Dynamic Correlations in Symmetric Multivariate SV Models - CiteSeerX

Dynamic Correlations in Symmetric Multivariate SV Models Manabu Asai1 and Michael McAleer2 1

Faculty of Economics, Soka University, Japan, 2School of Economics and Commerce, University of Western Australia

Keywords: Multivariate stochastic volatility, Constant correlation structure, Dynamic correlation structure, Markov chain Monte Carlo. EXTENDED ABSTRACT This paper proposes two types of stochastic correlation structures for Multivariate Stochastic Volatility (MSV) models, namely the constant correlation (CC) MSV and dynamic correlation (DC) MSV models, from which the stochastic covariance structures could be obtained easily. Both structures can be used for purposes of optimal portfolio and risk management, and for calculating Value-at-Risk (VaR) forecasts and optimal capital charges under the Basel Accord. The choice between the CC MSV and DC MSV models can be made using a deviance information criterion. A technique is developed to estimate the DC MSV model using the Markov Chain Monte Carlo (MCMC) procedure.

2202

1.

positive

Introduction

definiteness

of

the

conditional

covariance matrix. Recently, Engle (2002) Covariance and correlation structures are used

developed the Dynamic Conditional Correlation

routinely for optimal portfolio choice, risk

(DCC) model, which allows the conditional

management, obtaining Value-at-Risk (VaR)

correlation matrix to vary parsimoniously over

forecasts, and determining optimal capital

time. McAleer (2005) provides a comprehensive

charges under the Basel Accord. This issue does

comparison of a wide range of univariate and

not yet seem to have been examined in the

multivariate,

Multivariate

financial volatility models.

Stochastic

Volatility

(MSV)

conditional

and

stochastic,

literature. This paper proposes two types of stochastic For multivariate GARCH models, the most

correlation structures for MSV models, namely

general expression is called the ‘vec’ model (see

the constant correlation (CC) MSV and dynamic

Engle and Kroner (1995)). The vec model

correlation (DC) MSV models, from which the

parameterizes

stochastic

the

vector

of

conditional

covariance

structures

could

be

covariance matrix of the returns vector, which is

obtained easily. Both structures can be used for

determined by its lags and the vector of outer

purposes

products of the lagged returns vector. A serious

management, for calculating VaR forecasts, and

issue with the vec model is that it has many

for determining optimal Basel Accord capital

parameters to be estimated, and will not

charges. The choice between the CC MSV and

guarantee positive definiteness of the conditional

DC MSV models can be made using a deviance

covariance matrix without further restrictions.

information criterion (see Berg et al. (2004)). A

Bollerslev et al. (1988) suggested the diagonal

technique is developed to estimate the DC MSV

GARCH model, which restricts the off-diagonal

model using the Markov Chain Monte Carlo

elements of the parameters matrices to be zero,

(MCMC) procedure.

of

optimal

portfolio

and

risk

and also reduced the number of parameters 2

drastically. Engle and Kroner (1995) proposed

Constant and Dynamic Correlations

the Baba, Engle, Kraft and Kroner (or BEKK) positive

This section develops two types of correlation

definiteness of the conditional covariance matrix.

models for MSV models, namely the CC MSV

This is essential for obtaining sensible VaR

and DC MSV models, from which the stochastic

forecasts. Bollerslev (1990) proposed the

covariance structures may be obtained easily.

specification

that

guaranteed

the

Constant Conditional Correlation (CCC) model, where

the

proportional

time-varying to

the

covariances

conditional

2.1

are

Constant Correlation MSV Models

standard

deviation derived from univariate GARCH

Consider the constant correlation (CC) MSV

processes. This specification also guarantees the

model proposed by Harvey, Ruiz and Shephard

2203

yt be an M × 1 vector of

(1994). Let

2 E ( ε tηt′ ) = diag {κ1σ η1/,112 ,..., κ mσ η1/, mm },

stochastic process, as follows: yt = Dt ε t , ε t : N ( 0, Γ ) ,

(1)

