Serial Correlation Test And Empirical Likelihood In Semi-parametric Varying-coefficient Partially Linear Errors-in-variables Models

Semiparametric varying-coefficient partially linear model, which has recently been proposed as a new model, holds ample contents and extensive applications. It includes many usual parametric, semiparametric and nonparametric regression models. Linear regression model, partially linear regression model and varying-coefficient model are its degeneration cases. In comparison with parametric linear model or semiparametric partially linear models, semiparametric varying-coefficie -nt partially linear model allows more flexible function forms, and avoids many "curse of dimensionality" problems. In the context of classic regression models, we generally assume that the errors are mutually independent and homoscedastic. If the errors are not independent, we say that the models are serially correlated. If the error variances are not equal, we say that the models are heteroscedastic. If the errors are serially correlated, we will face with the following problems: the parameter estimate is not efficient; the significant test for variable is meaningless, and the same to other tests; the forecast for the model is inefficient; some important explanatory variables are omitted or the functional form is misspecified. When the model is heteroscedastic, we can face with the same problems as serial correlation. Therefore, it is necessary to test the presence of serial correlation and heteroscedasticity before statistic inference.The importance of testing for serial correlation in the error terms of a linear regression model has been recognized for many years. Some authors have investigated in detail this problem in the linear regression model. As far as I know, no one has tested for serial correlation in semiparametric and nonparametric regression models. Until recently some authors have begun testing for serial correlation in multivariate regression models, partial linear models, bounded nonparametric functions, semiparametric time series models and semiparametric panel data models. As regards semiparametric varying-coefficient partially linear models and semiparametric varying-coefficient partially linear time series models which are recently proposed, no one has investigated serial correlation test in these models. Hence, in this paper, we particularly investigate serial correlation test in these models, and obtain some useful results.In many applications, however, there often exist measurement errors because of the nature of measurement mechanism or man-made factor. So it is more practical to investigate the measurement error (errors-in-variables) models than the ordinary regression models. Estimators and properties on linear errors-in-variables models and partially linear errors-in-variables models have been extensively investigated, while the research on semiparametric varying-coefficient partially linear errors-in-variables model hasn't attached enough importance. To this day, the research on this model has mainly the paper of You and Chen(2006). Based on estimators and properties proposed by You and Chen(2006) in semiparametric varying-coefficient partially linear errors-in-variables model, in this paper we particularly investigate serial correlation test under the case where the covariates are measured with additive errors and the covariance of measurement error is known(i.e., the identifying condition 1). Then, we construct the estimators of the parameter components and nonparametric components, and obtain their asymptotic properties under the case where the covariates are measured with additive errors and the variance ratio of measurement error to regression equation error is assumed to be known(i.e., the identifying condition 2). Based on these estimators and properties, we investigate serial correlation test in the model.However, in this paper we don't investigate testing heteroscedasticity in the aforementioned models. But this is certainly worthy of effort for further research.Empirical likelihood, proposed by Owen(1988, 1990), is a nonparametric method of inference. Compared with other classic or modern statistic methods, empirical likelihood has many prominent merits. For example, the empirical likelihood ratio confidence region is range preserving and transformation respecting, and the shape and orientation of the resulting confidence regions are determined entirely by the data. Therefore, the empirical likelihood has aroused many statisticians' interest, and been applied to many fields and statistic models. In this paper we propose an empirical log-likelihood ratio to testing serial correlation in semiparametric varying-coefficient partially linear model and semiparametric varying-coefficient partially linear errors-in-variables model, and investigate empirical likelihood inference in semiparametric varying-coefficient partially linear errors-in-variables model under the case where the measurement error is assumed to be additive. Besides, we also consider the block empirical likelihood inference in the longitudinal semiparametric varying-coefficient partially linear model.This paper obtains the following five main results: the first result is that we introduce empirical likelihood to test serial correlation in semiparametric varying-coefficient partially linear models and propose the empirical log-likelihood ratio test. The second result is that we introduce empirical likelihood to test serial correlation in semiparametric varying-coefficient partially linear errors-in-variables models and obtain the nonparametric version of Wilks' theorem of the proposed empirical log-likelihood ratio statistics under the identifying condition 1 and 2, respectively. The third result is that we generalize the method proposed by Li and Hsiao(1998) to semiparametric varying-coefficient partially linear models, semiparametric varying-coefficient partially linear errors-in-variables model, semiparametric varying-coefficient partially linear time series models and semiparametric varying-coefficient partially linear panel data models. The fourth result is that we apply empirical likelihood to semiparametric varying-coefficient partially linear errors-in-variables models, enrich and develop the empirical likelihood theory. The fifth result is that we apply the block empirical likelihood to the longitudinal semiparametric varying-coefficient partially linear models, and obtain the chi-square limiting distribution of the proposed block empirical log-likelihood ratio.