Heteroscedasticity Test Based On The Partially Linear Model In The Empirical Likelihood Of Missing Data

The test heteroscedasticity in regression models is an important practice in data analysis. In classic regression analysis, we generally assume that the errors are homoscedastic. Under this assumption, we can do some ordinary statistic inference. Violating this assumption, i.e. the model is heteroscedastic, can lead to many problems, such as the parameter estimate is not efficient, the significant test for variates is meaningless, the forecast for the models is inefficient. So it is important to test heteroscedasticity before statistic inference.Empirical likelihood, proposed by Owen, is a nonparametric method of statistic inference with sampling properties similar to those of bootstrap. The essence of this method is maximizing nonparametric likelihood ratio under some restrictions, so that the relationship between the interested parameters and the likelihood ratio is constructed. The empirical likelihood ratio has a limiting chi-squared distribution, leading to estimate and hypothesis test. Furthermore, empirical likelihood need not to estimate the covariance matrix, which is hard to be estimated.Partial linear model, first introduced by Engle et al., is an important statistic model since 1980's. I t contains not only linear part, but also nonparametric part. Partial linear model have been applied to many fields including industry, agriculture, economics, biometrics and finance. In practice, covariates may be missing due to various reasons, the research of missing data is important and meaningful. In the presence of missing data, the standard inference procedures cannot be applied directly, this in turn enlarges the difficulty of research.In this paper, we propose a diagnostic technique for checking heteroscedasticity based on empirical likelihood for the linear model and partial linear model with the covariate data missing at random. Based on empirical likelihood, we construct an empirical likelihood ratio test for heteroscedasticity. Also, under the null hypothesis and some mild conditions, show that our proposed test has an asymptotic chi-square distribution, Wilks theorem is derived. Simulation results reveal that the finite sample performance of our proposed test is satisfactory in both size and power.