Semiparametric Regression Estimation in Null Recurrent Nonlinear Time Series

Estimation theory in a nonstationary environment has been very popular in recent years. Existing studies focus on nonstationarity in parametric linear, parametric nonlinear and nonparametric nonlinear models. In this paper, we consider a partially linear model of nonstationary time series, in which both parametric and nonparametric regressors are beta-null recurrent Markov chains. We propose to estimate both the unknown parameters and functions semiparametrically. We then show that the proposed estimator of the parameter vector is still asymptotically normal with the same rate as for the case of stationary time series. We also establish the asymptotic normality for the nonparametric estimator of the function and the uniform consistency of the nonparametric estimator. The simulated example is given to show that our theory and method work well in practice