Testing Serial Correlation In Regression Model With Validation Data

Semi-parametric regression models,especially for partially linear model,are important statistical models that have rich theory and wide applications.Because it has the advantages of both parametric models' efficient and non-parameter models' robustness,and it avoids the curse of dimensionality in a certain extent.In many applications,the exact measurement of variable may be costly or time consuming,so a surrogate variable is observed,and we can observe a small part of the exact measurement data,which is called validation data.In regression models,we request that the residuals are independent identical distribution.Violating this assumption,we can have adverse consequences for the efficiency of estimators.There has been much work on the serial correlation test,but there has been no work about the testing serial correlation in regression model with the aid of validation data.So,this paper investigate the testing serial correlation in linear model and partially linear model with validation data.This paper consider the explanatory variable that has measurement error in linear model with validation data.Firstly,under the null hypothesis and with the aid of validation data,we use the non-parametric kernel method and the least squares method to estimate the unknown parametric.Then,under the null hypothesis,we apply the empirical likelihood methods to construct the testing statistics,and we prove that the testing statistics has an asymptotic chi-square distribution.Finally,we use R to simulate,and simulation results show that our method performs well.On the base of linear model research,we further consider the serial correlation test in partially linear model with validation data.Firstly,under the null hypothesis and with the aid of validation data,we apply the empirical likelihood methods to construct the testing statistics,and estimate the unknown parametric in the testing statistics.Then,we derive the asymptotic property of the testing statistics.Simulation results show that our method performs well both in size and power with finite same size.