Dt = diag {exp ( ht / 2 )} ,

(2)

where σ η ,ii is the ( i, i )th element of Ση , and

κ i is restricted to be negative. The model is a multivariate extension of Harvey and Shephard

ht +1 = μ + φ o( ht − μ ) + ηt , ηt : N ( 0, Ση ) , (3)

(1996), and has been analysed in Asai and

where the operator ‘ o ’ denotes the Hadamard

incorporate yt and

element-by-element

follows:

product

of

McAleer (2004). The other approach is to

two

yt

in equation (3), as

ht +1 = μ + φ o( ht − μ ) + λ1 o yt

identically-sized matrices or vectors. Here, we use ‘exp’ as the operator for vectors to denote

+λ2 o yt + ηt , ηt : N ( 0, Ση ) .

element-by-element exponentiation, and ‘diag’

(4)

for vectors to create a diagonal matrix. This

This model was proposed by Asai and McAleer

model deals with two kinds of correlation

(2004) to develop the multivariate extension of

matrices, namely the correlations between mean

the SV model suggested by Danielsson (1994).

variables, Γ , and the covariance matrix of

In the Bayesian framework, Yu (2005) and

volatility, Ση . It should be noted that the model

Omori et al. (2004) estimated the univariate

is different from the Constant Conditional

model of Harvey and Shephard (1996) using the

Correlation (CCC) model of Bollerslev (1990)

MCMC technique.

with respect to two major points, namely (i) the volatilities

are

stochastic,

and

(ii)

the

As the purpose of the present paper is to consider

disturbances of the volatilities are correlated

the stochastic correlation MSV model, namely

simultaneously.

DC MSV, we do not consider asymmetric models,

In order to obtain the VaR forecasts for a given

although

such

extensions

are

straightforward for the CC MSV model.

portfolio and to determine the optimal capital charges for a portfolio under the Basel Accord, it

2.2

Dynamic Correlation MSV Models

would be necessary to calculate the stochastic covariance matrix, Σt , from the correlation

This subsection develops a dynamic correlation

matrix, as Σt = Dt ΓDt .

(DC) MSV model, as an extension of the CC MSV model, which is based on the differences

There are two ways in which to deal with the

between the ε it . Our approach is based on the

so-called ‘leverage effects’. One approach is to

Wishart distribution. It is known that the Wishart

extend the model (1)-(3) in order to assume

distribution of a sample variance covariance

negative correlations between returns and

matrix

changes of volatilities as follows:

Gaussian observation (see Anderson (1984) and

2204

computed

from

i.i.d.

multivariate

Stuart and Ord (1994)). By considering the

MSV model, it is necessary to calculate the

serially dependent Wishart Process, we have a

stochastic covariance matrix, Σt , from the

process of positive definite matrices, under

stochastic correlation matrix, as Σt = Dt Γt Dt .

appropriate conditions. A standardization of the process leads to a serially dependent process of

In order to investigate the properties of the DC

correlation matrices.

MSV model, we consider the case of ν = 1 , for simplicity. In this case, Ξ t can be expressed as Ξ t = ξt ξt′ , where ξ t : N ( 0, Λ ) . Then we have

Let ε t have a multivariate normal distribution, N ( 0, Γ t ) ,

conditional

on

the

stochastic Qt +1 = Ω + ψ Qt + ξt ξt′,

correlation matrix, Γt , where Γt = Qt∗−1Qt Qt∗−1 , Qt +1 = Ω + ψ Qt + Ξ t , Ξ t ~ Wm (ν , Λ ) , Qt∗ = ( diagonal {Qt } )

1/ 2

,

(5)

(6)

which is analogous to the DCC model of Engle (2002), namely

and Wm (ν , Λ ) denotes a Wishart distribution, Qt +1 = Ω + α Qt + βε t ε t′.

