Modelling discrete valued cross sectional time series with observation driven models

This paper develops computationally feasible methods for estimating random effects models in the context of regression modelling of multiple independent time series of discrete valued counts in which there is serial dependence. Given covariates, random effects and process history, the observed responses at each time in each series are independent and have an exponential family distribution. We develop maximum likelihood estimation of the mixed effects model using an observation driven generalized linear autoregressive moving average specification for the serial dependence in each series. The paper presents an easily implementable approach which uses existing single time series methods to handle the serial dependence structure in combination with adaptive Gaussian quadrature to approximate the integrals over the regression random effects required for the likelihood and its derivatives. The models and methods presented allow extension of existing mixed model procedures for count data by incorporating serial dependence which can differ in form and strength across the individual series. The structure of the model has some similarities to longitudinal data transition models with random effects. However, in contrast 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 cross sectional time series. The method is illustrated on time series of binary responses to musical features obtained from a panel of listeners.