[R-sig-finance] Multivariate GARCH

Brett F. Sumsion brett at scmi.com
Fri Feb 11 17:37:49 CET 2005

Dr. Burns,
I have read your procedure posted on the internet for using univariate garch
estimates to form a multivariate result.  I am a new to this stuff and just
learning. I follow the procedure very well until I get to the end of step 5,
where you need to "rotate" the diagonal variance matrix back into asset
co-ordinates.  I don't understand what this means? Can you clarify.
Step six appears to describing the same procedure outlined in step 5, is
that the case?  I appreciate any insight you can provide.  I have attached
the post.
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 
Step 4)  Perform a univariate GARCH estimate on each of the principal
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
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 
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 
in step 1 to arrive at the estimate of the variance matrix at a point in 
Predictions are straightforward -- just predict the principal component 
models, do the transformation into assets, then predict the asset GARCH 
and put them together.
Patrick Burns
Burns Statistics
patrick at burns-stat.com
+44 (0)20 8525 0696
(home of S Poetry and "A Guide for the Unwilling S User")



Brett F. Sumsion, CFA

Strategis Financial Group, Inc.

(800) 279-3377

brett at strategisfinancial.com


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