Object-oriented estimation of jackknife covariances, i.e., based on the centered outer product of leave-on-out estimates of the model coefficients/parameters.

vcovJK(x, ...)

# S3 method for default
vcovJK(x, cluster = NULL, center = "mean", ...)

## Arguments

x

a fitted model object.

cluster

a variable indicating the clustering of observations, a list (or data.frame) thereof, or a formula specifying which variables from the fitted model should be used (see examples). By default (cluster = NULL), either attr(x, "cluster") is used (if any) or otherwise every observation is assumed to be its own cluster.

center

character specifying how to center the coefficients from all jacknife samples (each dropping one observational unit/cluster). By default the coefficients are centered by their "mean" across the sample or, alternatively, by the original full-sample "estimate".

...

arguments passed to methods. For the default method, this is passed to vcovBS.

## Details

Jackknife covariance estimation is based on leave-one-out estimates of the coefficients/parameters of a model. This means that the model is reestimated after dropping each observational unit once, i.e., each individual observation in independent observations or each cluster in dependent data. The covariance matrix is then constructed from the scaled outer product of the centered jackknife estimates. Centering can either be done by the mean of the jackknife coefficients (default) or by the original full-sample estimates. Scaling is done by (N - 1)/N where N is the number of observational units.

Recent research has shown that the jackknife covariance estimate have particularly useful properties in practice: they are not downward biased and yield better coverage rates for confidence intervals compared to other "robust" covariance estimates. See MacKinnon et al. (2022) and Hansen (2022) for more details.

As jackknife covariances are also based on reestimation of the coefficients on subsamples, their computation is very similar to bootstrap covariances. Hence, the vcovBS methods provided in the package all offer an argument vcovBS(..., type = "jackknife"). This is called by the default vcovJK method. Therefore, see the arguments of vcovBS for further details, e.g., for leveraging multicore computations etc.

In the linear regression model, the jackknife covariance can actually be computed without reestimating the coefficients but using only the full-sample estimates and certain elements of the so-called hat matrix. Namly the diagonal elements or blocks of elements from the hat matrix are needed for independent observations and clustered data, respectively. These alternative computations of the jackknife covariances are available in vcovHC and vcovCL, respectively, in both cases with argument type = "HC3". To obtain HC3 covariances that exactly match the jackknife covariances, the jackknife has to be centered with the full-sample estimates and the right finite-sample adjustment has to be selected for the HC3.

In small to moderate sample sizes, the HC3 estimation techniques are typically much faster than the jackknife. However, in large samples it may become impossible to compute the HC3 covariances while the jackknife approach is still feasible.

## Value

A matrix containing the covariance matrix estimate.

## References

Bell RM, McCaffrey DF (2002). “Bias Reduction in Standard Errors for Linear Regression with Multi-Stage Samples”, Survey Methodology, 28(2), 169--181.

Hansen BE (2022). “Jackknife Standard Errors for Clustered Regression”, Working Paper, August 2022. https://www.ssc.wisc.edu/~bhansen/papers/tcauchy.html

MacKinnon JG, Nielsen MØ, Webb MD (2022). “Cluster-Robust Inference: A Guide to Empirical Practice”, Journal of Econometrics, Forthcoming. doi:10.1016/j.jeconom.2022.04.001

Zeileis A, Köll S, Graham N (2020). “Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R.” Journal of Statistical Software, 95(1), 1--36. doi:10.18637/jss.v095.i01

## See also

vcovJK, vcovHC, vcovCL

## Examples

## cross-section data
data("PublicSchools", package = "sandwich")
m1 <- lm(Expenditure ~ poly(Income, 2), data = PublicSchools)
vcovJK(m1, center = "estimate")
#>                  (Intercept) poly(Income, 2)1 poly(Income, 2)2
#> (Intercept)         97.84092         1055.131         1370.855
#> poly(Income, 2)1  1055.13101        25053.095        31336.158
#> poly(Income, 2)2  1370.85525        31336.158        46955.800
vcovHC(m1, type = "HC3") * (nobs(m1) - 1)/nobs(m1)
#>                  (Intercept) poly(Income, 2)1 poly(Income, 2)2
#> (Intercept)         97.84092         1055.131         1370.855
#> poly(Income, 2)1  1055.13101        25053.095        31336.158
#> poly(Income, 2)2  1370.85525        31336.158        46955.800

## clustered data
data("PetersenCL", package = "sandwich")
m2 <- lm(y ~ x, data = PetersenCL)

## jackknife estimator coincides with HC3 (aka CV3)
vcovJK(m2, cluster = ~ firm, center = "estimate")
#>               (Intercept)             x
#> (Intercept)  4.499186e-03 -6.714627e-05
#> x           -6.714627e-05  2.577098e-03
vcovCL(m2, cluster = ~ firm, type = "HC3", cadjust = FALSE)
#>               (Intercept)             x
#> (Intercept)  4.499186e-03 -6.714627e-05
#> x           -6.714627e-05  2.577098e-03