In this thesis, we present an expanded account of generalized linear models for time series data based on an article by I.Funatogawa, T. Funatogawa and Y. Ohashi (2007). In many studies, repeated assessments of a response are often performed at various time points from each individual under study. This type of data is referred to as longitudinal data. For the analysis of a continuous response, a popular approach is to adopt linear mixed effect models. This thesis investigates the existence of time effects in longitudinal data which create some kind of correlations between errors. The analysis of correlated data arising from repeated measurements when the measurements are assumed to be multivariate normal has been studied extensively. However, the normality assumption may not always be reasonable; for example, different methodology must be used in the data analysis when the responses are discrete and correlated. Generalized linear mixed model is an increasingly popular choice for modeling of this type of data. A number of methods are currently available for fitting a generalized linear mixed model including generalized estimating equations (GEEs), maximum likelihood and Monte-Carlo Markov-chain approaches in a Bayesian perspective. GEEs provide a practical method with reasonable statistical efficiency to analyze correlated data. We also take into account the application of suitable assumptions for error structure in order to obtain correct results in fitting various models; such that in modeling of both binary and count time-series data. Next, we describe the autoregressive linear mixed effects models, where the response in assumed to be a linear function of covariates and a lagged response variable. In addition, we consider an extension of the autoregressive models in two aspects, random effects and the error structure. Furthermore, various kinds of structures for error components and related covariates matrices are investigated. We also analyze some real data sets in order to highlight the important issues. Keywords: Dynamic Longitudinal data, Generalized estimating equations, Generalized linear models, Maximum likelihood, Random effects, Variance components.