  In the analysis of cluster data, the regression coefficients are frequently
assumed to be the same across all clusters. This hampers the ability to study
the varying impacts of factors on each cluster. In this paper, a semiparametric
model is introduced to account for varying impacts of factors over clusters by
using cluster-level covariates. It achieves the parsimony of parametrization
and allows the explorations of nonlinear interactions. The random effect in the
semiparametric model also accounts for within-cluster correlation. Local,
linear-based estimation procedure is proposed for estimating functional
coefficients, residual variance and within-cluster correlation matrix. The
asymptotic properties of the proposed estimators are established, and the
method for constructing simultaneous confidence bands are proposed and studied.
In addition, relevant hypothesis testing problems are addressed. Simulation
studies are carried out to demonstrate the methodological power of the proposed
methods in the finite sample. The proposed model and methods are used to
analyse the second birth interval in Bangladesh, leading to some interesting
findings.
