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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.

Suggested Citation

  • Søren Johansen & Morten Ørregaard Nielsen, 2012. "The role of initial values in conditional sum-of-squares estimation of nonstationary fractional time series models," Working Papers 1300, Queen's University, Department of Economics.
  • Handle: RePEc:qed:wpaper:1300

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    References listed on IDEAS

    1. 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.
    2. Johansen, Søren & Nielsen, Morten Ørregaard, 2010. "Likelihood inference for a nonstationary fractional autoregressive model," Journal of Econometrics, Elsevier, vol. 158(1), pages 51-66, September.
    3. Andrews, Donald W.K. & Lieberman, Offer & Marmer, Vadim, 2006. "Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes," Journal of Econometrics, Elsevier, vol. 133(2), pages 673-702, August.
    4. 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.
    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. Morten Ørregaard Nielsen, 2015. "Asymptotics for the Conditional-Sum-of-Squares Estimator in Multivariate Fractional Time-Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(2), pages 154-188, March.
    7. 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.
    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. 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, January.
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    Cited by:

    1. Søren Johansen & Morten Ørregaard Nielsen, 0111. "Testing the CVAR in the fractional CVAR model," CREATES Research Papers 2017-37, Department of Economics and Business Economics, Aarhus University.
    2. Gagnon, Marie-Hélène & Power, Gabriel J. & Toupin, Dominique, 2016. "International stock market cointegration under the risk-neutral measure," International Review of Financial Analysis, Elsevier, vol. 47(C), pages 243-255.
    3. repec:eee:econom:v:204:y:2018:i:1:p:54-65 is not listed on IDEAS
    4. Morten Ørregaard Nielsen & Sergei S. Shibaev, 2015. "Forecasting daily political opinion polls using the fractionally cointegrated VAR model," Working Papers 1340, Queen's University, Department of Economics.
    5. Søren Johansen & Morten Ørregaard Nielsen, 1705. "Nonstationary cointegration in the fractionally cointegrated VAR model," CREATES Research Papers 2018-17, Department of Economics and Business Economics, Aarhus University.
    6. Davide Delle Monache & Stefano Grassi & Paolo Santucci de Magistris, 0404. "Does the ARFIMA really shift?," CREATES Research Papers 2017-16, Department of Economics and Business Economics, Aarhus University.
    7. Sepideh Dolatabadi & Paresh Kumar Narayan & Morten Ørregaard Nielsen & Ke Xu, 2017. "Economic significance of commodity return forecasts from the fractionally cointegrated VAR model," Working Papers 1337, Queen's University, Department of Economics.

    More about this item


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

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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