Consistent model check of errors-in-variables varying-coefficient model with auxiliary variable

In this paper, we consider the adequacy check of the varying-coefficient model when covariates are measured with error and some auxiliary variable is available. With the help of auxiliary variable, we calibrate the measurement error and obtain an estimator of the unobservable true variable. The empirical-process-based test is built by applying the calibrated estimator of the model error. The asymptotic properties of the proposed test are rigorously investigated under the null hypothesis, local and global alternatives. It is shown that the proposed test is consistent and has good properties of power. We illustrate that the naive method cannot control Type I error and loses effect completely. But the proposed calibrated method performs well in terms of the empirical sizes close to the test level and high empirical powers. Simulation studies and two real data analyses are conducted to demonstrate the performance of the proposed approach.