Abstract We provide a detailed summary of the large and vibrant emergingliterature that deals with the multivariate modeling of conditional volatilityof financial time series within the framework of stochastic volatility. Thedevelopments and achievements in this area represent one of the great suc-cess stories of financial econometrics. Three broad classes of multivariatestochastic volatility models have emerged: one that is a direct extension ofthe univariate class of stochastic volatility model, another that is related tothe factor models of multivariate analysis and a third that is based on thedirect modeling of time-varying correlation matrices via matrix exponentialtransformations, Wishart processes and other means. We discuss each of thevarious model formulations, provide connections and differences and showhow the models are estimated. Given the interest in this area, further signif-icant developments can be expected, perhaps fostered by the overview anddetails delineated in this paper, especially in the fitting of high-dimensionalmodels.
Siddhartha ChibWashington University in St. Louis, Campus Box 1133, 1 Brookings Dr., St. Louis, MO63130, USA, e-mail: [email protected]
Yasuhiro OmoriFaculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-0033,Japan, e-mail: [email protected]
Manabu AsaiFaculty of Economics, Soka University, 1-236 Tangi-cho, Hachioji-shi Tokyo, 192-8577,Japan, e-mail: [email protected]
A considerable amount of recent literature on financial econometrics hasemerged on the modeling of conditional volatility, spurred by the demandfor such models in areas such as portfolio and risk management. Much ofthe early interest centered on multivariate versions of univariate generalizedautoregressive conditional heteroscedasticity (GARCH) models. These gen-eralizations have been ably summarized in recent surveys, for example, thoseof Bauwens et al. (2006) and Silvennoinen and Teräsvirta (2007). More re-cently, a large and prolific (parallel) body of literature has developed aroundgeneralizations of the univariate stochastic volatility (SV) model. A numberof multivariate SV (MSV) models are now available along with clearly ar-ticulated estimation recipes. Our goal in this paper is to provide a detailedsummary of these various model formulations, along with connections anddifferences, and discuss how the models are estimated. We aim to show thatthe developments and achievements in this area represent one of the greatsuccess stories of financial econometrics. We note that our treatment does notinclude any discussion of multivariate modeling of volatility that is relevantfor ultra-high-frequency data. Thus, there is no discussion of realized volatil-ity (Andersen et al. (2003) and Barndorff-Nielsen and Shephard (2004)).
To fix notation and set the stage for our developments, the univariate SVmodel that forms the basis for many MSV models is given by (Ghysels et al.(1996), Broto and Ruiz (2004) and Shephard (2004)):
yt = exp(ht/2)εt, t = 1, . . . , n, (1)ht+1 = μ+ φ(ht − μ) + ηt, t = 1, . . . , n− 1, (2)
h1 ∼ N(
μ, σ2η/(1 − φ2)
)
, (3)(
εtηt
)
|ht ∼ N2(0,Σ), Σ =(
1 00 σ2
η
)
, (4)
where yt is a univariate outcome, ht is a univariate latent variable andN (μ, σ2) and Nm(μ,Σ) denote, respectively, a univariate normal distribu-tion with mean μ and variance σ2, and an m-variate normal distributionwith mean vector μ and variance-covariance matrix Σ. In this model, condi-tioned on the parameters (μ, φ, σ2
η), the first generating equation representsthe distribution of yt conditioned on ht, and the second generating equationrepresents the Markov evolution of ht+1 given ht. The conditional mean of ytis assumed to be zero because that is a reasonable assumption in the settingof high-frequency financial data. The SV model is thus a state-space model,with a linear evolution of the state variable ht but with a nonlinear mea-surement equation (because ht enters the outcome model nonlinearly). Fur-thermore, from the measurement equation we see that Var(yt|ht) = exp(ht),which implies that ht may be understood as the log of the conditional vari-ance of the outcome. To ensure that the evolution of these log volatilities is
Multivariate Stochastic Volatility 367
stationarity, one generally assumes that |φ| < 1. Many other versions of theunivariate SV model are possible. For example, it is possible to let the modelerrors have a non-Gaussian fat-tailed distribution, to permit jumps, and in-corporate the leverage effect (through a nonzero off-diagonal element in Σ).The estimation of the canonical SV model and its various extensions was atone time considered difficult since the likelihood function of these models isnot easily calculable. This problem has been fully resolved by the creativeuse of Monte Carlo methods, primarily Bayesian Markov chain Monte Carlo(MCMC) methods (Jacquier et al. (1994), Kim et al. (1998), Chib et al.(2002) and Omori et al. (2007)). We refer the readers to Asai et al. (2006)for a discussion of how this problem can be addressed in some special casesby non-Bayesian methods. In this survey, on the other hand, we concentrateon Bayesian methods but mention the full range of methods (Bayesian andnon-Bayesian) that have been tried for the various models.
In the multivariate case, when one is dealing with a collection of financialtime series denoted by yt = (y1t, . . . , ypt)′, the main goal is to model thetime-varying conditional covariance matrix of yt. There are several ways inwhich this can be done. A typical starting point is the assumption of series-specific log volatilities htj (j ≤ p) whose joint evolution is governed by afirst-order stationary vector autoregressive process:
where ht = (h1t, . . . , hpt)′. To reduce the computational load, especially whenp is large, the log volatilities can be assumed to be conditionally independent.In that case,
are both diagonal matrices. We refer to the former specification as the VAR(1)model and the latter as the IAR(1) (for independent autoregressive) model.Beyond these differences, the various models primarily differ in the way inwhich the outcomes yt are modeled. In one formulation, the outcomes areassumed to be generated as
yt = V1/2t εt, V1/2
t = diag (exp(h1t/2), . . . , exp(hpt/2)) , t = 1, . . . , n,
with the additional assumptions that(
εtηt
)
|ht ∼ N2p(0,Σ), Σ =(
Σεε OO Σηη
)
and Σεε is a matrix in correlation (with units on the main diagonal). Thus,conditioned on ht, Var(yt) = V1/2
t ΣεεV1/2t is time-varying (as required), but
368 S. Chib, Y. Omori and M. Asai
the conditional correlation matrix is Σεε, which is not time-varying. In thesequel we refer to this model as the basic MSV model.
A second approach for modeling the outcome process is via a latent factorapproach. In this case, the outcome model is specified as
yt = Bf t + V1/2t εt, V1/2
t = diag (exp(h1t/2), . . . , exp(hpt/2)) ,
where B is a p × q matrix (q ≤ p) called the loading matrix, and ft =(f1t, ..., fqt) is a q × 1 latent factor at time t. For identification reasons, theloading matrix is subject to some restrictions (that we present later in thepaper), and Σεε is the identity matrix. The model is closed by assuming thatthe latent variables are distributed independently across time as
ft|ht ∼ Nq(0,Dt),
whereDt = diag (exp(hp+1,t), . . . , exp(hp+q,t))
is a diagonal matrix that depends on additional latent variables hp+k,t. Thefull set of log volatilities, namely,
ht = (h1t, . . . , hpt, hp+1,t, ..., hp+q,t),
are assumed to follow a VAR(1) or IAR(1) process. In this model, the varianceof yt conditional on the parameters and ht is
Var(yt|ht) = Vt + BDtB′
and as a result the conditional correlation matrix is time-varying.Another way to model time-varying correlations is by direct modeling of
the variance matrix Σt = Var(yt). One such model is the Wishart processmodel proposed by Philipov and Glickman (2006b), who assume that
yt|Σt ∼ Np(0,Σt),Σt|ν,St−1 ∼ IWp(ν,St−1),
where IWp(ν0,Q0) denotes a p-dimensional inverted Wishart distributionwith parameters (ν0,Q0), and St−1 is a function of Σt−1. Several modelsalong these lines have been proposed as we discuss in Section 4.
The rest of the article is organized as follows. In Section 2, we first discussthe basic MSV model along with some of its extensions. Section 3 is devotedto the class of factor MSV models, while Section 4 deals with models in whichthe dynamics of the covariance matrix are modeled directly and Section 5 hasour conclusions.
