1

PhD Program in Business Administration and Quantitative Methods

FINANCIAL ECONOMETRICS

2006-2007

ESTHER RUIZ

CHAPTER 5. MULTIVARIATE MODELS

Multivariate models are of central interest in several fields of Financial

Econometrics. First of all, international financial markets are dependent of each other

and one must consider them jointly to understand the dynamic structure of international

finance. For example, De Santis and Gerard (1997) test the CAPM for the world’s eight

largest equity markets using a multivariate GARCH model. Their results indicate that,

although severe market declines are contagious, the expected gains from international

diversification for a US investor average 2.11 percent per year and have not

significantly declined over the last two decades. Karolyi (1995) analyses the short-run

dynamics of stocks and volatility for stocks traded on the New York and Toronto stock

exchanges. He shows that inferences about the persistence of returns and the

transmission of effect depend importantly on how the dynamics in volatility are

modelled. He also discusses the implications for international asset pricing, hedging

strategies and regulatory policy. The results of Kearney and Patton (2000) on exchange

rate volatility transmission across the European Monetary System also indicate the

importance of checking for specification on multivariate GARCH models.

On the other hand, financial institutions usually hold multiple assets and the

dynamic relationships between them are important for risk management and asset

allocation. For example, Bollerslev, Engle and Wooldridge (1988) estimate a

multivariate GARCH model for returns to US Treasury Bills, gilts and stocks and

Brooks, Henry and Persand (2002) compared the effectiveness of hedging on the basis

of hedge ration derived from various multivariate GARCH specifications.

In this chapter, we analyse multivariate time series models that represent the

dynamic relationships between two or more variables observed sequentially in time.

2

Suppose that ( )',...,1 kttt yyY = is the vector of observations at time t. Although most of

the models and methods described before for univariate time series can be extended

directly to multivariate systems, there are some situations that require especial methods

and procedures.

5.1 Linear multivariate models

5.1.1 VARMA models for stationary series

The multivariate series tY is weakly stationary if its mean vector and covariance

matrices are constant over time:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

==

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

kkt

t

t

t

y

yy

EYE

μ

μμ

μ......

)( 2

1

2

1

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=−−=Γ −

)(...)()(............

)(...)()()(...)()(

)')((

21

2221

1121

hhh

hhhhhh

YYE

kkk

k

k

htth

γγγ

γγγγγγ

μμ

Note that with the exception of 0Γ , these matrices are not symmetric. From the cross-

covariance matrices, it is possible to compute the corresponding cross-correlations

matrices.

The simplest models to represent dynamic relationships between multiple time

series is the VAR(1) model:

ttt aYY +Φ+Φ= −110

where 0Φ is a kx1 vector of constants, 1Φ is a kxk matrix and ta is a sequence of

serially uncorrelated random vectors with mean zero and covariance matrix Σ . The

matrix 1Φ measures the dynamic dependence of tY while the contemporaneous relation

between the series appears in the off-diagonal elements of Σ .

Consider, for example, the bivariate case:

tttt

tttt

ayyyayyy

212221121202

112121111101

+++=+++=

−−

−−

φφφφφφ

3

The stationary condition is satisfied if the roots of the characteristic equation,

01 =Φ− zI are outside the unit circle. In these circumstances, ML and OLS are

asymptotically equivalent.

The adequacy of the model can be checked by applying the )(mQk statistic to

the residuals; see Hosking (1980, 1981) and Li and McLeod (1981). The )(mQk

statistic is given by

[ ]∑=

−− ΓΓΓΓ−

=m

lllk tr

lTTmQ

1

10

10

2 ˆˆˆˆ1)(

The statistic )(mQk is asymptotically distributed as a 2χ distribution with

gmk −2 degrees of freedom where g is the number of estimated parameters in the AR

coefficient matrices.

Unlike the VAR models, the estimation of VMA models is much more involved

and they are not usually fitted in empirical applications.

5.1.2 VAR models for non-stationary series

When dealing with multivariate non-stationary series, it is important to take into

account whether they are co-integrated. The models fitted to non-stationary cointegrated

and non-cointegrated series are different.

