Why use the particular test statistic for hypothesis testing linear regression coefficients There was a question asked here around why the t-test is appropriate for hypothesis testing linear regression coefficients: Why is a T distribution used for hypothesis testing a linear regression coefficient?. The answers on the page focus on demonstrating that if you take the deviation of the estimated coefficient, $\hat{\beta}$ from the true coefficient, $\beta$ and then divide by the residual sum of squares (RSS), then the distribution of that number is a t-distribution.
But why should I take that for granted? Why use that particular test statistic and not another? Is that particular test statistic special? Is it uniformly most powerful (UMP) among its peers?


Why use that particular test statistic and not another?

The t-test coincidences with the likelihood-ratio test and therefore has as good properties in being a powerful test.

Is it uniformly most powerful (UMP) among its peers?

Yes if you consider an alternative point hypothesis (think of Neyman's and Pearson's theorems), but not necessarily for composite hypotheses.
See for example the difference in power of one-sided versus two-sided tests. None of these tests are dominant everywhere

Image from the question: https://stats.stackexchange.com/a/548241/164061
However, for a one-sided composite hypothesis, e.g. $H_0: \beta = 0$ versus $H_a: \beta > 0$ the t-test is UMP among unbiased tests. You can argue this by considering that of all hypothesis tests with a simple alternative hypotheses inside the region of the one-sided composite hypothesis, the t-test is the same and UMP.
