Efficient estimation in a semiparametric longitudinal data model with unknown time-varying AR error structure

We explore a time-dependent auto-regressive error structure to model the within-subject dependence among longitudinal observations in partially linear varying coefficient model, where the auto-regressive order is completely unspecified. We propose a two-stage unified least squares approach to estimate both the parametric and nonparametric components. By incorporating polynomial spline approximations and penalized loss function, we identify the significant auto-regressive order in the error component and investigate the penalized estimators and their oracle property, based on which more efficient estimates are derived. Subsequently, we establish the asymptotic normality of the proposed estimators and the consistency of selecting auto-regressive orders. Finally, the efficacy of the proposed methodology is demonstrated through Monte Carlo simulations as well as its application to the analysis of meteorological data.