Distribution of the t-statistic

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.

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Are you assuming the fit is done with least squares? That would be a poor approach indeed, given that the residuals don't even have zero expectation! –  whuber Feb 16 '12 at 16:19
Yes, let's assume that estimation is made via OLS. –  marcin63 Feb 16 '12 at 16:21
At the very least, you should change the model formulation to $y = 1/2 + \beta_0 + \beta_1 x + u$ with $u$ uniformly distributed on $[-1/2, 1/2]$. Least-squares can still produce inadmissible results; e.g., consider $((1,2),(2,1),(3,4/3),(4,5/3),(5,2))$: the first residual (0.53) is too large. –  whuber Feb 16 '12 at 16:28
So, probably this problem can be restated somehow in order to make it sensible for solving(to make it a good exercise in econometrics)? –  marcin63 Feb 18 '12 at 14:26
I think you're right. However, linear combinations of uniform distributions tend to be nasty to calculate: their PDFs typically are only piecewise differentiable. Thus, finding an analytical expression for the distribution of such a statistic quickly becomes uninteresting and fruitless. Consider instead explaining why you want to know this distribution and how you hope to use it: that might open the question up to approximate solutions and alternative approaches. –  whuber Feb 18 '12 at 22:14