Pötscher, Benedikt M. and Preinerstorfer, David
(2021):
*Valid Heteroskedasticity Robust Testing.*

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## Abstract

Tests based on heteroskedasticity robust standard errors are an important technique in econometric practice. Choosing the right critical value, however, is not simple at all: conventional critical values based on asymptotics often lead to severe size distortions; and so do existing adjustments including the bootstrap. To avoid these issues, we suggest to use smallest size-controlling critical values, the generic existence of which we prove in this article for commonly used test statistics. Furthermore, sufficient and often also necessary conditions for their existence are given that are easy to check. Granted their existence, these critical values are the canonical choice: larger critical values result in unnecessary power loss, whereas smaller critical values lead to over-rejections under the null hypothesis, make spurious discoveries more likely, and thus are invalid. We suggest algorithms to numerically determine the proposed critical values and provide implementations in accompanying software. Finally, we numerically study the behavior of the proposed testing procedures, including their power properties.

Item Type: | MPRA Paper |
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Original Title: | Valid Heteroskedasticity Robust Testing |

Language: | English |

Keywords: | Heteroskedasticity, Robustness, Tests, Size of a test |

Subjects: | C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C20 - General |

Item ID: | 117855 |

Depositing User: | Benedikt Poetscher |

Date Deposited: | 08 Jul 2023 01:41 |

Last Modified: | 08 Jul 2023 01:41 |

References: | Bakirov, N. and Székely, G. (2005). Student's t-test for Gaussian scale mixtures. Zapiski Nauchnyh Seminarov POMI, 328 5-19. Bakirov, N. K. (1998). Nonhomogeneous samples in the Behrens-Fisher problem. J. Math. Sci. (New York), 89 1460-1467. Bell, R. M. and McCaffrey, D. (2002). Bias reduction in standard errors for linear regression with multi-stage samples. Survey Methodology, 28 169-181. Belloni, A. and Didier, G. (2008). On the Behrens-Fisher problem: a globally convergent algorithm and a finite-sample study of the Wald, LR and LM tests. Ann. Statist., 36 2377- 2408. Cattaneo, M. D., Jansson, M. and Newey, W. K. (2018). Inference in linear regression models with many covariates and heteroscedasticity. Journal of the American Statistical Association, 113 1350–1361. Chesher, A. and Jewitt, I. (1987). The bias of a heteroskedasticity consistent covariance matrix estimator. Econometrica, 55 1217-1222. Chesher, A. D. (1989). Hájek inequalities, measures of leverage, and the size of heteroskedasticity robust Wald tests. Econometrica, 57 971-977. Chesher, A. D. and Austin, G. (1991). The finite-sample distributions of heteroskedasticity robust Wald statistics. Journal of Econometrics, 47 153-173. Chu, J., Lee, T.-H., Ullah, A, and Xu, H. (2021). Exact distribution of the F-statistic under heteroskedasticity of unknown form for improved inference. Journal if Statistical Computation and Simulation, 91 1782-1801. Cragg, J. G. (1983). More efficient estimation in the presence of heteroscedasticity of unknown form. Econometrica, 51 751-763. Cragg, J. G. (1992). Quasi-Aitken estimation for heteroscedasticity of unknown form. Journal of Econometrics, 54 179-201. Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics & Data Analysis, 45 215-233. Davidson, R. and Flachaire, E. (2008). The wild bootstrap, tamed at last. Journal of Econometrics, 146 162-169. Davidson, R. and MacKinnon, J. G. (1985). Heteroskedasticity-robust tests in regressions directions. Ann. I.N.S.É.É. 183-218. Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of chi2 random variables. Journal of the Royal Statistical Society. Series C (Applied Statistics), 29 323-333. DiCiccio, C. J., Romano, J. P. and Wolf, M. (2019). Improving weighted least squares inference. Econometrics and Statistics, 10 96-119. Duchesne, P. and de Micheaux, P. L. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 858-862. Eicker, F. (1963). Asymptotic normality and consistency of the least squares estimators for families of linear regressions. Annals of Mathematical Statistics, 34 447-456. Eicker, F. (1967). Limit theorems for regressions with unequal and dependent errors. