Rank-Based Testing for Semiparametric VAR Models: a measure transportation approach

Abstract

We develop a class of tests for semiparametric vector autoregressive (VAR) models with unspecified innovation densities, based on the recent measure-transportation-based concepts of multivariate center-outward ranks and signs. We show that these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield novel testing procedures, which (a) are valid under a broad class of innovation densities (possibly non-elliptical, skewed, and/or with infinite moments), (b) are optimal (locally asymptotically maximin or most stringent) at selected ones, and (c) are robust against additive outliers. In order to do so, we establish a Hájek asymptotic representation result, of independent interest, for a general class of center-outward rank-based serial statistics. As an illustration, we consider the problems of testing the absence of serial correlation in multiple-output and possibly non-linear regression (an extension of the classical Durbin-Watson problem) and the sequential identification of the order p of a vector autoregressive (VAR(p)) model. A Monte Carlo comparative study of our tests and their routinely-applied Gaussian competitors demonstrates the benefits (in terms of size, power, and robustness) of our methodology; these benefits are particularly significant in the presence of asymmetric and leptokurtic innovation densities. A real data application concludes the paper.

Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics

“…Center-outward ranks and signs also have been used successfully in other statistical problems: construction of R-estimators (Hallin et al, 2021b(Hallin et al, , 2020b in VARMA models, rank tests for multiple-output regression and MANOVA (Hallin et al, 2020a), and two-sample goodness-of-fit tests Hallin and Mordant, 2021). We show here how center-outward ranks and signs naturally allow us to define distributionfree multivariate versions of the popular quadrant, Spearman, and Kendall tests.…”

Pitman Efficiency Lower Bounds for Multivariate Distribution-Free Tests Based on Optimal Transport

“…In particular, in Section 4.1 we will construct a class of distribution-free nonparametric tests of independence which are natural multivariate analogues of Spearman's rank correlation [110], that enjoy favorable ARE properties similar to T rank m,n,sc . In Section 4.2 we will discuss how our techniques provide direct improvements of the results in some related papers such as [107,45,25,46,47], including the resolution of an open question laid out in [107].…”