and ‘diagonal’ creates a diagonal matrix by setting the off-diagonal elements of matrices to be zero. The DC MSV model guarantees the positive

definiteness

of

Γt

the

given above, that of the DCC model is

definite,

predetermined and can be observed by using

0 < ψ < 1 , and that Ω and Λ are symmetric

estimated conditional volatility, while that of the

positive definite matrices. The second condition

DC MSV model is unobservable. Furthermore,

also implies that the time-varying stochastic

letting qt = vec ( Qt ) , we have a VAR(1) process

correlations are mean reverting. Given Q1 is

for qt , as follows:

assumption

that

Qt

is

under

Comparing the last terms in the two equations

positive

positive definite, Qt is guaranteed to be positive qt +1 = (ω + λ ) + ψ qt + vec (ξ t ξt′ − Λ ) ,

definite (the proof is obtained in a similar

(7)

manner to that of the DCC model - see below for the case of ν = 1 ). By the positive definiteness

where ω = vec ( Ω ) and λ = vec ( Λ ) . Since the

of Qt , Γt will also be positive definite. With

expectation of the last term is zero, it is possible

the above dynamic and stochastic structures, the

to

DC MSV model for yt = Dt ε t is defined by (2),

obtain

E ( qt ) = (1 −ψ )

(3) and (5).

E ( Qt ) = (1 −ψ )

As in the case of the CC MSV model, in order to

shown that

obtain the VaR forecasts for a given portfolio and

V ( qt ) = (1 −ψ )

to determine the optimal capital charges for a

−1

−2

−1

(ω + λ )

,

or

(Ω + Λ) .

Similarly, it can be

(I

)(Λ ⊗ Λ),

m2

+ K mm

where K ij is the commutation matrix. Noticing

portfolio under the Basel Accord for the DC

2205

that Ξ t = ξt ξt′ , the other moments can be

Gallant and Tauchen (1998). Asai and McAleer

obtained through tedious calculation by using the

(2004) and Liesenfeld and Richard (2003)

moment of the multivariate normal distribution.

estimated MSV models by using the importance

When ν = 1 , it may be natural to assume that

sampling procedures. Although we have to deal

Q1 = (1 −ψ )

−1

with the latent process of Qt , which has Wishart

( Ω + Ξ 0 ) , since its mean is equal

disturbances, the importance sampling approach

to the unconditional mean, and as it is a positive

is inapplicable when the latent process is

definite matrix.

non-Gaussian. Furthermore, the reprojection method needs to be extended further for

In the following, we consider ν = 1

for

application to MSV models.

convenience as (i) it is easy to interpret in this case, and (ii) an approximated Gaussian process

Let θ1 denote the vector of parameters to be

of (7) based on ξt is used to develop the

estimated in the latent process for the dynamic

MCMC technique.

correlations,

Qt , and θ 2

the vector of

parameters for the volatility processes, ht . 3

Given the prior density, π (θ1 , θ 2 ) , the aim of

Bayesian MCMC: An Overview

Bayesian inference is to obtain the parameter In this section, we develop a technique to

vector, (θ1 , θ 2 ) , from the augmented posterior

estimate the DC MSV model via Markov Chain

distribution, namely.

Monte Carlo (MCMC) procedure (see, for example, Chib and Greenberg (1996)). For

π (θ1 , θ 2 , Q, h | y ) ∝ π (θ1 , θ 2 ) ×

univariate SV models, there are two kinds of

Πf (y

T

efficient and fast MCMC methods, namely (i)

t =1

t

| Qt , ht , θ1 , θ 2 ) ×

f ( Qt | Qt −1 , θ1 ) f ( ht | ht −1 , θ 2 ) .

the integration sampler suggested by Kim et al. (1998) and Chib et al. (2002), and (ii) the multi-move sampler proposed by Shephard and

In order to conduct inferences on the parameters,

Pitt (1997, 2004) and Watanabe and Omori

we produce a sample

(2004). For reasons explained in Subsection 3.2

(θ

(g) 1

, θ 2( g ) , Q ( g ) , h( g ) )

below, we develop the block samplers which are

from this density by the MCMC method. The

an extension of Shephard and Pitt (1997, 2004).

produced draws (θ1( g ) , θ 2( g ) ) are taken from the posterior density marginalized over ( Q, h ) .