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66). Univ.California Press, Berkeley, Calif., Vol. I: Statistics, pp. 59-82. Flachaire, E. (2005). More e¢ cient tests robust to heteroskedasticity of unknown form. Econometric Reviews, 24 219-241. Godfrey, L. G. (2006). Tests for regression models with heteroskedasticity of unknown form. Computational Statistics & Data Analysis, 50 2715-2733. Hansen, B. (2021). The exact distribution of the White t-ratio. Hinkley, D. V. (1977). Jackknifing in unbalanced situations. Technometrics, 19 285-292. Ibragimov, R. and Müller, U. K. (2010). t-statistic based correlation and heterogeneity robust inference. Journal of Business and Economic Statistics, 28 453-468. Ibragimov, R. and Müller, U. K. (2016). Inference with few heterogeneous clusters. The Review of Economics and Statistics, 98 83-96. Imbens, G. W. and Kolesár, M. (2016). Robust standard errors in small samples: Some practical advice. The Review of Economics and Statistics, 98 701-712. Kim, S.-H. and Cohen, A. S. (1998). On the Behrens-Fisher problem: A review. Journal of Educational and Behavioral Statistics, 23 356-377. Kolesár, M. (2019). dfadjust: Degrees of Freedom Adjustment for Robust Standard Errors. R package version 1.0.1, URL https://CRAN.R-project.org/package=dfadjust. Leeb, H. and Pötscher, B.M. (2017). Testing in the presence of nuisance parameters: some comments on tests post model selection and random critical calues. In Big and complex data analysis. Contrib. Stat., Springer, Cham, 69-82. Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses. 3rd ed. Springer Texts in Statistics, Springer, New York. Lin, E. S. and Chou, T.-S. (2018). Finite-sample refinement of GMM approach to nonlinear models under heteroskedasticity of unknown form. Econometric Reviews, 37 1-28. Loh, W.-Y. (1985). A new method for testing separate families of hypotheses. J. Amer. Statist. Assoc., 80 362-368. Long, J. S. and Ervin, L. H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician, 54 217-224. MacKinnon, J. G. (2013). Thirty years of heteroskedasticity-robust inference. In Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis (X. Chen and N. R. E. Swanson, eds.). Springer, 437-462. MacKinnon, J. G. and White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29 305-325. Mickey, M. R. and Brown, M. B. (1966). Bounds on the distribution functions of the Behrens-Fisher statistic. Ann. Math. Statist., 37 639-642. Nelder, J. A. and Mead, R. (1965). A simplex method for function minimization. The Computer Journal, 7 308-313. Phillips, P. C. (1993). Operational algebra and regression t-tests. In Models, Methods and Applications of Econometrics: Essays in Honor of A.R. Bergstrom (P. C. Phillips, ed.). Oxford: Basil Blackwell, 140–152. Pötscher, B. M. and Preinerstorfer, D. (2018). Controlling the size of autocorrelation robust tests. Journal of Econometrics, 207 406-431. Pötscher, B. M. and Preinerstorfer, D. (2019). Further results on size and power of heteroskedasticity and autocorrelation robust tests, with an application to trend testing. Electronic Journal of Statistics, 13 3893-3942. Pötscher, B. M. and Preinerstorfer, D. (2020). How reliable are bootstrap-based heteroskedasticity robust tests? Econometric Theory, forthcoming. Preinerstorfer, D. (2021). hrt: Heteroskedasticity Robust Testing. R package version 1.0.0. Preinerstorfer, D. and Pötscher, B. M. (2016). On size and power of heteroskedasticity and autocorrelation robust tests. Econometric Theory, 32 261-358. Robinson, G. (1979). Conditional properties of statistical procedures. Annals of Statistics, 7 742-755. Romano, J. P. and Wolf, M. (2017). Resurrecting weighted least squares. Journal of Econometrics, 197 1-19. Rothenberg, T. J. (1988). Approximate power functions for some robust tests of regression coefficients. Econometrica, 56 997-1019. Ruben, H. (2002). A simple conservative and robust solution of the Behrens-Fisher problem. Sankhya Ser. A, 64 139-155. Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2 110-114. Welch, B. L. (1938). The significance of the difference between two means when the population variances are unequal. Biometrika, 29 350-362. Welch, B. L. (1951). On the comparison of several mean values: an alternative approach. Biometrika, 38 330-336. White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48 817-838. Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. 2nd ed. MIT Press, Cambridge, MA. Wooldridge, J. M. (2012). Introductory Econometrics. 5th ed. South-Western, OH. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/117855 |