In the least square linear regression model, if the explanatory variables and the error term are independent, and the error term is normal, why does the t-statistic of $\hat{\beta}$ follow a t distribution? I understand that if x is nonrandom, then thet-statistic of $\hat{\beta}$ follow a t distribution.

In other words, $\hat{\beta}$ is Gaussian conditional on X. The unconditional distribution of $\hat{\beta}$ is non-gaussian. How can the test statistic still follow a t distribution?

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    $\begingroup$ Are you asking why, in the model $Y = X\beta + \varepsilon$ with $\varepsilon \sim \mathcal N(0, \sigma^2 I)$, we have $\hat \beta_j \sim t$? Because if so, that's not correct; $\hat \beta \sim \mathcal N(\beta, \sigma^2 (X^T X)^{-1})$. Maybe you're thinking of the test statistic? $\endgroup$ – jld Jul 1 '17 at 16:36
  • $\begingroup$ @Chaconne I believe that OP is asking: Why do we use a test statistic that follows a t-distribution (i.e., we use the t-statistic) to test $\hat{\beta}$? Why not, for instance, something that follows a normal distribution (i.e., z-score)? $\endgroup$ – Mark White Jul 1 '17 at 18:36
  • $\begingroup$ The estimate $\hat\beta$ doesn't follow a t distribution. Are you perhaps referring to the estimate divided by its estimated standard error? $\endgroup$ – whuber Jul 2 '17 at 14:01

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