On testing for independence between generalized error models of several time series
Pith reviewed 2026-05-23 18:44 UTC · model grok-4.3
The pith
Independence tests for generalized error models of time series have Gaussian limits independent of parameter estimates.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Generalized innovations are defined for generalized error models having arbitrary innovation distributions. Families of empirical processes constructed from lagged generalized errors have joint asymptotic distributions that 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, allowing construction of several test statistics based on Cramer-von Mises functionals and dependence measures.
What carries the argument
Empirical processes from lagged generalized errors together with their Moebius transformations; these deliver the joint Gaussian limits that do not depend on parameter estimates.
If this is right
- The tests apply to stochastic volatility models and regime-switching models without extra regularity conditions.
- The Gaussian limits continue to hold when innovations are mixtures of continuous and discrete distributions.
- Cramer-von Mises statistics and graphical methods become available for detecting dependence at various lags.
- The methods cover both continuous financial returns and discrete crime count series.
Where Pith is reading between the lines
- The same empirical-process construction could be applied to test cross-dependence among more than two series.
- The graphical diagnostics might reveal lag-specific patterns not captured by scalar dependence measures.
- Because no additional regularity is required, the tests could be used in settings where standard parametric assumptions are hard to verify.
Load-bearing premise
The individual time series are correctly specified as generalized error models.
What would settle it
A Monte Carlo experiment that deliberately misspecifies the fitted generalized error models while keeping the innovations independent would show whether the test statistics still follow the claimed Gaussian limits.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines generalized innovations for generalized error models with arbitrary (mixed continuous-discrete) distributions, including stochastic volatility and regime-switching models. It extends Duchesne, Ghoudi and Rémillard (2012) by constructing families of empirical processes from lagged generalized errors and showing that their joint asymptotic distributions are Gaussian and independent of the estimated parameters. Möbius transformations are used to obtain tractable covariances, leading to test statistics based on Cramér-von Mises and dependence measures, with numerical experiments and applications to financial and crime data. The methods are implemented in the CRAN package IndGenErrors.
Significance. If the central asymptotic result holds, this work offers a significant extension for independence testing in time series models that accommodate discrete or mixed innovation distributions, broadening applicability beyond continuous cases. The parameter-free limiting distribution and the accompanying R package are notable strengths that enhance reproducibility and usability. The applications to real data demonstrate practical relevance.
major comments (2)
- [Asymptotic theory section (around the statement of joint asymptotics)] The claim that the joint limiting distributions are Gaussian and independent of estimated parameters for innovations with possible atoms is load-bearing. The tightness of the empirical process and the effect of the Möbius transformation need explicit verification when the innovation distribution function has discontinuities, as the locations of jumps depend on the parameter estimates and could introduce dependence in the limit.
- [Regularity conditions] The manuscript asserts that the extension from the 2012 results requires no additional regularity conditions. However, for mixed distributions, conditions ensuring that the atoms do not affect the convergence (e.g., no atoms at the relevant lagged points or bounded variation) may be necessary to maintain the parameter-free property.
minor comments (2)
- [Abstract] The abstract mentions 'numerical experiments are performed to assess the power' but does not specify the simulation design; this should be clarified in the main text for reproducibility.
- [Introduction or methods] Ensure that the definition of generalized innovations is clearly distinguished from standard residuals to avoid confusion with prior work.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, providing clarifications and indicating where revisions will be made to strengthen the presentation.
read point-by-point responses
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Referee: [Asymptotic theory section (around the statement of joint asymptotics)] The claim that the joint limiting distributions are Gaussian and independent of estimated parameters for innovations with possible atoms is load-bearing. The tightness of the empirical process and the effect of the Möbius transformation need explicit verification when the innovation distribution function has discontinuities, as the locations of jumps depend on the parameter estimates and could introduce dependence in the limit.
Authors: We agree that explicit verification of tightness and the impact of the Möbius transformation is warranted when the innovation distribution has atoms, as the referee notes. Our extension defines generalized innovations to handle mixed continuous-discrete distributions uniformly, building directly on the empirical process construction in Duchesne et al. (2012). The joint Gaussian limit independent of parameters follows from the same martingale arguments and the fact that parameter estimation affects the processes only through a term that vanishes in the limit. However, to make this fully rigorous for discontinuous cases, we will add an explicit verification of tightness (via the properties of the generalized empirical processes) and confirm that the Möbius transformation, being a fixed functional of the marginals, does not introduce parameter dependence in the limit. This will be incorporated as a remark or short appendix subsection. revision: partial
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Referee: [Regularity conditions] The manuscript asserts that the extension from the 2012 results requires no additional regularity conditions. However, for mixed distributions, conditions ensuring that the atoms do not affect the convergence (e.g., no atoms at the relevant lagged points or bounded variation) may be necessary to maintain the parameter-free property.
Authors: The manuscript states that the extension requires no additional regularity conditions because the generalized innovation definition already accommodates atoms without altering the convergence properties established in the 2012 paper. The atoms are mapped through the generalized errors in a manner that preserves the necessary measurability and bounded variation for the empirical processes. That said, we recognize the referee's point that a brief clarification would be helpful. We will revise the regularity conditions section to include a short paragraph explaining why the existing assumptions suffice for mixed distributions and why no extra conditions (such as restrictions on atom locations) are needed to retain the parameter-free limiting distribution. revision: partial
Circularity Check
Minor self-citation to overlapping authors' 2012 paper; central extension to mixed distributions presented as independent result
full rationale
The paper cites Duchesne, Ghoudi and Remillard (2012) for the base construction on stochastic volatility models and extends it to generalized error models with arbitrary (possibly mixed discrete-continuous) distributions. The abstract and reader's summary state that the joint asymptotic distributions are Gaussian and independent of estimated parameters, with the extension holding without additional regularity conditions. No quoted equations or steps in the provided text exhibit self-definition (e.g., a quantity defined in terms of itself), a fitted parameter renamed as a prediction, or any reduction of the limiting distribution to the inputs by construction. The self-citation supports the starting point but is not load-bearing for the claimed extension; the novel content concerns the broader innovation class and Möbius-transformed statistics. This matches the default expectation of no significant circularity (score 0-2) for a self-contained statistical derivation against external empirical-process benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The individual series follow correctly specified generalized error models whose innovations admit arbitrary distributions (continuous, discrete, or mixed).
- standard math Standard regularity conditions of empirical-process theory hold so that the joint limiting distribution remains Gaussian and parameter-free.
discussion (0)
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