Anon

232 The Basics of financial economeTrics

It is possible to define theoretically a multivariate ARCH/GARCH
model called the VEC-GARCH model.^15 The model replaces the volatility
parameter with the covariance matrix and models the covariance matrix.
The problem is that in a multivariate GARCH model each individual ele-
ment of the covariance matrix is regressed over every other element and
every other product between past returns. To see this, consider a simple
bivariate return process:

R

R

tt

tt

11

22

,

,

=

=

ε

ε

(^15) The VEC model in the ARCH framework was proposed in Robert F. Engle, Clive
Granger, and Dennis Kraft, “Combining Competing Forecasts of Inflation Using a
Bivariate ARCH Model,” Journal of Economic Dynamics and Control 8 (1984):
151–165; and Dennis Kraft and Robert F. Engle, “Autoregressive Conditional Het-
eroscedasticity in Multiple Times Series,” (Unpublished manuscript, Department of
Economics, University of California, San Diego, 1983). The first GARCH version
was proposed in Tim Bollerslev, Robert F. Engle, and Jeffrey M. Wooldridge, “A
Capital Asset Pricing Model with Time-Varying Covariances,” Journal of Political
Economy 96 (1998): 116–131.
0 1,000 2,000 3,000
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Av
erage Correlation
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FIGure 11.9 Plot of Average Correlations between Stocks in the S&P 500 Universe
in the Period from May 25, 1989, to December 30, 2011