Cointegrated series

The series contained in the vector tY are cointegrated if each of the series is I(d)

but there exists at least one linear combination of them which is I(b) where b<d. Note

that if we deal with I(1) variables, as it is often the case in financial prices, then the

linear combination is stationary.

To understand the concept of cointegration, consider that we are modelling a

bivariate system with variables that have a level that evolves locally over time. In this

case,

ttt

ttty

11

11

ημμεμ+=+=

−

and

ttt

ttty

21

22

ηδδεδ+=+=

−

4

If the underlying levels of both variables are related, then ty1 and ty2 are related

in the long run. In this case, consider that, for example,

tt ba μδ +=

Then, the linear combination tttttt bbaybya 2121 εδεμ −−++=−+ is stationary.

Furthermore, we can establish the following long-run relationship between both

variables:

ttt uyy ++= 2101 ββ

where ),1( β− is known as the cointegration vector and β is the parameter that

measures the long-run relationship between the variables.

It is important to note that the stationary linear combination is not unique. In the

example above, we can consider any other linear combination )( 21 tt ybyac −+ which is

also stationary. For this reason, the cointegration vector is always normalized for one of

the variables in the model.

Consider a VAR(p) model to represent the dynamic evolution of a system of I(1)

cointegrated variables

tptptt aYYY +Φ++Φ+Φ= −− ...110

Given that the model is not stationary, we know that 0...1 =Φ−−Φ− pI .

Therefore, the matrix pI Φ−−Φ− ...1 has not full rank. Consider the following

reparametrization of the model:

tptpttt aYYYY +ΔΓ++ΔΓ+Π=Δ −−−−− 11111 ...

where pI Φ−−Φ−=Π ...1 . Suppose that pI Φ++Φ≠ ...1 , so the rank of Π is

different from zero although smaller than k. Tests of cointegration are based on

estimates of the rank of the matrix Π . When the rank is k, there are not unit roots and

the model is stationary. If the rank is zero, the model is not stationary and the variables

are non-cointegrated. Finally, when the rank is between zero and k, the model is not

stationary and the variables are cointegrated. In this case, the rank of the matrix is the

number of cointegrated relationships. The matrix Π can be decomposed as

'αβ=Π

Therefore, the model adequate to represent their dynamic evolution is a VAR with error

correction model given by VAR-ECM:

tptpttt aYYYY +ΔΓ++ΔΓ+=Δ −−−−− 11111 ...'αβ

5

where α is a nxr matrix of coefficients that measure the speed of the adjustment of the

disequilibrium deviations and β is a kxr matrix of long-run coefficients.

Consider, for example, a bivariate system

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡ΔΔ

⎥⎦

⎤⎢⎣

⎡+−−⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ΔΔ

−

−−−

t

t

t

ttt

t

t

aa

yy

yyyy

2

1

12

11

2221

1211121011

2

1

2

1 )(φφφφ

ββαα

The VAR-ECM can be estimated jointly by ML as proposed by Johansen.

Alternatively, Engle and Granger proposed a two-step estimator based on estimating

first the long run relationships by OLS and then estimating the model using the

residuals from this estimation as proxy variables of the deviations from the long-run

equilibrium.

Example: Logarithms of monthly prices of beef in US (LUSA) y Argentina (LARG) from 1977-1997

6.4

6.8

7.2

7.6

8.0

8.4

78 80 82 84 86 88 90 92 94 96LARG LUSA

Augmented Dickey-Fuller tests

Test ADF Statistic Critical values (1%, 5%, 10%)

LUSA (4) -2.833254 -3.9984, -3.4292, -3.1378

DLUSA (4) -7.572660 -3.4585, -2.8734, -2.5730

LARG (4) -2.610202 -3.9984, -3.4292, -3.1378

DLARG (4) -7.482174 -3.4585, -2.8734, -2.5730

6

To analyse the dynamic relationship between both prices, we have included two

dummy variables in the VAR-VEC model, D1t y D2t. D1t is zero up to December and

then takes values 1, 2, 3… up to December 1994, then is zero again up to the end of the

sample. The variable D2t is zero up to December 1994 and one after words. The

estimated model is:

