig. 16). O n average, the expect­ e d tim e of the M arkov chain rem aining in a particular state (once it has been achieved) differs from its posterior m ean b y about 805 to 5251 w eekdays (see Tab. 8). As regards the ergodic probabilities, it is noticed th at their posterior densities, th o u g h of a regular shape, are fairly diffused over the u n it interval, therefore preclud in g precise inference on approxim ately h o w long27 the chain rem ains in a particular state.28 On the other h an d , posterior distributions of the rem aining regim e characteristics (i.e. state-dependent log-volatility m eans and v ariances) e v id e n tly clu ste r a ro u n d th e ir p o ste rio r m ean s, a lth o u g h slight asym m etries in their profiles m ay be observed (see Fig. 17 an d 18).

27 In term s of a part of the entire period over w hich the data is analyzed.

28 We draw attention to the fact, that such an interpretation o f the ergodic probabilities of a Mar­ kov chain is valid once the chain has converged to its stationary (ergodic, invariant) distribution.

Table 8 Posterior means and standard deviations (in parentheses) of selected regim e characteristics

in m odel M 2 and M 3 Regim e characteristics lviuuei 7l\ 7tl Dur1 _{Dur2 Ei Ei Vi} V2 M 2 0.4736 0.5264 467.60 555.61 -0.8396 -2.9871 2.4658 2.4643 [MSSV(//)] _{(0.1873) (0.1873) (1149.07) (5251.38) (0.1639) (0.1615) (0.1967) . (0.1964)} M3 0.4104 0.5896 337.52 484.60 -1.0254 -2.4847 . 1.8301 3.8422 [M SSV ($] _{(0.1799) (0.1799) (80524) (979.22) (0.1600) (01959) (02428) (03792)} p K i y, M2) p ( ^ i y , M2) p ( ^ ly , M 3) 1 p(TT2ly , M3)

Fig. 15. Marginal prior (solid line) and posterior distributions of ergodic probabilities , in m odel M 2 and M , •

p(Dur,\y, M2) 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 W C O C M C O C O C O C O C O C O_{CO ▼- CD,} T-p(Dur,\y,M.) 0.045 0.040 0.035 • 0.030 0.025 0.020 0.015 0.010 ' °-005 0.000 in to cm ffl r-c m m co co co eg co eg O CM K r~ CT> CM r- t- t cm rg 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 p(Dur,\y, M3) W CO CO CO CO_{* CO CD CO} <r-CN CD O) CO CO S8 CO S ' 0.045 ■ 0.040 ■ 0.035 • 0.030 ■ 0.025 • 0.020 ■ 0.015 - 0.010 p(Dur1|y, M,) 0.005 • 0.000 IldlflnpirtTmp-CM CM ■ CM l O O O c O C Q C O C O C O C O C O C O - J C O C D O O T - C O C D C O r - 0 ^ N ^ l O f f l ’- C O W C O O t- t- CM CO CO CD < CO a " a

Fig. 16. M arginal prior (solid line) and posterior distributions o f expected duration of each state in m odel M 2 and M 3

3.0 2.5 2.0 1.5 1.0 0.5 p(E,\y,M2) feta o o o o o o o o C M C O ^ t O C D C M C » r t 05 . O CO T- r*- CM CO m >3- CO v- O - CO h- U V 7 7 7 *? *? 3.0 2.5 2.0 1.5 1.0 0.5 o-i p(E2\y,M2) tferr T- CO . T-c«p op Oj> Cj> CM in CM 3

Fig. 17a. M arginal prior (solid line) and posterior distributions of state-conditional log-volatility m eans in m odel M 2

-2

.2

0

7

P(E,\y,M3) 2.5 2.0 1.5 1.0 0.5 0 P(E2\y, M.) 7 o o o O O ' o T- CO io 7 7 o o o o o o o o o o o o o o Is- 03 T- CO in h- 03 S I f l ^ (VI O CO ( D

