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Thirty Years of Heteroskedasticity-Robust Inference

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Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis

Abstract

White (Econometrica, 48:817–838, 1980) marked the beginning of a new era for inference in econometrics. It introduced the revolutionary idea of inference that is robust to heteroskedasticity of unknown form, an idea that was very soon extended to other forms of robust inference and also led to many new estimation methods. This paper discusses the development of heteroskedasticity-robust inference since 1980. There have been two principal lines of investigation. One approach has been to modify White’s original estimator to improve its finite-sample properties, and the other has been to use bootstrap methods. The relation between these two approaches, and some ways in which they may be combined, are discussed. Finally, a simulation experiment compares various methods and shows how far heteroskedasticity-robust inference has come in just over 30 years.

Research for this paper was supported, in part, by a grant from the Social Sciences and Humanities Research Council of Canada. I am grateful to Dimitris Politis, Patrik Guggenberger, and an anonymous referee for comments.

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References

  • Andrews, D.W.K. (1991). “Heteroskedasticity and autocorrelation consistent covariance matrix estimation”, Econometrica, 59:817–858.

    Article  Google Scholar 

  • Andrews, D.W.K., and J. C. Monahan (1992). “An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator”, Econometrica, 60, 953–966.

    Article  Google Scholar 

  • Cameron, A. C., J. B. Gelbach, and D. L. Miller (2008). “Bootstrap-based improvements for inference with clustered errors”, Review of Economics and Statistics, 90, 414–427.

    Article  Google Scholar 

  • Chesher, A. (1989). “Hájek inequalities, measures of leverage and the size of heteroskedasticity robust tests”, Journal of Econometrics, 57, 971–977.

    Article  Google Scholar 

  • Chesher, A., and G. Austin (1991). “The finite-sample distributions of heteroskedasticity robust Wald statistics”, Journal of Econometrics, 47, 153–173.

    Article  Google Scholar 

  • Chesher, A., and I. Jewitt (1987). “The bias of a heteroskedasticity consistent covariance matrix estimator”, Econometrica, 55, 1217–1222.

    Article  Google Scholar 

  • Cribari-Neto, F. (2004). “Asymptotic inference under heteroskedasticity of unknown form”, Computational Statistics and Data Analysis, 45, 215–233.

    Article  Google Scholar 

  • Cribari-Neto, F., and M. G. A. Lima (2009). “Heteroskedasticity-consistent interval estimators”, Journal of Statistical Computation and Simulation, 79, 787–803.

    Article  Google Scholar 

  • Cribari-Neto, F., and M. G. A. Lima (2010). “Sequences of bias-adjusted covariance matrix estimators under heteroskedasticity of unknown form”, Annals of the Institute of Mathematical Statistics, 62, 1053–1082.

    Article  Google Scholar 

  • Cribari-Neto, F., T. C. Souza, and K. L. P. Vasconcellos (2007). “Inference under heteroskedasticity and leveraged data”, Communications in Statistics: Theory and Methods, 36, 1977–1988 [see also Erratum (2008), 37, 3329–3330.].

    Google Scholar 

  • Davidson, R., and E. Flachaire (2008). “The wild bootstrap, tamed at last”, Journal of Econometrics, 146, 162–169.

    Article  Google Scholar 

  • Davidson, R. and J. G. MacKinnon (1993). Estimation and Inference in Econometrics, New York, Oxford University Press.

    Google Scholar 

  • Davidson, R., and J. G. MacKinnon (1999). “The size distortion of bootstrap tests”, Econometric Theory, 15, 361–376.

    Article  Google Scholar 

  • Davidson, R., and J. G. MacKinnon (2000). “Bootstrap tests: How many bootstraps?” Econometric Reviews, 19, 55–68.

    Article  Google Scholar 

  • Davidson, R., and J. G. MacKinnon (2006). “The power of bootstrap and asymptotic tests”, Journal of Econometrics, 133, 421–441.

    Article  Google Scholar 

  • Davidson, R., and J. G. MacKinnon (2010). “Wild bootstrap tests for IV regression”, Journal of Business and Economic Statistics, 28, 128–144.

    Article  Google Scholar 

  • Davidson, R., and J. G. MacKinnon (2011). “Confidence sets based on inverting Anderson-Rubin tests”, Queen’s University, QED Working Paper No. 1257.

    Google Scholar 

  • Driscoll, J. C., and A. C. Kraay (1998). “Consistent covariance matrix estimation with spatially dependent panel data”, Review of Economics and Statistics, 80, 549–560.

    Article  Google Scholar 

  • Efron, B. (1979). “Bootstrap methods: Another look at the jackknife”, Annals of Statistics, 7, 1–26.

    Article  Google Scholar 

  • Efron, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans. Philadelphia, Society for Industrial and Applied Mathematics.

    Book  Google Scholar 

  • Eicker, F. (1963). “Asymptotic normality and consistency of the least squares estimators for families of linear regressions”, Annals of Mathematical Statistics, 34, 447–456.

    Article  Google Scholar 

  • Eicker, F. (1967). “Limit theorems for regressions with unequal and dependent errors”, in Fifth Berkeley Symposium on Mathematical Statistics and Probability, ed. L. M. Le Cam and J. Neyman, Berkeley, University of California Press, 1, 59–82.

    Google Scholar 

  • Flachaire, E. (1999). “A better way to bootstrap pairs”, Economics Letters, 64, 257–262.

