The generalization of univariate GARCH models to the multivariate
case is straightforward. For the error term
of a
-dimensional time series model we assume that the conditional
mean is zero and the conditional covariance matrix is given by the
positive definite
matrix
, i.e.,
Let
vech denote the operator that stacks the lower
triangular part of a symmetric
matrix into a
dimensional vector. Furthermore we use the notation
vech
and
vech
. The
Vec specification of a multivariate GARCH(
) model is
then given by
For the bivariate case and we can write the model
explicitly as
In the Vec representation (12.37), the multivariate
GARCH() process
is covariance stationary if
and only if all eigenvalues of the matrix
In order to illustrate the prediction of volatility, let us
consider in the following the often used GARCH(1,1) model. The
optimal prediction with respect to the mean squared prediction
error is the conditional expectation of volatility. Due to the law
of iterated expectations, the -step prediction of
is identical to the
-step prediction of
, that is,
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In the bivariate case () and with
, there are already
21 parameters that characterize the dynamics of volatility. In
order to obtain a feasible model for empirical work, one often
imposes restrictions on the parameter matrices of the Vec model.
Bollerslev et al. (1988) propose to use diagonal parameter matrices
such that the conditional variance of one variable only depends on
lagged squared values of the same variable, and the conditional
covariances between two variables only depend on lagged values of
cross-products of these variables. This model reduces
substantially the number of parameters (in the above case from 21
to 9), but potentially important causalities are excluded.
For parameter estimation the Quasi Maximum Likelihood Method (QML)
is suitable. The conditional likelihood function for a sample time
series of observations is given by
with
If the conditional distribution of
is not normal,
then (12.40) is interpreted as quasi likelihood
function, which serves merely as target function in the numerical
optimization, but which does not say anything about the true
distribution. In the multivariate case, the QML estimator is
consistent and asymptotically normal under the main assumptions
that the considered process is strictly stationary and ergodic
with finite eighth moment. Writing all parameters in one vector,
, we obtain the following standard result.
In empirical work one often finds that estimated standardized residuals are not normally distributed. In this case the QML likelihood function would be misspecified and provides only consistent, not efficient parameter estimators. Alternatively, one can assume that the true innovation distribution is given by some specific non-normal parametric distribution, but in general this does not guarantee that parameter estimates are consistent in the case that the assumption is wrong.
Engle and Kroner (1995) discuss the following specification of a multivariate GARCH model.
In (12.43), is a lower triangular matrix and
and
are
parameter matrices. For
example, in the bivariate case with
,
and
, the
conditional variance of
can be written as
The so-called BEKK specification in (12.43) guarantees
under weak assumptions that is positive definite. A
sufficient condition for positivity is for example that at least
one of the matrices
or
have full rank and the
matrices
are positive definite. The BEKK
model allows for dependence of conditional variances of one
variable on the lagged values of another variable, so that
causalities in variances can be modelled. For the case of diagonal
parameter matrices
and
, the BEKK model is a
restricted version of the Vec model with diagonal matrices.
Due to the quadratic form of the BEKK model, the parameters are
not identifiable without further restriction. However, simple sign
restrictions will give identifiability. For example, in the often
used model and
, it suffices to assume that the upper
left elements of
and
are positive. The number of
parameters reduces typically strongly when compared to the Vec
model. For the above mentioned case, the number of parameters
reduces from 21 to 11.
For each BEKK model there is an equivalent Vec representation, but
not vice versa, so that the BEKK model is a special case of the
Vec model. To see this, just apply the
vech operator to
both sides of (12.43) and define
vech
,
, and
. Here
denotes the
Kronecker matrix product, and
and
are the elementary
elimination and duplication matrices. Therefore, one can derive
the stochastic properties of the BEKK model by those of the Vec
model. For the empirical work, the BEKK model will be preferable,
because it is much easier to estimate while being sufficiently
general.
Bollerslev (1990) suggested a multivariate GARCH model in which all conditional correlation are constant and the conditional variances are modelled by univariate GARCH models. This so-called CCC model (constant conditional correlation) is not a special case of the Vec model, but belongs to another, nonlinear model class. For example, the CCC(1,1) model is given by
We consider a bivariate exchange rates example, two European
currencies, DEM and GBP, with respect to the US Dollar. The sample
period is 01/01/1980 to 04/01/1994 with altogether
observations. Figure 12.9 shows the time series
of returns on both exchange rates. Table 12.4
provides some simple descriptive statistics of returns
. Apparently, the empirical mean of both processes
is close to zero.
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As can be seen in Figure 12.9, the exchange rate returns follow a pattern that resembles a GARCH process: there is a clustering of volatilities in both series, and the cluster tend to occur simultaneously. This motivates an application of a bivariate GARCH model.
A first simple method to estimated the parameters of a BEKK model
is the BHHH algorithm. This algorithm uses the first derivatives
of the QML likelihood with respect to the 11 parameters that are
contained in
and
, recalling equation
(12.43). As this is an iterative procedure, the BHHH
algorithm needs suitable initial parameters. For the diagonal
elements of the matrices
and
, values between 0.3
and 0.9 are sensible, because this is the range often obtained in
estimations. For the off-diagonal elements there is no rule of
thumb, so one can try different starting values or just set them
to zero. The starting values for
can be obtained by the
starting values for
and
using the formula for
the unconditional covariance matrix and matching the sample
covariance matrix with the theoretical version.
For the bivariate exchange rate example, we obtain the following estimates:
the previous value represents the computed minimum of the negative
log likelihood function. The displayed vector contains in the
first three components the parameters in , the next four
components are the parameters in
, and the last four
components are the parameters in
.
In this example we thus obtain as estimated parameters of the BEKK model:
Estimates for the conditional covariances are obtained by applying successively the difference equation (12.43), where the empirical covariance matrix
In Figure 12.10 estimated conditional variance and covariance processes are compared. The upper and lower plots show the variance of the DEM/USD and GBP/USD returns and the plot in the middle shows the estimated conditional covariance process. Apart from a very short period at the beginning of the sample, the covariance is positive and of not negligible magnitude.
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The estimated parameters can also be used to simulate volatility.
This can be done by drawing at every time step one realization of
a multivariate normal distribution with mean zero and variance
. With these realizations one updates
according to equation (12.43). Next, a new realization
is obtained by drawing from
N
, and so on. We
will now apply this method with
. The results of the
simulation in Figure 12.11 show similar patterns as in
the original process (Figure 12.10). For a further
comparison, we include two independent univariate GARCH processes
fitted to the two exchange rate return series. This corresponds to
a bivariate Vec representation with diagonal parameter matrices.
Obviously, both methods capture the clustering of volatilities.
However, the more general bivariate model also captures spill over
effect, that is, the increased uncertainty in one of the returns
due to increased volatility in the other returns. This has an
important impact on the amplitude of volatility.