## 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, making an interpretation as an integer

valued GARCH process possible. 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

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 their asymptotic covariance matrix. 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.

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, making an interpretation as an integer

valued GARCH process possible. 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

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 their asymptotic covariance matrix. 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.

Original language | English |
---|---|

Place of publication | Aarhus |

Publisher | Institut for Økonomi, Århus Universitet |

Number of pages | 34 |

Publication status | Published - 2009 |

## Keywords

- asymptotic theory, count data, generalized linear models, geometric ergodicity, integer GARCH, likelihood, noncanonical link function, observation driven models, Poisson regression, ø-irreducibility