Multivariate Stochastic Volatility 369
2 Basic MSV Model
2.1 No-leverage model
As in the preceding section, let yt = (y1t, . . . , ypt)′ denote a set of observa-tions at time t on p financial variables and let ht = (h1t, . . . , hpt)′ be thecorresponding vector of log volatilities. Then one approach to modeling theconditional covariance matrix of yt is to assume that
Of course, for identification purposes, the diagonal elements of Σεε must be 1,which means that the matrix Σεε is a correlation matrix.
Analyses of this model are given by Harvey et al. (1994), Daníelsson (1998),Smith and Pitts (2006) and Chan et al. (2006). Actually, Harvey et al. (1994)dealt with a special case of this model in which Φ = diag(φ1, . . . , φp). To fitthe model, the measurement equation (5) is linearized by letting wit = log y2
it.Because
E(log ε2it) = −1.27, Var(log ε2it) = π2/2, (8)
one now has (a non-Gaussian) linear measurement equation:
wt = (−1.27)1 + ht + ξt, (9)
where wt = (w1t, . . . , wpt)′, ξt = (ξ1t, . . . , ξpt)′, ξit = log ε2it + 1.27 and1 = (1, . . . , 1)′. Although the new state error ξt does not follow a normaldistribution, approximate or quasi maximum likelihood (QML) estimates canbe obtained by assuming Gaussianity. Calculation of the (misspecified) Gaus-sian likelihood also requires the covariance matrix of ξt. Harvey et al. (1994)showed that the (i, j)th element of the covariance matrix of ξt = (ξ1t, . . . , ξpt)′
is given by (π2/2)ρ∗ij , where ρ∗ii = 1 and
ρ∗ij =2π2
∞∑
n=1
(n− 1)!{∏n
k=1(1/2 + k − 1)}nρ2nij . (10)
370 S. Chib, Y. Omori and M. Asai
The model was applied to four daily foreign exchange rates (pound/dollar,Deutschemark/dollar, yen/dollar and Swiss franc/dollar). As mentioned inHarvey et al. (1994), the preceding fitting method cannot be extended to theleverage model considered below.
So et al. (1997) provide a similar analysis, but unlike Harvey et al.(1994) the nondiagonal elements of Φ are not assumed to equal zero. Es-timation of the parameters is again by the QML method which is imple-mented through a computationally efficient and numerically well-behavedexpectation-maximization (EM) algorithm. The asymptotic variance-covari-ance matrix of the resulting estimates is based on the information matrix.Another related contribution is that of Daníelsson (1998), where the model
is analyzed. The parameters of this model are estimated by the simulatedmaximum likelihood method. The model and fitting method should be ap-plied in the estimation of a bivariate model for foreign exchange rates(Deutschemark/dollar, yen/dollar) and stock indices (S&P500 and Tokyostock exchange). On the basis of the log-likelihood values, they concluded thatthe MSV model is superior to alternative GARCH models such as the vectorGARCH, diagonal vector GARCH (Bollerslev et al. (1988)), Baba–Engle–Kraft–Kroner (BEKK) model (Engle and Kroner (1995)) and the constantconditional correlation (CCC) model (Bollerslev (1990)).
Smith and Pitts (2006) considered a bivariate model without leverage thatis similar to the model of Daníelsson (1998). The model is given by
where the (i, j)th element of Σ0 is the (i, j)th element of Σηη divided by1 − φiφj to enforce the stationarity of ht − Ztα. To measure the effect ondaily returns in the yen/dollar foreign exchange of intervention by the Bankof Japan, they included in Zt a variable that represents central bank inter-vention which they modeled by a threshold model. The resulting model wasfit by Bayesian MCMC methods (Chib and Greenberg (1996), Chib (2001)).Because the likelihood of the parameters is complex, sampling of the poste-rior distributions in all applications of MCMC methods in MSV models isindirectly achieved by sampling the posterior distribution of the parametersand each of the latent variables. This tactic circumvents the computation ofthe likelihood. For this tactic to work it is necessary to efficiently sample theresulting high-dimensional posterior distribution. This is the challenge thathas to be surmounted on a model-by-model basis.
Multivariate Stochastic Volatility 371
To improve the efficiency of the MCMC algorithm, Smith and Pitts (2006)sampled ht’s in blocks, as in Shephard and Pitt (1997); see also Watanabe andOmori (2004). For simplicity, we describe their algorithm without the thresh-old specification and without missing observations. Let Yt = {y1, . . . ,yt}denote the set of observations until time t. Then the MCMC algorithm ofSmith and Pitts (2006) is given by:
1. Sample {ht}nt=1|ρ12, φ1, φ2, α,Σηη, Yn. Divide {ht}nt=1 into several blocks,and sample a block at a time given other blocks. Let ha:b = (h′
a, . . . ,h′b)
′.To sample a block ha:b given other hj ’s, we conduct a Metropolis–Hastings(M-H) algorithm using a proposal density of the type introduced by Chiband Greenberg (1994, 1998) and Chib (2001):
The proposal density is a Gaussian approximation of the conditional pos-terior density based on a Taylor expansion of the conditional posteriordensity around the mode ha:b. The mode is found numerically by theNewton–Raphson method.
2. Sample ρ12|{ht}nt=1, φ1, φ2, α,Σηη, Yn using the M-H algorithm.3. Sample φ1, φ2|{ht}nt=1, ρ12, α,Σηη, Yn using the M-H algorithm.4. Sample α|{ht}nt=1, ρ12, φ1, φ2,Σηη, Yn ∼ N2(δ,Σ), where
δ = Σn
∑
t=2
(Zt − ΦZt−1)′Σ−1ηη (ht − Φht−1) + Z′
1Σ−10 h1,
Σ−1 =n
∑
t=2
(Zt − ΦZt−1)′Σ−1ηη (Zt − ΦZt−1) + Z′
1Σ−10 Z1.
5. Sample Σηη|{ht}nt=1, ρ12, φ1, φ2, α, Yn using the M-H algorithm.
Bos and Shephard (2006) considered a similar model but with the meanin the outcome specification driven by an r × 1 latent process vector αt:
where Gtut and Htut are independent and the off-diagonal element of Φmay be nonzero. Given {ht}nt=1, this is a linear Gaussian state-space model,
yt = Ztαt + u∗t , u∗
t ∼ Np(0,GtVtG′t),
αt+1 = Ttαt + v∗t , v∗
t ∼ Nr(0,HtVtH′t),
where u∗t and v∗
t are independent. Bos and Shephard (2006) took a Bayesianapproach and conducted the MCMC simulation in two blocks. Let θ = (ψ, λ),where ψ indexes the unknown parameters in Tt,Zt,Gt,Ht, and λ denotesthe parameter of the SV process of ut.
1. Sample θ, {αt}nt=1|{ht}nt=1, Yn.
– Sample θ|{ht}nt=1, Yn using a M-H algorithm or a step from the adap-tive rejection Metropolis sampler by Gilks et al. (1995); see Bos andShephard (2006).
– Sample {αt}nt=1|θ, {ht}nt=1, Yn using a simulation smoother for a lin-ear Gaussian state-space model; see de Jong and Shephard (1995)and Durbin and Koopman (2002). We first sample disturbances ofthe linear Gaussian state-space model and obtain samples of αt re-cursively.
2. Sample {ht}nt=1|θ, {αt}nt=1, Yn. For t = 1, . . . , n, we sample ht one at atime by the M-H algorithm with the proposal distribution
Although the sampling scheme which samples ht at a time is expectedto produce highly autocorrelated MCMC samples, the adaptive rejectionMetropolis sampling of θ seems to overcome some of the inefficiencies. Yuand Meyer (2006) provide a survey of MSV models that proceed along theselines and illustrate how the Bayesian software program WinBUGS can beused to fit bivariate models.
