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:

• Numerator:

$Z(\widehat{{\beta }_1})=\frac{\widehat{{\beta }_1}-{\beta }_1}{se(\widehat{{\beta }_1})}=\frac{(\widehat{{\beta }_1}-{\beta }_1)(\sum X_i^2{)}^1/2}{\sigma }$

• Denominator:

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

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

Therefore, putting it all together we get,

$t(\widehat{{\beta }_1})=\frac{Z(\widehat{{\beta }_1})}{\hat{\sigma }/\sigma }=\frac{(\widehat{{\beta }_1-{\beta }_1})(\sum X_i^2{)}^{1/2}/\sigma }{{\sigma }^2/\sigma }=\frac{\widehat{{\beta }_1}-{\beta }_1}{\hat{\sigma }/(\sum X_i^2{)}^{1/2}}=\frac{\widehat{{\beta }_1}-{\beta }_1}{\widehat{se}(\widehat{{\beta }_1})}\sim t[N-2]$

Notes

• $se(\widehat{{\beta }_1})=\frac{\sigma }{(\sum X_i^2{)}^{1/2}}$
• $\widehat{se}(\widehat{{\beta }_1})=\frac{\hat{\sigma }}{(\sum X_i^2{)}^{1/2}}$