Heteroscedasticity and autocorrelation consistent (HAC) estimation of the covariance matrix of the coefficient estimates in a (generalized) linear regression model.

vcovHAC(x, ...)

# S3 method for default
vcovHAC(x, order.by = NULL, prewhite = FALSE, weights = weightsAndrews,
  adjust = TRUE, diagnostics = FALSE, sandwich = TRUE, ar.method = "ols",
  data = list(), ...)

meatHAC(x, order.by = NULL, prewhite = FALSE, weights = weightsAndrews,
  adjust = TRUE, diagnostics = FALSE, ar.method = "ols", data = list(), ...)



a fitted model object.


Either a vector z or a formula with a single explanatory variable like ~ z. The observations in the model are ordered by the size of z. If set to NULL (the default) the observations are assumed to be ordered (e.g., a time series).


logical or integer. Should the estimating functions be prewhitened? If TRUE or greater than 0 a VAR model of order as.integer(prewhite) is fitted via ar with method "ols" and demean = FALSE.


Either a vector of weights for the autocovariances or a function to compute these weights based on x, order.by, prewhite, ar.method and data. If weights is a function it has to take these arguments. See also details.


logical. Should a finite sample adjustment be made? This amounts to multiplication with \(n/(n-k)\) where \(n\) is the number of observations and \(k\) the number of estimated parameters.


logical. Should additional model diagnostics be returned? See below for details.


logical. Should the sandwich estimator be computed? If set to FALSE only the meat matrix is returned.


character. The method argument passed to ar for prewhitening.


an optional data frame containing the variables in the order.by model. By default the variables are taken from the environment which vcovHAC is called from.


arguments passed to sandwich (in vcovHAC) and estfun (in meatHAC), respectively.


The function meatHAC is the real work horse for estimating the meat of HAC sandwich estimators -- the default vcovHAC method 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 in Zeileis (2004).

Both functions construct weighted information sandwich variance estimators for parametric models fitted to time series data. These are basically constructed from weighted sums of autocovariances of the estimating functions (as extracted by estfun). The crucial step is the specification of weights: the user can either supply vcovHAC with some vector of weights or with a function that computes these weights adaptively (based on the arguments x, order.by, prewhite and data). Two functions for adaptively choosing weights are implemented in weightsAndrews implementing the results of Andrews (1991) and in weightsLumley implementing the results of Lumley (1999). The functions kernHAC and weave respectively are to more convenient interfaces for vcovHAC with these functions.

Prewhitening based on VAR approximations is described as suggested in Andrews & Monahan (1992).

The covariance matrix estimators have been improved by the addition of a bias correction and an approximate denominator degrees of freedom for test and confidence interval construction. See Lumley & Heagerty (1999) for details.


A matrix containing the covariance matrix estimate. If diagnostics was set to TRUE this has an attribute "diagnostics" which is a list with


multiplicative bias correction


Approximate denominator degrees of freedom


Andrews DWK (1991). “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Econometrica, 59, 817--858.

Andrews DWK & Monahan JC (1992). “An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator.” Econometrica, 60, 953--966.

Lumley T & Heagerty P (1999). “Weighted Empirical Adaptive Variance Estimators for Correlated Data Regression.” Journal of the Royal Statistical Society B, 61, 459--477.

Newey WK & West KD (1987). “A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica, 55, 703--708.

Zeileis A (2004). “Econometric Computing with HC and HAC Covariance Matrix Estimators.” Journal of Statistical Software, 11(10), 1--17. doi: 10.18637/jss.v011.i10

Zeileis A (2006). “Object-Oriented Computation of Sandwich Estimators.” Journal of Statistical Software, 16(9), 1--16. doi: 10.18637/jss.v016.i09

See also


x <- sin(1:100) y <- 1 + x + rnorm(100) fm <- lm(y ~ x) vcovHAC(fm)
#> (Intercept) x #> (Intercept) 0.009716252 0.004740023 #> x 0.004740023 0.023346936
#> (Intercept) x #> (Intercept) 1.114254e-02 2.818886e-05 #> x 2.818886e-05 2.216611e-02