The joint distribution of repeated binary observations is multinomial, and can be specified using a representation first suggested by Bahadur (1961, in Studies in Item Analysis and Prediction,158-168. Stanford, California: Stanford University Press), and later by Cox (1972, Applied Statistics 21, 13-120). Using the Bahadur representation, the marginal probabilities of success can be related to a set of covariates using the logistic link function, or any other suitable link function. Besides the parameters of the marginal regression model, we may also have interest in the probability of success on any of the repeated measures. For example, in the Six Cities study, a longitudinal study of the health effects of air pollution, we have interest in both the marginal probability of a child wheezing at age t (t = 10, 11, 12), and the union probability of wheezing at any of the three ages. This "union" probability can be specified in terms of the joint probabilities and the second higher-order correlations. We discuss several methods of estimating the parameters of the Bahadur model.