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The role of initial values in conditional sum-of-squares estimation of nonstationary fractional time series models

  • Søren Johansen

    ()

    (University of Copenhagen and CREATES)

  • Morten Ørregaard Nielsen

    ()

    (Queen's University and CREATES)

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, for which data exist from some point in time, which we call -N_0+1, but we only start observing it at a later time, t=1. The parameter (d,μ,σ²) is estimated by CSS based on the model Δ_0^d (X_t-μ)=ε_t, t=N+1,…,N+T, conditional on X_1,…,X_N. We derive an expression for the second-order bias of d as a function of the initial values, X_t, t=-N_0+1,…,N, and we investigate the effect on the bias of setting aside the first N observations as initial values. We compare d with an estimator, d_c, derived similarly but by choosing μ=C. We find, both theoretically and using a data set on voting behavior, that in many cases, the estimation of the parameter μ picks up the effect of the initial values even for the choice N=0. If N_0=0, we show that the second-order bias can be completely eliminated by a simple bias correction. If, on the other hand, N_0>0, it can only be partly eliminated because the second-order bias term due to the initial values can only be diminished by increasing N.

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File URL: http://qed.econ.queensu.ca/working_papers/papers/qed_wp_1300.pdf
File Function: First version 2012
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Paper provided by Queen's University, Department of Economics in its series Working Papers with number 1300.

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Length: 39 pages
Date of creation: Nov 2012
Date of revision:
Handle: RePEc:qed:wpaper:1300
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  1. Rolf Tschernig & Enzo Weber & Roland Weigand, 2013. "Long-Run Identification in a Fractionally Integrated System," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(4), pages 438-450, October.
  2. 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.
  3. Søren Johansen & Morten Ørregaard Nielsen, 2010. "Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model," Discussion Papers 10-15, University of Copenhagen. Department of Economics.
  4. 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.
  5. 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.
  6. Søren Johansen & Morten Ørregaard Nielsen, 2010. "Likelihood inference for a nonstationary fractional autoregressive model," Working Papers 1172, Queen's University, Department of Economics.
  7. 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.
  8. 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.
  9. 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.
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