F-test

The F-test is the OLS workhorse for jointly testing multiple linear restrictions on coefficients. The statistic compares the residual sum of squares of the restricted and unrestricted models: $F = \frac{(SSR_R - SSR_{UR}) / q}{SSR_{UR} / (n - k - 1)}$, where $q$ is the number of restrictions. Under $H_0$, $F \sim F(q, n - k - 1)$; reject if $F$ exceeds the critical value (or p < $\alpha$).

When to use

Standard uses: testing whether a group of regressors is jointly insignificant, testing whether two coefficients are equal, testing the overall fit of a regression ($H_0$: all slopes = 0). In R: linearHypothesis(model, c("x1 = 0", "x2 = 0")) from car. The first-stage F is the standard Instrument Relevance diagnostic in IV (F > 10 = not weak). The MLE analogue for non-linear models is the Likelihood Ratio Test.