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Center-Outward R-Estimation for Semiparametric VARMA Models

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  • 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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    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 & 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.
    3. Marc Hallin, 1986. "Nonstationary q-dependent processes and time-varying moving average models: invertibility properties and the forecasting problem," ULB Institutional Repository 2013/2005, ULB -- Universite Libre de Bruxelles.
    4. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    5. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    6. Hallin, Marc & La Vecchia, Davide, 2020. "A Simple R-estimation method for semiparametric duration models," Journal of Econometrics, Elsevier, vol. 218(2), pages 736-749.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Mukherjee, Kanchan, 2007. "Generalized R-estimators under conditional heteroscedasticity," Journal of Econometrics, Elsevier, vol. 141(2), pages 383-415, December.
    13. 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.
    14. Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
    15. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, April.
    16. 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.
    17. Hallin, Marc & La Vecchia, Davide, 2017. "R-estimation in semiparametric dynamic location-scale models," Journal of Econometrics, Elsevier, vol. 196(2), pages 233-247.
    18. 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.
    19. 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.
    20. Andrews, Beth, 2012. "Rank-Based Estimation For Garch Processes," Econometric Theory, Cambridge University Press, vol. 28(5), pages 1037-1064, October.
    21. 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.
    22. 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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    Citations

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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. Hallin, Marc & La Vecchia, Davide, 2020. "A Simple R-estimation method for semiparametric duration models," Journal of Econometrics, Elsevier, vol. 218(2), pages 736-749.
    3. Yihui He & Fang Han, 2023. "On propensity score matching with a diverging number of matches," Papers 2310.14142, arXiv.org, revised Nov 2023.
    4. Hongjian Shi & Mathias Drton & Marc Hallin & Fang Han, 2023. "Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics," Working Papers ECARES 2023-03, ULB -- Universite Libre de Bruxelles.
    5. Hongjian Shi & Marc Hallin & Mathias Drton & Fang Han, 2020. "Rate-Optimality of Consistent Distribution-Free Tests of Independence Based on Center-Outward Ranks and Signs," Working Papers ECARES 2020-23, ULB -- Universite Libre de Bruxelles.
    6. del Barrio, Eustasio & González-Sanz, Alberto & Hallin, Marc, 2020. "A note on the regularity of optimal-transport-based center-outward distribution and quantile functions," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    7. Guy Melard, 2020. "An Indirect Proof for the Asymptotic Properties of VARMA Model Estimators," Working Papers ECARES 2020-10, ULB -- Universite Libre de Bruxelles.

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    Keywords

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