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Asymptotics for the conditional-sum-of-squares estimator in multivariate fractional time series models

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  • Morten Ørregaard Nielsen

    () (Queen?s University and CREATES)

Abstract

This paper proves consistency and asymptotic normality for the conditional-sum-of-squares estimator, which is equivalent to the conditional maximum likelihood estimator, in multivariate fractional time series models. The model is parametric and quite general, and, in particular, encompasses the multivariate non-cointegrated fractional ARIMA model. The novelty of the consistency result, in particular, is that it applies to a multivariate model and to an arbitrarily large set of admissible parameter values, for which the objective function does not converge uniformly in probablity, thus making the proof much more challenging than usual. The neighborhood around the critical point where uniform convergence fails is handled using a truncation argument.

Suggested Citation

  • Morten Ørregaard Nielsen, 2014. "Asymptotics for the conditional-sum-of-squares estimator in multivariate fractional time series models," CREATES Research Papers 2014-34, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2014-34
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    References listed on IDEAS

    as
    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. Nielsen, Morten rregaard, 2004. "Efficient Likelihood Inference In Nonstationary Univariate Models," Econometric Theory, Cambridge University Press, vol. 20(01), pages 116-146, February.
    3. Hosoya, Yuzo, 1996. "The quasi-likelihood approach to statistical inference on multiple time-series with long-range dependence," Journal of Econometrics, Elsevier, vol. 73(1), pages 217-236, July.
    4. Newey, Whitney K, 1991. "Uniform Convergence in Probability and Stochastic Equicontinuity," Econometrica, Econometric Society, vol. 59(4), pages 1161-1167, July.
    5. 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.
    6. Johansen, Søren & Ørregaard Nielsen, Morten, 2012. "A Necessary Moment Condition For The Fractional Functional Central Limit Theorem," Econometric Theory, Cambridge University Press, vol. 28(03), pages 671-679, June.
    7. 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.
    8. Morten Ørregaard Nielsen & Per Houmann Frederiksen, 2005. "Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration," Econometric Reviews, Taylor & Francis Journals, vol. 24(4), pages 405-443.
    9. Peter M Robinson, 2006. "Conditional-Sum-of-Squares Estimation ofModels for Stationary Time Series with Long Memory," STICERD - Econometrics Paper Series 505, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    10. Robinson, Peter, 2006. "Conditional-sum-of-squares estimation of models for stationary time series with long memory," LSE Research Online Documents on Economics 4536, London School of Economics and Political Science, LSE Library.
    11. Yuzo Hosoya, 2005. "Fractional Invariance Principle," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(3), pages 463-486, May.
    12. 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.
    13. Tanaka, Katsuto, 1999. "The Nonstationary Fractional Unit Root," Econometric Theory, Cambridge University Press, vol. 15(04), pages 549-582, August.
    14. Shao, Xiaofeng, 2010. "Nonstationarity-Extended Whittle Estimation," Econometric Theory, Cambridge University Press, vol. 26(04), pages 1060-1087, August.
    15. repec:dau:papers:123456789/7622 is not listed on IDEAS
    16. Lieberman, Offer & Rosemarin, Roy & Rousseau, Judith, 2012. "Asymptotic Theory For Maximum Likelihood Estimation Of The Memory Parameter In Stationary Gaussian Processes," Econometric Theory, Cambridge University Press, vol. 28(02), pages 457-470, April.
    17. Robinson, Peter M. & Velasco, Carlos, 2000. "Whittle pseudo-maximum likelihood estimation for nonstationary time series," LSE Research Online Documents on Economics 2273, London School of Economics and Political Science, LSE Library.
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    Citations

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    Cited by:

    1. Torben G. Andersen & Rasmus T. Varneskov, 2702. "Consistent Inference for Predictive Regressions in Persistent VAR Economies," CREATES Research Papers 2018-09, Department of Economics and Business Economics, Aarhus University.
    2. 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.
    3. Rasmus T. Varneskov & Pierre Perron, 2018. "Combining long memory and level shifts in modelling and forecasting the volatility of asset returns," Quantitative Finance, Taylor & Francis Journals, vol. 18(3), pages 371-393, March.
    4. Cavaliere, Giuseppe & Nielsen, Morten Ørregaard & Taylor, A.M. Robert, 2015. "Bootstrap score tests for fractional integration in heteroskedastic ARFIMA models, with an application to price dynamics in commodity spot and futures markets," Journal of Econometrics, Elsevier, vol. 187(2), pages 557-579.
    5. Ergemen, Yunus Emre & Velasco, Carlos, 2017. "Estimation of fractionally integrated panels with fixed effects and cross-section dependence," Journal of Econometrics, Elsevier, vol. 196(2), pages 248-258.
    6. Yunus Emre Ergemen, 2016. "System Estimation of Panel Data Models under Long-Range Dependence," CREATES Research Papers 2016-02, Department of Economics and Business Economics, Aarhus University.
    7. Cavaliere, Giuseppe & Nielsen, Morten Ørregaard & Taylor, A.M. Robert, 2017. "Quasi-maximum likelihood estimation and bootstrap inference in fractional time series models with heteroskedasticity of unknown form," Journal of Econometrics, Elsevier, vol. 198(1), pages 165-188.
    8. Johansen, Søren & Nielsen, Morten Ørregaard, 2016. "The Role Of Initial Values In Conditional Sum-Of-Squares Estimation Of Nonstationary Fractional Time Series Models," Econometric Theory, Cambridge University Press, vol. 32(05), pages 1095-1139, October.
    9. Søren Johansen & Morten Ørregaard Nielsen, 2012. "The role of initial values in nonstationary fractional time series models," CREATES Research Papers 2012-47, Department of Economics and Business Economics, Aarhus University.
    10. Cavaliere, Giuseppe & Nielsen, Morten Ørregaard & Taylor, A.M. Robert, 2017. "Quasi-maximum likelihood estimation and bootstrap inference in fractional time series models with heteroskedasticity of unknown form," Journal of Econometrics, Elsevier, vol. 198(1), pages 165-188.
    11. Yunus Emre Ergemen & Carlos Velasco, 1203. "Persistence Heterogeneity Testing in Panels with Interactive Fixed Effects," CREATES Research Papers 2018-11, Department of Economics and Business Economics, Aarhus University.

    More about this item

    Keywords

    Asymptotic normality; conditional-sum-of-squares estimator; consistency; fractional integration; fractional time series; likelihood inference; long memory; nonstationary; uniform convergence.;

    JEL classification:

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