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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.

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Consistency and asymptotic normality are established for the highly applied quasimaximum likelihood estimator in the GARCH~1,1! model+ Contrary to existing literature we allow the parameters to be in the region where no stationary version of the process exists+ This has the important implication that the likelihood-based estimator for the GARCH parameters is consistent and asymptotically normal in the entire parameter region including both stationary and explosive behavior+ In particular, there is no "knife edge result like the unit root case" as hypothesized in Lumsdaine~1996, Econometrica 64, 575-596!+ with t ϭ 1, + + + , T and z t an independent and identically distributed~i+i+d+!~0,1! sequence+ As to initial values the analysis is conditional on the observed value y 0 , whereas the unobserved variance, h 0~u !, is parametrized by g, h 0~u ! ϭ g+ The parameter u of the GARCH model is therefore u ϭ~a, b, v, g! (3) with a, b, v, and g all positive+ Denote henceforth the positive true parameter values by u 0 ϭ~a 0 , b 0 , v 0 , g 0 !+ The GARCH model was introduced by Bollerslev~1986!, extending the autoregressive conditional heteroskedastic~ARCH! model of Engle~1982!+ Asymp-Anders Rahbek is grateful for support from the Danish Social Sciences Research Council, the Centre for Analytical Finance~CAF!, and the EU network DYNSTOCH+ Both authors thank the two anonymous referees and the editor for highly valuable and detailed comments that have, we believe, led to a much improved version of the paper, both in terms of the econometric theory

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