This article proposes an autoregressive model for time series of counts with non-stationary means, variances and covariances as functions of certain time-dependant covariates. For the estimation of the regression, overdispersion and correlation index parameters, a conditional generalized quasilikelihood (CGQL) approach is developed under the assumption that the count responses marginally satisfy the first two moments of a negative binomial distribution. Thus this CGQL approach avoids the use of the likelihood or so-called partial likelihood of the data which are known to be extremely complicated in the present non-stationary time series set-up. It is shown through an extensive simulation study that the proposed CGQL approach performs very well in estimating the parameters of the model. This is also shown that the CGQL approach performs better than an existing GQL approach, especially for the estimation of the overdispersion parameter of the model. Copyright 2008 The Authors
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