Nonparametric Optimal Tests for Independence in the Elliptical VAR Model Maria Caterina Bramati * Marc Hallin* Davy Paindaveine* draft † Abstract In this work we construct a class of locally asymptotically most stringent (in the Le Cam sense) tests for independence between two sets of variables in the VAR models. These tests are based on multivariate ranks of distances and multivariate signs of the observations and are shown to be asymptotically distribution-free under very mild assumptions on the noise, which is obtained by applying a linear transformation to a block of spherical innovation. The class of tests derived is invariant with respect to the group of block affine transformations and asymptotically invariant with respect to the group of continuous monotone marginal radial transformations. Key Words: VAR models, Independence, Local asymptotic normality, Locally asymptotically most stringent tests, Elliptical densities, Multivariate ranks and signs. JEL codes: C12, C14, C32 1 Introduction The issue of independence between blocks of variables in i.i.d. samples has been treated in many papers. However, there is still very little literature on tests for independence in multivariate time series. The importance of testing for block exogeneity between two vector series, as defined in Hamilton (1994), is crucial when trying to explain relationships between economic variables in one economy or among different economies. VAR models are widely used in macroeconomic studies, since they allow to study the dynamic mechanisms of variables/economies; they have also appeared in the microeconometrics literature as well (see Chamberlain, 1983). Moreover, those tests represent also the main tool in the study of Granger causality (see Geweke (1982), Boudjellaba, Dufour and Roy (1992)). Therefore, several parametric tests have been constructed, as in Haugh (1976), which is basically a portmanteau type test, and Koch and Yang (1986) for the univariate case; El Himdi and Roy (1997) and Hallin and Saidi (2004), respectively, for their multivariate version. Unfortunately, those tests suffer of low power (especially Haugh test) and they do not give any optimality results, except for the latter, which applies the Le Cam Local Asymptotic Normality approach, providing locally asymptotically optimal tests. Le Cam’s work (Le Cam, 1986) has indeed clarified the Fisherian claim of asymptotic efficiency of MLE in the parametric models casting the traditional ’regularity assumptions’ into a single re- quirement called Local Asymptotic Normality, LAN for short. Indeed, he explains the Fisher’s notion of efficiency of MLE’s as the local asymptotic minimaxity of the MLE using the fact that certain asymptotic problems can, under LAN, be reduced to the solutions of the corresponding Gaussian shift problems. Therefore, the power of LAN is that, once we have the LAN approx- imation, we can forget all about the underlying modelling assumptions, and concentrate on the consequences of approximate normality in a local sense. * Universit´ e Libre de Bruxelles, ECARES, ISRO and Department of Mathematics, Campus de la Plaine, C.P. 210 - Bd. du Triomphe, B-1050 Brussels, Belgium. Corresponding author: mbramati@ulb.ac.be † Not to be quoted without permission. Not to be reproduced 1