# What is special about covariance estimation from statistical factor models?

If you were to compare the usual sample covariance estimate to a robust covariance estimate (such as MCD), you can say that the robust estimate is more tolerant to outliers in the data and will not be influenced as much by the presence of these outliers. In other words, the robust estimate is not "jumping at shadows".

Is there a similar statement you can make about covariance estimation using a statistical factor model?

For the sake of completeness, the procedure for estimating covariance using a statistical factor model is as follows:

1. Estimate an initial covariance matrix via some other means (sample or robust)
2. Decompose into Eigen values and Eigen vectors and apply the rotation.
3. Discard all but the most significant eigen vectors (you can use a Marcenko-Pastur distribution to calculate a cutoff)
4. Regress the returns on the remaining factors.
5. Calculate the covariance in the usual way for a factor model.

It seems to me that by discarding the non significant eigen vectors, you are ignoring a lot of the noise and the resulting covariance matrix represents more signal on the off diagonal entries, with more noise represented on the diagonal entries. This is not the same as being robust to outliers though.

Does the above seem like a fair statement? Is there anything else we can say about it?

• Factor model helps reduce overfitting, i.e. fitting signal+noise rather than just the signal. (Fitting just the signal is infeasible but desirable.) So I think your statement is fine. – Richard Hardy Sep 15 '16 at 12:47