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 $[Z(0,1)]^{2}=\chi ^{2}[1]$, 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: $F({\hat {\beta _{1}}})={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}(\sum X_{i}^{2})/\sigma ^{2}}{{\hat {\sigma ^{2}}}/\sigma ^{2}}}={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}}{{\hat {\sigma ^{2}}}/\sum X_{i}^{2}}}={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}}{{\hat {Var}}({\hat {\beta _{1}}})}}\sim F[1,N-2]$