Center-Outward R-Estimation for Semiparametric VARMA Models

Abstract

We propose a new class of R-estimators for semiparametric VARMA models in which the innovation density plays the role of the nuisance parameter. Our estimators are based on the novel concepts of multivariate center-outward ranks and signs. We show that these concepts, combined with Le Cam’s asymptotic theory of statistical experiments, yield a class of semiparametric estimation procedures, which are efficient (at a given reference density), root-n consistent, and asymptotically normal under a broad class of (possibly non-elliptical) actual innovation densities. No kernel density estimation is required to implement our procedures. A Monte Carlo comparative study of our R-estimators and other routinely applied competitors demonstrates the benefits of the novel methodology, in large and small sample. Proofs, computational aspects, and further numerical results are available in the supplementary materials. Supplementary materials for this article are available online.

Keywords:

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

Supplementary Materials

Appendices A-E are available online: https://doi.org/10.1080/01621459.2020.1832501 Appendices A and B contain proofs and technical details. Appendix C provides details on the numerical implementation of our R-estimators, including a description of the algorithm used. Appendix D gives supplementary material for Section 5: center-outward quantile contours for the Gaussian mixture and skew distributions used in simulations and further small- and large-sample VAR simulations. In Appendix E, we provide further information on the VARMA fitting of Section 6 and the concept of impulse response functions.

Acknowledgments

The authors thank the editor and two anonymous referees for helpful comments, which significantly helped improve the final version of the article. Marc Hallin is grateful to Marc Henry for introducing him to the subtleties of measure transportation.

Notes

1 Unless otherwise stated, “QMLE” throughout refers to “Gaussian QMLE.”

2 Actually, for $n_0>1$, the $n!/n_0!$ permutations with repetitions of the n gridpoints.

3 An order statistic ${\mathit{Z}}_{(·)}^{(n)}$ of the un-ordered n-tuple ${\mathit{Z}}^{(n)}$ is an arbitrarily ordered version of the same; see Appendix D and Hallin et al. (2020). For instance, one may consider the ordering based on the first components.

4 That is, a symmetric function of the ${\mathit{Z}}_t$’s.

5 Asymptotic discreteness is only a theoretical requirement since, in practice, ${\widehat{\mathit{\theta }}}^{(n)}$ anyway only has a bounded number of digits; see Le Cam and Yang (2000, chap. 6) and van der Vaart (1998, sec. 5.7) for details.

6 Confidence intervals based on the estimated IRFs can be constructed from the asymptotic distributions of these estimators: see Lütkepohl (1990) for details. One also can recur to computer-intensive methods (e.g., bootstrap or Monte Carlo integration), see Kilian (1998).

Additional information

Funding

Hang Liu gratefully acknowledges the ESRC North West Social Science Doctoral Training Partnership (NWSSDTP Grant Number ES/P000665/1) for funding his PhD studentship. Davide La Vecchia acknowledges the Swiss National Science Foundation grant number 100018 169559 for financial support. Following the mathematical tradition, the authors of this paper are listed alphabetically, meaning that they all are “first authors.” [Added upon request from the University of Science and Technology of China].