  Clustered binary data with a large number of covariates have become
increasingly common in many scientific disciplines. This paper develops an
asymptotic theory for generalized estimating equations (GEE) analysis of
clustered binary data when the number of covariates grows to infinity with the
number of clusters. In this "large $n$, diverging $p$" framework, we provide
appropriate regularity conditions and establish the existence, consistency and
asymptotic normality of the GEE estimator. Furthermore, we prove that the
sandwich variance formula remains valid. Even when the working correlation
matrix is misspecified, the use of the sandwich variance formula leads to an
asymptotically valid confidence interval and Wald test for an estimable linear
combination of the unknown parameters. The accuracy of the asymptotic
approximation is examined via numerical simulations. We also discuss the
"diverging $p$" asymptotic theory for general GEE. The results in this paper
extend the recent elegant work of Xie and Yang [Ann. Statist. 31 (2003)
310--347] and Balan and Schiopu-Kratina [Ann. Statist. 32 (2005) 522--541] in
the "fixed $p$" setting.