It is worth mentioning that it is possible to relax the assumption that thevolatility process is VAR of order 1. In one notable attempt, So and Kwok(2006) consider a MSV model where the volatility vector ht−μ follows a sta-tionary vector autoregressive fractionally integrated moving average process,
where B is a backward operator such that Bjht = ht−j . The εt and ηtare assumed to be independent. So and Kwok (2006) investigated statisticalproperties of the model and proposed a QML estimation method as in Har-vey et al. (1994). They linearized the measurement equation by taking thelogarithm of the squared returns and considered the linear state-space model
for i = 1, . . . , n. The covariance matrix of ξt can be obtained as in Harveyet al. (1994). To conduct the QML estimation, So and Kwok (2006) assumedthat ξt follows a normal distribution and obtained estimates based on thelinear Gaussian state-space model. However, since ht − μ follows a vectorARFIMA(p,d, q) process, the conventional Kalman filter is not applicable asthe determinant and inverse of a large covariance matrix is required to calcu-late the quasi-log-likelihood function. To avoid this calculation, So and Kwok(2006) approximated the quasi-log-likelihood function by using a spectrallikelihood function based on a Fourier transform.
2.2 Leverage effects
Another extension of the basic MSV model is to allow for correlation betweenεt and ηt by letting Σεη �= O. This extension is important because at leastfor returns on stocks there is considerable evidence that the measurementand volatility innovations are correlated (Yu (2005), Omori et al. (2007)).That this correlation (the leverage effect) should be modeled is mentionedby Daníelsson (1998) but this suggestion is not implemented in his empiricalstudy of foreign exchange rates and stock indices. One compelling work ona type of leverage model is due to Chan et al. (2006), who considered themodel
yt = V1/2t εt,
ht+1 = μ+ diag(φ1, . . . , φp)(ht − μ) + Ψ1/2ηt,
h1 ∼ Np(μ,Ψ1/2Σ0Ψ1/2),
374 S. Chib, Y. Omori and M. Asai
where the (i, j) element of Σ0 is the (i, j) element of Σηη divided by 1−φiφjsatisfying a stationarity condition such that
Σ0 = ΦΣ0Φ + Σηη
and
V1/2t = diag (exp(h1t/2), . . . , exp(hpt/2)) ,
Ψ1/2 = diag(√
ψ21 , . . . ,
√
ψ2p
)
,
(
εtηt
)
∼ N2p(0,Σ), Σ =(
Σεε Σεη
Σηε Σηη
)
.
Actually, the model considered in Chan et al. (2006) had correlation betweenεt and ηt−1, which is not correctly a model of leverage. Our discussion there-fore modifies their treatment to deal with the model just presented, whereεt and ηt are correlated. Note that Σ is a 2p × 2p correlation matrix withΣεη �= O. Now, following Wong et al. (2003) and Pitt et al. (2006), we repa-rameterize Σ such that
Σ−1 = TGT, T = diag(√
G11, . . . ,√Gpp
)
,
where G is a correlation matrix and Gii denotes the (i, i)th element of theinverse matrix of G. Under this parameterization, we can find the posteriorprobability that the strict lower triangle of the transformed correlation matrixG is equal to zero. Let Jij = 1 if Gij �= 0 and Jij = 0 if Gij = 0 fori = 1, . . . , 2p, j < i and S(J) denote the number of elements that are 1’s inJ = {Jij , i = 1, . . . , 2p, j < i}. Further let G{J=k} = {Gij : Jij = k ∈ J}(k = 0, 1) and A denote a class of 2p× 2p correlation matrices. Wong et al.(2003) proposed a hierarchical prior for G:
π(dG|J) = V (J)−1dG{J=1}I(G ∈ A), V (J) =∫
G∈AdG{J=1},
π(J|S(J) = l) =V (J)
∑
J∗:S(J∗)=l
V (J∗),
π(S(J) = l|ϕ) =(
p(2p− 1)l
)
ϕl(1 − ϕ)p(2p−1)−l.
If we assume ϕ ∼ U(0, 1), the marginal prior probability π(S(J) = l) =1/(p(2p − 1) + 1); see Wong et al. (2003) for the evaluation of V (J). Letφ = (φ1, . . . , φp)′ and ψ = (ψ1, . . . , ψp)′ (ψj > 0, j = 1, . . . , p).
1. Sample φ|μ, {ht}nt=1, ψ,Σ, Yn where Yn = {y1, . . . ,yn}. Let Σij denotethe (i, j)th block of the 2p×2pmatrix Σ−1 and d be a vector that consists
Multivariate Stochastic Volatility 375
of the diagonal elements
n−1∑
t=1
Ψ−1/2(ht − μ)(
y′tV
−1/2t Σ12 + Ψ−1/2(ht+1 − μ)′Σ22
)
.
Propose a candidate
φ ∼ T NR(μφ,Σφ), R = {φ : φj ∈ (−1, 1), j = 1, . . . , p},
Σ−1φ = Σ22 �
{
n−1∑
t=1
Ψ−1/2(ht − μ)(ht − μ)′Ψ−1/2
}
,
μφ = Σφd,
where � is the element-by-element multiplication operator (Hadamardproduct) and apply the M-H algorithm.
2. Sample μ|φ, {ht}nt=1, ψ,Σ, Yn ∼ Np(μ∗,Σ∗), where
3. Sample ψ|φ, μ, {ht}nt=1,Σ, Yn. Let v = (ψ−11 , . . . , ψ−1
p ) and l(v) denotethe logarithm of the conditional probability density of v and v denotethe mode of l(v). Then conduct the M-H algorithm using a truncatedmultivariate t distribution on the region R = {v : vj > 0, j = 1, . . . , p}with six degrees of freedom, location parameter v and a covariance matrix−{∂2l(v)/∂v∂v′}−1
v=v.4. Sample {ht}nt=1|φ, μ, ψ,Σ, Yn. We divide {ht}nt=1 into several blocks, and
sample a block at a time given other blocks as in Smith and Pitts (2006).Let ha:b = (h′
a, . . . ,h′b)
′. To sample a block ha:b given other hj ’s, weconduct a M-H algorithm using a Chib and Greenberg (1994) proposal,
ha:b ∼ Np(b−a+1)
(
ha:b,
[
− ∂l(ha:b)∂ha:b∂h′
a:b
]−1
ha:b=ha:b
)
,
l(ha:b) = const − 12
b∑
t=a
1′ht −12
b+1∑
t=a
r′tΣ−1rt,
rt =(
V−1/2t yt
Ψ−1/2{ht+1 − μ− Φ(ht − μ)}
)
,
376 S. Chib, Y. Omori and M. Asai
a Gaussian approximation of the conditional posterior density based onTaylor expansion of the conditional posterior density around the modeha:b. The mode is found using the Newton–Raphson method numerically.The analytical derivatives can be derived similarly as in the Appendix ofChan et al. (2006).
5. Sample Σ|φ, μ, ψ, {ht}nt=1, Yn. Using the parsimonious reparameteriza-tion proposed in Wong et al. (2003), we generate each element Gij oneat a time using the M-H algorithm.
Chan et al. (2006) applied the proposed estimation method to equities atthree levels of aggregation: (1) returns for eight different markets (portfolios ofstocks in NYSE, AMEX, NASDAQ and S&P500 indices); (2) returns for eightdifferent industries (portfolios of eight well-known and actively traded stocksin petroleum, food products, pharmaceutical, banks, industrial equipment,aerospace, electric utilities, and department/discount stores); (3) returns forindividual firms within the same industry. They found strong evidence ofcorrelation between εt and ηt−1 only for the returns of the eight differentmarkets and suggested that this correlation is mainly a feature of marketwiderather than firm-specific returns and volatility.
Asai and McAleer (2006) also analyzed a MSV model with leverage effects,letting
The cross-asset leverage effects are assumed to be 0 (Corr(εit, ηjt) = 0, fori �= j). As in Harvey and Shephard (1996), they linearized the measurementequations and considered the following state-space model conditional on st =(s1t, . . . , spt)′, where sit = 1 if yit is positive and sit = −1 otherwise:
log y2it = hit + ζit, ζit = log ε2it, i = 1, . . . , p, t = 1, . . . , n,
ht+1 = μ+ μ∗t + diag(φ1, . . . , φp)ht + η∗t ,
μ∗t =
√
2πΣεηΣ−1
εε st, η∗t ∼ Np(0,Ση∗t η
∗t),
where E(ζit) = −1.27, and Cov(ζit, ζjt) = (π2/2)ρ∗ij given in (10). The matrixΣη∗
t η∗t
and E(η∗t ζ′t) are given in Asai and McAleer (2006). They also consid-ered an alternative MSV model with leverage effects and size effects givenby
This model is a generalization of a univariate model given by Daníelsson(1994). It incorporates both leverage effects and the magnitude of the previ-ous returns through their absolute values. Asai and McAleer (2006) fit thesetwo models to returns of three stock indices—S&P500 Composite Index, theNikkei 225 Index, and the Hang Seng Index—by an importance samplingMonte Carlo maximum likelihood estimation method. They found that theMSV model with leverage and size effects is preferred in terms of the Akaikeinformation criterion (AIC) and Bayesian information criterion (BIC) mea-sures.
