Econometric Theory/F-Test

An F-test involves the computation of an F-statistic, which is then compared to the critical values of an F-distribution for a given significance and numerator and denominator degrees-of-freedom.

An F-statistic is calculated by dividing a chi-squared distribution divided by its degrees-of-freedom by another (independent) chi-squared distribution by its degrees-of-freedom. The resulting F-statistic has two degrees-of-freedom parameters, one each for the numerator and the denominator.

Therefore, the F-statistic for $$\hat{\beta_1} $$ would be: We know (somehow) that $$ [Z(0,1)]^2 = \chi^2 [1] $$, therefore we set the numerator equal to:
 * Numerator:

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

From the same implication of the last assumption of the CLRM as used by the t-test explanation,
 * Denominator:

$$ \frac{\chi^2 [N-2]}{N-2} \sim \frac{\hat{\sigma^2}}{\sigma^2} $$

Therefore, putting it all together gives us: $$ F(\hat{\beta_1}) = \frac{(\hat{\beta_1} - \beta_1 )^2 (\sum X_i^2) / \sigma^2}{\hat{\sigma^2} / \sigma ^2} = \frac{(\hat{\beta_1} - \beta_1)^2}{\hat{\sigma^2} / \sum X_i^2} = \frac{(\hat{\beta_1} - \beta_1)^2}{\hat{Var} (\hat{\beta_1})} \sim F[1,N-2] $$