Independent random effects in generalized linear models induce an exchangeable correlation structure, but long sequences of counts or binomial observations typically show correlations decaying with increasing lag. This dissertation introduces models with autocorrelated random effects for a more appropriate, parameter driven analysis of discrete-valued time series data. We present a Monte Carlo EM algorithm with Gibbs sampling to jointly obtain maximum likelihood estimates of regression parameters and variance components. Marginal mean, variance and correlation properties of the conditionally specified models are derived for Poisson, negative binomial and binary/binomial random components. They are used for constructing goodness of fit tables and checking the appropriateness of the modeled correlation structure. Our models define a likelihood and hence estimation of the joint probability of two or more events is possible and used in predicting future responses. Also, all methods are flexible enough to allow for multiple gaps or missing observations in the observed time series. The approach is illustrated with the analysis of a cross-sectional study over 30 years, where only observations from 16 unequally spaced years are available, a time series of 168 monthly counts of polio infections and two long binary time series
