This paper proposes a Bayesian generalized linear multilevel model with a pth-order autoregressive error process to analyze unbalanced binary time-series cross-sectional (TSCS) data. The model specification is motivated by the generic TSCS data structure and is intended to handle the associated inefficiency and endogeneity problems. It accommodates heterogeneity across units and between time periods in the form of random intercepts and random-effect coefficients. At the same time, its pth-order autoregressive error process, employed either by itself or in concert with other dynamic methods, adequately corrects serial correlation and improves statistical inference and forecasting. With a stationarity restriction on the error process, the model can also be used as a residual-based cointegration test on discrete TSCS data. This is especially valuable because cointegration testing on discrete TSCS data is methodologically challenging and rarely conducted in practice. To handle the estimation difficulties, I developed an efficient Markov chain Monte Carlo (MCMC) algorithm by orthogonalizing the error term with the Cholesky decomposition and adding an auxiliary variable. The parameter expansion method, that is, partial group move—multigrid Monte Carlo updating (PGM-MGMC), is employed to further improve MCMC mixing and speed up convergence. The paper also provides a computational scheme to approximate the Bayes's factor for the purposes of serial correlation diagnostics, lag order determination, and variable selection. Simulated and empirical examples are used to assess the model and techniques.