Fig. 17b. Marginal prior (solid line) and posterior distributions of state-conditional log-volatility means in m odel M 3 '

p(v,\y, m2) 2.50 2.00 1.50 1.00 0.50 0.00 P(V2\y, M2) tn CO CO CM CM CM CM CM CM co co 1.80 1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 Will iii

s- a

O O O O O O O O ) O O O O O O © „ 03 O CO CO ^ CM CO 00 CO t . l O S f f l r N ' t f f l (\i c\i csi (\i CN Csi CO CO

1.20 1.00 0.80 0.60 0.40 0.20 0.00 p(V2\y,M3) 7m I t t a s n o o o o o o O 0 - 0 o o o T- o h- L O ’ CO ■»­ W CO CO O CM Tt llllllllllllirl o o r r N N C M W N N C O C O C O

Fig. ,18. Marginal prior (solid line) and posterior distributions of state-conditional log-volatility variances in m odel M 2 and M 3 .

-1

.5

10

0

4.6. B a y e s i a n m o d e l c o m p a r i s o n

In Table 9 w e p re se n t selected q uan tities (obtained via the New ton-Raftery procedure) allow ing Bayesian com parison of the analyzed m odels in respect of their fit to the data. It is seen th at b o th sw itching SV specifications are strongly preferred over the basic stochastic volatility m odel. Posterior probability29 of

Table 9 Logs of the marginal likelihoods along with the posterior m odel probabilities and logs

. : ; of Bayes factors against m odel M r /

Model Number

of parameters lo g .o P O 'l^ ,) Pr(M,- 1 y) logio Bn Rank

Mi (BSV) 3 . -440.8213 6.215E-16 0 3

M 2 [MSSV(//)1 6 -425.6148 0.999997 15.2066 1

M 3[M SSV(f»)] , 6 -431.0917 0.000003 9.7296 2

■ log p [ y | M ( 1 ) ] -log p\y\M{2)] log p[y|M (3)]

Fig. 19. Logs of the marginal likelihoods of the models against the num ber of cycles

29 Posterior m odel probabilities, Pr(M ; I y), have been obtained under equal prior probabilities of the m odels, i.e. Pr(Al) = 1/3.

the m odel w ith a sw itching intercept is as m uch as ab o u t 1015 tim es the p o s­ terior chances of the BSV m odel, an d about 105 tim es the chances of the other sw itching m odel; These are com pelling a rg u m e n ts ag ain st the h o m o g en eity (i.e. the lack of stru ctu ral shifts) of the m o delled tim e series.

N evertheless, the results m ay be considered som ew hat d u b io u s in view of the no torious instability of the N ew ton-R aftery algorithm . T herefore, the logs of the m arginal likelihood in each of the m odels an d selected Bayes fac­ tors are plotted against the n u m b er of MCMC iterations (see Fig. 19 an d 20). We observe relative stabilization of these quantities only after ab o u t 650,000 cycles. M ore im portantly, how ever, the ranking of the m odels rem ains visibly u nchanged th ro u g h o u t (see Fig. 19).

I O 9 , 0 6 2 , 25 20 15 10 , 5 I* j— t j — H o o o o o o CO O i

? s s.

o o o o o o O CD O o O O_{O ^ O O O O / C3} h-‘ y> CO CD* K lO CD h- 00 00 O) N CO O) o t- CN '09,063, 15' 10

-£

-5 -1 0 -15 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 o 0 0 o 0 0 0 0 0 o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 20 15 10 5 lo9 , o e 2:

t

o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o o o h - 1 0 CO T - CM CO CM CO - t

s s

N Ifl t- O N in (fl S CO CO 05 S CO CD O f CM

Fig. 20. Logs of Bayes factors against the num ber of cycles

5. CONCLUSIONS

In the p a p er tw o special cases of a general M arkov sw itching SV m o d el are under consideration. O ne of them allows discrete shifts only in the intercept, w hereas the other — in the auto regression p a ram eter of the laten t log-vola- tility process. Both constructions are capable of accounting for su d d e n changes

in the m ean volatility level. Hence, w e aim to com pare these tw o specifications in respect of goodness of their fit to the data. : . ,r

