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Consider an ordinary least squares model, $$y = \beta X + \epsilon \qquad \epsilon\sim N(0, \sigma)$$

The Gauss-Markov theorem tells us that the ordinary least-squares (OLS) estimator is the minimum-variance linear unbiased estimator (BLUE) for the coefficients: $$ \beta \approx \hat\beta = (X^TX)^{-1}X^Ty $$

Does an unbiased, nonlinear estimator with lower variance, $\tilde\beta$, exist?

Based on my previous question.

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  • $\begingroup$ Not under normality. If the error is e.g. Laplace distributed, the mean absolute deviation estimator is more efficient. $\endgroup$ – hejseb Jul 4 '17 at 4:45
  • $\begingroup$ Can you suggest why no such estimator $\tilde\beta$ exists? $\endgroup$ – user126350 Jul 4 '17 at 4:49
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    $\begingroup$ If the underlying data generating process is i.i.d gaussian with a constant variance, and a linear mean model (basically the model matches the truth), in that case OLS is the minimum variance unbiased estimator, because it attains the Cramer Rao Lower Bound. Note that I didn't say minimum variance linear unbiased. Basically it is the best. econ.ohio-state.edu/dejong/note5.pdf. page 17. $\endgroup$ – Cowboy Trader Jul 4 '17 at 6:24

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