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Given the linear regression model $$y=X\beta + \epsilon$$ Where $E(y|X)=X\beta$, and $Var(\epsilon)=\sigma^2I$.

The Gauss Markov Theorem states that the OLS estimator $b$ is the smallest variance unbiased linear estimator of $\beta$.

When I first read this, I used to read this "linear" as referring to the fact that $\beta$ is a linear parameter in the model. However, "linear" actually refers to the estimator $b$, not to the parameter itself.

And indeed in the proof of the Gauss Markov theorem given in "Greene, Econometric analysis (p. 60)", $b_{OLS}$ is contrasted with another "linear estimator" $b_0$, and the property that $b_0$ is linear is written as $$b_0=Cy$$

So this made me wonder, what is the significance of the fact that OLS is a "linear estimator", given that the model itself is already linear?

  • i.e. can we have non-linear estimators of linear parameters, and does this make sense?

  • Are there non-linear estimators of $\beta$ (in the linear model given above), that are unbiased, and have smaller variance than $b_{OLS}$?

  • Can we have linear estimators of non-linear parameters, and why would we do this (presumably they cannot be unbiased).

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