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An Indirect Proof for the Asymptotic Properties of VARMA Model Estimators

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  • Guy Melard

Abstract

Strong consistency and asymptotic normality of a Gaussian quasi-maximum likelihood estimator for the parameters of a causal, invertible, and identiable vector autoregressive-moving average (VARMA) model are established in an indirect way. The proof is based on similar results for a much wider class of VARMA models with time-dependent coecients, hence in the context of non-stationary and heteroscedastic time series. For that reason, the proof avoids spectral analysis arguments and does not make use of ergodicity. The results presented are also applicable to ARMA models.

Suggested Citation

  • Guy Melard, 2020. "An Indirect Proof for the Asymptotic Properties of VARMA Model Estimators," Working Papers ECARES 2020-10, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/304272
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    1. Melard, Guy & Roy, Roch & Saidi, Abdessamad, 2006. "Exact maximum likelihood estimation of structured or unit root multivariate time series models," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 2958-2986, July.
    2. Francq, C. & Zakoian, J. -M., 2001. "Stationarity of multivariate Markov-switching ARMA models," Journal of Econometrics, Elsevier, vol. 102(2), pages 339-364, June.
    3. M. Hallin & D. La Vecchia & H. Liu, 2022. "Center-Outward R-Estimation for Semiparametric VARMA Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(538), pages 925-938, April.
    4. Gouriéroux, Christian & Monfort, Alain & Renne, Jean-Paul, 2017. "Statistical inference for independent component analysis: Application to structural VAR models," Journal of Econometrics, Elsevier, vol. 196(1), pages 111-126.
    5. Kascha, Christian & Mertens, Karel, 2009. "Business cycle analysis and VARMA models," Journal of Economic Dynamics and Control, Elsevier, vol. 33(2), pages 267-282, February.
    6. Ling, Shiqing & McAleer, Michael, 2003. "Asymptotic Theory For A Vector Arma-Garch Model," Econometric Theory, Cambridge University Press, vol. 19(2), pages 280-310, April.
    7. Newton, H. Joseph, 1978. "The information matrices of the parameters of multiple mixed time series," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 317-323, June.
    8. Francq, Christian & Gautier, Antony, 2004. "Estimation of time-varying ARMA models with Markovian changes in regime," Statistics & Probability Letters, Elsevier, vol. 70(4), pages 243-251, December.
    9. Yacouba Boubacar Maïnassara & Landy Rabehasaina, 2020. "Estimation of weak ARMA models with regime changes," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 1-52, April.
    10. Leon Wegge, 2012. "ARMAX(p,r,q) Parameter Identifiability Without Coprimeness," Working Papers 1217, University of California, Davis, Department of Economics.
    11. Boubacar Mainassara, Y. & Francq, C., 2011. "Estimating structural VARMA models with uncorrelated but non-independent error terms," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 496-505, March.
    12. André Klein & Guy Melard, 2020. "Invertibility Condition of the Fisher Information Matrix of a VARMAX Process and the Tensor Sylvester Matrix," Working Papers ECARES 2020-11, ULB -- Universite Libre de Bruxelles.
    13. Hannan, E. J. & Dunsmuir, W. T. M. & Deistler, M., 1980. "Estimation of vector ARMAX models," Journal of Multivariate Analysis, Elsevier, vol. 10(3), pages 275-295, September.
    14. Athanasopoulos, George & Vahid, Farshid, 2008. "VARMA versus VAR for Macroeconomic Forecasting," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 237-252, April.
    15. Christian Kascha, 2012. "A Comparison of Estimation Methods for Vector Autoregressive Moving-Average Models," Econometric Reviews, Taylor & Francis Journals, vol. 31(3), pages 297-324.
    16. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(1), pages 70-86, February.
    17. Yao, Qiwei & Brockwell, Peter J, 2006. "Gaussian maximum likelihood estimation for ARMA models. I. Time series," LSE Research Online Documents on Economics 57580, London School of Economics and Political Science, LSE Library.
    18. Alain Berlinet & Christian Francq, 1998. "On the Identifiability of Minimal VARMA Representations," Statistical Inference for Stochastic Processes, Springer, vol. 1(1), pages 1-15, January.
    19. Dufour, Jean-Marie & Jouini, Tarek, 2014. "Asymptotic distributions for quasi-efficient estimators in echelon VARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 73(C), pages 69-86.
    20. Guy Melard & Rajae Azrak, 2017. "Asymptotic Properties of Conditional Least-squares Estimators for Array Time Series," Working Papers ECARES ECARES 2017-49, ULB -- Universite Libre de Bruxelles.
    21. Qiwei Yao & Peter J. Brockwell, 2006. "Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(6), pages 857-875, November.
    22. Stelzer, Robert, 2009. "On Markov-Switching Arma Processes—Stationarity, Existence Of Moments, And Geometric Ergodicity," Econometric Theory, Cambridge University Press, vol. 25(1), pages 43-62, February.
    23. Kohn, R, 1979. "Asymptotic Estimation and Hypothesis Testing Results for Vector Linear Time Series Models," Econometrica, Econometric Society, vol. 47(4), pages 1005-1030, July.
    24. George Athanasopoulos & Farshid Vahid, 2008. "A complete VARMA modelling methodology based on scalar components," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(3), pages 533-554, May.
    25. Yao, Qiwei & Brockwell, Peter J., 2006. "Gaussian maximum likelihood estimation for ARMA models I: time series," LSE Research Online Documents on Economics 5825, London School of Economics and Political Science, LSE Library.
    26. D. S. Poskitt, 2005. "A Note on the Specification and Estimation of ARMAX Systems," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(2), pages 157-183, March.
    27. Peter Brockwell & Alexander Lindner & Bernd Vollenbröker, 2012. "Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(6), pages 1089-1119, December.
    28. Yacouba Boubacar Maïnassara & Bruno Saussereau, 2018. "Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(524), pages 1813-1827, October.
    29. Abdelkamel Alj & Rajae Azrak & Christophe Ley & Guy Mélard, 2017. "Asymptotic Properties of QML Estimators for VARMA Models with Time-dependent Coefficients," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 617-635, September.
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    Keywords

    non-stationary process; multivariate time series; time-varying models; identifiability; ARMA models;
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