The results of the Bayesian analysis of both sw itching m odels as well as a basic SV m o d el p ro vides com pelling evidence against hom ogeneity of the series of the A R (l)-residuals for, the daily WIBORIM interest rates, as evident su p e rio rity of the sw itching m odels over the BSV construction is observed. A m ong th em the one th at features a regim e-changing intercept is undoubtedly

p re fe rre d th e m o st. . , , ... y ,

The tw o regim es are distinguishable in term s of either only the m ean log- volatility level, (in the m odel w ith a sw itching intercept) or, additionally, the sta te -d ep en d en t variances of the log-volatility process (while the autoregres­ sion p a ra m ete r is allow ed regim e-changing).

. REFERENCES

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H w ang S., Satchell S. E., Pereira P. L. V. 2004. How Persistent is Volatility? An Answer with Stocha­ stic Volatility M odels with Markov Regime Switching State Equations, CEA@Cass Working Paper Series, http://www.cass.city.ac.uk/cea/index.html

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APPENDIX

U n d e r the no tatio n established in Section 3, the joint posterior distribution of all the u n k n o w n quantities of the MSSV m odel is decom posed as:

p ( 6 ,h ,S \ y ) <* p (y I h)p(h I S ,9 )p(S 10)p{6),

Individu al com ponents of the above factorization presents them selves as follow s:

— P ( y \ h ) = Y l f N(yt \Q,ht),

/=1

w here: f N{yt 10, ht) denotes the density function of a norm ally distributed ran­ d o m variable y t w ith m ean an d variance equal zero an d ht, respectively;

— p ( h \ S / 6) = Y [ f w (ht \ m l,(T2), 1

w here: f LN(ht I m t, cr2) denotes the den sity function of a log-norm ally distrib­ u te d ra n d o m variable ht w ith the scale p a ram eter equal a 2 an d the location p ara m ete r equal m t given as:

| u s + <p In ht_x fo r MSSV (ji)

1YI — < 1 '

* [/J. + (pSi ln/zf_, fo r MSSV (cp)'

— P(S I e ) oc p(s0)p(s 10) = p(s0) f [ p ( s ( 1 s M ;0), <=i

w here p(S0) denotes the probability distribution of a discrete ran d o m variable SQ. In the stu d y the latter does n o t constitute a quantity of interest, although it is straightforw ard to accom m odate the sam pling algorithm so th at inference on S0 is available.

U n d e r the p rio r structure presented in Section 3, the following conditional p o ste rio r d istrib u tio n s are obtained:

— V(Vu 10-R, / ^ S , y ) = / B(p„ Iai t b.) , for i = 1, 2,

w h e re

a\ = ci\+nn , b* = b x + nn ,

Cl1 = O-I +^22/ ^2 = ^2 ^"^21/ a n d

— p ( a 2 \ e _ a2/ h , S , x j ) = f IC{ a 2 \v'u v'2 ) , w h e re T W = 2 +V1' v2 = 2 v2 -1

-

f°r * ® v W

1 /n ifi IA / O’ A"1 ) / ( ^ < 1)I(R2 < 2) fo r MSSV (<p) w h ere

p. = A7l {azA 0p o + W ' In ft), A - = a2 A, + W ’W, ln /i = (In , In /z2 In h , )’,

a n d W = 1 1(S,= 2) ln h 0 1 I{S2 = 2) ln/ij 1 I(ST = 2 ) lnhj..! for MSSV (ji), W = 1 / ( ^ = l ) l n/;0 /(S1=2) l n/ z0 1 I(S2 = l ) l n V I(S2=2) l nf y 1 7(Sr = l ) l n / ir _1 I(ST = 2 ) In /zr _, p(h, \ e,h_n S,y)<=c-l-ex^

K

w h ere

for the MSSV(//) model:

