Unconditional Robust covariance matrix estimators a la Beck and Katz for panel models (a.k.a. Panel Corrected Standard Errors (PCSE)).
An object of class "matrix" containing the estimate of
the covariance matrix of coefficients.
vcovBK is a function for estimating a robust covariance matrix of
parameters for a panel model according to the
Beck and Katz (1995)
method, a.k.a. Panel
Corrected Standard Errors (PCSE), which uses an unconditional
estimate of the error covariance across time periods (groups)
inside the standard formula for coefficient
covariance. Observations may be clustered either by "group" to
account for timewise heteroskedasticity and serial correlation or
by "time" to account for cross-sectional heteroskedasticity and
correlation. It must be borne in mind that the Beck and Katz
formula is based on N- (T-) asymptotics and will not be appropriate
elsewhere.
The diagonal logical argument can be used, if set to
TRUE, to force to zero all non-diagonal elements in the
estimated error covariances; this is appropriate if both serial and
cross–sectional correlation are assumed out, and yields a
timewise- (groupwise-) heteroskedasticity–consistent estimator.
Weighting schemes specified by type are analogous to those in
sandwich::vcovHC() in package sandwich and are
justified theoretically (although in the context of the standard
linear model) by MacKinnon and White (1985)
and
Cribari–Neto (2004)
(see Zeileis 2004)
.
The main use of vcovBK (and the other variance-covariance estimators
provided in the package vcovHC, vcovNW, vcovDC, vcovSCC) is to pass
it to plm's own functions like summary, pwaldtest, and phtest or
together with testing functions from the lmtest and car packages. All of
these typically allow passing the vcov or vcov. parameter either as a
matrix or as a function, e.g., for Wald–type testing: argument vcov. to
coeftest(), argument vcov to waldtest() and other methods in the
lmtest package; and argument vcov. to
linearHypothesis() in the car package (see the
examples), see (see also Zeileis 2004)
, 4.1-2, and examples below.
Beck N, Katz JN (1995). “What to do (and not to do) with time-series cross-section data.” American Political Science Review, 89(03), 634–647.
Cribari–Neto F (2004). “Asymptotic Inference Under Heteroskedasticity of Unknown Form.” Computational Statistics & Data Analysis, 45, 215–233.
Greene WH (2003). Econometric Analysis, 5th edition. Prentice Hall.
MacKinnon JG, White H (1985). “Some Heteroskedasticity–Consistent Covariance Matrix Estimators With Improved Finite Sample Properties.” Journal of Econometrics, 29, 305–325.
Zeileis A (2004). “Econometric Computing With HC and HAC Covariance Matrix Estimators.” Journal of Statistical Software, 11(10), 1–17. https://www.jstatsoft.org/article/view/v011i10.
sandwich::vcovHC() from the sandwich
package for weighting schemes (type argument).
data("Produc", package="plm")
zz <- plm(log(gsp)~log(pcap)+log(pc)+log(emp)+unemp, data=Produc, model="random")
summary(zz, vcov = vcovBK)
#> Oneway (individual) effect Random Effect Model
#> (Swamy-Arora's transformation)
#>
#> Note: Coefficient variance-covariance matrix supplied: vcovBK
#>
#> Call:
#> plm(formula = log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp,
#> data = Produc, model = "random")
#>
#> Balanced Panel: n = 48, T = 17, N = 816
#>
#> Effects:
#> var std.dev share
#> idiosyncratic 0.001454 0.038137 0.175
#> individual 0.006838 0.082691 0.825
#> theta: 0.8888
#>
#> Residuals:
#> Min. 1st Qu. Median 3rd Qu. Max.
#> -0.1067230 -0.0245520 -0.0023694 0.0217333 0.1996307
#>
#> Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> (Intercept) 2.1354110 0.2562311 8.3339 < 2.2e-16 ***
#> log(pcap) 0.0044386 0.0494249 0.0898 0.9284424
#> log(pc) 0.3105484 0.0427824 7.2588 3.906e-13 ***
#> log(emp) 0.7296705 0.0497995 14.6522 < 2.2e-16 ***
#> unemp -0.0061725 0.0017190 -3.5908 0.0003296 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Total Sum of Squares: 29.209
#> Residual Sum of Squares: 1.1879
#> R-Squared: 0.95933
#> Adj. R-Squared: 0.95913
#> Chisq: 4175.52 on 4 DF, p-value: < 2.22e-16
summary(zz, vcov = function(x) vcovBK(x, type="HC1"))
#> Oneway (individual) effect Random Effect Model
#> (Swamy-Arora's transformation)
#>
#> Note: Coefficient variance-covariance matrix supplied: function(x) vcovBK(x, type = "HC1")
#>
#> Call:
#> plm(formula = log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp,
#> data = Produc, model = "random")
#>
#> Balanced Panel: n = 48, T = 17, N = 816
#>
#> Effects:
#> var std.dev share
#> idiosyncratic 0.001454 0.038137 0.175
#> individual 0.006838 0.082691 0.825
#> theta: 0.8888
#>
#> Residuals:
#> Min. 1st Qu. Median 3rd Qu. Max.