4tLUSA*(0.14)0.163tLUSA*

(0.11)0.152tLUSA*

(0.12)0.051tLUSA*

(0.11)0.02

4tLARG*(0.06)0.093tLARG*

(0.06)0.182tLARG*

(0.06)0.01 +1tLARG*

(0.06)0.19

)2-2tD*(0.11)0.52-1-1tD*

(0.002)0.008-1tLUSA1tLARG

(0.03)0.51(*

(0.03)0.13tLARG

−Δ−−Δ−−Δ+−Δ+

−Δ−−Δ+−Δ−Δ−

−−−+−=Δ

4tLUSA*(0.06)0.063tLUSA*

(0.07)0.042tLUSA*

(0.05)0.221tLUSA*

(0.06)0.46

4tLARG*(0.04)0.063tLARG*

(0.04)0.032tLARG*

(0.04)0.09 +1tLARG*

(0.04)0.0004

)2-2tD*(0.11)0.52-1-1tD*

(0.002)0.008-1tLUSA1tLARG

(0.03)0.51(*

(0.01)0.008tLUSA

−Δ−−Δ−−Δ−−Δ+

−Δ−−Δ+−Δ−Δ−

−−−+=Δ

Non-cointegrated variables

When the variables are not cointegrated, there are not long-run relationships

between the variables. In this case, the matrix Π is zero and the model appropriate to

represent the dynamic evolution of the variables in tY is a VAR model for the first

differences, i.e.

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡ΔΔ

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ΔΔ

−

−

t

t

t

t

t

t

aa

yy

yy

2

1

12

11

2221

1211

2

1

φφφφ

Finally, note that if we fit a VAR model to the first differences when the series

are cointegrated, we are introducing a non-invertible MA component. Consequently,

when using the AR approximation we need infinite lags to represent the long-run

dependency between the variables.

7

5.2 Multivariate GARCH models

Consider that the series of returns, tY , is given by

ttt HY ε2/1=

where 2/1tH is a kxk positive definite matrix such that tH is the conditional covariance

matrix of tY , tε is a vector of white noise processes such that 0)( =tE ε and

IE tt =)( 'εε .

There are two main branches in the literature related with multivariate models

with GARCH errors. The first type of models represents directly the evolution of the

time-varying covariances while the second type models conditional correlations.

Bauwens, Laurent and Rombouts (2005) and Silvennonen and Teräsvirta (2006) are

very complete and updated surveys on multivariate GARCH models.

Multivariate GARCH models have two additional limitations over the ones

commented in the univariate case. First of all, the number of parameters increases

rapidily with the number of series considered. Furthermore, since tH is a covariance

matrix, positive definiteness has to be ensured.

5.2.1 Models for conditional covariances

The most direct generalization of the univariate GARCH(1,1) model is the VEC model,

proposed by Bollerslev, Engle and Wooldridge (1988), and given by

)()()()( 1'

11 −−− ++= tttt HBvechAvechWvechHvech εε

where vech is the operator that applied to a symmetric matrix put the elements in the

lower diagonal in a column. Consider, for example, the bivariate model with k=2 given

by

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

−

−−

−

2122

2112

2111

333231

232221

131211

22

1211

211

333231

232221

131211

22

12

11

222

12

211

t

t

t

t

tt

t

t

t

t

bbbbbbbbb

yyy

y

aaaaaaaaa

www

σσσ

σσσ

From where,

212233

211232

211131

2233121132

2113122

222

212223

211222

211121

2223121122

2112112

212

212213

211212

211111

2213121112

2111111

211

−−−−−−

−−−−−−

−−−−−−

++++++=

++++++=

++++++=

tttttttt

tttttttt

tttttttt

bbbyayyayaw

bbbyayyayaw

bbbyayyayaw

σσσσ

σσσσ

σσσσ

The VEC model is covariance stationary if the modulus of the eigenvalues of

BA + are less than one. Hafner (2003) provides analytical expressions of the fourth

8

order moments while Gourieroux (1997) gives sufficient conditions for the positivity of

tH .

The number of parameters in a VEC model is 2

)1)1()(1( +++ kkkk . The

following table reports the number of parameters for 5,3,2=k and 10.

