  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 {\it center-outward
ranks} and {\it 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 H\' ajek
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.
