# Are there unbiased, non-linear estimators with lower variance than the OLS estimator?

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.

• Not under normality. If the error is e.g. Laplace distributed, the mean absolute deviation estimator is more efficient. Commented Jul 4, 2017 at 4:45
• Can you suggest why no such estimator $\tilde\beta$ exists? Commented Jul 4, 2017 at 4:49
• 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. Commented Jul 4, 2017 at 6:24

• While the question does specify $iid$ Gaussian errors, this answer would be improved by repeating that to clarify this additional requirement beyond the Gauss-Markov conditions. Without that, this answer seems to suggest that Gauss-Markov does not take it far enough.