This paper is concerned with regression modelling of multiple independent time series of counts in which there is serial dependence in each series and random effects terms across series. Given covariates, the observed response has an exponential family distribution. The talk will first briefly review current work on developing and comparing score type tests for detecting serial dependence in discrete values time series and use these to screen for correlation in the multiple series. We present a simple and easily implementable approach to estimation of the mixed model based on adaptive Gaussian Quadrature and the Laplace approximation used in conjunction with existing software for fitting observation driven models for univariate discrete valued time series. This approach allows extension of existing mixed model procedures for count data to incoporate serial dependence. The structure of the model has similarities to longitudinal data transition models with random effects. However, in constrast to that setting, where there are many cases and few to moderate observations per case, the time series setting has many observations per series and a few to moderate number of series. Application is made to assessing the impact on single vehicle nighttime fatalities of lowering the legal BAC limit for drivers in 17 US states. This example is typical of others that are encountered in public health policy. In examples such as these there are multiple time series in which shared regression effects need to be tested for equality and there is the potential for serial dependence. A second example, analysing binary responses by a panel of listeners to musical features is also presented.