The ANOVA in linear regression models is equivalent to the Wald test (and the likelihood ratio test) of the corresponding nested models. So when you want to conduct the corresponding test using heteroskedasticity-consistent (HC) standard errors, this cannot be obtained from a decomposition of the sums of squares but you can carry out the Wald test using a HC covariance estimate. This idea is used in both Anova() and linearHypothesis() from the car package and coeftest() and waldtest() from the lmtest package. The latter three can also be used with plm objects.
A simple (albeit not very interesting/meaningful) example is the following. We use the standard model from the ?plm manual page and want to carry out a Wald test for the significance of both log(pcap) and unemp. We need these packages:
library("plm")
library("sandwich")
library("car")
library("lmtest")
The model (under the alternative) is:
data("Produc", package = "plm")
mod <- plm(log(gsp) ~ log(pc) + log(emp) + log(pcap) + unemp,
data = Produc, index = c("state", "year"))
First, let's look at the marginal Wald tests with HC standard errors for all individual coefficients:
coeftest(mod, vcov = vcovHC)
t test of coefficients:
Estimate Std. Error t value Pr(>|t|)
log(pc) 0.2920069 0.0617425 4.7294 2.681e-06 ***
log(emp) 0.7681595 0.0816652 9.4062 < 2.2e-16 ***
log(pcap) -0.0261497 0.0603262 -0.4335 0.66480
unemp -0.0052977 0.0024958 -2.1226 0.03411 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
And then we carry out a Wald test for both log(pcap) and unemp:
linearHypothesis(mod, c("log(pcap)", "unemp"), vcov = vcovHC)
Linear hypothesis test
Hypothesis:
log(pcap) = 0
unemp = 0
Model 1: restricted model
Model 2: log(gsp) ~ log(pc) + log(emp) + log(pcap) + unemp
Note: Coefficient covariance matrix supplied.
Res.Df Df Chisq Pr(>Chisq)
1 766
2 764 2 7.2934 0.02608 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Alternatively, we can also fit the model under the null hypothesis (mod0 say) without the two coefficients and then call waldtest():
mod0 <- plm(log(gsp) ~ log(pc) + log(emp),
data = Produc, index = c("state", "year"))
waldtest(mod0, mod, vcov = vcovHC)
Wald test
Model 1: log(gsp) ~ log(pc) + log(emp)
Model 2: log(gsp) ~ log(pc) + log(emp) + log(pcap) + unemp
Res.Df Df Chisq Pr(>Chisq)
1 766
2 764 2 7.2934 0.02608 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The test statistic and p-value computed by linearHypothesis() and waldtest() is exactly the same. Just the interface and output formatting is somewhat different. In some cases one or the other is simpler or more intuitive.
Note: If you supply a covariance matrix estimate (i.e., a matrix like vocvHC(mod)) instead of a covariance matrix estimator (i.e., a function like vocvHC), make sure that you supply the HC covariance matrix estimate of the model under the alternative, i.e., the non-restricted model.
Best Answer
To make things simple, suppose you have 3 observations. Robust standard errors allow for a variance-covariance matrix of the errors to look like this: $$\Sigma = \begin{bmatrix} \sigma_{1} & 0 & 0\\ 0 & \sigma_{2} & 0 \\ 0 & 0 & \sigma_{3} \end{bmatrix} $$
The diagonal terms are the variances of the errors for each of the 3 observations. The covariance terms are all zero because we still assume that the errors are uncorrelated across observations. If you want to relax that, you will need cluster-robust errors.
Ordinary, non-robust errors assume that $\sigma_1=\sigma_2=\sigma_2=\sigma$: all observations have the same (unknown) error variance. Neither the robust nor the non-robust VCE make any assumptions about the distribution of the error, such as normality. They only make assumptions about the variance being the same (homoskedasticity) or different across observations (heteroskedasticity).
Thus, depending on what you are doing, you may still need to test for normality of the errors. Alternatively, you may be able to bootstrap or rely on asymptotics for inference.