Poisson Autoregression

Abstract

This paper considers geometric ergodicity and likelihood based inference for linear and nonlinear Poisson autoregressions. In the linear case the conditional mean is linked linearly to its past values as well as the observed values of the Poisson process. This also applies to the conditional variance, implying an interpretation as an integer valued GARCH process. In a nonlinear conditional Poisson model, the conditional mean is a nonlinear function of its past values and a nonlinear function of past observations. As a particular example an exponential 1 autoregressive Poisson model for time series is considered. Under geometric ergodicity the maximum likelihood estimators of the parameters are shown to be asymptotically Gaussian in the linear model. In addition we provide a consistent estimator of the asymptotic covariance, which is used in the simulations and the analysis of some transaction data. Our approach to verifying geometric ergodicity proceeds via Markov theory and irreducibility. Finding transparent conditions for proving ergodicity turns out to be a delicate problem in the original model formulation. This problem is circumvented by allowing a perturbation of the model. We show that as the perturbations can be chosen to be arbitrarily small, the differences between the perturbed and non-perturbed versions vanish as far as the asymptotic distribution of the parameter estimates is concerned.

Predicting Bid–Ask Spreads Using Long‐Memory Autoregressive Conditional Poisson Models

“…As shown by Ferland et al (2006), the series ¹S t º is covariance stationary as well as strictly stationary if P q iD1˛i C P p j D1ˇj < 1. Fokianos et al (2009) derive the ergodicity conditions for a covariance stationary ACP process in the case q D p D 1.…”

“…Ferland et al (2006), for instance, show that under common stationarity conditions all moments of a ACP(1,1) model exist-in contrast to corresponding GARCH and ACD specifications. Only recently, contributions of Fokianos et al (2009), Fokianos and Tjostheim (2011) and Neumann (2011, for example, established stationarity and ergodicity of simple ACP models. However, these results have been shown mostly for the low-order case q D p D 1.…”

“…However, these results have been shown mostly for the low-order case q D p D 1. Fokianos et al (2009) require the conditions of an approximation lemma to be fulfilled in order to prove that properties of a perturbed model carry over to the Poisson model (see Doukhan et al, 2012, for a short overview). Theoretical properties of a long-memory Poisson model are still widely unknown.…”

“…Though conceptually elegant, these approaches suffer from tedious estimation procedures which makes them intractable in extensive high-frequency applications. As an alternative, the more recent literature focuses rather on observationdriven models, such as the autoregressive conditional Poisson (ACP) model as introduced by Rydberg and Shephard (1999) and put forward by Heinen (2003), Fokianos et al (2009) andFerland et al (2006). In contrast to the aforementioned models, ACP models are straightforward to estimate and tractable even for a large number of observations.…”

PARX model for football match predictions

“…Our paper focuses on football match prediction using the Poisson autoregression with exogenous covariates (PARX) model introduced by Agosto, Cavaliere, Kristensen, and Rahbek (2016), which extends the Poisson autoregression (PAR) model originally proposed by Fokianos, Rahbek, and Tjostheim (2009) to include covariates in its specification. This model has been successfully used to predict corporate defaults and, within the framework of football betting, it is particularly useful, since the intensity of the goals scored by a team is characterized by an autoregressive persistence that the PARX model is able to account for.…”