On testing for independence between generalized error models of several time series

TL;DR

This paper introduces new statistical tests for independence between generalized error models in time series, applicable to models with mixed distributions, and demonstrates their effectiveness through simulations and real data applications.

Contribution

It extends existing independence testing methods to generalized error models with mixed distributions, providing new empirical processes and tractable test statistics.

Findings

Tests have Gaussian asymptotic distributions independent of parameter estimates.

Proposed methods effectively detect dependence in financial and crime data.

Numerical experiments show good power of the tests.

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. We also propose statistics for testing independence between the generalized errors of these models, extending previous results of Duchesne, Ghoudi and Remillard (2012) obtained 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. Moebius transformations of the empirical processes are used to obtain tractable covariances. Several tests statistics are then proposed, based on Cramer-von Mises 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. ll developed methodologies are implemented in the CRAN package IndGenErrors.