Orthogonalized Gnanadesikan-Kettenring Covariance

Estimating a covariance matrix robustly for many assets at once is harder than it looks. Robustly estimating each individual variance is easy — just apply a robust scale estimator like Qn to each asset's returns — but the off-diagonal covariance terms are the problem: a naive robust version of "average of $x_i y_i$" doesn't automatically stay a valid, positive-semidefinite matrix once contaminated data is present.

The Gnanadesikan-Kettenring (GK) approach sidesteps this using an algebraic identity: the covariance between two variables can be written entirely in terms of the variances of their sum and their difference, since $\text{Cov}(x,y) = \tfrac{1}{4}\big(\text{Var}(x+y) - \text{Var}(x-y)\big)$. Because this formula only ever needs a robust variance estimator — never a robust covariance directly — you can plug in Qn or MAD for each of $\text{Var}(x+y)$ and $\text{Var}(x-y)$ and get a robust pairwise covariance estimate built entirely from robust univariate pieces.

The catch is that running plain GK pairwise across many assets doesn't guarantee the resulting matrix is positive-semidefinite, which a covariance matrix must be to make sense for portfolio optimization. The orthogonalized version fixes this by first applying GK to get an initial robust estimate, then projecting the data onto its eigenvectors (which are, by construction, uncorrelated with each other), applying a robust scale estimator to each of those uncorrelated projections individually, and reassembling a matrix from the results — a procedure guaranteed to produce a valid, positive-semidefinite covariance matrix even when a meaningful fraction of the return history is contaminated by fat-fingered prints or stale marks.