Simple-regression output identities ($F=t^2=(n-2)R^2/(1-R^2)$)

Consider ordinary least squares with an intercept, one nonconstant predictor, $n>2$ cases, $S_{xx}={\sum }_i(x_i-\bar{x}{)}^2>0$ a nonconstant response, and a positive residual sum of squares. With the usual common-variance standard error, put $s^2=\mathrm{RSS}/(n-2)$. ThenThe least-squares normal equations give zero residual sum and zero residual inner product with the centred predictor, proving the first identity and $\hat{\beta }=S_{xy}/S_{xx}$. The fitted centred sum of squares is ${\hat{\beta }}^2{S}_{xx}$. Dividing it by $s^2$ gives the one-predictor F statistic and the squared zero-slope t statistic. Dividing fitted and residual sums of squares by their total gives the last expression. If $0\le R^2<1$, the adjusted coefficient of determination is $1-(1-R^2)(n-1)/(n-2)$.

These are algebraic consistency identities for one specified fit; the mean identity needs no normal-error assumption. A normal linear model supplies the usual finite-sample t and F reference distributions. Heteroscedasticity-robust standard errors need not satisfy the displayed F=t-squared relationship to the classical sums-of-squares F statistic. Rounding can create small discrepancies, while different analysed records or a changed model can create larger ones; neither should be silently treated as the same fit.^[ref]