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

Kilani Ghoudi, Bouchra R. Nasri, Bruno N. Remillard

TL;DR

This paper develops a general, copula-based framework to test independence between generalized error models across multiple time series with arbitrary margins, including discrete and mixed distributions. It constructs generalized innovations and lagged empirical processes, derives Gaussian limits that are invariant to parameter estimation, and employs Möbius transforms to obtain tractable covariances. A suite of test statistics (Cramér--von Mises, generalized cross-correlations, and other copula-based measures) are extended and combined, with two- and three-series variants; the methods are shown to perform well in finite samples and are implemented in the R package IndGenErrors. Applications to NASDAQ financial data and Pittsburgh crime data demonstrate practical utility for diagnosing residual dependencies and validating conditional independence after modeling, highlighting the approach’s relevance for regime-switching and stochastic-volatility-like settings.

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.

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

TL;DR

This paper develops a general, copula-based framework to test independence between generalized error models across multiple time series with arbitrary margins, including discrete and mixed distributions. It constructs generalized innovations and lagged empirical processes, derives Gaussian limits that are invariant to parameter estimation, and employs Möbius transforms to obtain tractable covariances. A suite of test statistics (Cramér--von Mises, generalized cross-correlations, and other copula-based measures) are extended and combined, with two- and three-series variants; the methods are shown to perform well in finite samples and are implemented in the R package IndGenErrors. Applications to NASDAQ financial data and Pittsburgh crime data demonstrate practical utility for diagnosing residual dependencies and validating conditional independence after modeling, highlighting the approach’s relevance for regime-switching and stochastic-volatility-like settings.

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.

Paper Structure

This paper contains 31 sections, 11 theorems, 93 equations, 5 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Multivariate time series models
  3. Empirical processes for testing independence
  4. Möbius transforms
  5. Relationship with Kheifets/Velasco:2017
  6. Test statistics
  7. Cramér--von Mises test statistics
  8. Generalized cross-correlations
  9. Other dependence measures
  10. Combining test statistics
  11. Numerical experiments
  12. Test of independence between the generalized innovations of two series
  13. Test of independence between the generalized innovations of three time series
  14. Applications
  15. Application for financial data from the NASDAQ market
  16. ...and 16 more sections

Key Result

Theorem 1

Let $L$ be a finite subset of $\mathbb{Z}^d$. Then, under the null hypothesis of independence and Assumptions A0--A6, $\left\{(\beta_{n,{\boldsymbol\ell}},\alpha_{n,{\boldsymbol\ell}},\mathbb{K}_{n,{\boldsymbol\ell}}); {\boldsymbol\ell}\in L\right\}$ converge jointly in $\mathcal{D}[0,1]^d \times \m where ${\boldsymbol\Gamma}_j$ is defined in Assumption A3. In particular, for any $j\in\{1,\ldots,d

Figures (5)

  • Figure 1: Dependogram of Cramér-von Mises statistics for the Apple, Intel and Hewlett-Packard returns.
  • Figure 2: Dependogram of Cramér-von Mises statistics for the generalized errors of Apple, Intel and Hewlett-Packard AR-Gaussian HMM models with no covariate.
  • Figure 3: Dependogram of Cramér-von Mises statistics for the generalized errors of Apple, Intel and Hewlett-Packard AR-Gaussian HMM models, including NASDAQ as a covariate.
  • Figure 4: Dependograms for ROB and CMIS without fitting (left panel) and with INGARCH fitting (right panel).
  • Figure 5: Dependograms for ROB and MVT without fitting on the left panel and with INGARCH fitting on the right panel.

Theorems & Definitions (17)

  • Example 1
  • Remark 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Remark 3
  • Proposition 1
  • ...and 7 more