Estimator that maximizes t-value

Question

Consider the linear regression model:$$y=\beta_{0}+\beta_{1}x+u$$

The OLS estimators minimze the squared sum of residuals. By the Gauss-Markov theorem, OLS estimators give the lowest variance of estimates in the class of unbiased estimators. However, we also know that estimators such as Ridge estimators exploit the Bias-Variance tradeoff and are able to achieve higher efficiency. Is there an estimator that would maximizes the t-value of a test of the null:$$H_{0}:\beta_{1}=0$$ or the ratio$$\frac{\hat{\beta}}{SE(\hat{\beta})}$$

Answer 1

You do know that not all estimators of $\beta$ have a $t$-distribution when divided by an estimator of their sampling variation, right?

If you want to maximize that ratio, just use an estimator that is always equal to some constant, $c$ (pick any value for $c$ that you want). Then, the standard error is zero, and that ratio is infinite.