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Multiple Local Whittle Estimation in StationarySystems

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  • Peter M Robinson
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    Abstract

    Moving from univariate to bivariate jointly dependent long memory time series introduces a phase parameter (?), at the frequency of principal interest, zero; for shortmemory series ? = 0 automatically. The latter case has also been stressed under longmemory, along with the 'fractional differencing' case ( ) / 2; 2 1 ? = d - d p where 1 2 d , dare the memory parameters of the two series. We develop time domain conditionsunder which these are and are not relevant, and relate the consequent properties ofcross-autocovariances to ones of the (possibly bilateral) moving averagerepresentation which, with martingale difference innovations of arbitrary dimension,is used in asymptotic theory for local Whittle parameter estimates depending on asingle smoothing number. Incorporating also a regression parameter (ß) which, whennon-zero, indicates cointegration, the consistency proof of these implicitly-definedestimates is nonstandard due to the ß estimate converging faster than the others. Wealso establish joint asymptotic normality of the estimates, and indicate how thisoutcome can apply in statistical inference on several questions of interest. Issues ofimplementation are discussed, along with implications of knowing ß and of correct orincorrect specification of ? , and possible extensions to higher-dimensional systemsand nonstationary series.

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    Bibliographic Info

    Paper provided by Suntory and Toyota International Centres for Economics and Related Disciplines, LSE in its series STICERD - Econometrics Paper Series with number /2007/525.

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    Date of creation: Oct 2007
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    Handle: RePEc:cep:stiecm:/2007/525

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    Web page: http://sticerd.lse.ac.uk/_new/publications/default.asp

    Related research

    Keywords: Long memory; phase; cointegration; semiparametricestimation; consistency; asymptotic normality.;

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    References

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    1. Davidson, James & Hashimzade, Nigar, 2008. "Alternative Frequency And Time Domain Versions Of Fractional Brownian Motion," Econometric Theory, Cambridge University Press, vol. 24(01), pages 256-293, February.
    2. D. Marinucci & Peter M. Robinson, 2001. "Narrow-band analysis of nonstationary processes," LSE Research Online Documents on Economics 303, London School of Economics and Political Science, LSE Library.
    3. Javier Hualde & Peter M Robinson, 2006. "Semiparametric Estimation of Fractional Cointegration," STICERD - Econometrics Paper Series /2006/502, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    4. Clifford M. Hurvich & Eric Moulines & Philippe Soulier, 2005. "Estimating Long Memory in Volatility," Econometrica, Econometric Society, vol. 73(4), pages 1283-1328, 07.
    5. Shimotsu, Katsumi, 2007. "Gaussian semiparametric estimation of multivariate fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 137(2), pages 277-310, April.
    6. 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.
    7. Willa Chen & Clifford Hurvich, 2004. "Semiparametric Estimation of Fractional Cointegrating Subspaces," Econometrics 0412007, EconWPA.
    8. Morten Oerregaard Nielsen, . "Local Whittle Analysis of Stationary Fractional Cointegration," Economics Working Papers 2002-8, School of Economics and Management, University of Aarhus.
    9. repec:cup:etheor:v:24:y:2007:i:01:p:256-293 is not listed on IDEAS
    10. L. Giraitis & P.M. Robinson, 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.
    11. Nielsen, Morten Oe., . "Semiparametric Estimation in Time Series Regression with Long Range Dependence," Economics Working Papers 2002-17, School of Economics and Management, University of Aarhus.
    12. Carlos Velasco, 2003. "Gaussian Semi-parametric Estimation of Fractional Cointegration," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(3), pages 345-378, 05.
    13. D Marinucci & Peter Robinson, 2001. "Narrow-band analysis of nonstationary processes," LSE Research Online Documents on Economics 2015, London School of Economics and Political Science, LSE Library.
    14. Liudas Giraitis & Peter M Robinson, 2002. "Edgeworth Expansions for Semiparametric Whittle Estimation of Long Memory," STICERD - Econometrics Paper Series /2002/438, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    15. Peter M Robinson, 2004. "ROBUST COVARIANCE MATRIX ESTIMATION: "HAC" Estimates with Long Memory/Antipersistence Correction," STICERD - Econometrics Paper Series /2004/471, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    16. D Marinucci & Peter M Robinson, 2001. "Narrow-Band Analysis of Nonstationary Processes," STICERD - Econometrics Paper Series /2001/421, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    17. 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.
    18. Katsumi Shimotsu & Peter C.B. Phillips, 2000. "Local Whittle Estimation in Nonstationary and Unit Root Cases," Cowles Foundation Discussion Papers 1266, Cowles Foundation for Research in Economics, Yale University, revised Sep 2003.
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    Cited by:
    1. Peter C.B. Phillips, 2008. "Long Memory and Long Run Variation," Cowles Foundation Discussion Papers 1656, Cowles Foundation for Research in Economics, Yale University.

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