2=k 3=k 5=k 10=k

VEC 21 78 465 6105

DVEC 9 18 45 165

BEKK 11 24 65 255

F-GARCH 7 12 25 75

CCC 7 12 25 75

DCC 9 14 27 77

To overcome the problem of the extremely large number of parameters,

Bollerslev, Engle and Wooldridge (1988) propose to restrict the matrices A and B to be

diagonal. In this case, each element ijth depends only on its own lag and on the previous

value of jtitεε . Therefore, the conditional variances and covariances have a GARCH

specification. Consider, once more, the bivariate model

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

−

−−

−

2122

2112

2111

33

22

11

22

1211

211

33

22

11

22

12

11

222

12

211

000000

000000

t

t

t

t

tt

t

t

t

t

bb

b

yyy

y

aa

a

www

σσσ

σσσ

212233

2123313

222

21122212112212

212

211111

2111111

211

−−

−−−

−−

++=

++=

++=

ttt

tttt

ttt

byaw

byyaw

byaw

σσ

σσ

σσ

The number of parameters is reduced to 2

)1(3 +kk . Attanasio (1991) analyse the

conditions to guarantee that the conditional covariance matrices are positive definite

which can be imposed through a Cholesky decomposition. The dynamics of the DVEC

model are very restrictive. For example, it is not suitable to represent volatility

transmissions between assets.

There are other proposals based on the Cholesky decomposition to guarantee the

positiviness of tH in the VEC representation; see, for example, Tsay (2002).

9

Because, it is difficult to guarantee the positiviness of tH in the general VEC

model and the DVEC model is too restrictive, Engle and Kroner (1995) proposed the

BEKK specification based on quadratic forms, where

'1

1

''11 jtj

J

jjttjt BHBAAWH −

=−− ++= ∑ εε

Consider, the bivariate model with J=1,

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

+⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

−−

−

−

2212

21112

122112

2111

2221

1211

2212

21112221

211

2221

1211

2212

1122212

211

bbbb

bbbb

aaaa

yyyy

aaaa

www

tt

t

ttt

t

tt

t

σσσ

σσσ

2122

3221122221

2111

221

21222112122

2122

3221122221

2111

221

212

22212112221

211

22122

222

21221222

211222111221

21111121

2121222121122111221

211112112

212

212

2121121211

2111

211

21212111111

2122

2121121211

2111

211

212

21212111211

211

21111

211

2)(

22

)(

)(

2)(

22

−−−−−

−−−−−−−

−−−

−−−−

−−−−−

−−−−−−−

+++++=

++++++=

+++

+++++=

+++++=

++++++=

ttttt

tttttttt

ttt

ttttt

ttttt

tttttttt

bbbbyayaw

bbbbyayyaayaw

bbbbbbbb

yaayyaaaayaaw

bbbbyayaw

bbbbyayyaayaw

σσσ

σσσσ

σσσ

σ

σσσ

σσσσ

The number of parameters is 2

)15( +kk . DVEC and BEKK models are

parsimonious but very restrictive for the cross-dynamics.

Alternatively, Engle, Ng and Rothschild (1990) proposed the Factor-ARCH (F-

ARCH) model using the idea that co-movements of financial returns can depend on a

small number of underlying factors. The F-GARCH model is given by

)( 1'2

1

'11

'2'jtjj

J

jjttjjjjt HWH ωωβωεεωαλλ −

=−− ++= ∑

where ⎩⎨⎧

=≠

=jkjk

jk ,1,0' λω and ∑

=

=K

kjk

1

1ω .

