Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T17:58:08.812Z Has data issue: false hasContentIssue false
  • English
  • Français

ON SIZE AND POWER OF HETEROSKEDASTICITY AND AUTOCORRELATION ROBUST TESTS

Published online by Cambridge University Press:  26 February 2015

David Preinerstorfer  and
Benedikt M. Pötscher
David Preinerstorfer
Affiliation:
University of Vienna
Benedikt M. Pötscher*
Affiliation:
University of Vienna
*
*Address correspondence to Benedikt Pötscher, Department of Statistics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Testing restrictions on regression coefficients in linear models often requires correcting the conventional F-test for potential heteroskedasticity or autocorrelation amongst the disturbances, leading to so-called heteroskedasticity and autocorrelation robust test procedures. These procedures have been developed with the purpose of attenuating size distortions and power deficiencies present for the uncorrected F-test. We develop a general theory to establish positive as well as negative finite-sample results concerning the size and power properties of a large class of heteroskedasticity and autocorrelation robust tests. Using these results we show that nonparametrically as well as parametrically corrected F-type tests in time series regression models with stationary disturbances have either size equal to one or nuisance-infimal power equal to zero under very weak assumptions on the covariance model and under generic conditions on the design matrix. In addition we suggest an adjustment procedure based on artificial regressors. This adjustment resolves the problem in many cases in that the so-adjusted tests do not suffer from size distortions. At the same time their power function is bounded away from zero. As a second application we discuss the case of heteroskedastic disturbances.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 2 , April 2016 , pp. 261 - 358
Copyright
Copyright © Cambridge University Press 2015 

