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.
| Originalsprog | Engelsk |
|---|---|
| Udgivelsessted | Aarhus |
| Udgiver | Institut for Økonomi, Århus Universitet |
| Antal sider | 34 |
| Status | Udgivet - 2009 |