The Oracle estimator is suboptimal for global minimum variance portfolio optimisation

A common misconception is that the Oracle eigenvalue estimator of the covariance matrix yields the best realized portfolio performance. In reality, the Oracle estimator simply modifies the empirical covariance matrix eigenvalues so as to minimize the Frobenius distance between the filtered and the realized covariance matrices. This leads to the best portfolios only when the in-sample eigenvectors coincide with the out-of-sample ones. In all the other cases, the optimal eigenvalue correction can be obtained from the solution of a Quadratic-Programming problem. Solving it shows that the Oracle estimators only yield the best portfolios in the limit of infinite data points per asset and only in stationary systems.