Empirical likelihood of varying coefficient errors-in-variables models with longitudinal data

In this paper, we investigate the empirical likelihood inferences of varying coefficient errors-in-variables models with longitudinal data. The naive empirical log-likelihood ratios for the time-varying coefficient function based on the global and local variance structures are introduced. The corresponding maximum empirical likelihood estimators of the time-varying coefficients are derived, and their asymptotic properties are established. Wilks' phenomenon of the naive empirical log-likelihood ratio, which ignores the within subject correlation, is proven through the employment of undersmoothing. To avoid the undersmoothing, we recommend a residual-adjust empirical log-likelihood ratio and prove that its asymptotic distribution is standard chi-squared. Thus, this result can be used to construct the confidence regions of the time-varying coefficients. We also establish the asymptotic distribution theory for the corresponding residual-adjust maximum empirical likelihood estimator and find it to be unbiased even when an optimal bandwidth is used. Furthermore, we consider the construction of the pointwise confidence interval for a component of the time-varying coefficients and provide the simulation studies to assess the finite sample performance, while we conduct a real example to illustrate the proposed method.