This article considers the modeling of count data time series with a finite range having extra-binomial variation. We propose a beta-binomial autoregressive model using the concept of random coefficient thinning. We discuss the stationarity conditions, derive the moments and autocovariance function and consider approaches for parameter estimation. Furthermore, we develop two new tests for detecting extra-binomial variation, and we derive the asymptotic distributions of the test statistics under the null hypothesis of a binomial autoregressive model. The size and power performance of the two tests are analyzed under various alternatives taken from a beta-binomial autoregressive model with Monte Carlo experiments. The article ends with a real-data example about the Harmonised Index of Consumer Prices of the European Union. Copyright © 2013 John Wiley & Sons, Ltd.