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. Thanks, Brett 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 at burns-stat.com <https://stat.ethz.ch/mailman/listinfo/r-sig-finance> +44 (0)20 8525 0696 http://www.burns-stat.com (home of S Poetry and "A Guide for the Unwilling S User") Brett F. Sumsion, CFA Strategis Financial Group, Inc. (800) 279-3377 brett@strategisfinancial.com
Multivariate GARCH
1 message · Brett F. Sumsion