[Paper Review] Stationarity and Moment Properties of some Multivariate Count Autoregressions

Zinsou Max Debaly, Lionel Truquet|arXiv (Cornell University)|Sep 25, 2019

Stochastic processes and statistical mechanics9 references4 citations

TL;DR

This paper establishes a unified framework for analyzing stationarity and moment properties of univariate and multivariate count autoregressive processes using contraction and stability techniques for iterated random maps. It improves existing results by deriving optimal stationarity conditions and proving the existence of exponential moments for INGARCH and multivariate extensions, which are critical for regularized estimation in high-dimensional time series models.

ABSTRACT

We study stationarity and moments properties of some count time series models from contraction and stability properties of iterated random maps. Both univariate and multivariate processes are considered, including the recent multivariate count time series models introduced recently by Doukhan et al. (2017). We improve many existing results by providing optimal stationarity conditions or conditions ensuring existence of some exponential moments.

Motivation & Objective

• To develop a general, unified approach for studying stationarity and moment properties of univariate and multivariate count autoregressive processes.
• To overcome limitations of standard Markov chain techniques, which fail due to lack of irreducibility in the state space.
• To extend contraction-based methods for iterated random maps to higher-order and multivariate models, particularly for INGARCH-type processes.
• To establish sufficient conditions ensuring existence of exponential moments, which are essential for consistency of regularized estimators like LASSO in high-dimensional settings.
• To improve upon existing results in Doukhan et al. (2017) by providing tighter and more general stationarity and moment conditions.

Proposed method

• Adapting the contraction-on-average framework of Wu and Shao (2004) to higher-order autoregressive processes via backward iterations of random maps.
• Using measurable functions $ F_s(x) $ to define the evolution of the state $ X_t $, with $ X_{t+1} = F_{\epsilon_{t+1}}(X_t) $, where $ \epsilon_t $ are i.i.d. innovations.
• Applying moment and contraction conditions: $ \mathbb{E}[|f_0(x) - f_0(y)|] \leq C|x-y| $ and $ \mathbb{E}[|f_1^m(x) - f_1^m(y)|] \leq \kappa|x-y| $ with $ \kappa < 1 $, to ensure ergodicity and stationarity.
• Extending the framework to $ q $-th order multivariate autoregressions by defining companion matrices and using spectral radius conditions on coefficient matrices.
• Employing vectorized forms of coefficient matrices $ A_j, B_j $ and applying matrix norms to derive conditions on spectral radius less than 1 for stability.
• Using moment bounds for Poisson random variables via Sterling numbers of the second kind to control tail behavior and derive exponential moment conditions.

Experimental results

Research questions

• RQ1 What conditions ensure the existence of a stationary solution for multivariate count autoregressive processes using iterated random maps?
• RQ2 How can contraction techniques be generalized to higher-order and multivariate autoregressive models of count data?
• RQ3 Under what conditions do these processes possess exponential moments, and why is this important for high-dimensional inference?
• RQ4 Can the results from univariate INGARCH models be extended and improved in a multivariate setting using a unified framework?
• RQ5 How do the proposed conditions compare to existing ones in Doukhan et al. (2017), particularly in terms of tightness and applicability?

Key findings

• The paper establishes a general sufficient condition for stationarity of multivariate count autoregressions based on contraction of iterated random maps, applicable to both univariate and multivariate cases.
• It proves that the spectral radius of the matrix $ \sum_{j=1}^q \begin{pmatrix} |B_j|_{\text{vec}} & |A_j|_{\text{vec}} \\ |B_j|_{\text{vec}} & |A_j|_{\text{vec}} \end{pmatrix} $ less than 1 ensures stationarity and moment existence.
• The existence of exponential moments is established under a condition involving $ |M|_{\infty} < 1 $, where $ M = \sum_{j=1}^q (|B_j|_{\text{vec}} + |A_j|_{\text{vec}}) $, which is crucial for high-dimensional estimation.
• The authors improve upon results in Doukhan et al. (2017) by providing tighter and more general stationarity conditions for multivariate INGARCH processes.
• For Poisson-distributed innovations, the paper derives a uniform moment bound: $ \|X_\lambda\|_r \leq (1+\delta)\lambda + b_{r,\delta} $, independent of $ \lambda $, which supports exponential moment analysis.
• The framework allows for a clean, unified treatment of both univariate and multivariate count processes, subsuming standard INGARCH and INAR models as special cases.