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The role of initial values in nonstationary fractional time series models

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  • Søren Johansen

    ()
    (University of Copenhagen and CREATES)

  • Morten Ørregaard Nielsen

    ()
    (Queen's University and CREATES)

Abstract

In this paper we analyze the influence of observed and unobserved initial values on the bias of the conditional maximum likelihood or conditional sum-of-squares (CSS, or least squares) estimator of the fractional parameter, d, in a nonstationary fractional time series model. The CSS estimator is popular in empirical work due, at least in part, to its simplicity and its feasibility, even in very complicated nonstationary models. We consider a process, X_{t}, which exists from some time point which we call -N_{0}+1, and we start observing it at time t=1. We assume that the data centered at μ is generated by the fractional filter truncated at -N_{0}, and that we have T_{0}=N+T observations. We estimate (d,μ,σ²) based on the Gaussian likelihood conditional on the first N observations, for the model given by the fractional filter truncated at zero. We derive an expression for the second-order bias of d as a function of all initial values, and investigate the effect on the bias of setting aside N observations as initial values. We compare d with an estimator, d_{c}, derived from centering the data at C. We find, both theoretically and using a data set on voting behavior, that in many cases, the parameter μ picks up the effect of the initial values even for the choice N=0.

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File URL: http://qed.econ.queensu.ca/working_papers/papers/qed_wp_1300.pdf
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Bibliographic Info

Paper provided by Queen's University, Department of Economics in its series Working Papers with number 1300.

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Length: 31 pages
Date of creation: Nov 2012
Date of revision:
Handle: RePEc:qed:wpaper:1300

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Keywords: fractional integration; conditional inference; bias; Asymptotic expansion; initial values; likelihood inference;

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References

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  1. Juan J. Dolado & Jesus Gonzalo & Laura Mayoral, 2002. "A Fractional Dickey-Fuller Test for Unit Roots," Econometrica, Econometric Society, vol. 70(5), pages 1963-2006, September.
  2. Søren Johansen & Morten Ørregaard Nielsen, 2010. "Likelihood inference for a nonstationary fractional autoregressive model," Working Papers 1172, Queen's University, Department of Economics.
  3. Søren Johansen & Morten Ørregaard Nielsen, 2012. "Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model," Econometrica, Econometric Society, vol. 80(6), pages 2667-2732, November.
  4. Tschernig, Rolf & Weber, Enzo & Weigand, Roland, 2010. "Long-run Identification in a Fractionally Integrated System," University of Regensburg Working Papers in Business, Economics and Management Information Systems 447, University of Regensburg, Department of Economics.
  5. David Byers & James Davidson & David Peel, 1997. "Modelling Political Popularity: an Analysis of Long-range Dependence in Opinion Poll Series," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 160(3), pages 471-490.
  6. Eduardo Rossi & Paolo Santucci de Magistris, 2013. "A No‐Arbitrage Fractional Cointegration Model for Futures and Spot Daily Ranges," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(1), pages 77-102, 01.
  7. Morten Ørregaard Nielsen, 2011. "Asymptotics for the conditional-sum-of-squares estimator in multivariate fractional time series models," Working Papers 1259, Queen's University, Department of Economics.
  8. Donald W.K. Andrews & Offer Lieberman, 2002. "Higher-order Improvements of the Parametric Bootstrap for Long-memory Gaussian Processes," Cowles Foundation Discussion Papers 1378, Cowles Foundation for Research in Economics, Yale University.
  9. Johansen, SØren, 2008. "A Representation Theory For A Class Of Vector Autoregressive Models For Fractional Processes," Econometric Theory, Cambridge University Press, vol. 24(03), pages 651-676, June.
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Cited by:
  1. Rossi, Eduardo & Santucci de Magistris, Paolo, 2013. "Long memory and tail dependence in trading volume and volatility," Journal of Empirical Finance, Elsevier, vol. 22(C), pages 94-112.
  2. Bent Jesper Christensen & Robinson Kruse & Philipp Sibbertsen, 2013. "A unified framework for testing in the linear regression model under unknown order of fractional integration," CREATES Research Papers 2013-35, School of Economics and Management, University of Aarhus.

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