Econometric Theory/t-Test

A t-test involves the computation of a t-statistic, which is then compared to the critical values of a t-distribution for a given significance level.

A t-test is essentially the Z-statistic of a variable divided by the square root of an independent chi-square distribution divided by its own degrees-of-freedom. The resulting value is the t-statistic with the same degrees-of-freedom as the chi-squared distribution.

$$ t = \frac{Z}{\sqrt{V/m}} \sim t[m] $$

Therefore, the t-statistic of $$\beta_1$$ would be: $$ Z(\hat{\beta_1}) = \frac{\hat{\beta_1} - \beta_1}{se(\hat{\beta_1})} = \frac{(\hat{\beta_1} - \beta_1)(\sum X_{i}^2)^1/2}{\sigma} $$
 * Numerator:

We know (as an implication of the last assumption of the CLRM) that $$ \frac{(N-2)\hat{\sigma^2}}{\sigma^2} \sim \chi^2 [N-2] $$
 * Denominator:

Therefore, $$ \frac{\hat{\sigma^2}}{\sigma^2} \sim \frac{\chi^2 [N-2]}{[N-2]} \Rightarrow \sqrt{\frac{\chi^2 [N-2}{[N-2]}} \sim \frac{\hat{\sigma}}{\sigma} $$

Therefore, putting it all together we get,

$$ t(\hat{\beta_1}) = \frac{Z(\hat{\beta_1})}{\hat{\sigma}/\sigma} = \frac{(\hat{\beta_1 - \beta_1})(\sum X_i^2)^{1/2}/\sigma}{\sigma^2 / \sigma} = \frac{\hat{\beta_1} - \beta_1}{\hat{\sigma} / (\sum X_i^2)^{1/2}} = \frac{\hat{\beta_1} - \beta_1}{\hat{se}(\hat{\beta_1})} \sim t[N-2] $$