Negative eigenvalues in covariance matrix

[BWV]

Trying to run the factoran function in MATLAB on a large matrix of daily stock returns. The function requires the data to have a positive definite covariance matrix, but this data has many very small negative eigenvalues (< 10^-17), which I understand to be a floating point issue as 'real' covariance matrices are positive semi-definite. Does not make a difference whether or not I subtract the market return (which reduces the correlation). Any thoughts, I can find options to 'fix' the cov matrix, but nothing to tweak the data prior to calculating the cov matrix

 

[marcusl]

Add a small constant value σ² to the diagonal elements of your covariance matrix. This has the effect of increasing the noise floor in your data, eliminating the negative values arising from numerical errors. The smallest eigenvalues will now take the value σ² (approximately--but that's a topic for another thread). Choose σ² large compared to 1e-17 but small compared to the eigenvalues of interest. If your eigenvalues of interest are of order >=1, for instance, then σ²=1e-10 will suffice without affecting the validity of your data. This is known in the statistical signal processing literature as diagonal loading and in the mathematics literature as Tikhonov regularization.

 

[BWV]

Thanks, afraid of that, will have to hack the MATLAB function to do this, surprised there is not an option on the cov function or another built-in function to do this. Its just something I am playing around with, so not sure if I will invest the time

 

[marcusl]

Let us know how it turns out!