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

1. Problem Statement

The paper addresses the critical challenge of testing for conditional independence between multiple time series, particularly when the series exhibit arbitrary distributions (mixtures of continuous and discrete, or purely discrete).

• Limitations of Existing Methods: Previous literature (e.g., Duchesne et al., 2012) focused primarily on continuous time series, specifically stochastic volatility (SV) models. Traditional methods relying on cross-correlations often fail to detect non-linear or non-Gaussian dependencies. Furthermore, standard copula-based approaches struggle with discrete or mixed distributions because the probability integral transform (PIT) does not yield independent uniform variables when margins are discontinuous.
• The Gap: There was a lack of a unified framework to test for independence in "generalized error models" that accommodate zero-inflated regimes, count data (Poisson), and regime-switching behaviors without assuming continuous margins.

2. Methodology

The authors propose a two-stage methodology based on generalized innovations and empirical processes.

A. Generalized Innovations (The "Randomization" Technique)

To handle arbitrary distributions (discrete, continuous, or mixed), the authors define generalized errors (or innovations) $U_t$ using a randomization technique (Brockwell, 2007; Nešlehová, 2007).

• Let $G_t(y) = P(X_t \le y | \mathcal{F}_{t-1})$ be the conditional cumulative distribution function (CDF).
• Let $V_t \sim U(0,1)$ be an independent uniform random variable.
• The generalized innovation is defined as:
  $$U_t = G_t(X_t-) + V_t \Delta G_t(X_t)$$
  where $\Delta G_t(y) = G_t(y) - G_t(y-)$ is the jump size at $y$.
• Result: $U_t$ is uniformly distributed on $(0,1)$ and, under the null hypothesis of conditional independence, the vector of innovations $\mathbf{U}_t$ follows the independence copula $\Pi$. This transformation works for both continuous and discrete data.

B. Multivariate Copula Framework

The authors model the dependence between $d$ time series using a conditional Sklar's equation:
$$P(X_{1t} \le y_1, \dots, X_{dt} \le y_d | \mathcal{F}_{t-1}) = C_t^{\star} \{G_{1t}(y_1), \dots, G_{dt}(y_d)\}$$
where $C_t^{\star}$ is the multilinear copula. The null hypothesis ($H_0$) is that $C_t^{\star} = \Pi$ (independence copula) for all $t$.

C. Empirical Processes and Asymptotic Theory

The testing procedure relies on lagged empirical processes constructed from pseudo-observations (ranks) of the generalized errors.

1. Pseudo-observations: $e_{n,jt} = H_{\hat{\theta},jt}(X_{jt}, V_{jt})$, where $\hat{\theta}$ are estimated parameters.
2. Empirical Processes: The authors define processes $K_{n,\ell}$ and $C_{n,\ell}$ based on lagged pseudo-observations.
3. Möbius Transformations: To simplify covariance structures and improve power, the authors apply Möbius transformations to the empirical processes, focusing on subsets of variables $A \subset \{1, \dots, d\}$ with $|A| > 1$.
4. Asymptotic Results:
   • Theorem 1 & Corollary 1: Under the null hypothesis and specific regularity assumptions (A0–A6), the empirical processes converge weakly to Gaussian processes (specifically, Brownian bridges).
   • Key Finding: The limiting distribution of the test statistics is parameter-free and independent of the estimated parameters $\hat{\theta}$ and the randomization variables $V_t$. This allows for the use of standard critical values without re-estimation.

D. Test Statistics

Several test statistics are proposed based on the transformed empirical processes:

1. Cramér–von Mises (CvM) Statistics ($S_{n,A,\ell}$): Integrated squared deviations of the empirical copula process.
2. Generalized Cross-Correlations ($\hat{r}_{n,A,\ell}$): Analogous to Pearson correlations but applied to the transformed ranks.
3. Dependence Measures: Extensions of Spearman's $\rho$, van der Waerden's coefficient, and Savage's coefficient.
4. Combined Statistics ($W_n, F_n, H_n$): To increase power against a wide range of alternatives, the authors combine statistics across different lags and subsets using methods similar to Littell and Folks (1973) and Duchesne et al. (2012).