References

REFERENCES

Anderson, T.W. (1971) The Statistical Analysis of Time Series. Wiley Series in Probability and Mathematical Statistics. Wiley.Google Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Andrews, D.W.K. & Monahan, J.C. (1992) An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60, 953–966.Google Scholar
Bakirov, N. & Szkely, G. (2005) Student’s t-test for gaussian scale mixtures. Zapiski Nauchnyh Seminarov POMI 328, 5–19.Google Scholar
Banerjee, A.N. & Magnus, J.R. (2000) On the sensitivity of the usual t- and F-tests to covariance misspecification. Journal of Econometrics 95, 157–176.Google Scholar
Bartlett, M.S. (1950) Periodogram analysis and continuous spectra. Biometrika 37, 1–16.CrossRefGoogle ScholarPubMed
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489–502.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Cribari-Neto, F. (2004) Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics and Data Analysis 45, 215–233.Google Scholar
Deistler, M. & Pötscher, B.M. (1984) The behaviour of the likelihood function for ARMA models. Advances in Applied Probability 16, 843–866.Google Scholar
den Haan, W.J. & Levin, A.T. (1997) A practitioner’s guide to robust covariance matrix estimation. In Maddala, G. & Rao, C. (eds.), Robust Inference. Handbook of Statistics, vol. 15, pp. 299–342. Elsevier.CrossRefGoogle Scholar
Dufour, J.-M. (1997) Some impossibility theorems in econometrics with applications to structural and dynamic models. Econometrica 65, 1365–1387.Google Scholar
Dufour, J.-M. (2003) Identification, weak instruments, and statistical inference in econometrics. Canadian Journal of Economics/Revue canadienne d’économique 36, 767–808.Google Scholar
Eicker, F. (1963) Asymptotic normality and consistency of the least squares estimators for families of linear regressions. The Annals of Mathematical Statistics 34, 447–456.CrossRefGoogle Scholar
Eicker, F. (1967) Limit theorems for regressions with unequal and dependent errors. In Le Cam, L. M. & Neyman, J. (eds.), Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, Calif., 1965/66) vol. 1: Statistics, pp. 59–82. University of California Press.Google Scholar
Flegal, J.M. & Jones, G.L. (2010) Batch means and spectral variance estimators in Markov chain Monte Carlo. Annals of Statistics 38, 1034–1070.CrossRefGoogle Scholar
Grenander, U. & Rosenblatt, M. (1957) Statistical Analysis of Stationary Time Series. Wiley.CrossRefGoogle Scholar
Hannan, E.J. (1957) The variance of the mean of a stationary process. Journal of the Royal Statistical Society. Series B 19, 282–285.Google Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley Series in Probability and Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Hansen, B.E. (1992) Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica 60, 967–972.CrossRefGoogle Scholar
Heidelberger, P. & Welch, P.D. (1981) A spectral method for confidence interval generation and run length control in simulations. Communications of the ACM 24, 233–245.Google Scholar
Ibragimov, R. & Müller, U.K. (2010) t-statistic based correlation and heterogeneity robust inference. Journal of Business and Economic Statistics 28, 453–468.Google Scholar
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 1449–1459.Google Scholar
Jansson, M. (2004) The error in rejection probability of simple autocorrelation robust tests. Econometrica 72, 937–946.CrossRefGoogle Scholar
Jowett, G.H. (1955) The comparison of means of sets of observations from sections of independent stochastic series. Journal of the Royal Statistical Society. Series B 17, 208–227.Google Scholar
Keener, R.W., Kmenta, J., & Weber, N.C. (1991) Estimation of the covariance matrix of the least-squares regression coefficients when the disturbance covariance matrix is of unknown form. Econometric Theory 7, 22–45.Google Scholar
Kelejian, H.H. & Prucha, I.R. (2007) HAC estimation in a spatial framework. Journal of Econometrics 140, 131–154.CrossRefGoogle Scholar
Kelejian, H.H. & Prucha, I.R. (2010) Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics 157, 53–67.Google Scholar
Kiefer, N.M. & Vogelsang, T.J. (2002a) Heteroskedasticity-autocorrelation robust standard errors using the Bartlett kernel without truncation. Econometrica 70, 2093–2095.Google Scholar
Kiefer, N.M. & Vogelsang, T.J. (2002b) Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory 18, 1350–1366.CrossRefGoogle Scholar
Kiefer, N.M. & Vogelsang, T.J. (2005) A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory 21, 1130–1164.CrossRefGoogle Scholar
Kiefer, N.M., Vogelsang, T.J., & Bunzel, H. (2000) Simple robust testing of regression hypotheses. Econometrica 68, 695–714.Google Scholar
Krämer, W. (1989) On the robustness of the F-test to autocorrelation among disturbances. Economics Letters 30, 37–40.Google Scholar
Krämer, W. (2003) The robustness of the F-test to spatial autocorrelation among regression disturbances. Statistica (Bologna) 63, 435–440.Google Scholar
