Heteroscedasticity-Corrected Covariance Matrices

Calculates heteroscedasticity-corrected covariance matrices linear models fit by least squares or weighted least squares. These are also called “White-corrected” or “White-Huber” covariance matrices.

Usage

hccm(model, ...)

# S3 method for class 'lm'
hccm(model, type=c("hc3", "hc0", "hc1", "hc2", "hc4"), 
  singular.ok=TRUE, ...)

# Default S3 method
hccm(model, ...)

Arguments

model: a unweighted or weighted linear model, produced by lm.
type: one of "hc0", "hc1", "hc2", "hc3", or "hc4"; the first of these gives the classic White correction. The "hc1", "hc2", and "hc3" corrections are described in Long and Ervin (2000); "hc4" is described in Cribari-Neto (2004).
singular.ok: if FALSE (the default is TRUE), a model with aliased coefficients produces an error; otherwise, the aliased coefficients are ignored in the coefficient covariance matrix that's returned.
...: arguments to pass to hccm.lm.

Details

The original White-corrected coefficient covariance matrix ("hc0") for an unweighted model is $$V(b)=(X^{\prime }X)^{-1}X^{\prime }diag(e_{i}^{2})X(X^{\prime }X)^{-1}$$ where $e_{i}^{2}$ are the squared residuals, and $X$ is the model matrix. The other methods represent adjustments to this formula. If there are weights, these are incorporated in the corrected covariance matrix.

The function hccm.default simply catches non-lm objects.

See Freedman (2006) and Fox and Weisberg (2019, Sec. 5.1.2) for discussion of the use of these methods in generalized linear models or models with nonconstant variance.

Value

The heteroscedasticity-corrected covariance matrix for the model.

The function will return an error, rather than a matrix, under two circumstances. First, if any of the cases have hatvalue (leverage) equal to one, then the corresponding fitted value will always equal the observed value. For types hc2, hc3 and hc4 the hccm matrix is undefined. For hc0 and hc1 it is defined but it can be shown to be singular, and therefore not a consistent estimate of the covariance matrix of the coefficient estimates. A singular estimate of the covariance matrix may also be obstained if the matrix $X$ is ill-conditioned. In this latter case rescaling the model matrix may give a full-rank estimate.

References

Cribari-Neto, F. (2004) Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics and Data Analysis 45, 215–233.

Fox, J. (2016) Applied Regression Analysis and Generalized Linear Models, Third Edition. Sage.

Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression, Third Edition, Sage.

Freedman, D. (2006) On the so-called "Huber sandwich estimator" and "robust standard errors", American Statistician, 60, 299–302.

Long, J. S. and Ervin, L. H. (2000) Using heteroscedasity consistent standard errors in the linear regression model. The American Statistician 54, 217–224.

White, H. (1980) A heteroskedastic consistent covariance matrix estimator and a direct test of heteroskedasticity. Econometrica 48, 817–838.

Author

John Fox jfox@mcmaster.ca

Examples

mod <- lm(interlocks ~ assets + nation, data=Ornstein)
print(vcov(mod), digits=4)
#>             (Intercept)     assets  nationOTH   nationUK   nationUS
#> (Intercept)   1.079e+00 -1.588e-05 -1.037e+00 -1.057e+00 -1.032e+00
#> assets       -1.588e-05  1.642e-09  1.155e-05  1.362e-05  1.109e-05
#> nationOTH    -1.037e+00  1.155e-05  7.019e+00  1.021e+00  1.003e+00
#> nationUK     -1.057e+00  1.362e-05  1.021e+00  7.405e+00  1.017e+00
#> nationUS     -1.032e+00  1.109e-05  1.003e+00  1.017e+00  2.128e+00
##             (Intercept)     assets  nationOTH   nationUK   nationUS
## (Intercept)   1.079e+00 -1.588e-05 -1.037e+00 -1.057e+00 -1.032e+00
## assets       -1.588e-05  1.642e-09  1.155e-05  1.362e-05  1.109e-05
## nationOTH    -1.037e+00  1.155e-05  7.019e+00  1.021e+00  1.003e+00
## nationUK     -1.057e+00  1.362e-05  1.021e+00  7.405e+00  1.017e+00
## nationUS     -1.032e+00  1.109e-05  1.003e+00  1.017e+00  2.128e+00

print(hccm(mod), digits=4)             
#>             (Intercept)     assets  nationOTH   nationUK   nationUS
#> (Intercept)   1.664e+00 -3.957e-05 -1.569e+00 -1.611e+00 -1.572e+00
#> assets       -3.957e-05  6.752e-09  2.275e-05  3.051e-05  2.231e-05
#> nationOTH    -1.569e+00  2.275e-05  8.209e+00  1.539e+00  1.520e+00
#> nationUK     -1.611e+00  3.051e-05  1.539e+00  4.476e+00  1.543e+00
#> nationUS     -1.572e+00  2.231e-05  1.520e+00  1.543e+00  1.946e+00
##             (Intercept)     assets  nationOTH   nationUK   nationUS
## (Intercept)   1.664e+00 -3.957e-05 -1.569e+00 -1.611e+00 -1.572e+00
## assets       -3.957e-05  6.752e-09  2.275e-05  3.051e-05  2.231e-05
## nationOTH    -1.569e+00  2.275e-05  8.209e+00  1.539e+00  1.520e+00
## nationUK     -1.611e+00  3.051e-05  1.539e+00  4.476e+00  1.543e+00
## nationUS     -1.572e+00  2.231e-05  1.520e+00  1.543e+00  1.946e+00