The name factor-GARCH stems from the fact that the time-variation in the

conditional covariance matrix can be summarized by a few linear combinations of

returns. These linear combinations, known as “factor representing portfolios”, are

GARCH models. The factors are related to those linear combinations of the series that

summarize the co-movements in the conditional variances; see Bollerslev and Engle

(1993). Consider, for example, that there is just one factor in a bivariate system. Then

the F-GARCH model is given by

10

[ ] ( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

+⎥⎦

⎤⎢⎣

⎡

=++=

−−

−

−−−

−

−−−

2

12

122112

2111

212

2

12

121211

211

212

212

1

2212

11

1'2'

11'2' )(

ωω

σσσ

ωωβωω

ωωαλλλλ

ωωβωεεωαλλ

tt

t

ttt

t

tttt

yyyy

www

HWH

( )( )

( )2122211221

21111

22122111

22222

222

2122211221

21111

22122111

22112

212

2122211221

21111

22122111

22111

211

()(

()(

()(

−−−−−

−−−−−

−−−−−

+++++=

+++++=

+++++=

tttttt

tttttt

tttttt

yyw

yyw

yyw

σωσωωσωβωωαλσ

σωσωωσωβωωαλλσ

σωσωωσωβωωαλσ

When there is just one factor, the number of parameters is 2

)5( +kk . The F-

GARCH model is nested within the BEKK model and, consequently, it guarantees a

positive semi-definite covariance matrix. These models allow the conditional variances

and covariances to depend on the past of all variances and covariances but they imply

common persistence in all these elements. It is important to note that these models are

not the same as the factor models with heteroscedastic disturbances; see Sentana (1998).

5.2.2 Models for conditional correlations

These models are based on the decomposition of the covariances into

correlations and standard deviations. For these models, the conditions for stationarity

and the existence of moments are not easy to obtain. However, they are easier to

estimate.

Bollerslev (1990) proposes the Constant Conditional Correlation (CCC) model

which allows each series to follow a separate GARCH model while restricting the

conditional correlations to be constant. Consequently, the conditional covariances are

proportional to the product of the corresponding conditional standard deviations. The

CCC-GARCH model is given by

ttt RDDH =

where ( )kttt diagD σσ ,...,1= and ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1............

...1

1

1

k

k

Rρ

ρ. The elements of tD can be

defined as GARCH(1,1) models 2

1,2

1,2, −− ++= tiitiiiti y σβαωσ

11

The number of parameters is the same as in the F-GARCH model with one

factor, 2

)5( +kk . The estimation of this model is very simple and, consequently, the

CCC-GARCH model is very popular between practitioners. However, when the

constant correlation hypothesis is tested with real data, it is often concluded that it is not

an adequate assumption; see Tse (2000) and Bera and Kim (2002) for tests for constant

correlations. Consequently, Jeantheau (1998) introduced an extension of the CCC-

GARCH model that allows dynamic interactions between the conditional variance

equations. The properties of this model have been analysed by He and Teräsvirta

(2004).

Given that it seems that the conditional correlations increase during periods of

market disturbance, there are several proposals of models where the correlations have a

regime-switching structure driven by an unobserved state variable; see, for example,

Berben and Jansen (2005) and Silvennoinen and Teräsvirta (2005).

Tse and Tsui (2002) and Engle (2002) have proposed the Dynamic Conditional

Correlation (DCC) GARCH models that impose GARCH-type dynamics on the

conditional correlations as well as on the conditional variances. In both cases, the

conditional covariance matrix is given by

tttt DRDH =

Tse and Tsui (2002) propose to model tR as

11)1( −− +Ψ+−−= ttt RRR βαβα

where α and β are non-negative parameters satisfying 1<+ βα , R is a symmetric kxk

positive definite matrix with 1=iiρ and tΨ is the kxk conditional correlation matrix of

tY where the conditional correlations are computed at each time t using the M previous

observations, i.e.

∑∑

∑−

=−

−

=−

−

=−−

=Ψ1

0

21

0

2

1

0

ˆˆ

ˆˆ

M

mmjt

M

mmit

M

mmjtmit

ijt

εε

εε

A necessary condition to ensure the positivity of tΨ and, consequently, of tR is

that kM ≥ .

Alternatively, the model proposed by Engle (2002) is

12

( ) ( ) 2/12/1 )()( −−= tttt QdiagQQdiagR

where 1'

11 ˆˆ)1( −−− ++−−= tttt QQQ βεεαβα , Q is the marginal covariance matrix of

tε̂ and α and β are non-negative parameters such that 1<+ βα .

In both models, all the conditional correlations abbey the same dynamics. This is

necessary to ensure that tR is positive definite.

The number of parameters is 2

)4)(1( ++ kk and the model is relatively easy to

estimate by a consistent two-steps procedure even if the number of series in the system

is large.

