The multivariate (G)ARCH model

All the models described earlier are univariate. However, risk analysis of speculative
prices examines both an asset’s return volatility and its co-movement with other
securities in the market. Modeling the co-movement among assets in a portfolio is
best archived using a multivariate conditional heteroskedastic model which accounts
for the non-normality in the multivariate distribution of speculative price changes.
Hence, the ARCH models can find more prominent use in empirical finance if they
could describe risk in a multivariate context. There are several reasons for examining
the variance parameter of a multivariate distribution of financial time series after
modeling within the ARCH framework. For example, covariances and the beta
coefficient which in finance theory is used as a measure of the risk could be
represented and forecasted in the same way as variances.
The ARCH model has been extended to a multivariate case using different parameterizations.
The most popular is the diagonal one where each element of the
conditional variance–covariance matrix Ht is restricted to depend only on its own
lagged squared errors.4
Thus, a diagonal bivariate GARCH(1, 1) is written as:
Y1,tó'T
1,td1òe1,t (2.15a)
Y2,tó'T
2,td2òe2,t (2.15b)
with
e1,t
e2,t ~N(0,Ht )
where Y1,t , Y2,t is the return on the two assets over the period (tñ1, t). Conditional on
the information available up to time (tñ1), the vector with the surprise errors et is
assumed to follow a bivariate normal distribution with zero mean and conditional
variance–covariance matrix Ht . Considering a two-asset portfolio, the variance–
covariance matrix Ht can be decomposed as
h1,tóu1òa1e2
1,tñ1òb1h1,tñ1 (2.16a)
h12,tóu12òa12e1,tñ1e2,tñ1òb12h12,tñ1 (2.16b)
h2,tóu2òa2e2
2,tñ1òb2h2,tñ1 (2.16c)
Here h1,t and h2,t can be seen as the conditional variances of assets 1 and 2
respectively. These are expressed as past realizations of their own squared disturbances
denoted as e2
1,tñ1. The covariance of the two return series, h12,t , is a function
of the cross-product between past disturbances in the two assets. The ratio
h1,th2,t /h12,t forms the correlation between assets 1 and 2.
However, using the ARCH in a multivariate context is subject to limitations in
modeling the variances and covariances in a matrix, most notably, the number of variances and covariances that are required to be estimated. For example, in a widely
diversified portfolio containing 100 assets, there are 4950 conditional covariances
and 100 variances to be estimated. Any model used to update the covariances must
keep to the multivariate normal distribution otherwise the risk measure will be
biased. Given the computationally intensive nature of the exercise, there is no
guarantee that the multivariate distribution will hold.