Linear Single-index Model Diagnosis Technology

Partial linear single-index model is an important semi-parametric statistic model. Thismodel includes many models, such as partial linear model, single-index model,varying-coefficient model, the classical linear model and so on. So that the model hasbeen applied to many fields. Partial linear single-index model overcomes the “dimensiondisaster” problem of the nonparametric model, at the same time, the model combines theparametric model and nonparametric model together organically, such that the model’sexplanation ablitily can be enhanced. The research of partial linear single-index model aremainly concentrated in the model’s estimation of unknown parameters and the unknownconnections function. However serial correlation test and heteroscedastic test for the modelare less been involved.Testing serial correlation and heteroscedasticity in residuals for regression models areimportant practice in econometric data analysis. We request the fitted residuals are whitenoise for a good fitted models, that is to say the residuals don’t include any information ofthe models. So the assumptions of independent and homoscedasticity are some basicassumptions. Under those assumptions, we can do some ordinary statistic inference such asparameter estimate, hypothesis test and further to forecast. If the error is serially correlated,the model of the statistical inference, such as the parameter estimation, etc will lose theireffectiveness; If the model exists very intense serial correlation, it is very likely the modelhas been misused.So, it’s necessary to diagnose heteroscedasticity and serial correlationbefore statistic inference.In this paper, we mainly investigate the test for serial correlation in partial linearsingle-index model. We use two methods to test serial correlation in partial linearsingle-index model, firstly extend the methodV_T,P to partial linear single-index model,and get the asymptotic distribution of theV_T,Ptest statistics; the second method isempirical likelihood ratio test, established empirical likelihood ratio test statistics and thenderive it’s asymptotic distribution under null hypothesis, built a nonparametric version ofWilk’s theorem. We also through simulation investigate the finite sample properties of ourstatistics. The simulation results show that the both tests have good effect and anapplication to a real dataset is illustrated.