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HIGHER-ORDER ACCURATE, POSITIVE SEMIDEFINITE ESTIMATION OF LARGE-SAMPLE COVARIANCE AND SPECTRAL DENSITY MATRICES

Published online by Cambridge University Press:  03 March 2011

Dimitris N. Politis
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Abstract

A new class of large-sample covariance and spectral density matrix estimators is proposed based on the notion of flat-top kernels. The new estimators are shown to be higher-order accurate when higher-order accuracy is possible. A discussion on kernel choice is presented as well as a supporting finite-sample simulation. The problem of spectral estimation under a potential lack of finite fourth moments is also addressed. The higher-order accuracy of flat-top kernel estimators typically comes at the sacrifice of the positive semidefinite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semidefinite (even strictly positive definite) while maintaining its higher-order accuracy. In addition, an easy (and consistent) procedure for optimal bandwidth choice is given; this procedure estimates the optimal bandwidth associated with each individual element of the target matrix, automatically sensing (and adapting to) the underlying correlation structure.

Type
Research Article
Information
Econometric Theory , Volume 27 , Issue 4 , August 2011 , pp. 703 - 744
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

This research was partially supported by NSF grants SES-04-18136 and DMS-07-06732. Many thanks are due to Arthur Berg and Tucker McElroy for numerous helpful interventions, to Peter Robinson for a critical reading and suggestions of some key early references, and to Dimitrios Gatzouras for his help with the proof of Lemma B.1. The S+ software for the practical computation of the different spectral density estimators was compiled with the invaluable help of Isheeta Nargis and Arif Dowla of www.stochasticlogic.com, and is now publicly available from www.math.ucsd.edu/∼politis/SOFT/SfunctionsFLAT-TOPS.html. Finally, the author is grateful to the co-editor and three anonymous reviewers for their insightful suggestions.

