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 $\widehat{{\beta }_1}$ would be:

• Numerator:

We know (somehow) that $[Z(0,1){]}^2={\chi }^2[1]$, therefore we set the numerator equal to:

$Z(\widehat{{\beta }_1}{)}^2=\frac{(\widehat{{\beta }_1}-{\beta }_1{)}^2(\sum X_i^2)}{{\sigma }^2}\sim {\chi }^2[1]=\frac{{\chi }^2[1]}{1}$

• Denominator:

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

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

Therefore, putting it all together gives us: $F(\widehat{{\beta }_1})=\frac{(\widehat{{\beta }_1}-{\beta }_1{)}^2(\sum X_i^2)/{\sigma }^2}{\widehat{{\sigma }^2}/{\sigma }^2}=\frac{(\widehat{{\beta }_1}-{\beta }_1{)}^2}{\widehat{{\sigma }^2}/\sum X_i^2}=\frac{(\widehat{{\beta }_1}-{\beta }_1{)}^2}{\widehat{Var}(\widehat{{\beta }_1})}\sim F[1,N-2]$