Empirical likelihood for a varying coefficient model with longitudinal data

In this article local empirical likelihood-based inference for a varying coefficient model with longitudinal data is investigated. First, we show that the naive empirical likelihood ratio is asymptotically standard chi-squared when undersmoothing is employed. The ratio is self-scale invariant and the plug-in estimate of the limiting variance is not needed. Second, to enhance the performance of the ratio, mean-corrected and residual-adjusted empirical likelihood ratios are recommended. The merit of these two bias corrections is that without undersmoothing, both also have standard chi-squared limits. Third, a maximum empirical likelihood estimator (MELE) of the time-varying coefficient is defined, the asymptotic equivalence to the weighted least-squares estimator (WLSE) is provided, and the asymptotic normality is shown. By the empirical likelihood ratios and the normal approximation of the MELE/WLSE, the confidence regions of the time-varying coefficients are constructed. Fourth, when some components are of particular interest, we suggest using mean-corrected and residual-adjusted partial empirical likelihood ratios to construct the confidence regions/intervals. In addition, we also consider the construction of the simultaneous and bootstrap confidence bands. A simulation study is undertaken to compare the empirical likelihood, the normal approximation, and the bootstrap methods in terms of coverage accuracies and average areas/widths of confidence regions/bands. An example in epidemiology is used for illustration.