    Article  Google Scholar 

  • Flachaire, E. (2005). “Bootstrapping heteroskedastic regression models: Wild bootstrap vs pairs bootstrap”, Computational Statistics and Data Analysis, 49, 361–376.

    Article  Google Scholar 

  • Freedman, D. A. (1981). “Bootstrapping regression models”, Annals of Statistics, 9, 1218–1228.

    Article  Google Scholar 

  • Froot, K. A. (1989). “Consistent covariance matrix estimation with cross-sectional dependence and heteroskedasticity in financial data”, Journal of Financial and Quantitative Analysis, 24, 333–355.

    Article  Google Scholar 

  • Furno, M. (1996). “Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator”, Journal of Statistical Computation and Simulation, 54, 115–128.

    Article  Google Scholar 

  • Hansen, L. P. (1982). “Large sample properties of generalized method of moments estimators”, Econometrica, 50, 1029–1054.

    Article  Google Scholar 

  • Hinkley, D.V. (1977). “Jackknifing in unbalanced situations”, Technometrics, 19, 285–292.

    Article  Google Scholar 

  • Horn, S. D., R. A. Horn, and D. B. Duncan (1975). “Estimating heteroskedastic variances in linear models”, Journal of the American Statistical Association, 70, 380–385.

    Article  Google Scholar 

  • Horowitz, J. L. (2001). “The bootstrap”, in Handbook of Econometrics, Vol. 5, ed. J. J. Heckman and E. E. Leamer. Amsterdam, North-Holland, 3159–3228.

    Google Scholar 

  • Jöckel, K.-H. (1986). “Finite sample properties and asymptotic efficiency of Monte Carlo tests”, Annals of Statistics, 14, 336–347.

    Article  Google Scholar 

  • Lancaster, T. (2006). “A note on bootstraps and robustness”, Brown University Working Paper 2006–6.

    Google Scholar 

  • Liu, R.Y. (1988). “Bootstrap procedures under some non-I.I.D. models”, Annals of Statistics, 16, 1696–1708.

    Article  Google Scholar 

  • MacKinnon, J. G. (2002). “Bootstrap inference in econometrics”, Canadian Journal of Economics, 35, 615–645.

    Article  Google Scholar 

  • MacKinnon, J. G., and H. White (1985). “Some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties”, Journal of Econometrics, 29, 305–325.

    Article  Google Scholar 

  • Mammen, E. (1993). “Bootstrap and wild bootstrap for high dimensional linear models”, Annals of Statistics, 21, 255–285.

    Article  Google Scholar 

  • Newey, W.K., and K. D. West (1987). “A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix”, Econometrica, 55, 703–708.

    Article  Google Scholar 

  • Newey, W. K., and K. D. West (1994). “Automatic lag selection in covariance matrix estimation”, Review of Economic Studies, 61,631–653.

    Article  Google Scholar 

  • Paparoditis, E., and D. N. Politis (2005). “Bootstrap hypothesis testing in regression models”, Statistics and Probability Letters, 74, 356–365.

    Article  Google Scholar 

  • Poirier, D. J. (2010). “Bayesian interpretations of heteroskedastic consistent covariance estimators using the informed Bayesian bootstrap”, Econometric Reviews, 30, 457–468.

    Article  Google Scholar 

  • Politis, D.N. (2010). “Model-free model-fitting and predictive distributions”, Department of Economics, Univ. of California-San Diego.

    Google Scholar 

  • Qian, L., and S. Wang (2001). “Bias-corrected heteroscedasticity robust covariance matrix (sandwich) estimators”, Journal of Statistical Computation and Simulation, 70, 161–174.

    Article  Google Scholar 

  • Racine, J. S., and J. G. MacKinnon (2007). “Simulation-based tests that can use any number of simulations”, Communications in Statistics: Simulation and Computation, 36, 357–365.

    Article  Google Scholar 

  • Rogers, W. H. (1993). “Regression standard errors in clustered samples”, STATA Technical Bulletin, 13, 19–23.

    Google Scholar 

  • Stine, R. A. (1985). “Bootstrap prediction intervals for regression”, Journal of the American Statistical Association, 80, 1026–1031.

    Article  Google Scholar 

  • White, H. (1980). “A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity”, Econometrica, 48, 817–838.

    Article  Google Scholar 

  • White, H. (1982). “Maximum likelihood estimation of misspecified models”, Econometrica, 50, 1–25.

    Article  Google Scholar 

  • White, H., and I. Domowitz (1984). “Nonlinear regression with dependent observations”, Econometrica, 52, 143–161.

    Article  Google Scholar 

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Acknowledgments

Research for this paper was supported, in part, by a grant from the Social Sciences and Humanities Research Council of Canada. I am grateful to Dimitris Politis, Patrik Guggenberger, and an anonymous referee for comments.

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Authors and Affiliations

  1. Department of Economics, Queen’s University, Kingston, ON, K7L 3N6, Canada

    James G. MacKinnon

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  1. James G. MacKinnon
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Correspondence to James G. MacKinnon .

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  1. Yale University, 30 Hillhouse Ave, New Haven, 06501, USA

    Xiaohong Chen

  2. Hamilton St 75, NEW BRUNSWICK, 08901-1248, New Jersey, USA

    Norman R. Swanson

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MacKinnon, J.G. (2013). Thirty Years of Heteroskedasticity-Robust Inference. In: Chen, X., Swanson, N. (eds) Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1653-1_17

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