#StackBounty: #generalized-least-squares #blue #gauss-markov-theorem Generalized Least Square When Disturbance Covariance Matrix Is Ran…

Bounty: 50

I cannot find any general results on the following Generalized Least Square (GLS) problem.

Let $Y = Xbeta + E$, where $X$ is deterministic and of full column rank $k$, and $E$ is of zero mean, with a $n$-by-$n$ covariance matrix $V$ with rank $r < n$. Let $V^+$ be the unique Moore-Penrose inverse of $V$. It is further assumed that $X’V^+X$ is invertible. Question: what is the best linear unbiased estimator (BLUE) of $beta$?

You would think the answer must be $(X’V^+X)^{-1}X’V^+Y$. But that actually would be wrong.

Please note: This has nothing to do multicollinearity. This problem arises when we have "redundant" observations and/or infinitely precise observations. For the former, we generally just drop the redundant observations. For the latter, which are basically linear restrictions of the parameters, we generally re-parameterize to incorporate such restrictions directly. I would like to find a unified approach. My guess is $(X’V^+X)^{-1}X’V^+Y$ works for the former case only.

Get this bounty!!!

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