Cluster-robust variance-covariance matrix

This is a generic function, with specific methods defined for lm, plm, glm, gls, lme, robu, rma.uni, and rma.mv objects.

vcovCR returns a sandwich estimate of the variance-covariance matrix of a set of regression coefficient estimates.

Usage

vcovCR(obj, cluster, type, target, inverse_var, form, ...)

# Default S3 method
vcovCR(
  obj,
  cluster,
  type,
  target = NULL,
  inverse_var = FALSE,
  form = "sandwich",
  ...
)

Arguments

obj: Fitted model for which to calculate the variance-covariance matrix
cluster: Expression or vector indicating which observations belong to the same cluster. For some classes, the cluster will be detected automatically if not specified.
type: Character string specifying which small-sample adjustment should be used, with available options "CR0", "CR1", "CR1p", "CR1S", "CR2", or "CR3". See "Details" section of vcovCR for further information.
target: Optional matrix or vector describing the working variance-covariance model used to calculate the CR2 and CR4 adjustment matrices. If a vector, the target matrix is assumed to be diagonal. If not specified, vcovCR will attempt to infer a value.
inverse_var: Optional logical indicating whether the weights used in fitting the model are inverse-variance. If not specified, vcovCR will attempt to infer a value.
form: Controls the form of the returned matrix. The default "sandwich" will return the sandwich variance-covariance matrix. Alternately, setting form = "meat" will return only the meat of the sandwich and setting form = B, where B is a matrix of appropriate dimension, will return the sandwich variance-covariance matrix calculated using B as the bread. form = "estfun" will return the (appropriately scaled) estimating function, the transposed crossproduct of which is equal to the sandwich variance-covariance matrix.
...: Additional arguments available for some classes of objects.

Value

An object of class c("vcovCR","clubSandwich"), which consists of a matrix of the estimated variance of and covariances between the regression coefficient estimates. The matrix has several attributes:

type: indicates which small-sample adjustment was used
cluster: contains the factor vector that defines independent clusters
bread: contains the bread matrix
v_scale: constant used in scaling the sandwich estimator
est_mats: contains a list of estimating matrices used to calculate the sandwich estimator
adjustments: contains a list of adjustment matrices used to calculate the sandwich estimator
target: contains the working variance-covariance model used to calculate the adjustment matrices. This is needed for calculating small-sample corrections for Wald tests.

Details

vcovCR returns a sandwich estimate of the variance-covariance matrix of a set of regression coefficient estimates.

Several different small sample corrections are available, which run parallel with the "HC" corrections for heteroskedasticity-consistent variance estimators, as implemented in vcovHC. The "CR2" adjustment is recommended (Pustejovsky & Tipton, 2017; Imbens & Kolesar, 2016). See Pustejovsky and Tipton (2017) and Cameron and Miller (2015) for further technical details. Available options include:

"CR0": is the original form of the sandwich estimator (Liang & Zeger, 1986), which does not make any small-sample correction.
"CR1": multiplies CR0 by m / (m - 1), where m is the number of clusters.
"CR1p": multiplies CR0 by m / (m - p), where m is the number of clusters and p is the number of covariates.
"CR1S": multiplies CR0 by (m (N-1)) / [(m - 1)(N - p)], where m is the number of clusters, N is the total number of observations, and p is the number of covariates. Some Stata commands use this correction by default.
"CR2": is the "bias-reduced linearization" adjustment proposed by Bell and McCaffrey (2002) and further developed in Pustejovsky and Tipton (2017). The adjustment is chosen so that the variance-covariance estimator is exactly unbiased under a user-specified working model.
"CR3": approximates the leave-one-cluster-out jackknife variance estimator (Bell & McCaffrey, 2002).

References

Bell, R. M., & McCaffrey, D. F. (2002). Bias reduction in standard errors for linear regression with multi-stage samples. Survey Methodology, 28(2), 169-181.

Cameron, A. C., & Miller, D. L. (2015). A Practitioner's Guide to Cluster-Robust Inference. Journal of Human Resources, 50(2), 317-372. doi:10.3368/jhr.50.2.317

Imbens, G. W., & Kolesar, M. (2016). Robust standard errors in small samples: Some practical advice. Review of Economics and Statistics, 98(4), 701-712. doi:10.1162/rest_a_00552

Liang, K.-Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73(1), 13-22. doi:10.1093/biomet/73.1.13

Pustejovsky, J. E. & Tipton, E. (2018). Small sample methods for cluster-robust variance estimation and hypothesis testing in fixed effects models. Journal of Business and Economic Statistics, 36(4), 672-683. doi:10.1080/07350015.2016.1247004

Examples


# simulate design with cluster-dependence
m <- 8
cluster <- factor(rep(LETTERS[1:m], 3 + rpois(m, 5)))
n <- length(cluster)
X <- matrix(rnorm(3 * n), n, 3)
nu <- rnorm(m)[cluster]
e <- rnorm(n)
y <- X %*% c(.4, .3, -.3) + nu + e
dat <- data.frame(y, X, cluster, row = 1:n)

# fit linear model
lm_fit <- lm(y ~ X1 + X2 + X3, data = dat)
vcov(lm_fit)
#>              (Intercept)            X1            X2           X3
#> (Intercept)  0.024119296 -0.0033629563  0.0030753483  0.001678683
#> X1          -0.003362956  0.0175124039 -0.0001519176 -0.002512630
#> X2           0.003075348 -0.0001519176  0.0166577695  0.003156097
#> X3           0.001678683 -0.0025126298  0.0031560972  0.024070803

# cluster-robust variance estimator with CR2 small-sample correction
vcovCR(lm_fit, cluster = dat$cluster, type = "CR2")
#>              (Intercept)           X1           X2          X3
#> (Intercept)  0.118258531 -0.058199675 -0.019174051 0.002266691
#> X1          -0.058199675  0.043149660  0.013660753 0.002410578
#> X2          -0.019174051  0.013660753  0.011604149 0.001719445
#> X3           0.002266691  0.002410578  0.001719445 0.001876608

# compare small-sample adjustments
CR_types <- paste0("CR",c("0","1","1S","2","3"))
sapply(CR_types, function(type) 
       sqrt(diag(vcovCR(lm_fit, cluster = dat$cluster, type = type))))
#>                    CR0       CR1       CR1S        CR2        CR3
#> (Intercept) 0.31083475 0.3322963 0.34011613 0.34388738 0.38123090
#> X1          0.18390490 0.1966026 0.20122919 0.20772496 0.23542350
#> X2          0.09753920 0.1042738 0.10672762 0.10772256 0.11989761
#> X3          0.04068856 0.0434979 0.04452152 0.04331983 0.04774566