Polynomial Spline Estimation and Inference 
for Varying Coefficient Models with Longitudinal Data

Jianhua Z. Huang, Colin O. Wu and  Lan Zhou

We consider nonparametric estimation of coefficient functions in
a varying coefficient model of the form 
$Y_{ij}=X_i^T(t_{ij}) \bbeta(t_{ij}) + \epsilon_i(t_{ij})$ based
on longitudinal observations $\{(Y_{ij}, X_i(t_{ij}), t_{ij}), 
\ i=1, \ldots, n, \ j=1, \ldots, n_i\}$, where $t_{ij}$ and 
$n_i$ are the time of the $j$th measurement and the number of 
repeated measurements for the $i$th subject, and $Y_{ij}$ and 
$X_i(t_{ij})=(X_{i0}(t_{ij}), \ldots, X_{iL}(t_{ij}))^T$ for 
$L \geq 0$ are the $i$th subject's observed outcome and covariates 
at $t_{ij}$. We approximate each coefficient function by a polynomial
spline and employ the least squares method to do the estimation. 
An asymptotic theory for the resulting estimates is established 
including consistency, rate of convergence and asymptotic distribution. 
The asymptotic distribution results are used as a guideline to
construct approximate confidence intervals and confidence bands
for components of $\bbeta(t)$. We also propose a polynomial spline 
estimate of the covariance structure of $\epsilon(t)$, which is used 
to estimate the variance of the spline estimate $\hat\bbeta(t)$.
A real data example in epidemiology and a simulation study are
used to demonstrate our methods.