2.3 Heavy-tailed measurement error models
It has by now been quite well established that the tails of the distributionof asset returns are heavier than those of the Gaussian. To deal with thissituation it has been popular to employ the Student t distribution as a re-placement for the default Gaussian assumption. One reason for the popularityof the Student t distribution is that it has a simple hierarchical form as a scalemixture of normals. Specifically, if T is distributed as standard Student t withν degrees of freedom then T can be expressed as
T = λ−1/2Z, Z ∼ N (0, 1), λ ∼ G(ν/2, ν/2).
This representation can be exploited in the fitting, especially in the Bayesiancontext. One early example of the use of the Student t distribution occurred inHarvey et al. (1994), who assumed that in connection with the measurementerror εit that
and (8) and (10), where ψ and ψ′ are the digamma and trigamma functions.On the other hand, Yu and Meyer (2006) considered a multivariate Studentt distribution for εt, in which case the measurement error has the form
T = λ−1/2t εt, εt ∼ Np(0, I), λt ∼ G(ν/2, ν/2).
They mentioned that this formulation was empirically better supported thanthe formulation in Harvey et al. (1994). The model was fit by BayesianMCMC methods.
Another alternative to the Gaussian distribution is the generalized hyper-bolic (GH) distribution introduced by Barndorff-Nielsen (1977). This familyis also a member of the scale mixture of normals family of distributions. Inthis case, the mixing distribution is a generalized inverse Gaussian distribu-tion. The GH distribution is a rich class of distributions that includes thenormal, normal inverse Gaussian, reciprocal normal inverse Gaussian, hyper-bolic, skewed Student’s t, Laplace, normal gamma, and reciprocal normal hy-perbolic distributions (Barndorff-Nielsen and Shephard (2001)). Aas and Haff(2006) employed the univariate GH distributions (normal inverse Gaussiandistributions and univariate GH skew Student’s t distributions) and estimatedin the analysis of the total index of Norwegian stocks (TOTX), the SSBWGhedged bond index for international bonds, the Norwegian kroner/euro ex-change rate and the EURIBOR five-year interest rate. They found that theGH skew Student’s t distribution is superior to the normal inverse Gaussiandistribution for heavy-tailed data, and superior to the skewed t distributionproposed by Azzalini and Capitanio (2003) for very skewed data.
The random variable x ∼ GH(ν, α, β,m, δ,S) follows a multivariate GHdistribution with density
f(x) =(γ/δ)νKν− p
2
(
α√
δ2 + (x − m)′S−1(x − m))
exp{β′(x − m)}
(2π)p2Kν(δγ)
{
α−1√
δ2 + (x − m)′S−1(x − m)}
p2−ν
,
(15)
γ ≡√
α2 − β′Sβ ≥ 0, α2 ≥ β′Sβ,
ν, α ∈ R, β,m ∈ Rn, δ > 0,
where Kν is a modified Bessel function of the third kind, and S is a p × ppositive-definite matrix with determinant |S| = 1 (Protassov (2004), Schmidtet al. (2006)). It can be shown that x can be expressed as
x = m + ztSβ +√ztS
1/2εt,
Multivariate Stochastic Volatility 379
where S1/2 is a p × p matrix such that S = S1/2S1/2′and ε ∼ Np(0, I) and
zt ∼ GIG(ν, δ, γ) follows a generalized inverse Gaussian distribution whichwe denote z ∼ GIG(ν, δ, γ) and whose density is given by
f(z) =(γ/δ)ν
2Kν(δγ)zν−1 exp
{
−12(
δ2z−1 + γ2z)
}
, γ, δ ≥ 0, ν ∈ R, z > 0,
where the range of the parameters is given by
δ > 0, γ2 ≥ 0, if ν < 0,δ > 0, γ2 > 0, if ν = 0,δ ≥ 0, γ2 > 0, if ν > 0.
For the generation of a random sample from GIG(ν, a, b), see Dagpunar(1989), Doornik (2002) and Hörmann et al. (2004). The estimation of such amultivariate distribution would be difficult and Protassov (2004) relied on theEM algorithm with ν fixed and fit the five-dimensional normal inverse Gaus-sian distribution to a series of returns on foreign exchange rates (Swiss franc,Deutschemark, British pound, Canadian dollar and Japanese yen). Schmidtet al. (2006) proposed an alternative class of distributions, called the multi-variate affine GH class, and applied it to bivariate models for various assetreturns data (Dax, Cac, Nikkei and Dow returns). Other multivariate skewdensities have also been proposed, for example, in Arellano-Valle and Azzalini(2006), Bauwens and Laurent (2005), Dey and Liu (2005) Azzalini (2005),Gupta et al. (2004) and Ferreira and Steel (2004).
3 Factor MSV Model
3.1 Volatility factor model
A weakness of the preceding MSV models is that the implied conditionalcorrelation matrix does not vary with time. One approach for generatingtime-varying correlations is via factor models in which the factors follow aSV process. One type of factor SV model (that however does not lead to time-varying correlations) was considered by Quintana and West (1987) and byJungbacker and Koopman (2006), who utilized a single factor to decomposethe outcome into two multiplicative components, a scalar common volatilityfactor and a vector of idiosyncratic noise variables, as
yt = exp(
ht2
)
εt, εt ∼ Np(0,Σεε),
ht+1 = μ+ φ(ht − μ) + ηt, ηt ∼ N (0, σ2η),
380 S. Chib, Y. Omori and M. Asai
where ht is a scalar. The first element in Σεε is assumed to be 1 for identifica-tion reasons. By construction, the positivity of the variance of yt is ensured.In comparison with the basic MSV model, this model has fewer parameters,which makes it more convenient to fit. The downside of the model, however,is that unlike the mean factor MSV model which we discuss below, the condi-tional correlations in this model are time-invariant. Moreover, the correlationbetween log volatilities is 1, which is clearly limiting.
In order to estimate the model, Jungbacker and Koopman (2006) applieda Monte Carlo likelihood method to fit data on exchange rate returns of theBritish pound, the Deutschemark and the Japanese yen against the US dollar.They found that the estimate of φ is atypically low, indicating that the modelis inappropriate for explaining the movements of multivariate volatility.
A more general version of this type was considered by Harvey et al. (1994),who introduced a common factor in the linearized state-space version of thebasic MSV model by letting
where wt = (w1t, . . . , wpt)′, ξt = (ξ1t, . . . , ξpt)′ and ht = (h1t, . . . , hqt)′ (q ≤p). Furthermore, one assumes that
Θ =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
θ11 0 · · · 0
θ21 θ22. . .
......
. . . . . . 0θq1 · · · θq,q−1 θqq...
......
θp,1 · · · θp,q−1 θp,q
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, h =
⎛
⎜
⎜
⎜
⎝
0hq+1
...hp
⎞
⎟
⎟
⎟
⎠
.
Harvey et al. (1994) estimated the parameters by the QML method. To makethe factor loadings interpretable, the common factors are rotated such thatΘ∗ = ΘR′ and h∗
t = Rht, where R is an orthogonal matrix.Tims and Mahieu (2006) considered a similar but simpler model for the
logarithm of the range of the exchange rates in the context of an appli-cation involving four currencies. Let wij denote a logarithm of the rangeof foreign exchange rate of the currency i relative to the currency j, andw = (w12, w13, w14, w23, w24, w34). Now assume that
where c is a 6 × 1 mean vector, Σηη is diagonal, ht = (h1t, . . . , h4t)′ and hjtis a latent factor for the jth currency at time t and
Multivariate Stochastic Volatility 381
Z =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 1 0 01 0 1 01 0 0 10 1 1 00 1 0 10 0 1 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
Since this is a linear Gaussian state-space model, the estimation of the pa-rameters is straightforward by Kalman filtering methods.