Apart from MCMC, competing approaches include the likelihood approach based on importance sampling, such as Sandmann and

The proposed MCMC algorithm is based on the

Koopman (1998) and Liesenfeld and Richard

two blocks, (θ1 , Q ) and (θ 2 , h ) :

(2003), and the reprojection method proposed by

2206

sampler proposed by Shephard and Pitt (1997,

Algorithm

1.

Initialize Q and (θ1 , θ 2 ) .

2.

Sample

h

and

θ2

2004) and Watanabe and Omori (2004).

from

θ 2 , h y, Q

Although

(a) h from h y, θ 2 , Q ; (b) θ 2 from θ 2 | h . Sample

Q

and

was

developed

θ1

from

extension to MSV models is straightforward. As

θ1 , Q y, h

the model for ht can be interpreted as a seemingly unrelated regression (SUR) model, in

through drawing: (a) Q from Q y, θ1 , h ; (b) θ1 from θ1 | Q . 4.

approach

specifically for univariate SV models, the

through drawing:

3.

their

that SUR can be written as a regression model with parameter restrictions, Step 2b follows from

Go to Step 2.

the updates of a VAR(1) model with parameter restrictions.

The remainder of this section proposes the sampling method for each step mentioned above. Step 3a is most important. Our method is based

5

Conclusion

on the Metropolis-Hastings (MH) algorithm (see Chib and Greenberg (1995)). Since the process

As covariance and correlation structures are used

of qt is linear but non-Gaussian, we consider a

routinely for optimal portfolio choice, risk

linear approximation of the equation, and apply

management, obtaining Value-at-Risk (VaR)

the simulation smoother proposed by de Jong

forecasts, and determining optimal capital

and Shephard (1995) in order to generate

charges under the Basel Accord, it is essential to

candidates for the MH algorithm. Sampling all

model

latent variables as one block will run into large

accurately. This issue does not yet seem to have

very

been examined in the Multivariate Stochastic

large

rejection

frequencies,

while

the

covariances

and

correlations

Volatility (MSV) literature.

generating a single state at a time, which yields a highly autocorrelated sample, and needs a huge number of samples to conduct a statistical

This paper proposed two types of stochastic

inference. We will take an intermediate approach,

correlation structures for MSV models, namely

namely a block sampler such as Shephard and

the constant correlation (CC) MSV and dynamic

Pitt (1997) and Elerian et al. (2001).

correlation (DC) MSV models, from which the stochastic

covariance

structures

could

be

Step 3b is based on the MH algorithm. We will

obtained easily. The choice between the CC

use

generate

MSV and DC MSV models was shown to be

candidates for Ω and Λ , while we use the gamma distribution for ψ . In order to

based on a deviance information criterion. A

implement Step 2a, we extend the multi-move

MSV model using the Markov Chain Monte

the Wishart distribution

to

−1

technique was developed to estimate the DC

Carlo (MCMC) procedure.

2207

“Understanding

Acknowledgements

the

Metropolis-Hastings

Algorithm”, American Statistician, 49, 327–335. The authors wish to acknowledge very helpful

Chib, S. and E. Greenberg (1996), “Markov

discussions with Christian Gourieroux, Neil

Chain Monte Carlo Simulation Methods in

Shephard, George Tauchen, Bernardo da Veiga

Econometrics”,

and Jun Yu. The first author appreciates the

409-431.

financial support of the Japan Society for the

Danielsson, J. (1994), “Stochastic Volatility in

Promotion of Science and the Australian

Asset

Academy of Science. The second author is

Maximum

grateful for the financial support of the

Econometrics, 61, 375-400.

Australian Research Council. An earlier version

Elerian, O, S. Chib and N. Shephard (2001),

of the paper was presented to the First

“Likelihood Inference for Discretely Observed

Symposium

Diffusions”, Econometrica, 69, 959-993.

on

Econometric

Theory

and

Prices:

Econometric

Estimation

Theory,

with

Simulated

Journal

Likelihood”,

12,

of

Applications (SETA), Institute of Economics,

Engle, R.F. (2002), “Dynamic Conditional

Academia Sinica, Taipei, Taiwan.

Correlation: A Simple Class of Multivariate Generalized

Autoregressive

Conditional

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