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:

• Numerator:

We know (somehow) that [Z(0,1)]² = χ²[1], therefore we set the numerator equal to:

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

• Denominator:

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

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

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