Ray and Tsay (2000) introduced long-range dependence into the volatilityfactor model by supposing that ht follows a fractionally integrated processsuch that
yt = V1/2t εt, V1/2
t = diag(exp(z′1ht/2), . . . , exp(z′qht/2)),
(1 − L)dht = ηt, εt ∼ Np(0,Σεε), ηt ∼ Nq(0,Σηη),
where zi (i = 1, . . . , q) are q × 1 vectors with q < p. In the fitting, themeasurement equation is linearized as in Harvey et al. (1994).
Calvet et al. (2006) generalized the univariate Markov-switching multifrac-tal (MSM) model proposed by Calvet and Fisher (2001) to the multivariateMSM and factor MSM models. The univariate model is given by
yt = (M1,tM2,t · · ·Mk,t)1/2εt, εt ∼ N (0, σ2),
where Mj,t (j ≤ k) are random volatility components, satisfying E(Mj,t) = 1.Given Mt = (M1,t,M2,t, . . . ,Mk,t), the stochastic volatility of return yt isgiven by σ2M1,tM2,t · · ·Mk,t. Each Mj,t follows a hidden Markov chain asfollows:
Mj,t drawn from distribution M,with probability γj ,Mj,t = Mj,t−1,with probability 1 − γj ,
where γj = 1 − (1 − γ)(bj−k), (0 < γ < 1, b > 1) and the distribution of M
is binomial, giving values m or 2 − m (m ∈ [1, 2]) with equal probability.Thus, the MSM model is governed by four parameters (m,σ, b, γ), which areestimated by the maximum likelihood method.
For the bivariate MSM model, we consider the vector of the random volatil-ity component Mj,t = (M1
where � denotes the element-by-element product. For each component Mj,t
in the bivariate model, Calvet et al. (2006) assumed that volatility arrivalsare correlated but not necessarily simultaneous. For details, let sij,t (i = 1, 2)denote the random variable equal to 1 if there is an arrival on M i
j,t with prob-
382 S. Chib, Y. Omori and M. Asai
ability γj , and equal to 0 otherwise. Thus, each sij,t follows the Bernoulli dis-tribution. At this stage, Calvet et al. (2006) introduced the correlation coeffi-cient λ, giving the conditional probability P (s2j,t = 1|s1j,t = 1) = (1−λ)γj+λ.They showed that arrivals are independent if λ = 0, and simultaneous ifλ = 1. Given the realization of the arrival vectors s1j,t and s2j,t, the construc-tion of the volatility components Mj,t is based on a bivariate distributionM = (M1,M2). If arrivals hit both series (s1j,t = s2j,t = 1), the state vectorMj,t is drawn from M. If only one series i (i = 1, 2) receives an arrival, thenew component M i
j,t is sampled from the marginal M i of the bivariate distri-bution M. Finally, Mj,t = Mj,t−1 if there is no arrival (s1j,t = s2j,t = 0). Theyassumed that M has a bivariate binomial distribution controlled by m1 andm2, in parallel fashion to the univariate case. Again, the closed-form solutionof the likelihood function is available. This approach can be extended to ageneral multivariate case. As the number of parameters therefore grows atleast as fast as a quadratic function of p, Calvet et al. (2006) proposed notonly the multivariate MSM model but also the factor MSM model.
The factor MSM model based on q volatility factors f lt = (f l1,t, . . . , f lk,t)′,
where the weights are nonnegative and add up to 1, and the constant Ci ischosen to guarantee that E(M i
j,t) = 1, and is thus not a free parameter. Cal-vet et al. (2006) specified the model as follows. For each vector f lt , f lj,t followsa univariate MSM process with parameters (b, γ,ml). The volatility of eachasset i is also affected by an idiosyncratic shock ui
t = (ui1,t, . . . , uik,t)
′, whichis specified by parameters (b, γ,mq+i). Draws of the factors f lj,t and idiosyn-cratic shocks uij,t are independent, but timing of arrivals may be correlated.Factors and idiosyncratic components thus follow the univariate MSM withidentical frequencies.
3.2 Mean factor model
Another type of factor MSV model is considered in Pitt and Shephard (1999),who, following a model proposed in Kim et al. (1998), worked with the spec-ification
and ht = (h1t, . . . , hpt, hp+1,t, ..., hp+q,t). For identification purposes, the p×qloading matrix B is assumed to be such that bij = 0 for (i < j, i ≤ q) andbii = 1 (i ≤ q) with all other elements unrestricted. Thus, in this model,each of the factors and each of the errors evolve according to univariate SVmodels. A similar model was also considered by Jacquier et al. (1999) andLiesenfeld and Richard (2003) but under the restriction that Vt is not time-varying. Jacquier et al. (1999) estimated their model by MCMC methods,sampling hit one at a time from its full conditional distribution, whereasLiesenfeld and Richard (2003) showed how the maximum likelihood estima-tion can be obtained by the efficient importance sampling method. For themore general model described above, Pitt and Shephard (1999) also employeda MCMC-based approach, now sampling ht along the lines of Shephard andPitt (1997). An even further generalization of this factor model was devel-oped by Chib et al. (2006), who allowed for jumps in the observation modeland a fat-tailed t distribution for the errors εt. The resulting model and itsfitting are explained in Section 3.3. Alternative interesting approaches werealso proposed by Diebold and Nerlove (1989) and King et al. (1994) in theframework of GARCH models using Kalman filter algorithms, but we omitthe details to focus on the MSV models.
Lopes and Carvalho (2007) considered a general model which nests themodels of Pitt and Shephard (1999) and Aguilar and West (2000), and ex-tended it in two directions by (1) letting the matrix of factor loadings Bbe time-dependent and (2) allowing Markov switching in the common factorvolatilities. The general model is given by (19)–(22), with
. . . , φp+q) and Σfηη is the nondiagonal covariance matrix. Letting the pq −
q(q + 1)/2 unconstrained elements of vec(Bt) be bt = (b21,t, b31,t, . . . , bpq,t)′,they assumed that each element of bt follows an AR(1) process. Following
384 S. Chib, Y. Omori and M. Asai
So et al. (1998), where the fitting was based on the work of Albert andChib (1993), they assumed μst followed a Markov switching model, where stfollows a multistate first-order Markovian process. Lopes and Carvalho (2007)applied this model to two datasets: (1) returns on daily closing spot ratesfor six currencies relative to the US dollar (Deutschemark, British pound,Japanese yen, French franc, Canadian dollar, Spanish peseta), and (2) returnson daily closing rates for four Latin American stock markets indices. In theformer application, they used q = 3 factors and in the latter case q = 2factors.
Han (2006) modified the model of Pitt and Shephard (1999) and Chibet al. (2006) by allowing the factors to follow an AR(1) process:
ft = c + Aft−1 + D1/2t γt, γt ∼ Nq(0, I). (25)
The model was fit by adapting the approach of Chib et al. (2006) and appliedto a collection of 36 arbitrarily chosen stocks to examine the performance ofvarious portfolio strategies.
3.3 Bayesian analysis of mean factor MSV model
We describe the fitting of factor models in the context of the general modelof Chib et al. (2006). The model is given by
yt = Bft + Ktqt + V1/2t Λ−1
t εt, εt ∼ Np(0, I), (26)
where Λt = diag(λ1t, . . . , λpt), qt is p independent Bernoulli “jump" randomvariables and Kt = diag(k1t, . . . , kpt) are jump sizes. Assume that each el-ement qjt of qt takes the value 1 with probability κj and the value 0 withprobability 1 − κj , and that each element ujt of ut = V1/2
t Λ−1t εt follows an
independent Student t distribution with degrees of freedom νj > 2, which weexpress in hierarchical form as
ujt = λ−1/2jt exp(hjt/2)εjt, λjt
i.i.d.∼ G(νj
2,νj2
)
, t = 1, 2, . . . , n. (27)
The εt and ft are assumed to be independent and(
V1/2t εtft
)
|Vt,Dt,Kt,qt ∼ Np+q
{
0,(
Vt OO Dt
)}
are conditionally independent Gaussian random vectors. The time-varyingvariance matrices Vt and Dt are defined by (20) and (21). Chib et al. (2006)assumed that the variables ζjt = ln(1 + kjt), j ≤ p, are distributed asN (−0.5δ2j , δ
2j ), where δ = (δ1, . . . , δp)′ are unknown parameters.
