Abstract
In linear regression with heteroscedastic errors, the Generalized Least Squares (GLS) estimator is optimal, i.e., it is the Best Linear Unbiased Estimator (BLUE). The Ordinary Least Squares (OLS) estimator is suboptimal but still valid, i.e., unbiased and consistent. White, in his seminal paper (White, Econometrica 48:817–838, 1980) used the OLS residuals in order to obtain an estimate of the standard error of the OLS estimator under an unknown structure of the underlying heteroscedasticity. The GLS estimator similarly depends on the unknown heteroscedasticity, and is thus intractable. In this paper, we introduce two different approximations to the optimal GLS estimator; the starting point for both approaches is in the spirit of White’s correction, i.e., using the OLS residuals to get a rough estimate of the underlying heteroscedasticity. We show how the new estimators can benefit from the Wild Bootstrap both in terms of optimising them, and in terms of providing valid standard errors for them despite their complicated construction. The performance of the new estimators is compared via simulations to the OLS and to the exact (but intractable) GLS.
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Chatterjee, S., Mächler, M.: Robust regression: a weighted least squares approach, Commun. Stat. Theory Methods 26(6), 1381–1394 (1997)
Carol, J.R.: Adapting for heteroscedasticity in linear models. Ann. Stat. 10(4), 1224–1233 (1982)
Davidson, R., Flachaire, E.: The wild bootstrap, tamed at last. J. Econometrics 146, 162–169 (2008)
Le, Q.V., Smola, A.J., Canu, S.: Heteroscedastic gaussian process regression. In: Proceedings of the 22nd International Conference on Machine Learning, Bonn (2005)
Liu, R.Y.: Bootstrap procedures under some non-iid models. Ann. Stat. 16, 1696–1708 (1988)
Mammen, E.: When Does Bootstrap Work? Asymptotic Results and Simulations. Springer Lecture Notes in Statistics. Springer, New York (1992)
MacKinnon, J.G.: Thirty years of heteroskedasticity-robust inference. In: Chen, X., Swanson, N.R. (eds.) Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis, pp. 437–462. Springer, New York (2012)
Politis, D.N.: Model-free model-fitting and predictive distributions. Discussion Paper, Department of Economics, Univ. of California—San Diego. Retrievable from http://escholarship.org/uc/item/67j6s174 (2010)
Politis, D.N.: Model-free model-fitting and predictive distributions. Invited Discuss. Pap. J. TEST 22(2), 183–221 (2013)
White, H.: A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48, 817–838 (1980)
Wu, C.F.J.: Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Stat. 14, 1261–1295 (1986)
Yuan, M., Wahba, G.: Doubly penalized likelihood estimator in heteroscedastic regression. Stat. Probab. Lett. 34, 603–617 (2004)
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Politis, D.N., Poulis, S. (2014). Heteroskedastic Linear Regression: Steps Towards Adaptivity, Efficiency, and Robustness. In: Akritas, M., Lahiri, S., Politis, D. (eds) Topics in Nonparametric Statistics. Springer Proceedings in Mathematics & Statistics, vol 74. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0569-0_26
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DOI: https://doi.org/10.1007/978-1-4939-0569-0_26
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