Abstract
Homoskedasticity is an important assumption in ordinary least squares (OLS) regression. Although the estimator of the regression parameters in OLS regression is unbiased when the homoskedasticity assumption is violated, the estimator of the covariance matrix of the parameter estimates can be biased and inconsistent under heteroskedasticity, which can produce significance tests and confidence intervals that can be liberal or conservative. After a brief description of heteroskedasticity and its effects on inference in OLS regression, we discuss a family of heteroskedasticity-consistent standard error estimators for OLS regression and argue investigators should routinely use one of these estimators when conducting hypothesis tests using OLS regression. To facilitate the adoption of this recommendation, we provide easy-to-use SPSS and SAS macros to implement the procedures discussed here.
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Bera, A. K., Suprayitno, T., &Premaratne, G. (2002). On some heteroskedasticity-robust estimators of variance-covariance matrix of the least-squares estimators.Journal of Statistical Planning & Inference,108, 121–136.
Berry, W. D. (1993).Understanding regression assumptions. Newbury Park, CA: Sage.
Box, G. E. P., &Cox, D. R. (1964). An analysis of transformations.Journal of the Royal Statistical Society B,26, 211–243.
Breusch, T. S., &Pagan, A. R. (1979). A simple test for heteroskedasticity and random coefficient variation.Econometrica,47, 1287–1294.
Cai, L., & Hayes, A. F. (in press). A new test of linear hypotheses in OLS regression under heteroskedasticity of unknown form.Journal of Educational & Behavioral Statistics.
Carroll, R. J. (2003). Variances are not always nuisance parameters.Biometrics,59, 211–220.
Carroll, R. J., &Ruppert, D. (1988).Transformation and weighting in regression. New York: Chapman and Hall.
Chesher, A., &Jewitt, I. (1987). The bias of a heteroskedasticity consistent covariance matrix estimator.Econometrica,55, 1217–1222.
Cook, R. D., &Weisberg, S. (1983). Diagnostics for heteroskedasticity in regression.Biometrika,70, 1–10.
Cook, R. D., &Weisberg, S. (1999).Applied regression including computing and graphics. New York: Wiley.
Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form.Computational Statistics & Data Analysis,45, 215–233.
Cribari-Neto, F., Ferrari, S. L. P., &Cordeiro, G. M. (2000). Improved heteroskedasticity-consistent covariance matrix estimators.Biometrika,87, 907–918.
Cribari-Neto, F., Ferrari, S. L. P., &Oliveira, W. A. S. C. (2005). Numerical evaluation of tests based on different heteroskedasticity-consistent covariance matrix estimators.Journal of Statistical Computation & Simulation,75, 611–628.
Cribari-Neto, F., &Zarkos, S. G. (2001). Heteroskedasticity-consistent covariance matrix estimation: White’s estimator and the bootstrap.Journal of Statistical Computation & Simulation,68, 391–411.
Darlington, R. B. (1990).Regression and linear models. New York: McGraw-Hill.
Davidson, R., &MacKinnon, J. G. (1993).Estimation and inference in econometrics. Oxford: Oxford University Press.
Downs, G. W., &Rocke, D. M. (1979). Interpreting heteroskedasticity.American Journal of Political Science,23, 816–828.
Draper, N. R., &Smith, H. (1981).Applied regression analysis (2nd ed.). New York: Wiley.
Duncan, G. T., &Layard, W. M. J. (1973). A Monte-Carlo study of asymptotically robust tests for correlation coefficients.Biometrika,60, 551–558.
Edgell, S. E., &Noon, S. M. (1984). Effect of violation of normality on the t test of the correlation coefficient.Psychological Bulletin,95, 576–583.
Eicker, F. (1963). Asymptotic normality and consistency of the least squares estimator for families of linear regression.Annals of Mathematical Statistics,34, 447–456.
Eicker, F. (1967). Limit theorems for regression with unequal and dependent errors. In L. M. Le Cam & J. Neyman (Eds.),Proceedings of the fifth Berkeley symposium on mathematical statistics and probability. Berkeley, CA: University of California Press.
Furno, M. (1996). Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator.Journal of Statistical Computation & Simulation,54, 115–128.
Godfrey, L. G. (2006). Tests for regression models with heteroskedasticity of unknown form.Computational Statistics & Data Analysis,50, 2715–2733.
Godfrey, L. G., &Orne, C. D. (2004). Controlling the finite sample significance levels of heteroskedasticity-robust tests of several linear restrictions on regression coefficients.Economics Letters,82, 281–287.
Goldfeld, S. M., &Quandt, R. E. (1965). Some tests for homoskedasticity.Journal of the American Statistical Association,60, 539–547.
Hayes, A. F. (1996). Permutation test is not distribution-free: TestingHo: π=0.Psychological Methods,1, 184–198.
Hinkley, D. V. (1977). Jackknifing in unbalanced situations.Technometrics,19, 285–292.
Huber, P. J. (1967). The behavior of maximum likelihood estimation under nonstandard conditions. In L. M. Le Cam & J. Neyman (Eds.),Proceedings of the fifth Berkeley symposium on mathematical statistics and probability. Berkeley, CA: University of California Press.
Kauermann, G., &Carroll, R. J. (2001). A note on the efficiency of sandwich covariance matrix estimation.Journal of the American Statistical Association,96, 1387–1396.
Kowalski, C. J. (1973). On the effects of non-normality on the distribution of the sample product-moment correlation coefficient.Applied Statistics,21, 1–12.
Long, J. S., &Ervin, L. H. (2000). Using heteroskedasticity consistent standard errors in the linear regression model.American Statistician,54, 217–224.
MacCallum, R. C. (2003). Working with imperfect models.Multivariate Behavioral Research,38, 113–139.
MacKinnon, J. G., &White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties.Journal of Econometrics,29, 305–325.
Mardia, K. V., Kent, J. T., &Bibby, J. M. (1979).Multivariate analysis. New York: Academic Press.
Moore, D. S., &McCabe, G. P. (2003).Introduction to the practice of statistics (4th ed.). New York: Freeman.
Olejnik, S. F. (1988). Variance heterogeneity: An outcome to explain or a nuisance factor to control?Journal of Experimental Education,56, 193–197.
Perry, D. K. (1986). Looking for heteroskedasticity: A means of searching for neglected conditional relationships. In M. L. McLaughlin (Ed.),Communication yearbook 9 (pp. 658–670). Beverly Hills, CA: Sage.
Rasmussen, J. L. (1989). Computer-intensive correlational analysis: Bootstrap and approximate randomization techniques.British Journal of Mathematical & Statistical Psychology,42, 103–111.
Sudmant, W., &Kennedy, P. (1990). On inference in the presence of heteroskedasticity without replicated observations.Communication in Statistics-Simulation & Computation,19, 491–504.
Weisberg, S. (1980).Applied linear regression. New York: Wiley.
White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity.Econometrica,48, 817–838.
Wilcox, R. R. (2001). Comment.The American Statistician,55, 374–375.
Wilcox, R. R. (2005).Introduction to robust estimation and hypothesis testing. New York: Academic Press.
Wooldridge, J. M. (2000).Introductory econometrics: A modern approach. Cincinnati, OH: South-Western College Publishing.
Wu, C. F. J. (1986). Jackknife bootstrap and other resampling methods in regression analysis.Annals of Statistics,14, 1261–1295.
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Hayes, A.F., Cai, L. Using heteroskedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation. Behavior Research Methods 39, 709–722 (2007). https://doi.org/10.3758/BF03192961
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DOI: https://doi.org/10.3758/BF03192961