In Classic linear regression model, under the single restriction

$$y=X\beta +u$$ $$H_0: \beta_j=0$$

If I apply t test, then the t statistic is derived as follows


Where $b_j$ is OLS estimate of beta. And $s\sqrt{a_{jj}}$ is the estimated standard error of $b_j$.

I can use F statistic

And F statistic is as follows:

$$\frac{e_R’e_R - e’e}{e’e}.(n-k)/k_2$$

where $y= X_1\beta_1+X_2\beta_2+u$

I write the same regression in the partitioned form.

My question is that how I can show these two are equivalent for single restriction?



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