# Heteroscedasticity-consistent F-test

Why is the F-test for overall significance (OLS regression analysis) invalid when residuals are heteroscedastic? Is there a way to calculate it in a consistent way under heteroscedasticity? Is there any function in R to accomplish that?

• I think it can be calculated in R through waldtest function. Example: waldtest(model_restricted, model_unrestricted, vcov=vcovHC(model_unrestricted)) Is it correct? – Baumann Oct 22 '13 at 18:59
• About the invalidity of the test, the Wooldridge (Introductory Econometrics, 2nd edition, p. 253) says: "If heteroskedasticity is present, this version of the [F] test is invalid. The heteroskedasticity-robust version has no simple form, but it can be computed using certain statistical packages." And the mystery continues! – Baumann Oct 22 '13 at 19:05

$F=\frac{\left(Rb-r\right)^T\left[R\left(X^T X\right)^{-1}R^T\right]^{-1}\left(Rb-r\right)/ \rho}{s^2}$
(notation: we're testing the hypothesis $R\beta = r$, $R\in\mathbb{R}^{\rho\times K}$, rank($R$)=$\rho$, $s^2$ is the variance of the error term)
We can immediately see why this won't apply to the heteroskedastic case--in particular, we definitely don't expect $s^2$ to be a constant.
Intuitively, we know there's a (nontrivial) relationship between $t$ and $F$ statistics--so given that $t$ statistics are heteroskedasticity-dependent, we should expect $F$-statistics to be so as well.
See here (pp 3-4) for why the Wald test you suggested is correct & hence the waldtest function in R is appropriate.