vcovHC(x,
type = c("HC3", "const", "HC", "HC0", "HC1", "HC2", "HC4"),
omega = NULL, sandwich = TRUE, ...)meatHC(x, type = , omega = NULL)
"lm"
.residuals
(the residuals of the linear model), diaghat
(the diagonal
of the corresponding hat matrix) and df
(the residual degrees of
freedom). FFALSE
only the meat matrix is returned.sandwich
.meatHC
is the real work horse for estimating
the meat of HC sandwich estimators -- vcovHC
is a wrapper calling
sandwich
and bread
. See Zeileis (2006) for
more implementation details. The theoretical background, exemplified
for the linear regression model, is described below and in Zeileis (2004). When type = "const"
constant variances are assumed and
and vcovHC
gives the usual estimate of the covariance matrix of
the coefficient estimates:
$$\hat \sigma^2 (X^\top X)^{-1}$$
All other methods do not assume constant variances and are suitable in case of
heteroskedasticity. "HC"
(or equivalently "HC0"
) gives White's
estimator, the other estimators are refinements of this. They are all of form
$$(X^\top X)^{-1} X^\top \Omega X (X^\top X)^{-1}$$
and differ in the choice of Omega. This is in all cases a diagonal matrix whose
elements can be either supplied as a vector omega
or as a
a function omega
of the residuals, the diagonal elements of the hat matrix and
the residual degrees of freedom. For White's estimator
omega <- function(residuals, diaghat, df) residuals^2
Instead of specifying of providing the diagonal omega
or a function for
estimating it, the type
argument can be used to specify the
HC0 to HC4 estimators. If omega
is used, type
is ignored.
Long & Ervin (2000) conduct a simulation study of HC estimators in
the linear regression model, recommending to use HC3 which is thus the
default in vcovHC
. Cribari-Neto (2004) suggests the HC4 type
estimator which is tailored to take into account the effect of leverage
points in the design matrix. For more details see the references.
Long J. S., Ervin L. H. (2000), Using Heteroscedasticity Consistent Standard Errors in the Linear Regression Model. The American Statistician, 54, 217--224.
MacKinnon J. G., White H. (1985), Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics 29, 305--325.
White H. (1980), A heteroskedasticity-consistent covariance matrix and a direct test for heteroskedasticity. Econometrica 48, 817--838.
Zeileis A (2004), Econometric Computing with HC and HAC Covariance Matrix
Estimators. Journal of Statistical Software, 11(10), 1--17.
URL
Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.
Journal of Statistical Software, 16(9), 1--16.
URL
lm
, hccm
,
bptest
, ncv.test
## generate linear regression relationship
## with homoskedastic variances
x <- sin(1:100)
y <- 1 + x + rnorm(100)
## compute usual covariance matrix of coefficient estimates
fm <- lm(y ~ x)
vcovHC(fm, type="const")
vcov(fm)
sigma2 <- sum(residuals(lm(y~x))^2)/98
sigma2 * solve(crossprod(cbind(1,x)))
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