  In this paper, we present a generalized estimating equations based estimation
approach and a variable selection procedure for single-index models when the
observed data are clustered. Unlike the case of independent observations,
bias-correction is necessary when general working correlation matrices are used
in the estimating equations. Our variable selection procedure based on
smooth-threshold estimating equations \citep{Ueki-2009} can automatically
eliminate irrelevant parameters by setting them as zeros and is computationally
simpler than alternative approaches based on shrinkage penalty. The resulting
estimator consistently identifies the significant variables in the index, even
when the working correlation matrix is misspecified. The asymptotic property of
the estimator is the same whether or not the nonzero parameters are known (in
both cases we use the same estimating equations), thus achieving the oracle
property in the sense of \cite{Fan-Li-2001}. The finite sample properties of
the estimator are illustrated by some simulation examples, as well as a real
data application.
