robust.coef: Unstandardized Coefficients with Heteroscedasticity-Consistent Standard Errors

Description

This function computes heteroscedasticity-consistent standard errors and significance values for linear models estimated by using the lm() function and generalized linear models estimated by using the glm() function. For linear models the heteroscedasticity-robust F-test is computed as well. By default, the function uses the HC4 estimator.

Usage

robust.coef(model, type = c("HC0", "HC1", "HC2", "HC3", "HC4", "HC4m", "HC5"),
            digits = 3, p.digits = 4, write = NULL, append = TRUE, check = TRUE,
            output = TRUE)

Value

Returns an object of class misty.object, which is a list with following entries:

call: function call
type: type of analysis
model: model specified in model
args: specification of function arguments
result: list with results, i.e., coef for the unstandardized regression coefficients with heteroscedasticity-consistent standard errors, F.test for the heteroscedasticity-robust F-Test, and sandwich for the sandwich covariance matrix

Arguments

model: a fitted model of class lm or glm.
type: a character string specifying the estimation type, where "H0" gives White's estimator and "H1" to "H5" are refinement of this estimator. See help page of the vcovHC() function in the R package sandwich for more details.
digits: an integer value indicating the number of decimal places to be used for displaying results. Note that information criteria and chi-square test statistic are printed with digits minus 1 decimal places.
p.digits: an integer value indicating the number of decimal places to be used for displaying p-values.
write: a character string naming a file for writing the output into either a text file with file extension ".txt" (e.g., "Output.txt") or Excel file with file extension ".xlsx" (e.g., "Output.xlsx"). If the file name does not contain any file extension, an Excel file will be written.
append: logical: if TRUE (default), output will be appended to an existing text file with extension .txt specified in write, if FALSE existing text file will be overwritten.
check: logical: if TRUE (default), argument specification is checked.
output: logical: if TRUE (default), output is shown.

Author

Takuya Yanagida takuya.yanagida@univie.ac.at

Details

The family of heteroscedasticity-consistent (HC) standard errors estimator for the model parameters of a regression model is based on an HC covariance matrix of the parameter estimates and does not require the assumption of homoscedasticity. HC estimators approach the correct value with increasing sample size, even in the presence of heteroscedasticity. On the other hand, the OLS standard error estimator is biased and does not converge to the proper value when the assumption of homoscedasticity is violated (Darlington & Hayes, 2017).

White (1980) introduced the idea of HC covariance matrix to econometricians and derived the asymptotically justified form of the HC covariance matrix known as HC0 (Long & Ervin, 2000). Simulation studies have shown that the HC0 estimator tends to underestimate the true variance in small to moderately large samples (\(N \leq 250\)) and in the presence of leverage observations, which leads to an inflated type I error risk (e.g., Cribari-Neto & Lima, 2014). The alternative estimators HC1 to HC5 are asymptotically equivalent to HC0 but include finite-sample corrections, which results in superior small sample properties compared to the HC0 estimator. Long and Ervin (2000) recommended routinely using the HC3 estimator regardless of a heteroscedasticity test. However, the HC3 estimator can be unreliable when the data contains leverage observations. The HC4 estimator, on the other hand, performs well with small samples, in the presence of high leverage observations, and when errors are not normally distributed (Cribari-Neto, 2004). In summary, it appears that the HC4 estimator performs the best in terms of controlling the type I and type II error risk (Rosopa, 2013). As opposed to the findings of Cribari-Neto et al. (2007), the HC5 estimator did not show any substantial advantages over HC4. Both HC5 and HC4 performed similarly across all the simulation conditions considered in the study (Ng & Wilcox, 2009).

Note that the F-test of significance on the multiple correlation coefficient R also assumes homoscedasticity of the errors. Violations of this assumption can result in a hypothesis test that is either liberal or conservative, depending on the form and severity of the heteroscedasticity.

Hayes (2007) argued that using a HC estimator instead of assuming homoscedasticity provides researchers with more confidence in the validity and statistical power of inferential tests in regression analysis. Hence, the HC3 or HC4 estimator should be used routinely when estimating regression models. If a HC estimator is not used as the default method of standard error estimation, researchers are advised to at least double-check the results by using an HC estimator to ensure that conclusions are not compromised by heteroscedasticity. However, the presence of heteroscedasticity suggests that the data is not adequately explained by the statistical model of estimated conditional means. Unless heteroscedasticity is believed to be solely caused by measurement error associated with the predictor variable(s), it should serve as warning to the researcher regarding the adequacy of the estimated model.

References

Darlington, R. B., & Hayes, A. F. (2017). Regression analysis and linear models: Concepts, applications, and implementation. The Guilford Press.

Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics & Data Analysis, 45, 215-233. https://doi.org/10.1016/S0167-9473(02)00366-3

Cribari-Neto, F., & Lima, M. G. (2014). New heteroskedasticity-robust standard errors for the linear regression model. Brazilian Journal of Probability and Statistics, 28, 83-95.

Cribari-Neto, F., Souza, T., & Vasconcellos, K. L. P. (2007). Inference under heteroskedasticity and leveraged data. Communications in Statistics - Theory and Methods, 36, 1877-1888. https://doi.org/10.1080/03610920601126589

Hayes, A.F, & Cai, L. (2007). Using heteroscedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation. Behavior Research Methods, 39, 709-722. https://doi.org/10.3758/BF03192961

Long, J.S., & Ervin, L.H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician, 54, 217-224. https://doi.org/10.1080/00031305.2000.10474549

Ng, M., & Wilcoy, R. R. (2009). Level robust methods based on the least squares regression estimator. Journal of Modern Applied Statistical Methods, 8, 284-395. https://doi.org/10.22237/jmasm/1257033840

Rosopa, P. J., Schaffer, M. M., & Schroeder, A. N. (2013). Managing heteroscedasticity in general linear models. Psychological Methods, 18(3), 335-351. https://doi.org/10.1037/a0032553

White, H. (1980). A heteroskedastic-consistent covariance matrix estimator and a direct test of heteroskedasticity. Econometrica, 48, 817-838. https://doi.org/10.2307/1912934

Zeileis, A., & Hothorn, T. (2002). Diagnostic checking in regression relationships. R News, 2(3), 7–10. http://CRAN.R-project.org/doc/Rnews/

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. https://doi.org/10.18637/jss.v095.i01

Examples

Run this code

dat <- data.frame(x1 = c(3, 2, 4, 9, 5, 3, 6, 4, 5, 6, 3, 5),
                  x2 = c(1, 4, 3, 1, 2, 4, 3, 5, 1, 7, 8, 7),
                  x3 = c(0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1),
                  y1 = c(2, 7, 4, 4, 7, 8, 4, 2, 5, 1, 3, 8),
                  y2 = c(0, 1, 0, 2, 0, 1, 0, 0, 1, 2, 1, 0))

#-------------------------------------------------------------------------------
# Example 1: Linear model

mod1 <- lm(y1 ~ x1 + x2 + x3, data = dat)
robust.coef(mod1)

#-------------------------------------------------------------------------------
# Example 2: Generalized linear model

mod2 <- glm(y2 ~ x1 + x2 + x3, data = dat, family = poisson())
robust.coef(mod2)

if (FALSE) {
#----------------------------------------------------------------------------
# Write Results

# Example 3a: Write Results into a text file
robust.coef(mod1, write = "Robust_Coef.txt", output = FALSE)

# Example 3b: Write Results into an Excel file
robust.coef(mod1, write = "Robust_Coef.xlsx", output = FALSE)

result <- robust.coef(mod1, output = FALSE)
write.result(result, "Robust_Coef.xlsx")
}

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