Multivariate Stochastic Volatility 385
We may calculate the number of parameters and latent variables as fol-lows. Let β denote the free elements of B after imposing the identifyingrestrictions. Let Σηη = diag(σ2
1 , . . . , σ2p) and Σf
ηη = diag(σ2p+1, . . . , σ
2p+q).
Then there are pq − (q2 + q)/2 elements in β. The model has 3(p + q) pa-rameters θj = (φj , μj , σj) (1 ≤ j ≤ p + q) in the autoregressive processes(20) of {hjt}. We also have p degrees of freedom ν = (ν1, . . . , νp), p jumpintensities κ = (κ1, . . . , κp) and p jump variances δ = (δ1, . . . , δp). If we letψ = (β, θ1, . . . , θp, ν, δ, κ) denote the entire list of parameters, then the di-mension of ψ is 688 when p = 50 and q = 8. Furthermore, the model containsn(p+ q) latent volatilities {ht} that appears nonlinearly in the specificationof Vt and Dt, 2np latent variables {qt} and {kt} associated with the jumpcomponent and np scaling variables {λt}.
To conduct the prior-posterior analysis of this model, Chib et al. (2006) fo-cused on the posterior distribution of the parameters and the latent variables:
where the notation zj. is used to denote the collection (zj1, . . . , zjn). They
sampled this distribution by MCMC methods through the following steps:
1. Sample β. The full conditional distribution of β is given by
π(β|Yn, {hj.}, {ζj.}, {qj.}, {λj.}) ∝ p(β)n∏
t=1
Np(yt|Ktqt,Ωt),
where p(β) is the normal prior,
Ωt = V∗t + BDtB′ and V∗
t = Vt � diag(λ−11t , . . . , λ
−1pt ).
To sample from this density, Chib et al. (2006) employed the M-H algo-rithm (Chib and Greenberg (1995)), following Chib and Greenberg (1994)and taking the proposal density to be multivariate-t, T (β|m,Σ, v), wherem is the approximate mode of l = ln{
∏nt=1 Np(yt|Ktqt,Ωt)}, and Σ is
minus the inverse of the second derivative matrix of l. Then, a proposalvalue β∗ is drawn from T (m,Σ, v) and accepted with probability
α(β, β∗|y, {hj.}, {λj.})
= min{
1,p(β∗)
∏nt=1 Np (yt|0,V∗
t + B∗DtB∗′)T (β|m,Σ, v)p(β)
∏nt=1 Np ((yt|0,V∗
t + BDtB′)T (β∗|m,Σ, v)
}
,
where β is the current value. If the proposal value is rejected, the nextitem of the chain is taken to be the current value β.
2. Sample {ft}. The distribution {ft}|Yn,B,h, λ can be divided into theproduct of the distributions ft|yt,ht,h
ft , λt,B, which have Gaussian dis-
tribution with mean ft = FtB′(V∗t )−1yt and variance Ft =
{
B′(V∗t )−1B
+D−1t
}−1.
386 S. Chib, Y. Omori and M. Asai
3. Sample {θj} and {hj.}. Given {ft} and the conditional independenceof the errors in (20), the model separates into q conditionally Gaussianstate-space models. Let
where c is an “offset" constant that is set to 10−6. Then from Kim et al.(1998) it follows that the p+ q state-space models can be subjected to anindependent analysis for sampling the {θj} and {hj.}. In particular, thedistribution of zjt, which is hjt plus a logχ2 random variable with onedegree of freedom, may be approximated closely by a seven-componentmixture of normal distributions:
zjt|sjt, hjt ∼ N(
hjt +msjt , v2sjt
)
,
hj,t+1 − μj = φj (hj,t − μj) + ηjt, j ≤ p+ q,
where sjt is a discrete component indicator variable with mass functionPr(sjt = i) = qi, i ≤ 7, t ≤ n, and msjt , v2
sjtand qi are parameters
that are reported in Chib et al. (2002). Thus, under this representation,conditioned on the transformed observations
p ({sj.}, θ, {hj.}|z) =p+q∏
j=1
p (sj., θj ,hj.|zj.) ,
which implies that the mixture indicators, log volatilities and series-specific parameters can be sampled series by series. Now, for each j, onecan sample (sj., θj ,hj.) by the univariate SV algorithm given by Chibet al. (2002). Briefly, sj. is sampled from
p (sj.|zj.,hj.) =n∏
t=1
p (sjt|zjt, hjt) ,
where p(sjt|zjt, hjt) ∝ p(sjt)N(
zjt|hjt +msjt , v2sjt
)
is a mass functionwith seven points of support. Next, θj is sampled by the M-H algorithmfrom the density π(θj |zj., sj.) ∝ p(θj)p(zj.|sj., θj), where
p (zj.|sj., θj) = p (zj1|sj., θj)n∏
t=2
p(
zjt|F∗j,t−1, sj., θj
)
(29)
and p(zjt|F∗j,t−1, sj., θj) is a normal density whose parameters are ob-
tained by the Kalman filter recursions, adapted to the differing compo-nents, as indicated by the component vector sj.. Finally, hj. is sampled
Multivariate Stochastic Volatility 387
from [hj.|zj., sj., θj ] by the simulation smoother algorithm of de Jong andShephard (1995).
4. Sample {νj}, {qj.} and {λj.}. The degrees-of-freedom parameters, jumpparameters and associated latent variables are sampled independently foreach time series. The full conditional distribution of νj is given by
Pr(νj |yj.,hj ,B, f ,qj., ζj.) (30)
∝ Pr(νj)n∏
t=1
T (yjt|αjt + {exp(ζjt) − 1}qjt, exp(hjt), νj),
and one can apply the Metropolis-Hastings algorithm in a manner anal-ogous to the case of β. Next, the jump indicators {qj.} are sampled fromthe two-point discrete distribution,
followed by the components of the vector {λj.} from the density
λjt|yjt, hjt,B, f , νj , qjt, ψjt
∼ G(
νj + 12
,νj + (yjt − αjt − (exp(ζjt) − 1)qjt))2
2 exp(hjt)
)
.
5. Sample {δj} and {ζj.}. For simulation efficiency reasons, δj and ζj. mustalso be sampled in one block. The full conditional distribution of δj isgiven by
π(δj)n∏
t=1
N(αjt − 0.5δ2j qjt, δ2j q
2jt + exp(hjt)λ−1
jt ) (31)
by the M-H algorithm. Once δj has been sampled, the vectors ζj.are sampled, bearing in mind that their posterior distribution is up-dated only when qjt is 1. Therefore, when qjt is 0, we sample ζjt fromN (−0.5δ2j , δ
2j ), otherwise we sample from the distribution N (Ψjt(−0.5 +
exp(−hjt)λjtyjt), Ψjt), where Ψjt = (δ−2j + exp(−hjt)λjt)−1. The algo-
rithm is completed by sampling the components of the vector κ indepen-dently from κj |qj. ∼ β(u0j + n1j , u1j + n0j), where n0j is the count ofqjt = 0 and n1j = n− n0j is the count of qjt = 1.
388 S. Chib, Y. Omori and M. Asai
A complete cycle through these various distributions completes one tran-sition of our Markov chain. These steps are then repeated G times, where Gis a large number, and the values beyond a suitable burn-in of say a 1,000cycles are used for the purpose of summarizing the posterior distribution.
4 Dynamic Correlation MSV Model
Another way to model time-varying correlations is by constructing modelsthat model the correlations (or functions of correlations) directly. We describeseveral such approaches in this section.
