Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear EV Models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = Xτβ+ Zτα(T) +ε,ξ= X +ηwith the identifying condition E[(ε,ητ)τ] = 0, Cov[(ε,ητ)η] =σ2Ip+1. The estimators of interested regression parametersβ, and the model error varianceσ2, as well as the nonparametric componentsα(T), are constructed. Under some regular conditions, we show that the estimators of the unknown vectorβand the unknown parameterσ2 are strongly consistent and asymptotically normal and that the estimator ofα(T) achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.