In this paper we study time-varying coefficient models with time trend function and serially correlated errors to characterize nonlinear, nonstationary and trending phenomenon in time series. Compared with the Nadaraya-Watson method, the local linear approach is developed to estimate the time trend and coefficient functions. The consistency of the proposed estimators is obtained without any specification of the error distribution and the asymptotic normality of the proposed estimators is established under the alpha-mixing conditions. The explicit expressions of the asymptotic bias and variance are given for both estimators. The asymptotic bias is just in a regular nonparametric form but the asymptotic variance is shared by parametric estimators. Also, the asymptotic behaviors at both interior and boundary points are studied for both estimators and it shows that two estimators share the exact same asymptotic properties at the interior points but not at the boundaries. Moreover, proposed are a new bandwidth selector based on the nonparametric version of the Akaike information criterion, a consistent estimator of the asymptotic variance, and a simple nonparametric version of bootstrap (i.e. wild bootstrap) test for testing the misspecification and stationarity. Finally, we conduct some Monte Carlo experiments to examine the finite sample performances of the proposed modeling procedures and test.
Bandwidth selection Boundary effects Fixed design Functional coefficient models Local linear fitting Misspecification test