References

REFERENCES

Andrews, D. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Andrews, D. (1992) Equivalence of the higher order asymptotic efficiency of k–step and extremum statistics. Econometric Theory 18, 1040–1085.CrossRefGoogle Scholar
Andrews, D. & Monahan, J. (1992) An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60, 953–966.CrossRefGoogle Scholar
Berg, A. & Politis, D.N. (2009) Higher-order polyspectral estimation with flat-top lag-windows. Annals of the Institute of Statististical Mathematics 61, 477–498.CrossRefGoogle Scholar
Brillinger, D.R. (1979) Confidence intervals for the crosscovariance function. Selecta Statistica Canadiana 5, 3–16.Google Scholar
Brillinger, D.R. (1981) Time Series: Data Analysis and Theory. Holden-Day.Google Scholar
Brillinger, D.R. & Rosenblatt, M. (1967) Asymptotic theory of kth order spectra. In Harris, B. (ed.), Spectral Analysis of Time Series, pp. 153–188. Wiley.Google Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed. Springer.CrossRefGoogle Scholar
Choi, B.-S. (1992) ARMA Model Identification. Springer.CrossRefGoogle Scholar
Davis, R.A. & Mikosch, T. (2000) The sample autocorrelations of financial time series models. In Nonlinear and Nonstationary Signal Processing, pp. 247–274. Cambridge University Press.Google Scholar
Donoho, D.L. & Johnstone, I.M. (1995) Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association 90, 1200–1224.CrossRefGoogle Scholar
Eaton, M.E. & Tyler, D.E. (1991) On Weilandt’s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix. Annals of Statistics 19, 260–271.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997) Modelling Extremal Events. Springer.CrossRefGoogle Scholar
Epanechnikov, V.A. (1969) Non-parametric estimation of a multivariate probability density. Theory of Probability and Its Applications 14, 153–158.CrossRefGoogle Scholar
Gallant, A.R. (1987) Nonlinear Statistical Models. Wiley.CrossRefGoogle Scholar
Gallant, A.R. & White, H. (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Basil Blackwell.Google Scholar
Gao, J. & Gijbels, I. (2008) Bandwidth selection in nonparametric kernel testing. Journal of the American Statistical Association 103, 1584–1594.CrossRefGoogle Scholar
Hall, P. & Yao, Q. (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.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. (1958) The estimation of the spectral density after trend removal. Journal of the Royal Statistical Society, Series B 20, 323–333.Google Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley.CrossRefGoogle Scholar
Hannan, E.J. & Deistler, M. (1988) The Statistical Theory of Linear Systems. Wiley.Google Scholar
Hansen, B.E. (1992) Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica 60, 967–972.CrossRefGoogle Scholar
Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica. 50 1029–1054.CrossRefGoogle Scholar
Hashimzade, N. & Vogelsang, T.J. (2008) Fixed-b asymptotic approximations of the sampling behavior of nonparametric spectral density estimators. Journal of Time Series Analysis 29, 142–162.CrossRefGoogle Scholar
Inoue, A. & Shintani, M. (2006) Bootstrapping GMM estimators for time series. Journal of Econometrics 133, 531–555.CrossRefGoogle Scholar
Jowett, G.H. (1954) 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
Kiefer, N.M. & Vogelsang, T.J. (2002) Heteroscedasticity-autocorrelation robust standard errors using the Bartlett kernel without truncation. Econometrica 70, 1350–1366.CrossRefGoogle Scholar
Kiefer, N.M. & Vogelsang, T.J. (2005) A new asymptotic theory for heteroscedasticity-autocorrelation robust tests. Econometric Theory 21, 1130–1164.CrossRefGoogle Scholar
Lahiri, S.N. (2003) Resampling Methods for Dependent Data. Springer Verlag.CrossRefGoogle Scholar
Leadbetter, M.R., Lindgren, G., & Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag.CrossRefGoogle Scholar
Lepski, O.V. (1990) On a problem of adaptive estimation in Gaussian white noise. Theory of Probability and Its Applications 35, 454–466.CrossRefGoogle Scholar
Lepski, O.V. (1991) Asymptotically minimax adaptive estimation I: Upper bounds, optimally adaptive estimates. Theory of Probability and its Applications 36, 682–697.CrossRefGoogle Scholar
Lepski, O.V. (1992) Asymptotically minimax adaptive estimation II: Schemes without optimal adaptation. Theory of Probability and Its Applications 37, 433–468.CrossRefGoogle Scholar
McMurry, T. & Politis, D.N. (2004) Nonparametric regression with infinite order flat-top kernels. Journal of Nonparametric Statistics 16, 549–562.CrossRefGoogle Scholar
Newey, W. & West, K. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica. 55, 703–708.CrossRefGoogle Scholar
Newey, W. & West, K. (1994) Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61, 631–653.CrossRefGoogle Scholar
Ng, S. & Perron, P. (1996) The exact error in estimating the spectral density at the origin. Journal of Time Series Analysis 17, 379–408.CrossRefGoogle Scholar
Parzen, E. (1957) On consistent estimates of the spectrum of a stationary time series. Annals of Mathematical Statistics 28, 329–348.CrossRefGoogle Scholar
Parzen, E. (1961) Mathematical considerations in the estimation of Spectra. Technometrics 3, 167–190.CrossRefGoogle 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
Politis, D.N. (2001) On nonparametric function estimation with infinite-order flat-top kernels. In Charalambides, C. Koutras, M.V., & Balakrishnan, N. (eds.), Probability and Statistical Model with Applications, pp. 469–483. Chapman & Hall/CRC.Google Scholar
Politis, D.N. (2003) Adaptive bandwidth choice. Journal of Nonparametric Statistics 15, 517–533.CrossRefGoogle Scholar
Politis, D.N. (2004) A heavy-tailed distribution for ARCH residuals with application to volatility prediction. Annals of Economics and Finance 5, 283–298.Google Scholar
Politis, D.N. (2009) Higher-Order Accurate, Positive Semi-definite Estimation of Large-Sample Covariance and Spectral Density Matrices. Discussion paper 2005-03R, UCSD; available at http://escholarship.org/uc/item/66w826hz.Google Scholar
Politis, D.N. & Romano, J.P. (1995) Bias-corrected nonparametric spectral estimation. Journal of Time Series Analysis 16, 67–103.CrossRefGoogle Scholar
Politis, D.N., Romano, J.P., & Wolf, M. (1999) Subsampling. Springer Verlag.CrossRefGoogle Scholar
Politis, D.N. & White, H. (2004) Automatic block-length selection for the dependent bootstrap. Econometric Reviews 23, 53–70.CrossRefGoogle Scholar
Priestley, M.B. (1962) Basic considerations in the estimation of spectra. Technometrics 4, 551–564.CrossRefGoogle Scholar
Priestley, M.B. (1981). Spectral Analysis and Time Series. Adademic Press.Google Scholar
Robinson, P. (1991) Automatic frequency domain inference on semiparametric and nonparametric models. Econometrica 59, 1329–1363.CrossRefGoogle Scholar
Robinson, P.M. & Velasco, C. (1997) Autocorrelation-robust inference. In Maddala, G.S. & Rao, C.R. (eds.), Handbook of Statistics, vol. 15, pp. 267–298. North Holland.Google Scholar
Romano, J.P. & Thombs, L. (1996) Inference for autocorrelations under weak assumptions. Journal of the American Statistical Association 91, 590–600.CrossRefGoogle Scholar
Rosenblatt, M. (1984) Asymptotic normality, strong mixing and spectral density estimates. Annals of Probability 12, 1167–1180.CrossRefGoogle Scholar
Rosenblatt, M. (1985) Stationary Sequences and Random Fields. Birkhäuser.CrossRefGoogle Scholar
Samarov, A. (1977) Lower bound for the risk of spectral density estimates. Problems of Informmation Transmission 13, 67–72.Google Scholar
Shao, X. & Wu, W.B. (2007) Asymptotic spectral theory for nonlinear time series. Annals of Statistics 35, 1773–1801.CrossRefGoogle Scholar
Sun, Y. & Phillips, P.C.B. (2008) Optimal Bandwidth Choice for Interval Estimation in GMM Regression. Cowles Foundation Discussion paper 1661, Yale University.Google Scholar
Sun, Y., Phillips, P.C.B., & Jin, S. (2008) Optimal bandwidth selection in heteroscedasticity-autocorrelation robust testing. Econometrica 76, 175–194.CrossRefGoogle Scholar
Velasco, C. & Robinson, P.M. (2001) Edgeworth expansions for spectral density estimates and studentized sample mean. Econometric Theory 17, 497–539.10.1017/S0266466601173019CrossRefGoogle Scholar
West, K.D. (1997) Another heteroskedasticity— and autocorrelation—consistent covariance matrix estimator. Journal of Econometrics 76, 171–191.CrossRefGoogle Scholar