Generating a positive semi-definite covariance matrix (using KL transform)

Question

I have a set of input data X consisting of S&P 500 returns, that provides me with a covariance matrix C that is non positive semi-definite. The reason for the non-semi definite nature of the covariance matrix is that S&P returns data are asynchronous or incomplete. Is it possible to 'correct' this covariance matrix by kicking out the negative eigenvalues of C (or setting them equal to zero)? Could the Karhunen Loeve Transform be used, as it provides the smallest mean squared error approximation?

I used this standard procedure for the covariance matrix which is performed in Matlab using cov(): https://www.statlect.com/images/covariance-matrix__97.png

Example set: C = \begin{bmatrix}0.99&0.78&0.59&0.44\\0.78&0.92&0.28&0.81\\0.59&0.28&1.12&0.23\\0.44&0.81&0.23&0.99\end{bmatrix}

The eigenvalues in this case are -0.0004, 0.4214, 0.9988, 2.6001. In other cases the negative value is more significant than -0.0004.

Answer 1

Yes: if you view your observation $\hat C$ as a noisy estimate of the true covariance matrix, you could find the closest positive semidefinite matrix to your estimate $\hat C$ by discarding the negative eigenvalues. That is, letting the eigendecomposition of $\hat C$ be $Q \Lambda Q^T$, we have that $$ \operatorname*{argmin}_{\tilde C : \tilde C \succeq 0} \lVert \tilde C - \hat C \rVert_F = Q \max(\Lambda, 0) Q^T ,$$ where the max is elementwise.

For a proof, as well as an alternating algorithm for the same problem with correlation matrices, see

Nicholas J. Higham (2002). Computing the nearest correlation matrix — a problem from finance. IMA J Numer Anal (2002) 22 (3): 329-343. (doi, free version)

This corresponds to, for example, an observation model where $\hat C$ is the true correlation matrix $C$ plus isotropic Gaussian noise. Whether that's an appropriate assumption in your case is unclear to me, not totally understanding your setting; it seems that your discussion with whuber should help sort that out.