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I am reading a paper by Ledoit and Wolf,2001 on Improved Estimation of the Covariance Matrix of Stock Returns. I am a little confused by some points, and will appreciate a broader explanation.

The paper is attached here

Specifically, a paragraph states:

The traditional estimator — the sample covariance matrix — is seldom used because it imposes too little structure.

Then it proceeds to say

The cure is to impose some structure on the estimator. Ideally, the particular form of the structure should be dictated by the problem at hand. In the case of stock returns, a low-dimensional factor structure seems natural.

Then another paragraph states

One possible way is to specify a K-factor model with uncorrelated residuals. Then K controls how much structure we impose: the fewer the factors, the stronger the structure

My questions are the following:

How do you define the structure of the covariance matrix? How does one measure how good this structure is?

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This other answer here explains why is it problematic the estimation of the covariance of the matrix, in the same sense as they mention at the beginning of the paper.

They talk in terms of structure to say that you have no assumptions on your data. What they do is to impose some regularization, so that the resulting covariance matrix is not singular (is well conditioned).

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  • $\begingroup$ Thanks for your response juampa. From your explanations, it appears that what they refer to as structure is related to the assumptions i make about my data - so could you explain why the following statement must be true : "the fewer the factors, the stronger the structure" How can i prove this? $\endgroup$
    – Jaja
    Jun 26, 2015 at 12:46
  • $\begingroup$ I personally do not feel very comfortable with this way of taking about it. For me, if you talk about the structure of a matrix, I think of triangular matrices, Vandermonde matrices, or some other stuff like that. But that does not SEEM to be the case for me. They just look for a stable and meaningful estimate of the covariance matrix for that particular case. $\endgroup$
    – jpmuc
    Jun 26, 2015 at 14:09

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