I have found Weighted Least Squares with Endogenous Weights but the answers primarily tackle the question of when $w_i$ correlates with $\epsilon_i$. I would like to ask if we use $w_i$ as a control variable, i.e. running the regression $y_i = \beta x_i + \epsilon_i$ weighted by $w_i$, where $x_i$ contains $w_i$, whether the OLS estimator still retains properties such as unbiasedness, consistency, and whether this might skew our results.
1 Answer
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If you mean "weighted" in the sense of the variance of the vector $y=(y_1,...,y_n)$ (or the vector of $\epsilon_i$'s) is proportional to $\mbox{diag}(1/w_i)$, then the OLS solution will still be unbiased, which you can check via the usual formula, but you lose other properties in general, such as the linear combinations of the estimator being the best linear unbiased estimator (BLUE).
