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, ...)
# S3 method for default
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
```