#> -0.1067230 -0.0245520 -0.0023694 0.0217333 0.1996307
#>
#> Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> (Intercept) 2.1354110 0.2570198 8.3084 < 2.2e-16 ***
#> log(pcap) 0.0044386 0.0495770 0.0895 0.9286614
#> log(pc) 0.3105484 0.0429141 7.2365 4.604e-13 ***
#> log(emp) 0.7296705 0.0499528 14.6072 < 2.2e-16 ***
#> unemp -0.0061725 0.0017242 -3.5798 0.0003439 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Total Sum of Squares: 29.209
#> Residual Sum of Squares: 1.1879
#> R-Squared: 0.95933
#> Adj. R-Squared: 0.95913
#> Chisq: 4149.93 on 4 DF, p-value: < 2.22e-16
## standard coefficient significance test
library(lmtest)
#> Loading required package: zoo
#>
#> Attaching package: ‘zoo’
#> The following objects are masked from ‘package:base’:
#>
#> as.Date, as.Date.numeric
coeftest(zz)
#>
#> t test of coefficients:
#>
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.13541100 0.13346149 16.0002 < 2.2e-16 ***
#> log(pcap) 0.00443859 0.02341732 0.1895 0.8497
#> log(pc) 0.31054843 0.01980475 15.6805 < 2.2e-16 ***
#> log(emp) 0.72967053 0.02492022 29.2803 < 2.2e-16 ***
#> unemp -0.00617247 0.00090728 -6.8033 1.986e-11 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
## robust significance test, cluster by group
## (robust vs. serial correlation), default arguments
coeftest(zz, vcov.=vcovBK)
#>
#> t test of coefficients:
#>
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.1354110 0.2562311 8.3339 3.327e-16 ***
#> log(pcap) 0.0044386 0.0494249 0.0898 0.9284646
#> log(pc) 0.3105484 0.0427824 7.2588 9.153e-13 ***
#> log(emp) 0.7296705 0.0497995 14.6522 < 2.2e-16 ***
#> unemp -0.0061725 0.0017190 -3.5908 0.0003495 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
## idem with parameters, pass vcov as a function argument
coeftest(zz, vcov.=function(x) vcovBK(x, type="HC1"))
#>
#> t test of coefficients:
#>
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.1354110 0.2570198 8.3084 4.059e-16 ***
#> log(pcap) 0.0044386 0.0495770 0.0895 0.9286835
#> log(pc) 0.3105484 0.0429141 7.2365 1.068e-12 ***
#> log(emp) 0.7296705 0.0499528 14.6072 < 2.2e-16 ***
#> unemp -0.0061725 0.0017242 -3.5798 0.0003643 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
## idem, cluster by time period
## (robust vs. cross-sectional correlation)
coeftest(zz, vcov.=function(x) vcovBK(x, type="HC1", cluster="time"))
#>
#> t test of coefficients:
#>
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.1354110 0.2776015 7.6924 4.188e-14 ***
#> log(pcap) 0.0044386 0.0395528 0.1122 0.9106772
#> log(pc) 0.3105484 0.0458243 6.7769 2.360e-11 ***
#> log(emp) 0.7296705 0.0562371 12.9749 < 2.2e-16 ***
#> unemp -0.0061725 0.0018531 -3.3309 0.0009049 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
## idem with parameters, pass vcov as a matrix argument
coeftest(zz, vcov.=vcovBK(zz, type="HC1"))
#>
#> t test of coefficients:
#>
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.1354110 0.2570198 8.3084 4.059e-16 ***
#> log(pcap) 0.0044386 0.0495770 0.0895 0.9286835
#> log(pc) 0.3105484 0.0429141 7.2365 1.068e-12 ***
#> log(emp) 0.7296705 0.0499528 14.6072 < 2.2e-16 ***
#> unemp -0.0061725 0.0017242 -3.5798 0.0003643 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
## joint restriction test
waldtest(zz, update(zz, .~.-log(emp)-unemp), vcov=vcovBK)
#> Wald test
#>
#> Model 1: log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp
#> Model 2: log(gsp) ~ log(pcap) + log(pc)
#> Res.Df Df Chisq Pr(>Chisq)
#> 1 811
#> 2 813 -2 411.05 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
if (FALSE) { # \dontrun{
## test of hyp.: 2*log(pc)=log(emp)
library(car)
linearHypothesis(zz, "2*log(pc)=log(emp)", vcov.=vcovBK)
} # }