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Hi I am reviewing the example of Barra risk model in the following document page 23 there is the statement:

"Estimating a covariance matrix for, say, 3,000 stocks requires data for at least 3,000 periods.

Why the number of periods has to be [greater than or] equal to the number of stocks?

Covariances and variances of the stocks can be measured for any number of periods, cannot they? I don't get the point here. Can anybody please clarify?

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  • $\begingroup$ If you use less than 3000 periods the resulting variance covariance matrix will not be positive definite (it will be only positive semidefinite). This causes some problems when working with such a $\Sigma$ matrix (although there are also ways to get around them). For example $\Sigma^{-1}$ will not exist :(( $\endgroup$ – Alex C May 14 '18 at 0:50
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Look at the matrix A=[X1, X2, ... Xn]. Xn is a time series with m points for stock n, then A is a matrix with size mxn. Think A as a information transformation from time space to stock space. if m < n, obviously you cannot recover all information of n stock. Or put it in a another way, you cannot identify which stock with any given time series if m < n.

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