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Center-Outward R-Estimation for Semiparametric VARMA Models

Author

Listed:
  • Marc Hallin
  • Davide La Vecchia
  • H Liu

Abstract

We propose a new class of estimators for semiparametric VARMA models with the innovation density playing the role of nuisance parameter. Our estimators are R-estimators based on the multivariate concepts of center-outward ranks and signs recently proposed by Hallin~(2017). We show how these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield a robust yet flexible and powerful class of estimation procedures for multivariate time series. We develop the relevant asymptotic theory of our R-estimators, establishing their root-n consistency and asymptotic normality under a broad class of innovation densities including, e.g. multimodal mixtures of Gaussians or and multivariate skew-t distributions. An implementation algorithm is provided in the supplementary material, available online. A Monte Carlo study compares our R-estimators with the routinely-applied Gaussian quasi-likelihood ones; the latter appear to be quite significantly outperformed away from elliptical innovations. Numerical results also provide evidence of considerable robustness gains. Two real data examples conclude the paper.

Suggested Citation

  • Marc Hallin & Davide La Vecchia & H Liu, 2019. "Center-Outward R-Estimation for Semiparametric VARMA Models," Working Papers ECARES 2019-25, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/294809
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    File URL: https://dipot.ulb.ac.be/dspace/bitstream/2013/294809/3/2019-25-HALLIN_LAVECCHIA_LIU-center.pdf
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    References listed on IDEAS

    as
    1. Marc Hallin & Madan Lal Puri, 1994. "Aligned rank tests for linear models with autocorrelated errors," ULB Institutional Repository 2013/2045, ULB -- Universite Libre de Bruxelles.
    2. Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
    3. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, September.
    4. Marc Hallin & Ramon van den Akker & Bas Werker, 2013. "On Quadratic Expansions of Log-Likelihoods and a General Asymptotic Linearity Result," Working Papers ECARES ECARES 2013-34, ULB -- Universite Libre de Bruxelles.
    5. Bernard Garel & Marc Hallin, 1995. "Local asymptotic normality of multivariate ARMA processes with a linear trend," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(3), pages 551-579, September.
    6. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    7. Marc Hallin & Davide La Vecchia, 2017. "A Simple R-Estimation Method for Semiparametric Duration Models," Working Papers ECARES ECARES 2017-01, ULB -- Universite Libre de Bruxelles.
    8. Ernst R. Berndt & Bronwyn H. Hall & Robert E. Hall & Jerry A. Hausman, 1974. "Estimation and Inference in Nonlinear Structural Models," NBER Chapters, in: Annals of Economic and Social Measurement, Volume 3, number 4, pages 653-665, National Bureau of Economic Research, Inc.
    9. Marc Hallin, 2017. "On Distribution and Quantile Functions, Ranks and Signs in R_d," Working Papers ECARES ECARES 2017-34, ULB -- Universite Libre de Bruxelles.
    10. Hallin, Marc & La Vecchia, Davide, 2017. "R-estimation in semiparametric dynamic location-scale models," Journal of Econometrics, Elsevier, vol. 196(2), pages 233-247.
    11. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
    12. Mukherjee, Kanchan & Bai, Z. D., 2002. "R-estimation in Autoregression with Square-Integrable Score Function," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 167-186, April.
    13. Marc Hallin & Jean-François Ingenbleek & Madan Lal Puri, 1984. "Linear serial rank tests for randomness against ARMA alternatives," ULB Institutional Repository 2013/2167, ULB -- Universite Libre de Bruxelles.
    14. Andrews, Beth, 2012. "Rank-Based Estimation For Garch Processes," Econometric Theory, Cambridge University Press, vol. 28(5), pages 1037-1064, October.
    15. Beth Andrews, 2008. "Rank‐based estimation for autoregressive moving average time series models," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(1), pages 51-73, January.
    16. Marc Hallin & Guy Melard, 1988. "Rank-based tests for randomness against first-order serial dependence," ULB Institutional Repository 2013/2015, ULB -- Universite Libre de Bruxelles.
    17. Mukherjee, Kanchan, 2007. "Generalized R-estimators under conditional heteroscedasticity," Journal of Econometrics, Elsevier, vol. 141(2), pages 383-415, December.
    18. Hallin, M. & Puri, M. L., 1994. "Aligned Rank Tests for Linear Models with Autocorrelated Error Terms," Journal of Multivariate Analysis, Elsevier, vol. 50(2), pages 175-237, August.
    19. Terpstra, Jeffrey T. & McKean, Joseph W. & Naranjo, Joshua D., 2001. "GR-estimates for an autoregressive time series," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 165-172, January.
    20. Delphine Cassart & Marc Hallin & Davy Paindaveine, 2010. "On the estimation of cross-information quantities in rank-based inference," Working Papers ECARES ECARES 2010-010, ULB -- Universite Libre de Bruxelles.
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    Cited by:

    1. Eustasio Del Barrio & Alberto Gonzalez-Sanz & Marc Hallin, 2019. "A Note on the Regularity of Center-Outward Distribution and Quantile Functions," Working Papers ECARES 2019-33, ULB -- Universite Libre de Bruxelles.
    2. Guy Melard, 2020. "An Indirect Proof for the Asymptotic Properties of VARMA Model Estimators," Working Papers ECARES 2020-10, ULB -- Universite Libre de Bruxelles.

    More about this item

    Keywords

    Multivariate ranks; Distribution-freeness; Local asymptotic normality; Measure transportation; Quasi likelihood estimation; Skew innovation density;

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