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 the form Yt = Xτ t α+g(Vt)+ t, t = 1, · · · , n, where {Vt} is a sequence of β–null recurrent Markov chains, {Xt} is a sequence of either strictly stationary or nonstationary regressors and { t} is a stationary sequence. We propose to estimate both α and g(·) semiparametrically. We then show that the proposed estimator of α 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 g(·) and the uniform consistency of the nonparametric estimator. The simulated example is given to show that our theory and method work well in practice. Keyword: Asymptotic normality; β–null recurrent Markov chain; consistency; kernel estimator; partially linear model