/ 2 A J L 2h, exp fo r MSSV ((p) ; - i a n ^ - w , ) 2 2cr,

for t = 1/ 2, T - l: a 2 = O'2 _ Vs, ~ W s ,+i + l ( ln ht+i + ln K i ) l + <p2 '2 ' I f , = 1 + ( P 2

for t = T:

for the MSSV(^) m odel:

O j - a 2, w T = (iSt+ (p ln hT_j ;

for I = 1, 2, T - l: g> = — ay - + ^ Inh,_, + . ^ , ln l.,,,

f . -1 , 2 * * ■ - , 2 : »

1 + <pt _1+<pL

W ith reg ard to the conditional posterior distribution of vector S we only note here th at it can be decom posed as:

p ( S \ 0 , h / y) = p(ST \ d , h / y ) f [ p ( S l ISM / 0 / ht, y 1)

'=1 . ' , ' ' .

w here y f a n d h‘. denotes the history of the observable process .and the volatility process, respectively, u p to m om ent t. The idea has been suggested by Carter a n d K ohn (1994) an d it is there th at w e refer for fu rth er details.

T h e M C M C s a m p l i n g p r o c e d u r e

A full single cycle of the MCMC algorithm requires sam pling each quantity of interest from its conditional posterior distribution. We em ploy the Gibbs pro­ cedure to sam ple the m odel param eters an d vector S. For sam pling the con­ ditional volatilities, ht's, w e a d a p t the M etropolis-H astings algorithm used by Pajor (2003) in the case of non-sw itching SV m odels.

Let denote the outcom e on a from the q-th iteration30 (q = 1, 2, ..., M, M + l, ..., M + N), a n d o)_a — a vector consisting of the elem ents of co without its com ponent a . U n d er this notation, a single full step of the sam pling scheme p ro ceed s as follows:

Step 1: sam ple S(<7 + 1^ from p(S \ 6 (q),h {q), S i'q),y )

Note: For a detailed description of the algorithm of sam pling from

p(S I 0, h, y) — see C arter an d Kohn (1994);

Step 2: sam ple p ^ +1) fro m p(pa I 0 ^ , / z ('7),S ('7+1),i/), i = 1, 2; Step 3: sam ple + from p(fi l(cr2)(‘,),p ^ +1),p ^ +1),/j(‘,),S ('7+1),t/);

Step 3* — only for the MSSV(£>) m odel31 (permutation sampler, see F ruhw irth-Schnatter, 2001):

-» if (p\q+1) < (p^+l) is violated, th en:32

30 For q = 0 we obtain the set of starting values of the algorithm.

31 A note should be m ade here on sampling the model-specific parameters, P, when conside­ ring the MSSV(^>) model, i.e. Step 3* in the sampling scheme. In Section 2.2 the indentifiability con­ straint ipx < ip1 is imposed. To guarantee that the restriction holds an additional step, called the

permutation sampler (see Fruhwirth-Schnatter, 2001), is introduced to the Gibbs procedure. Once a new P has been sampled from its full conditional posterior distribution, we check whether the inequality q>{ < <p\ is violated. If so, the subscripts '1 ' and '2 ' are simply interchanged so that the restriction is valid again. Since prior to sampling P all the state variables, S/s, and the transition probabilities are generated, they must be 'updated' (if <0, and <p2 required switching), that is all the ones and twos in vector S as well as the subscripts of the probabilities need to be interchanged. Relevant assum ptions, theorems and proofs o f the validity of such an algorithm are found in Fruhw irth-Schnatter (2001).

<pS'?+1) := <P2+}), <P2+:'> := (p\q+X} (so ^ a t (p\l,+:) < (p^+l) is guaranteed), S(,,+1) := 3iT - S(,,+1) w here iT ’ = (1,1,..., l)(lxT),

-i for i, j = 1 , 2 ;

Step 4: sam ple (ct2) ^ from p(cr21 / x (<7>,S (<7+1),j/) ;

Step 5: (the M etropolis-H astings step): sam ple each from

p(ht \ e ^ \ h \ ^ y hlf+VTyS(^ \ y ) , w here his:t)= ( h s,hs+, ht)‘