4.1 Modeling by reparameterization
One approach is illustrated by Yu and Meyer (2006) in the context of thebivariate SV model:
yt = V1/2t εt, εt ∼ N2(0,Σεε,t), Σεε,t =
(
1 ρtρt 1
)
,
ht+1 = μ+ diag(φ1, φ2)(ht − μ) + ηt, ηt ∼ N2
(
0, diag(σ21 , σ
22))
,
qt+1 = ψ0 + ψ1(qt − ψ0) + σρvt, vt ∼ N (0, 1),
ρt =exp(qt) − 1exp(qt) + 1
,
where h0 = μ and q0 = ψ0. The correlation coefficient ρt is then obtainedfrom qt by the Fisher transformation. Yu and Meyer (2006) estimated thismodel by MCMC methods with the help of the WinBUGS program and foundthat it was superior to other models, including the mean factor MSV model.However, the generalization of this bivariate model to higher dimensions is noteasy because it is difficult to ensure the positive-definiteness of the correlationmatrix Σεε,t.
Another approach, introduced by Tsay (2005), is based on the Choleskidecomposition of the time-varying correlation matrix. Specifically, one canconsider the Choleski decomposition of the correlation matrix Σεε,t such thatCov(yt|ht) = LtVtL′
t. The outcome model is then given by yt = LtV1/2t εt,
εt ∼ Np(0, I). As an example, when bivariate outcomes are involved we have
which shows that the distribution of yt is modeled sequentially. We first lety1t ∼ N (0, exp(h1t)) and then we let y2t|y1t ∼ N (qty1t, exp(h2t)). Thus qt isa slope of conditional mean and the correlation coefficient between y1t andy2t is given by
qi,t+1 = ψi,0 + ψi,1(qi,t − ψ0) + σi,ρvit, vit ∼ N (0, 1).
Jungbacker and Koopman (2006) considered a similar model with Lt = L andestimated the parameters of the model by the Monte Carlo likelihood method.As in the one-factor case, they used the data set for the daily exchange ratereturns of the British pound, the Deutschemark and the Japanese yen againstthe US dollar.
4.2 Matrix exponential transformation
For any p× p matrix A, the matrix exponential transformation is defined bythe following power-series expansion:
exp(A) ≡∞∑
s=0
1s!
As,
where A0 is equal to a p × p identity matrix. For any real positive-definitematrix C, there exists a real symmetric p× p matrix A such that
C = exp(A).
Conversely, for any real symmetric matrix A, C = exp(A) is a positive-definite matrix; see Lemma 1 of Chiu et al. (1996) and Kawakatsu (2006). IfAt is a p× p real symmetric matrix, there exists a p× p orthogonal matrixBt and a p× p real diagonal matrix Ht of eigenvalues of A such that At =BtHtB′
t and
exp(At) = Bt
( ∞∑
s=0
1s!
Hst
)
B′t = Bt exp(Ht)B′
t .
Thus, we consider the matrix exponential transformation for the covariancematrix Var(yt) = Σt = exp(At), where At is a p× p real symmetric matrixsuch that At = BtHtB′
as suggested in Asai et al. (2006). The estimation of this model can be doneusing a MCMC or a simulated maximum likelihood estimation, but it is notstraightforward to interpret the parameters.
4.3 Wishart process
4.3.1 Standard model
Another way to obtain a time-varying correlation matrix is by the approachof Philipov and Glickman (2006a, 2006b), who assumed that the conditionalcovariance matrix Σt follows an inverted Wishart distribution with parame-ters that depend on the past covariance matrix Σt−1. In particular,
yt|Σt ∼ Np(0,Σt),Σt|ν,St−1 ∼ IWp(ν,St−1),
where IW(ν0,Q0) denotes an inverted Wishart distribution with parameters(ν0,Q0),
St−1 =1νA1/2
(
Σ−1t−1
)dA1/2′, (32)
A = A1/2A1/2′,
and A1/2 is a Choleski decomposition of a positive-definite symmetric matrixA and −1 < d < 1. Asai and McAleer (2007) pointed out that it is alsopossible to parameterize St−1 as ν−1
(
Σ−1t−1
)d/2A
(
Σ−1t−1
)d/2′.The conditional expected values of Σ−1
t and Σt are
392 S. Chib, Y. Omori and M. Asai
E(
Σ−1t |ν,St−1
)
= νSt−1 = A1/2(
Σ−1t−1
)dA1/2′,
E (Σt|ν,St−1) =1
ν − p− 1S−1t−1 =
ν
ν − p− 1A−1/2 (Σt−1)
d A−1/2′,
respectively. Thus, the scale parameter d expresses the overall strength ofthe serial persistence in the covariance matrix over time. On the basis ofthe process of the logarithm of the determinant and asymptotic behavior ofexpectation of the determinant, they assumed that |d| < 1, although it isnatural to assume that 0 < d < 1. Notice that when d = 0, for example, theserial persistence disappears and we get
E(
Σ−1t |ν,St−1
)
= A,
E (Σt|ν,St−1) =ν
ν − p− 1A−1.
The matrix A in this model is a measure of the intertemporal sensitivityand determines how the elements of the current period covariance matrix Σt
are related to the elements of the previous period covariance matrix. WhenA = I, we note that
E(
Σ−1t |ν,St−1
)
=
⎧
⎨
⎩
Σ−1t−1, d = 1,
I, d = 0,Σt−1, d = −1.
Philipov and Glickman (2006b) estimated this model from a Bayesian ap-proach and proposed a MCMC algorithm to estimate their models usingmonthly return data of five industry portfolios (manufacturing, utilities, re-tail/wholesale, financial and other) in NYSE, AMEX and NASDAQ stocks.Under the prior
A ∼ IWp(ν0,Q0), d ∼ π(d), ν − p ∼ G(α, β)
with Σ0 assumed known, the MCMC algorithm is implemented as follows:
1. Sample Σt|{Σs}s�=t,A, ν, d, Yn (t = 1, . . . , n−1), where Yn = {y1, . . . ,yn}.Given a current sampler Σt, we generate a candidate Σ∗
t ∼ Wp(ν, St−1),where Wp(ν, St−1) denotes a Wishart distribution with parameters (ν,St−1),
ν = ν(1 − d) + 1,
St−1 = S−1t−1 + yty′
t,
St−1 =1ν
(A1/2)(
Σ−1t−1
)d(A1/2)′,
and accept it with probability
Multivariate Stochastic Volatility 393
min
⎧
⎨
⎩
|Σ∗t |(νd−1)/2 exp
[
− 12 tr
{
νA−1 (Σ∗t )
−d Σ−1t+1
}]
|Σt|(νd−1)/2 exp[
− 12 tr
{
νA−1 (Σt)−d Σ−1
t+1
}] , 1
⎫
⎬
⎭
.
2. Sample Σn|{Σt}n−1t=1 ,A, ν, d, Yn ∼ Wp(ν, Sn−1).
3. Sample A|{Σt}nt=1, ν, d,y ∼ IWp(γ, Q), where γ = nν + ν0, and
Q−1 = ν
{
n∑
t=1
(
Σ−1t
)−d/2Σ−1
t
(
Σ−1t−1
)−d/2
}
+ Q−10 .
4. Sample d from
π(d|{Σt}nt=1,A, ν,y)
∝ π(d) exp
[
νd
2
n∑
t=1
log |Σt| −12
n∑
t=1
tr{
S−1t
(
Σ−1t−1
)−d}
]
.
To sample d, Philipov and Glickman (2006b) suggested discretizing theconditional distribution; see Appendix A.2 of Philipov and Glickman(2006b). Alternatively, we may conduct an independent M-H algorithmusing a candidate from a truncated normal distribution T N (0,1)(d, Vd),where T N (a,b)(μ, σ2) denote a normal distribution with mean μ and vari-ance σ2 truncated on the interval (a, b), d is a mode of conditional pos-terior probability density π(d|{Σt}nt=1,A, ν,y) and
Vd ={
− ∂2 log π(d|{Σt}nt=1,A, ν, Yn)∂d2
∣
∣
∣
∣
d=d
}−1
.
