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Multiple local whittle estimation in stationary systems

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  • Robinson, Peter M.

Abstract

Moving from univariate to bivariate jointly dependent long memory time series introduces a phase parameter (γ), at the frequency of principal interest, zero; for short memory series γ = 0 automatically. The latter case has also been stressed under long memory, along with the "fractional differencing" case ( ) / 2; 2 1 γ = δ − δ π where 1 2 δ , δ are the memory parameters of the two series. We develop time domain conditions under which these are and are not relevant, and relate the consequent properties of cross-autocovariances to ones of the (possibly bilateral) moving average representation which, with martingale difference innovations of arbitrary dimension, is used in asymptotic theory for local Whittle parameter estimates depending on a single smoothing number. Incorporating also a regression parameter (β) which, when non-zero, indicates cointegration, the consistency proof of these implicitly-defined estimates is nonstandard due to the β estimate converging faster than the others. We also establish joint asymptotic normality of the estimates, and indicate how this outcome can apply in statistical inference on several questions of interest. Issues of implementation are discussed, along with implications of knowing β and of correct or incorrect specification of γ , and possible extensions to higher-dimensional systems and nonstationary series.

Suggested Citation

  • Robinson, Peter M., 2007. "Multiple local whittle estimation in stationary systems," LSE Research Online Documents on Economics 4436, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:4436
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    File URL: http://eprints.lse.ac.uk/4436/
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    References listed on IDEAS

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    1. Morten Ørregaard Nielsen, 2005. "Semiparametric Estimation in Time‐Series Regression with Long‐Range Dependence," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(2), pages 279-304, March.
    2. Nielsen, Morten Orregaard, 2007. "Local Whittle Analysis of Stationary Fractional Cointegration and the ImpliedRealized Volatility Relation," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 427-446, October.
    3. Marinucci, D. & Robinson, Peter M., 2001. "Narrow-band analysis of nonstationary processes," LSE Research Online Documents on Economics 303, London School of Economics and Political Science, LSE Library.
    4. Lobato, Ignacio N., 1999. "A semiparametric two-step estimator in a multivariate long memory model," Journal of Econometrics, Elsevier, vol. 90(1), pages 129-153, May.
    5. Andrew Harvey (ed.), 1994. "Time Series," Books, Edward Elgar Publishing, volume 0, number 599.
    6. Javier Hualde & Peter Robinson, 2006. "Semiparametric Estimation of Fractional Cointegration," Faculty Working Papers 07/06, School of Economics and Business Administration, University of Navarra.
    7. Clifford M. Hurvich & Eric Moulines & Philippe Soulier, 2005. "Estimating Long Memory in Volatility," Econometrica, Econometric Society, vol. 73(4), pages 1283-1328, July.
    8. Marinucci, D & Robinson, Peter, 2001. "Narrow-band analysis of nonstationary processes," LSE Research Online Documents on Economics 2015, London School of Economics and Political Science, LSE Library.
    9. Shimotsu, Katsumi, 2007. "Gaussian semiparametric estimation of multivariate fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 137(2), pages 277-310, April.
    10. Robinson, P. M., 2005. "Robust covariance matrix estimation : 'HAC' estimates with long memory/antipersistence correction," LSE Research Online Documents on Economics 323, London School of Economics and Political Science, LSE Library.
    11. Christensen, Bent Jesper & Nielsen, Morten Orregaard, 2006. "Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting," Journal of Econometrics, Elsevier, vol. 133(1), pages 343-371, July.
    12. Robinson, P.M., 2005. "Robust Covariance Matrix Estimation: Hac Estimates With Long Memory/Antipersistence Correction," Econometric Theory, Cambridge University Press, vol. 21(1), pages 171-180, February.
    13. D Marinucci & Peter M Robinson, 2001. "Narrow-Band Analysis of Nonstationary Processes," STICERD - Econometrics Paper Series 421, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    14. Giraitis, L. & Robinson, P.M., 2003. "Edgeworth expansions for semiparametric Whittle estimation of long memory," LSE Research Online Documents on Economics 291, London School of Economics and Political Science, LSE Library.
    15. Carlos Velasco, 2003. "Gaussian Semi‐parametric Estimation of Fractional Cointegration," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(3), pages 345-378, May.
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    Cited by:

    1. Morten Ø. Nielsen & Per Houmann Frederiksen, 2008. "Fully Modified Narrow-band Least Squares Estimation Of Stationary Fractional Cointegration," Working Paper 1171, Economics Department, Queen's University.

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    More about this item

    Keywords

    Long memory; phase; cointegration; semiparametric estimation; consistency; asymptotic normality.;
    All these keywords.

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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