Smoothed partially linear varying coefficient quantile regression with nonignorable missing response

In this paper, we propose a smoothed quantile regression estimator and variable selection procedure for partially linear varying coefficient models with nonignorable nonresponse. To avoid the computational problem caused by the non-smooth quantile loss function, we employ the kernel smoothing method. To address the identifiability issue, we use an instrument and estimate the parametric propensity function based on the generalized method of moments. Once the propensity is estimated, we construct the bias-corrected estimating equations utilizing the inverse probability weighting approach. Then, we apply the empirical likelihood method to obtain an unbiased estimator. The asymptotic properties of the proposed estimators are established for both the parametric and nonparametric parts. Meanwhile, variable selection is considered by using the SCAD penalty. The finite-sample performance of the estimators is studied through simulations, and a real-data example is also presented.