Suppose we have a simple regression model $y=\beta_0+\beta_1x+u$ under Gauss-Markov assumptions, and $u$ has a uniform distribution on $[0,1]$. Find sampling distribution of the $t_{\beta_1}$-statistic. $t_{\hat{\beta_1}}=\frac{\hat{\beta_1}}{se(\hat{\beta_1})}$, where $se(\hat{\beta_1})=\frac{[(\sum\limits_{i=1}^n{\hat{u_i}}^2)/(n-2)]^{1/2}}{ (\sum\limits_{j=1}^n(x_i-\bar{x})^2)^{1/2}}$
We have that ${\hat\beta_1}=\beta_1+\sum\limits_{i=1}^nd_iu_i$, where $d_i=\frac{x_i-\bar{x}}{\sum\limits_{j=1}^n(x_i-\bar{x})^2}$. Conditionally on the sample $d_i$ can be treated as nonrandom variables, so $\hat{\beta_1}$ is constant plus the linear combination of independent uniform variables and it seems possible to find its distribution. Even if it is found, there still exists denominator of the $t$-statistic.
Probably another approach should be taken here.