3. Key Contributions

1. Extension to Arbitrary Distributions: The paper generalizes independence testing from continuous stochastic volatility models to generalized error models that include discrete, mixed, and zero-inflated distributions.
2. Unified Framework: It provides a single theoretical framework (using randomization and multilinear copulas) that handles both continuous and discrete time series simultaneously.
3. Theoretical Rigor: The authors prove that the asymptotic distributions of the proposed statistics are invariant to parameter estimation and the randomization step, a non-trivial result for discrete data.
4. Handling Discrete Data: By introducing the randomization vector $V_t$, the method effectively "smooths" discrete jumps, allowing the application of continuous copula theory to discrete data while maintaining distribution-free properties under $H_0$.
5. Software Implementation: All methodologies are implemented in the CRAN package IndGenErrors, making the complex methodology accessible to practitioners.

4. Results

Numerical Experiments (Simulation Study)

• Data Generating Processes (DGPs): The authors simulated data using Gaussian Hidden Markov Models (HMM), Poisson HMMs, and various copulas (Frank, Gaussian, Clayton, Tent Map) with varying lag structures.
• Size and Power:
  • The tests maintain correct empirical levels (close to the nominal 5%) even for small sample sizes ($n=100$).
  • The Cramér–von Mises statistic ($W_n$) generally demonstrated the best or near-best power across various dependence structures.
  • Discrete vs. Continuous: When one series was discrete, there was a slight loss in power compared to the continuous case due to the randomization step. However, this loss was mitigated by averaging over multiple randomizations ($M > 1$), which restored power to levels comparable to continuous models (though at the cost of losing the distribution-free property, requiring bootstrapping).
  • Marginal Sensitivity: Tests based on $W_n$ and $F_n$ were robust to the choice of marginal distributions (Gaussian, Exponential, Pareto), whereas cross-correlation based tests ($H_n$) were more sensitive to non-Gaussian margins.

Real Data Applications

1. Financial Data (NASDAQ):
   • Data: Log-returns of Apple, Intel, and Hewlett-Packard.
   • Modeling: Used AR-Gaussian HMMs (with zero-inflation for Intel due to 0 returns).
   • Finding: Without covariates, the series were dependent. However, after including the NASDAQ index as a covariate, the generalized errors became conditionally independent. This confirms that the NASDAQ index explains the cross-dependence among these stocks.
2. Crime Data (Pittsburgh):
   • Data: Counts of Robbery (ROB) and Crime Mischief (CMIS).
   • Modeling: Used INGARCH(1,1) and AR-Poisson HMMs.
   • Finding: The raw series showed no significant dependence. After fitting individual models, the generalized errors remained independent, suggesting the series are conditionally independent given their own past.
   • Motor Vehicle Theft (MVT): A dependence was found between Robbery and MVT. After fitting INGARCH models to both, the dependence vanished, indicating the dynamic models captured the interdependence.

5. Significance

This paper represents a significant advancement in time series econometrics and statistics by:

• Bridging the Gap: It unifies the treatment of continuous and discrete time series, a common real-world scenario often ignored in favor of purely continuous approximations.
• Diagnostic Power: It provides a robust diagnostic tool to determine if a multivariate time series model (including covariates) has successfully captured all cross-dependencies. If the test rejects independence, it suggests the need for a more complex multivariate structure or additional covariates.
• Practical Utility: The availability of the IndGenErrors package allows researchers to apply these sophisticated tests to financial, environmental, and social science data involving mixed data types (e.g., stock returns with zero trades, daily crime counts).

In summary, the authors successfully extended the theory of empirical processes for independence testing to the realm of generalized error models, providing a rigorous, flexible, and implementable solution for modern multivariate time series analysis.