The number of parameters in the DCC-GARCH model of Engle remains

relatively low because all conditional correlations are generated by GARCH models

with identical parameters.

5.2.3 An encompassing multivariate GARCH model

Kroner and Ng (1998) have proposed a general dynamic covariance model that

encompasses many multivariate GARCH models. In particular, it encompasses the DCC

model of Engle (2002) and Tse and Tsui (2002), the CCC, BEKK, DVEC and F-

GARCH models. This model is not practical from an empirical point of view as it

requires a very large number of parameters, namely, 2

4)17( +−kk . This number is

smaller than in the VEC model but larger than in the BEKK model. However, this

model can be interesting for theoretical comparisons.

5.2.4 Estimation of multivariate GARCH models

Assuming normality, the likelihood function can be written as

∑∑=

−

=

−−=T

tttt

T

ttT YHYHL

1

1'

1 21||log

21)(θ

The asymptotic properties of ML and QML estimators in multivariate GARCH

models are not yet well established. Jeantheau (1998) proves the consistency of the

Gaussian QML estimator. However, asymptotic normality has not being established in

general. Gourieroux (1997) has some results under high level conditions which are

difficult to check in practice. Comte and Lieberman (2003) has proven it for the BEKK

model.

13

Lin (1992) reviews and compares the finite sample properties of several

alternative methods to estimate F-GARCH models.

With respect to DCC models, they can be consistently estimated in two steps;

see Engle and Sheppard (2001). First, the univariate GARCH(1,1) model is estimated

for each of the series. Then, the observations are standardized using these estimates and,

in the second step, estimates of the parameters corresponding to the correlations are

estimated by minimizing

∑∑=

−

=

−−=T

tttt

T

ttT RRL

1

1'

12 ˆˆ

21||log

21)( εεθ

Brooks, Burke and Persand (2003) review the software available to estimate

multivariate GARCH models.

Model Reference

VEC Bollerslev, Engle and Wooldridge (1988),

DVEC Bollerslev, Engle and Wooldridge (1988),

F-GARCH Engle, Ng and Rothschild (1990)

CCC Bollerslev (1990)

BEKK Engle and Kroner (1995)

E-CCC Jeantheau (1998)

O-GARCH van der Weide (2002)

DCC Tse and Tsui (2002)

Engle (2002)

FF-GARCH Vrontos, Dellaportas and Politis (2003)

ST-CCC Berben and Jansen (2005)

Silvennoinen and Teräsvirta (2005)

14

5.3 Multivariate Stochastic Volatility models

Compared with the multivariate GARCH literature, the literature on multivariate

SV models is much limited; see Chib (2007) for a survey. The basic multivariate SV

models, proposed by Harvey, Ruiz and Shephard (1994), is specified as k univariate SV

models for the conditional variances

ititiit

ititiity

ησφσ

σεσ

+=

=

−2

12

*

loglog

where ( )',...,1 kttt εεε = is a white noise vector such that 0)( =tE ε and εεε Σ=)( 'ttE in

which the elements on the leading diagonal are unity and the off-diagonal elements are

ijρ . The vector of noises ( )',...,1 kttt ηηη = is multivariate normal with 0)( =tE η and

ηηη Σ=)( 'ttE . In the bivariate case,

{ }{ }2/exp

2/exp

222*2

111*1

ttt

ttt

hyhy

εσεσ

==

⎥⎦

⎤⎢⎣

⎡=Σ

11

ε

εε ρ

ρ

ttt

ttt

hhhh

21222

11111

ηφηφ+=+=

−

− ⎥⎥⎦

⎤

⎢⎢⎣

⎡=Σ 2

2

212

121

ηη

ηηη σσ

σσ

The model can be easily generalized to a VAR model. However, we focus on the

especial case when )'log,...,(log 221 kttth σσ= is a multivariate random walk. The state

space model in this case is given by

ttt

ttt

hhhYη

ξμ+=

++=

−1

*

where )log,...,(log 221

*kttt yyY = and ))(loglog),...,(log(log 1

21

21 tktttt EE ξξξξξ −−= . In

this model, common factor can be incorporated easily as follows:

ttt

ttt

hhhYη

ξθμ+=

++=

−1

*

where θ is a kxn matrix of factor loadings with n<k and th and tη are nxn vectors. In

this case, the log-volatilities are cointegrated. To identify the model, we require that

ijij >= ,0θ and that ηΣ is an identity matrix.

