# [S] Estimation problems with large covariance matrices

andrey@utstat.toronto.edu
Fri, 17 Jul 98 12:43 EDT

Frank Haertel inquired:

> We want to estimate covariance matrices with approximately 1000-2000
> variables (e.g. securities). One question now is how many observations
> are needed for estimating the full covariance matrix?

> Does anybody have experiences with estimating such large covariance
> matrices? Do special problems arise (positive-definiteness, many small
> eigenvalues, "noisy" results)...

I teach a course on `Statistical Methods for Investment and Finance'
at the University of Toronto, and this type of question comes up in a number
of contexts. Briefly, an n x n covariance matrix requires n + 1 vector--
observations in order that the ordinary sample covariance matrix should be
nonsingular (assuming continuously distributed data).

A common problem is that the covariance often cannot be assumed
to remain constant over the time interval required for such data to be
collected. There is a certain amount of detailed (but elementary)
discussion concerning such matters in the `RiskMetrics Technical Document'
which is put out by J.P. Morgan/Reuters. They mainly deal with this using
various (exponentially) weighted moving average methods and approaches
similar in spirit to cross-validation to choose the weight parameter(s).

In general positive-definiteness usually fails for large covariance
matrices computed by such methods, and more generally, the joint sampling
distribution of eigenalues and their corresponding eigenvectors is highly
variable. As a general rule, in dealing seriously with issues of this type,
it is necessary to examine carefully the manner in which the estimated
covariance matrix is actually going to be used, and which of its essential
features will most influence the `sensitivity' of the results in question.
Further, simplifying the structure of the covariance matrix using something
like factor analysis will usually lead to estimated matrices having much
lower overall (mean squared) error.

Andrey Feuerverger
University of Toronto
andrey@utstat.toronto.edu
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