In page 3, of Asymptotic Theory for Econometricians, the following assumptions of OLS are defined:

  1. OLS model: $Y= X\beta+\epsilon$
  2. $X$ is a nonstochastic and finite n x k matrix, n > k.
  3. $X'X$ is nonsingular.
  4. $E(\epsilon)=0$.
  5. $\epsilon \sim N(0, \sigma^2I)$, $\sigma^2 < \infty$


Given 1-5, $Efficiency$ of $ \hat \beta $ is the maximum likelihood estimator and is the best unbiased estimator in the sense that the variance covariance matrix of any other unbiased estimator exceeds that of $ \hat \beta $ by a positive semidefinite matrix, regardless of the value of $ \hat \beta $.

Could anynone provide an intuitive explanation for this definition?


Given that assumptions $(1) - (5)$ hold, then you can show that the OLS estimator of $\beta$ is the same as MLE, and the same as UMVUE and BLUE. The equivalence between OLS and MLE is straightforward for the multivariate set, as minimizing the square errors is the same as maximizing the $\exp$ of minus the square errors (which is the MLE). For the BLUE and UMVUE you can solve the corresponding Lagrangian problem.

Regarding the meaning of "exceeds that of $\hat{\beta}$ by a positive semidefinite matrix" - note that for the OLS you have that $$ var(\hat{\beta})=\sigma^2 (X'X)^{-1}, $$ now, it is possible to show that for any $\tilde{\beta}$ that is not OLS/MLE you can show that $$ var(\tilde{\beta})=\Sigma_{\beta} = \sigma^2 (X'X)^{-1} + \sigma^2\Sigma'_{\beta}, $$ such that $\Sigma'_{\beta}$ is a positive semidefinite matrix. This is a multidimensional generalization of the fact that if $\hat{\theta}$ is MVUE of $\theta$ with $var({\tilde{\theta}})=\sigma^2_{\hat{\theta}}$, thus for every other unbiased estimator $\tilde{\theta}$ you'll have $$ var(\tilde{\theta})=\sigma^2_{\hat{\theta}}+a^2 . $$


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