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A NEW ASYMPTOTIC THEORY FOR HETEROSKEDASTICITY-AUTOCORRELATION ROBUST TESTS

Published online by Cambridge University Press:  23 September 2005

Nicholas M. Kiefer  and
Timothy J. Vogelsang
Nicholas M. Kiefer
Affiliation:
Cornell University
Timothy J. Vogelsang
Affiliation:
Cornell University
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Abstract

A new first-order asymptotic theory for heteroskedasticity-autocorrelation (HAC) robust tests based on nonparametric covariance matrix estimators is developed. The bandwidth of the covariance matrix estimator is modeled as a fixed proportion of the sample size. This leads to a distribution theory for HAC robust tests that explicitly captures the choice of bandwidth and kernel. This contrasts with the traditional asymptotics (where the bandwidth increases more slowly than the sample size) where the asymptotic distributions of HAC robust tests do not depend on the bandwidth or kernel. Finite-sample simulations show that the new approach is more accurate than the traditional asymptotics. The impact of bandwidth and kernel choice on size and power of t-tests is analyzed. Smaller bandwidths lead to tests with higher power but greater size distortions, and large bandwidths lead to tests with lower power but smaller size distortions. Size distortions across bandwidths increase as the serial correlation in the data becomes stronger. Overall, the results clearly indicate that for bandwidth and kernel choice there is a trade-off between size distortions and power. Finite-sample performance using the new asymptotics is comparable to the bootstrap, which suggests that the asymptotic theory in this paper could be useful in understanding the theoretical properties of the bootstrap when applied to HAC robust tests.We thank an editor and a referee for constructive comments on a previous version of the paper. Helpful comments provided by Cliff Hurvich, Andy Levin, Jeff Simonoff, and seminar participants at NYU (Statistics), U. Texas Austin, Yale, U. Montreal, UCSD, UC Riverside, UC Berkeley, U. of Pittsburgh, SUNY Albany, U. Aarhus, Brown U., NBER/NSF Time Series Conference, and 2003 Winter Meetings of the Econometrics Society are gratefully acknowledged. We gratefully acknowledge financial support from the National Science Foundation through grant SES-0095211. We thank the Center for Analytic Economics at Cornell University.

Type
Research Article
Information
Econometric Theory , Volume 21 , Issue 6 , December 2005 , pp. 1130 - 1164
Copyright
© 2005 Cambridge University Press

