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I come across a problem about finding the least square estimator of A$\beta$, where $\beta$ is the parameter vector in linear model ($Y=X\beta+\epsilon$). My question is, would the least square estimator of A$\beta$ be Ab, where b is the least square estimator of $\beta$? I know that Ab is BLUE (best linear unbiased estimator) of A$\beta$, but I'm not sure if this property guarantees it to be the least square estimator of A$\beta$. Can you guys help me prove/disprove it? Many thanks.

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Assume $A$ to be an invertible matrix of constants, Let $\gamma = A\beta$ then for the model $Y = X\beta + \epsilon = XA^{-1}\gamma + \epsilon$. Hence $\hat{\gamma} = ({A'}^{-1}X'XA^{-1})^{-1}A'^{-1}X'Y = A(X'X)^{-1}(A'A'^{-1})X'Y = A(X'X)^{-1}X'Y = A\hat{\beta}$

Hence this will be true under all non-singular transformations.

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  • $\begingroup$ The problem here is A is not necessarily a square matrix and as I stated, I'm particularly looking at the linear combination, which means A is a row vector. $\endgroup$ – diidoobiib Jul 16 at 13:32

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