It is well known that the standard estimator of the covariance matrix can lose the property of being positive-semidefinite if the number of variables (e.g. number of stocks) exceeds the number of observations (e.g. trading days). I think the matrix can become singular. I have a clear idea why (inspired by the geometry of the problem) but does anybody have a short but rigorous proof for this fact?
Answer
The standard estimator of the covariance matrix is: ^cov(X)=1n−1n∑i=1(Xi−ˉX)(Xi−ˉX)T,
Edit: regarding positive-semidefiniteness: ^cov(X) is always positive-semidefinite because it is Gramian, even if its rank is not full. It loses the property of being positive-definite if and only if it is singular.
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