On Testing for Independence Between Generalized Error Models of Several Time Series

ABSTRACT We define generalized innovations associated with generalized error models having arbitrary distributions, that is, distributions that can be mixtures of continuous and discrete distributions. These models include stochastic volatility models and regime‐switching models with possibly zero‐inflated regimes. The main novelty of this article is to be able to test for conditional independence between several time series with arbitrary distributions, extending previous results of Duchesne, Ghoudi, and Rémillard (2012), obtained only for stochastic volatility models. We define families of empirical processes constructed from lagged generalized errors, and we show that their joint asymptotic distributions are Gaussian and independent of the estimated parameters of the individual time series. Möbius transformations of the empirical processes are used to obtain tractable covariances. Several tests statistics are then proposed, based on Cramér–von Mises type statistics and dependence measures, as well as graphical methods to visualize the dependence. In addition, numerical experiments are performed to assess the power of the proposed tests. Finally, to show the usefulness of our methodologies, examples of applications for financial data and crime data are given to cover both discrete and continuous cases. All developed methodologies are implemented in the CRAN package IndGenErrors (Ghoudi et al., 2025).