  In semivarying coefficient models for longitudinal/clustered data, usually of
primary interest is usually the parametric component which involves unknown
constant coefficients. First, we study semiparametric efficiency bound for
estimation of the constant coefficients in a general setup. It can be achieved
by spline regression provided that the within-cluster covariance matrices are
all known, which is an unrealistic assumption in reality. Thus, we propose an
adaptive estimator of the constant coefficients when the covariance matrices
are unknown and depend only on the index random variable, such as time, and
when the link function is the identity function. After preliminary estimation,
based on working independence and both spline and local linear regression, we
estimate the covariance matrices by applying local linear regression to the
resulting residuals. Then we employ the covariance matrix estimates and spline
regression to obtain our final estimators of the constant coefficients. The
proposed estimator achieves the semiparametric efficiency bound under normality
assumption, and it has the smallest covariance matrix among a class of
estimators even when normality is violated. We also present results of
numerical studies. The simulation results demonstrate that our estimator is
superior to the one based on working independence. When applied to the CD4
count data, our method identifies an interesting structure that was not found
by previous analyses.

Methodology