Krämer, W. & Hanck, C. (2009) More on the F-test under nonspherical disturbances. In Schipp, B. & Krämer, W. (eds.), Statistical Inference, Econometric Analysis and Matrix Algebra, pp. 179–184. Physica-Verlag HD.Google Scholar
Krämer, Kiviet and Breitung Krämer, W., Kiviet, J., & Breitung, J. (1990) The null distribution of the F-test in the linear regression model with autocorrelated disturbances. Statistica (Bologna) 50, 503–509.Google Scholar
Lehmann, E.L. & Romano, J.P. (2005) Testing Statistical Hypotheses, 3rd ed. Springer Texts in Statistics. Springer.Google Scholar
Long, J.S. & Ervin, L.H. (2000) Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician 54, 217–224.Google Scholar
Magee, L. (1989) An Edgeworth test size correction for the linear model with AR(1) errors. Econometrica 57, 661–674.Google Scholar
Martellosio, F. (2010) Power properties of invariant tests for spatial autocorrelation in linear regression. Econometric Theory 26, 152–186.Google Scholar
Neave, H.R. (1970) An improved formula for the asymptotic variance of spectrum estimates. The Annals of Mathematical Statistics 41, 70–77.CrossRefGoogle Scholar
Newey, W.K. & West, K.D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708.Google Scholar
Newey, W.K. & West, K.D. (1994) Automatic lag selection in covariance matrix estimation. The Review of Economic Studies 61, 631–653.Google Scholar
Park, R.E. & Mitchell, B.M. (1980) Estimating the autocorrelated error model with trended data. Journal of Econometrics 13, 185–201.CrossRefGoogle Scholar
Perron, P. & Ren, L. (2011) On the irrelevance of impossibility theorems: the case of the long-run variance. Journal of Time Series Econometrics 3, Article 1, 34.Google Scholar
Phillips, P.C.B. (2005) HAC estimation by automated regression. Econometric Theory 21, 116–142.Google Scholar
Phillips, P.C.B., Sun, Y., & Jin, S. (2006) Spectral density estimation and robust hypothesis testing using steep origin kernels without truncation. International Economic Review 47, 837–894.CrossRefGoogle Scholar
Phillips, P.C.B., Sun, Y., & Jin, S. (2007) Long run variance estimation and robust regression testing using sharp origin kernels with no truncation. Journal of Statistical Planning and Inference 137, 985–1023.Google Scholar
Politis, D. (2011) Higher-order accurate, positive semidefinite estimation of large-sample covariance and spectral density matrices. Econometric Theory 27, 703–744.Google Scholar
Pötscher, B.M. (2002) Lower risk bounds and properties of confidence sets for ill-posed estimation problems with applications to spectral density and persistence estimation, unit roots, and estimation of long memory parameters. Econometrica 70, 1035–1065.Google Scholar
Preinerstorfer, D. (2014) Finite Sample Properties of Tests Based on Prewhitened Nonparametric Covariance Estimators. Working Paper, Department of Statistics, University of Vienna.Google Scholar
Preinerstorfer, D. & Pötscher, B.M. (2014) On the Power of Invariant Tests for Hypotheses on a Covariance Matrix. Working Paper, Department of Statistics, University of Vienna.Google Scholar
Robinson, G. (1979) Conditional properties of statistical procedures. Annals of Statistics 7, 742–755.Google Scholar
Sun, Y. (2013) A heteroskedasticity and autocorrelation robust F test using an orthonormal series variance estimator. Econometrics Journal 16, 1–26.Google Scholar
Sun, Y. & Kaplan, D.M. (2012) Fixed-Smoothing Asymptotics and Accurate F Approximation Using Vector Autoregressive Covariance Matrix Estimators. Working Paper, Department of Economics, UC San Diego.Google Scholar
Sun, Y., Phillips, P.C.B., & Jin, S. (2008) Optimal bandwidth selection in heteroskedasticity-autocorrelation robust testing. Econometrica 76, 175–194.Google Scholar
Sun, Y., Phillips, P.C.B., & Jin, S. (2011) Power maximization and size control in heteroskedasticity and autocorrelation robust tests with exponentiated kernels. Econometric Theory 27, 1320–1368.CrossRefGoogle Scholar
Thomson, D.J. (1982) Spectrum estimation and harmonic analysis. Proceedings of the IEEE 70, 1055–1096.Google Scholar
Velasco, C. & Robinson, P.M. (2001) Edgeworth expansions for spectral density estimates and studentized sample mean. Econometric Theory 17, 497–539.Google Scholar
Vogelsang, T.J. (2012) Heteroskedasticity, autocorrelation, and spatial correlation robust inference in linear panel models with fixed-effects. Journal of Econometrics 166, 303–319.CrossRefGoogle Scholar
White, H. (1980) A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48, 817–838.CrossRefGoogle Scholar
Zhang, X. & Shao, X. (2013a) Fixed-smoothing asymptotics for time series. Annals of Statistics 41, 1329–1349.CrossRefGoogle Scholar
Zhang, X. & Shao, X. (2013b) On a general class of long run variance estimators. Economics Letters 120, 437–441.Google Scholar