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Higher-Order Accurate, Positive Semi-definite Estimation of Large-Sample Covariance and Spectral Density Matrices

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  • Politis, D N
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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 semi-definite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semi-definite (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.

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    Paper provided by Department of Economics, UC San Diego in its series University of California at San Diego, Economics Working Paper Series with number qt66w826hz.

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    Date of creation: 02 Mar 2009
    Handle: RePEc:cdl:ucsdec:qt66w826hz
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    Keywords: spectral density matrices; large-sample covariance;

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    1. Yixiao Sun & Peter C. B. Phillips & Sainan Jin, 2008. "Optimal Bandwidth Selection in Heteroskedasticity-Autocorrelation Robust Testing," Econometrica, Econometric Society, vol. 76(1), pages 175-194, 01.
    2. Nigar Hashimzade & Timothy J. Vogelsang, 2008. "Fixed-b asymptotic approximation of the sampling behaviour of nonparametric spectral density estimators," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(1), pages 142-162, 01.
    3. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    4. Robinson, P M, 1991. "Automatic Frequency Domain Inference on Semiparametric and Nonparametric Models," Econometrica, Econometric Society, vol. 59(5), pages 1329-1363, September.
    5. Gao, Jiti & Gijbels, Irène, 2008. "Bandwidth Selection in Nonparametric Kernel Testing," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1584-1594.
    6. Andrews, Donald W.K., 2002. "EQUIVALENCE OF THE HIGHER ORDER ASYMPTOTIC EFFICIENCY OF k-STEP AND EXTREMUM STATISTICS," Econometric Theory, Cambridge University Press, vol. 18(05), pages 1040-1085, October.
    7. West, Kenneth D., 1997. "Another heteroskedasticity- and autocorrelation-consistent covariance matrix estimator," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 171-191.
    8. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2005. "A New Asymptotic Theory For Heteroskedasticity-Autocorrelation Robust Tests," Econometric Theory, Cambridge University Press, vol. 21(06), pages 1130-1164, December.
    9. Whitney K. Newey & Kenneth D. West, 1994. "Automatic Lag Selection in Covariance Matrix Estimation," Review of Economic Studies, Oxford University Press, vol. 61(4), pages 631-653.
    10. Nicholas M. Kiefer & Timothy J. Vogelsang, 2002. "Heteroskedasticity-Autocorrelation Robust Standard Errors Using The Bartlett Kernel Without Truncation," Econometrica, Econometric Society, vol. 70(5), pages 2093-2095, September.
    11. Dimitris N. Politis, 2004. "A Heavy-Tailed Distribution for ARCH Residuals with Application to Volatility Prediction," Annals of Economics and Finance, Society for AEF, vol. 5(2), pages 283-298, November.
    12. Velasco, Carlos & Robinson, Peter M., 2001. "Edgeworth Expansions For Spectral Density Estimates And Studentized Sample Mean," Econometric Theory, Cambridge University Press, vol. 17(03), pages 497-539, June.
    13. Peter C. B. Phillips & Yixiao Sun & Sainan Jin, 2006. "Spectral Density Estimation And Robust Hypothesis Testing Using Steep Origin Kernels Without Truncation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 47(3), pages 837-894, 08.
    14. Hansen, Bruce E, 1992. "Consistent Covariance Matrix Estimation for Dependent Heterogeneous Processes," Econometrica, Econometric Society, vol. 60(4), pages 967-972, July.
    15. Inoue, Atsushi & Shintani, Mototsugu, 2006. "Bootstrapping GMM estimators for time series," Journal of Econometrics, Elsevier, vol. 133(2), pages 531-555, August.
    16. Dimitris Politis & Halbert White, 2004. "Automatic Block-Length Selection for the Dependent Bootstrap," Econometric Reviews, Taylor & Francis Journals, vol. 23(1), pages 53-70.
    17. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    18. Peter Hall & Qiwei Yao, 2003. "Inference in Arch and Garch Models with Heavy--Tailed Errors," Econometrica, Econometric Society, vol. 71(1), pages 285-317, January.
    19. Andrews, Donald W K & Monahan, J Christopher, 1992. "An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator," Econometrica, Econometric Society, vol. 60(4), pages 953-966, July.
    20. Politis, Dimitris N., 2004. "A heavy-tailed distribution for ARCH residuals with application to volatility prediction," University of California at San Diego, Economics Working Paper Series qt7r89639x, Department of Economics, UC San Diego.
    21. Yixiao Sun & Peter C.B. Phillips, 2008. "Optimal Bandwidth Choice for Interval Estimation in GMM Regression," Cowles Foundation Discussion Papers 1661, Cowles Foundation for Research in Economics, Yale University.
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