5. Sample ν from
π(ν|{Σt}nt=1,A, d,y)
∝(ν − p)α−1 exp{−β(ν − p)}{
|νA−1|ν/2
2νp∏p
j=1 Γ (ν+j−12 )
}n
× exp
[
−ν2
n∑
t=1
{
log |Qt| + tr(
A−1Q−1t
)}
]
.
As in the previous step, we may discretize the conditional distribution orconduct an independent M-H algorithm using a candidate from a trun-cated normal distribution T N (p,∞)(ν, Vν), where ν is a mode of condi-tional posterior probability density π(ν|{Σt}nt=1,A, d,y) and
Vν ={
− ∂2 log π(ν|{Σt}nt=1,A, d, Yn)∂ν2
∣
∣
∣
∣
ν=ν
}−1
.
394 S. Chib, Y. Omori and M. Asai
Asai and McAleer (2007) proposed two further models that are especiallyuseful in higher dimensions. Let Qt be a sequence of positive-definite matrices,which is used to define the correlation matrix Σεε,t = Q∗−1/2
t QtQ∗−1/2t ,
where Q∗t is a diagonal matrix whose (i, i)th element is the same as that of
Qt. Then the first of their dynamic correlation (DC) MSV models is givenby
Thus, in this model the MSV shocks are assumed to follow a Wishart pro-cess, where Wp(ν,Λ) denotes a Wishart distribution with degrees-of-freedomparameter ν and scale matrix Λ. The model guarantees that Pt is sym-metric positive-definite under the assumption that Q is positive-definite and|ψ| < 1. It is possible to consider a generalization of the model by lettingQt+1 = (11′ −Ψ)� Q+Ψ�Qt +Ξt, which corresponds to a generalizationof the dynamic conditional correlation (DCC) model of Engle (2002).
The second DC MSV model is given by
Qt+1|ν,St ∼ IWp(ν,St), St =1νQ−d/2
t AQ−d/2t ,
where ν and St are the degrees of freedom and the time-dependent scaleparameter of the Wishart distribution, respectively, A is a positive-definitesymmetric parameter matrix, d is a scalar parameter and Q−d/2
t is definedby using a singular value decomposition. The quadratic expression, togetherwith ν ≥ p, ensures that the covariance matrix is symmetric and positive-definite. For convenience, it is assumed that Q0 = Ip. Although their modelis closely related to the models of Philipov and Glickman (2006a, 2006b), theMCMC fitting procedures are different. Asai and McAleer (2007) estimatedthese models using returns of the Nikkei 225 Index, the Hang Seng Index andthe Straits Times Index.
Gourieroux et al. (2004) and Gourieroux (2006) used an alternative ap-proach and derived a Wishart autoregressive process. Let Yt and Γ denote,respectively, a stochastic symmetric positive-definite matrix of dimension p×pand a deterministic symmetric matrix of dimension p× p. A Wishart autore-gressive process of order 1 is defined to be a matrix process (denoted byWAR(1) process) with a conditional Laplace transform:
Ψt(Γ) = Et [exp{tr(ΓYt+1)}] =exp
[
tr{
M′−1MYt
}]
|I − 2ΣΓ|k/2 , (33)
where k is a scalar degree of freedom (k < p − 1), M is a p × p matrix ofautoregressive parameters and Σ is a p × p symmetric and positive-definite
Multivariate Stochastic Volatility 395
matrix such that the maximal eigenvalue of 2ΣΓ is less than 1. Here Et
denotes the expectation conditional on {Yt,Yt−1, . . . , }. It can be shownthat
Yt+1 = MYtM′ + kΣ + ηt+1,
where E(ηt+1) = O. The conditional probability density function of Yt+1 isgiven by
f(Yt+1|Yt) =|Yt+1|(k−p−1)/2
2kp/2Γp(k/2)|Σ|k/2 exp[
−12tr
{
Σ−1(Yt+1 + MYtM′)}
]
×0 F1(k/2; (1/4)MYtM′Yt+1),
where Γp is the multidimensional gamma function and 0F1 is the hypergeo-metric function of matrix augment; see Gourieroux et al. (2004) for details.When K is an integer and Yt is a sum of outer products of k independentvector AR(1) processes such that
Yt =k
∑
j=1
xjtx′jt, (34)
xjt = Mxj,t−1 + εjt, εjt ∼ Np(0,Σ),
we obtain the Laplace transform Ψt(Γ) given by (33). Gourieroux et al. (2004)also introduced a Wishart autoregressive process of higher order. They esti-mated the WAR(1) using a series of intraday historical volatility–covolatilitymatrices for three stocks traded on the Toronto Stock Exchange. Finally,Gourieroux (2006) introduced the continuous-time Wishart process as themultivariate extension of the Cox–Ingersoll–Ross (CIR) model in Cox et al.(1985).
4.3.2 Factor model
Philipov and Glickman (2006a) proposed an alternative factor MSV modelthat assumes that the factor volatilities follow an unconstrained Wishartrandom process. Their model has close ties to the model in Philipov andGlickman (2006b), and is given by
where St−1 is defined by (32). In other words, the conditional covariancematrix Σt of the factor ft follows an inverse Wishart distribution whose pa-rameter depends on the past covariance matrix Σt−1. They implemented themodel with q = 2 factors on return series data of 88 individual companiesfrom the S&P500.
396 S. Chib, Y. Omori and M. Asai
In another development, Carvalho and West (2006) proposed dynamicmatrix-variate graphical models, which are based on dynamic linear modelsaccommodated with the hyperinverse Wishart distribution that arises in thestudy of graphical models (Dawid and Lauritzen (1993), Carvalho and West(2006)). The starting point is the dynamic linear model
y′t = X′
tΘt + u′t, ut ∼ Np(0, vtΣ),
Θt = GtΘt−1 + Ωt, Ωt ∼ Nq×p(O,Wt,Σ),
where yt is the p × 1 vector of observations, Xt is a known q × 1 vector ofexplanatory variables, Θt is the q × p matrix of states, ut is the p× 1 inno-vation vector for observation, Ωt is the q × p innovation matrix for states,Gt is a known q× q matrix and Σ is the p× p covariance matrix. Ωt followsa matrix-variate normal with mean O (q × p), left covariance matrix Wt
and right covariance matrix Σ; in other words, any column ωit of Ωt has amultivariate normal distribution Nq(0, σiiWt), while any row ωit of Ωt, ωi′thas a multivariate normal distribution Np(0, wii,tΣ). Next, we suppose thatΣ ∼ HIWp(b,D), the hyperinverse Wishart distribution with a degree-of-freedom parameter b and location matrix D. It should be noted that the dy-namic linear model with Σ ∼ HIWp(b,D) can be handled from the Bayesianperspective without employing simulation-based techniques. Finally, insteadof time-invariant Σ, Carvalho and West (2006) suggested a time-varying pro-cess given by
Σt ∼ HIWp(bt,St),bt = δbt−1 + 1,St = δSt−1 + vtv′
t,
where vt is defined by Theorem 1 of Carvalho and West (2006). Intuitively,vt is the residual from the observation equation. As Σt appears in both ofthe observation and state equations, the proposed dynamic matrix-variategraphical model can be considered as a variation of the “factor MSV modelwith MSV error.” Setting δ = 0.97, Carvalho and West (2006) applied thedynamic matrix-variate graphical models to two datasets; namely, (1) 11international currency exchange rates relative to the US dollar and (2) 346securities from the S&P500 stock index.
5 Conclusion
We have conducted a comprehensive survey of the major current themes inthe formulation of MSV models. In time, further significant developments canbe expected, perhaps fostered by the overview and details delineated in thispaper, especially in the fitting of high-dimensional models. Open problems
Multivariate Stochastic Volatility 397
remain, primarily in the modeling of leverage effects, especially in relation togeneral specifications of cross-leverage effects embedded within multivariateheavy-tailed or skewed error distributions. We also expect that interest inthe class of factor-based MSV models and DC models will grow as theseapproaches have shown promise in the modeling of high-dimensional data.
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