Multivariate GARCH

Below I will outline a method of getting multivariate GARCH estimates
by using only univariate GARCH estimates.  I actually did it (years ago)
not for lack of a multivariate GARCH estimator, but to get estimates for
large problems (that is, a large number of assets) in a reasonable amount
of time.  For being ad hoc, it performs remarkably well.

Here is the recipe.  Assume there are n observations (dates) for each of
the p assets.

Step 1)  Perform a univariate GARCH estimation on each asset.

Step 2)  Form the standardized residuals of all of the assets.  This is 
an n by p
matrix where each value theoretically has mean 0 and variance 1.

Step 3)  Perform a principal component rotation on the standardized 
residuals.

Step 4)  Perform a univariate GARCH estimate on each of the principal
components.

Step 5)  At each point in time we have a variance for each of the principal
components.  If we cross our fingers real hard, we can assume that there is
no correlation between the principal components at each of the times.  (On
average throughout the sample period, this is true, but it is very 
doubtful that
it is always true.)

With our assumption the variance matrix for the principal components at a
point in time is diagonal.  Rotate this diagonal matrix back into asset
co-ordinates.

Step 6)  The end result of step 5 is conceptually the correlation matrix 
of the
assets at the point in time.  In actuality the diagonals will not all be 
1.  Perform
the transformation of a variance matrix into a correlation matrix on the 
result
of step 5.  (This may or may not undo some of the damage from the assumption
of constant zero correlation of the principal components.)

Step 7)  Scale the correlation matrix created in step 6 by the variances 
estimated
in step 1 to arrive at the estimate of the variance matrix at a point in 
time.

Predictions are straightforward -- just predict the principal component 
GARCH
models, do the transformation into assets, then predict the asset GARCH 
models
and put them together.

Patrick Burns

Burns Statistics
patrick@burns-stat.com
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and "A Guide for the Unwilling S User")

Multivariate GARCH

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