The basic model has also been extended by Yu and Meyer (2004) to introduce

Granger-causality

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

−

−

t

t

t

t

t

t

hh

hh

2

1

12

11

2221

1211

2

1

ηη

φφφφ

15

In this case, the number of parameters is 2

4 2k . They also propose a model with

time-varying correlations:

)()( 2/12/1 −−=Σ tttt QdiagQQdiagε

)()( '111 SvvASQBSQ tttt −+−+= −−− oo

where ),0( INvt → and o is the Hadamard product (matrix whose elements are

obtained by element-by-element multiplication).

There are several methods implemented in the literature to estimate the

parameters of MSV models:

· QML as in Harvey, Ruiz and Shephard (1994)

· MCMC as in Chi, Nardari and Shephard (2005)

·SML as in Danielsoon (1998)

5.4 Applications

Dynamic asset pricing models:

De Santis et al. (1998)

Volatility transmission between assets and markets:

Bollerslev (1990)

Kerney and Patton (2000)

Koutmos and Booth (1995)

Future hedging

Bera et al. (1997)

Sephton (1993)

Impact of exchange rate volatility on trade and output

Value-at-Risk

References

Bauwens, L., S. Laurent and J.V.K. Rombouts (2006), Multivariate GARCH models: A survey, Journal of Applied Econometrics, 21, 79-109. Bollerslev, T. (1990), Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach, Review of Economics and Statistics, 72, 498-505.

16

Bollerslev, T., R.F. Engle and J.M. Wooldridge (1988), A capital-Asset Pricing Model with time-varying covariances, Journal of Political Economy, 96, 116-31. Brooks, C., S. Burke and G. Persand (2003), Multivariate GARCH models: software choice and estimation issues, Journal of Applied Econometrics, 18, 725-734. Brooks, C., O.T. Henry and G. Persan (2002), Optimal hedging and the value of news, Journal of Business, 75, 333-52. Chib, S. (2007), Multivariate Stochastic Volatility models, in Andersen, T.G., R.A. Davis, J.-P. Kreiss and T. Mikosch (eds.), Handbook of Financial Econometrics, Springer, New York. Chib, S., F. Nardari and N.G. Shephard (2005), Analysis of high dimensional multivariate stochastic volatility models, Journal of Econometrics, forthcoming. Christodoulakis, G. and S.E. Satchell (2002), Correlated ARCH (corr-ARCH): modelling the time-varying conditional correlation between financial asset returns, European Journal of Operational Research, 139, 351-370. Danielsson, J. (1998), Multivariate stochastic volatility models: estimation and comparison with VGARCH models, Journal of Empirical Finance, 5, 155-173. De Santis, G. and B. Gerard (1997), International Asset Pricing and Portfolio Diversification with Time-Varying Risk, The Journal of Finance, 52, 1881-1912. Engle, R.F. (2002), Dynamic conditional correlation – A simple class of multivariate GARCH models, Journal of Business and Economic Statistics, 17, 239-250. Engle, R.F. and Kroner (1995), Multivariate simultaneous GARCH, Econometric Theory, 11, 122-150. Engle, R.F., V.K. Ng and M. Rothschild (1990), Asset pricing with a factor-ARCH covariance structure, Journal of Econometrics, 45, 213-237. Harvey, A.C., E. Ruiz and N.G. Shephard (1994), Multivariate stochastic variance models, Review of Economic Studies, 61, 247-264. He, C. and T. Teräsvirta (2004), An extended constant conditional correlation GARCH model and its fourth-moment structure, Econometric Theory, 20, 904-926. Hosking (1980), JASA, 75, 602-608. Hosking (1981), JRSS, B, 43, 219-230. Jeantheau, T. (1998), Strong consistency of estimators for multivariate ARCH models, Econometric Theory, 14, 70-86.

17

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