The Annals of Statistics

Edgeworth expansions for semiparametric Whittle estimation of long memory

L. Giraitis and P.M. Robinson

Full-text: Open access

Abstract

The semiparametric local Whittle or Gaussian estimate of the long memory parameter is known to have especially nice limiting distributional properties, being asymptotically normal with a limiting variance that is completely known. However in moderate samples the normal approximation may not be very good, so we consider a refined, Edgeworth, approximation, for both a tapered estimate and the original untapered one. For the tapered estimate, our higher-order correction involves two terms, one of order $m^{-1/2}$ (where m is the bandwidth number in the estimation), the other a bias term, which increases in m; depending on the relative magnitude of the terms, one or the other may dominate, or they may balance. For the untapered estimate we obtain an expansion in which, for m increasing fast enough, the correction consists only of a bias term. We discuss applications of our expansions to improved statistical inference and bandwidth choice. We assume Gaussianity, but in other respects our assumptions seem mild.

Article information

Source
Ann. Statist., Volume 31, Number 4 (2003), 1325-1375.

Dates
First available in Project Euclid: 31 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1059655915

Digital Object Identifier
doi:10.1214/aos/1059655915

Mathematical Reviews number (MathSciNet)
MR2001652

Zentralblatt MATH identifier
1041.62012

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Edgeworth expansion long memory semiparametric approximation

Citation

Giraitis, L.; Robinson, P.M. Edgeworth expansions for semiparametric Whittle estimation of long memory. Ann. Statist. 31 (2003), no. 4, 1325--1375. doi:10.1214/aos/1059655915. https://projecteuclid.org/euclid.aos/1059655915


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References

  • ANDREWS, D. W. K. and GUGGENBERGER, P. (2003). A bias-reduced log-periodogram regression estimator for the long memory parameter. Econometrica 71 675-712.
  • ANDREWS, D. W. K. and SUN, Y. (2001). Local poly nomial Whittle estimation of long range dependence. Preprint. Cowles Foundation for Research in Economics, Yale Univ.
  • BENTKUS, R. Y. and RUDZKIS, R. A. (1982). On the distribution of some statistical estimates of a spectral density. Theory Probab. Appl. 27 795-814.
  • BHATTACHARy A, R. N. and GHOSH, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434-451.
  • BHATTACHARy A, R. N. and RANGA RAO, R. (1976). Normal Approximation and Asy mptotic Expansions. Wiley, New York.
  • BRILLINGER, D. R. (1975). Time Series: Data Analy sis and Theory. Holden Day, San Francisco.
  • CHIBISOV, D. M. (1972). An asy mptotic expansion for the distribution of a statistic admitting an asy mptotic expansion. Theory Probab. Appl. 17 620-630.
  • FOX, R. and TAQQU, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517-532.
  • GEWEKE, J. and PORTER-HUDAK, S. (1983). The estimation and application of long memory time series models. J. Time Ser. Anal. 4 221-238.
  • GIRAITIS, L., ROBINSON, P. M. and SAMAROV, A. (1997). Rate optimal semiparametric estimation of the memory parameter of the Gaussian time series with long range dependence. J. Time Ser. Anal. 18 49-60.
  • GIRAITIS, L., ROBINSON, P. M. and SAMAROV, A. (2000). Adaptive semiparametric estimation of the memory parameter. J. Multivariate Anal. 72 183-207.
  • HALL, P. (1991). Edgeworth expansions for nonparametric density estimators, with applications. Statistics 22 215-232.
  • HANNAN, E. J. (1973). The asy mptotic theory of linear time series models. J. Appl. Probab. 10 130-145.
  • HENRY, M. and ROBINSON, P. M. (1996). Bandwidth choice in Gaussian semiparametric estimation of long range dependence. Athens Conference on Applied Probability and Time Series Analy sis. Time Series Analy sis. Lecture Notes in Statist. 115 220-232. Springer, New York.
  • HURVICH, C. M. and BRODSKY, J. (2001). Broadband semiparametric estimation of the memory parameter of a long memory time series using fractional exponential models. J. Time Ser. Anal. 22 221-249.
  • HURVICH, C. M. and CHEN, W. W. (2000). An efficient taper for potentially overdifferenced long memory time series. J. Time Ser. Anal. 21 155-180.
  • HURVICH, C. M. and RAY, B. (1995). Estimation of the memory parameter for nonstationary or noninvertable fractionally integrated processes. J. Time Ser. Anal. 16 17-42.
  • IBRAGIMOV, I. A. and HAS'MINSKII, R. Z. (1981). Statistical Estimation. Asy mptotic Theory. Springer, New York.
  • JANACEK, G. J. (1982). Determining the degree of differencing for time series via the log spectrum. J. Time Ser. Anal. 3 177-183.
  • KÜNSCH, H. (1987). Statistical aspects of self-similar processes. In Proc. First World Congress of the Bernoulli Society (Yu. A. Prokhorov and V. V. Sazonov, eds.) 1 67-74. VNU Science Press, Utrecht.
  • LIEBERMAN, O., ROUSSEAU, J. and ZUCKER, D. M. (2001). Valid Edgeworth expansion for the sample autocorrelation function under long range dependence. Econometric Theory 17 257-275.
  • MOULINES, E. and SOULIER, P. (1999). Broadband log-periodogram regression of time series with long range dependence. Ann. Statist. 27 1415-1439.
  • NISHIy AMA, Y. and ROBINSON, P. M. (2000). Edgeworth expansions for semiparametric averaged derivatives. Econometrica 68 931-979.
  • ROBINSON, P. M. (1995a). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048-1072.
  • ROBINSON, P. M. (1995b). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630-1661.
  • ROBINSON, P. M. (1995c). The approximate distribution of nonparametric regression estimates. Statist. Probab. Lett. 23 193-201.
  • ROBINSON, P. M. and HENRY, M. (1999). Long and short memory conditional heteroscedasticity in estimating the memory parameter of levels. Econometric Theory 15 299-336.
  • ROBINSON, P. M. and HENRY, M. (2003). Higher-order kernel semiparametric M-estimation of long memory. J. Econometrics 114 1-27.
  • TANIGUCHI, M. (1991). Higher-Order Asy mptotic Theory for Time Series Analy sis. Lecture Notes in Statist. 68. Springer, New York.
  • VELASCO, C. (1999a). Nonstationary log-periodogram regression. J. Econometrics 91 325-371.
  • VELASCO, C. (1999b). Gaussian semiparametric estimation of nonstationary time series. J. Time Ser. Anal. 20 87-127.
  • VELASCO, C. and ROBINSON, P. M. (2001). Edgeworth expansions for spectral density estimates and studentized sample mean. Econometric Theory 17 497-539.