In the linear regression $Y=X\beta+\epsilon$, with $E(\epsilon_i|x_i)=0$, it is known that the true $\beta$ satisfies the restriction $M\beta=0$, where $M$ is a $q \times k$ matrix with $q<k$. $\hat\beta$ is estimator for $\beta$, where $\hat\beta=(X^TX)^{-1}X^TY$.

Consider the estimator: $\tilde\beta=\hat\beta-(X^TX)^{-1}M^T[M(X^TX)^{-1}M^T]^{-1}M\hat\beta$. How to find $V(\tilde\beta|X)$? Also, what is an expression for a valid standard error for the elements of $\tilde\beta$?

I think I have to first write $\tilde\beta$ as a linear function of $\hat\beta$, but I am stuck on how to proceed. I have also shown that $M\tilde\beta=0$.


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