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References

REFERENCES

Abadir, K.M. & P. Paruolo (2002) Simple robust testing of regression hypotheses: A comment. Econometrica 70, 2097–2099.Google Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–854.Google Scholar
Andrews, D.W.K. & J.C. Monahan (1992) An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60, 953–966.Google Scholar
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489–502.Google Scholar
Bunzel, H., N.M. Kiefer, & T.J. Vogelsang (2001) Simple robust testing of hypotheses in non-linear models. Journal of the American Statistical Association 96, 1088–1098.Google Scholar
Bunzel, H. & T.J. Vogelsang (2005) Powerful trend function tests that are robust to strong serial correlation with an application to the Prebisch-Singer hypothesis. Journal of Business & Economic Statistics, forthcoming.Google Scholar
Cushing, M.J. & M.G. McGarvey (1999) Covariance matrix estimation. In L. Matyas (ed.), Generalized Method of Moments Estimation, pp. 63–95. Cambridge University Press.
Davison, A.C. & P. Hall (1993) On Studentizing and blocking methods for implementing the bootstrap with dependent data. Australian Journal of Statistics 35, 215–224.Google Scholar
de Jong, R.M. & J. Davidson (2000) Consistency of kernel estimators of heteroskedastic and autocorrelated covariance matrices. Econometrica 68, 407–424.Google Scholar
den Haan, W.J. & A. Levin (1997) A practitioner's guide to robust covariance matrix estimation. In G. Maddala and C. Rao (eds.), Handbook of Statistics: Robust Inference, vol. 15, pp. 291–341. Elsevier.
den Haan, W.J. & A. Levin (1998) Vector Autoregressive Covariance Matrix Estimation. Working paper, International Finance Division, FED Board of Governors.
Gallant, A. (1987) Nonlinear Statistical Models. Wiley.
Gallant, A. & H. White (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Blackwell.
Goncalves, S. & T.J. Vogelsang (2004) Block Bootstrap Puzzles in HAC Robust Testing: The Sophistication of the Naive Bootstrap. Working paper, Department of Economics, Cornell University.
Götze, F. & H.R. Künsch (1996) Second-order correctness of the blockwise bootstrap for stationary observations. Annals of Statistics 24, 1914–1933.Google Scholar
Hall, P. & J.L. Horowitz (1996) Bootstrap critical values for tests based on generalized method of moments estimators. Econometrica 64, 891–916.Google Scholar
Hansen, B.E. (1992) Consistent covariance matrix estimation for dependent heterogenous processes. Econometrica 60, 967–972.Google Scholar
Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054.Google Scholar
Hashimzade, N., N.M. Kiefer, & T.J. Vogelsang (2003) Moments of HAC Robust Covariance Matrix Estimators under Fixed- b Asymptotics. Working paper, Department of Economics, Cornell University.
Hashimzade, N. & T.J. Vogelsang (2003) A New Asymptotic Approximation for the Sampling Behavior of Spectral Density Estimators. Working paper, Department of Economics, Cornell University.
Heyde, C. (1997) Quasi-Likelihood and Its Application. A General Approach to Optimal Parameter Estimation. Springer.
Inoue, A. & M. Shintani (2004) Bootstrapping GMM estimators for time series. Journal of Econometrics, forthcoming.Google Scholar
Jansson, M. (2002) Consistent covariance estimation for linear processes. Econometric Theory 18, 1449–1459.Google Scholar
Jansson, M. (2004) The error rejection probability of simple autocorrelation robust tests. Econometrica 72, 937–946.Google Scholar
Kiefer, N.M. & T.J. Vogelsang (2002a) Heteroskedasticity-autocorrelation robust standard errors using the Bartlett kernel without truncation. Econometrica 70, 2093–2095.Google Scholar
Kiefer, N.M. & T.J. Vogelsang (2002b) Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory 18, 1350–1366.Google Scholar
Kiefer, N.M. & T.J. Vogelsang (2005) A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Working paper 05-08, Center for Analytic Economics, Cornell University.
Kiefer, N.M., T.J. Vogelsang, & H. Bunzel (2000) Simple robust testing of regression hypotheses. Econometrica 68, 695–714.Google Scholar
Magnus, J.R. & H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley.
Müller, U.K. (2004) A Theory of Robust Long-Run Variance Estimation. Mimeo, Department of Economics, Princeton University.
Neave, H.R. (1970) An improved formula for the asymptotic variance of spectrum estimates. Annals of Mathematical Statistics 41, 70–77.Google Scholar
Newey, W.K. & D.L. McFadden (1994) Large sample estimation and hypothesis testing. In R. Engle and D.L. McFadden (eds.), Handbook of Econometrics, vol. 4, pp. 2113–2247. Elsevier.
Newey, W.K. & K.D. West (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708.Google Scholar
Newey, W.K. & K.D. West (1994) Automatic lag selection in covariance estimation. Review of Economic Studies 61, 631–654.Google Scholar
Ng, S. & P. Perron (1996) The exact error in estimating the spectral density at the origin. Journal of Time Series Analysis 17, 379–408.Google Scholar
Phillips, P.C.B. & S.N. Durlauf (1986) Multiple regression with integrated processes. Review of Economic Studies 53, 473–496.Google Scholar
Phillips, P.C.B., Y. Sun, & S. Jin (2003) Consistent HAC Estimation and Robust Regression Testing Using Sharp Origin Kernels with No Truncation. Working paper, Department of Economics, Yale University.
Phillips, P.C.B., Y. Sun, & S. Jin (2004) Improved HAR Inference Using Power Kernels without Truncation. Working paper, Department of Economics, Yale University.
Phillips, P.C.B., Y. Sun, & S. Jin (2005) Spectral density estimation and robust hypothesis testing using steep origin kernels without truncation. International Economic Review, forthcoming.Google Scholar
Priestley, M.B. (1981) Spectral Analysis and Time Series, vol. 1. Academic Press.
Ravikumar, B., S. Ray, & N.E. Savin (2004) Robust Wald Tests and the Curse of Dimensionality. Working paper, Department of Economics, University of Iowa.
Robinson, P. (1998) Inference without smoothing in the presence of nonparametric autocorrelation. Econometrica 66, 1163–1182.Google Scholar
Simonoff, J. (1993) The relative importance of bias and variability in the estimation of the variance of a statistic. Statistician 42, 3–7.Google Scholar
Velasco, C. & P.M. Robinson (2001) Edgeworth expansions for spectral density estimates and studentized sample mean. Econometric Theory 17, 497–539.Google Scholar
Vogelsang, T.J. (2003) Testing in GMM models without truncation. In T. Fomby & R. Carter Hill (eds.), Advances in Econometrics: Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later, vol. 17, pp. 199–233. Elsevier.
White, H. (1984) Asymptotic Theory for Econometricians. Academic Press.