$X$ is a $T\times k$ random matrix with finite second moments, how to compute the expectation of the projection matrix $E[I_T-X(X'X)^{-1}X']$? (Assume that $X'X$ is positive definite.)
2026-04-04 19:56:21.1775332581
How to compute the expectation of a projection matrix?
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I've been tackling a very similar problem by expanding the matrix inverse in a power series and applying Isserlis' theorem to each of the resulting terms. The resulting series has convergence issues, but it has nice Padé approximants. If your elements are not jointly gaussian, you'll need some other technique to evaluate the expectations at each order.