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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?

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    $\begingroup$ I think it can be calculated in R through waldtest function. Example: waldtest(model_restricted, model_unrestricted, vcov=vcovHC(model_unrestricted)) Is it correct? $\endgroup$
    – Baumann
    Commented Oct 22, 2013 at 18:59
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    $\begingroup$ 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! $\endgroup$
    – Baumann
    Commented Oct 22, 2013 at 19:05

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Using the F-statistic formula from Hayashi:

$F=\frac{\left(Rb-r\right)^T\left[R\left(X^T X\right)^{-1}R^T\right]^{-1}\left(Rb-r\right)/ \rho}{s^2}$

(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.

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