Estimation in semi-parametric regression with non-stationary regressors

Abstract

In this paper, we consider a partially linear model of the form $Y_t = X_t^τθ_0 + g(V_t) + ϵ_t, t = 1, …, n$, where $\{V_t\}$ is a $β$ null recurrent Markov chain, {$X_t$} is a sequence of either strictly stationary or non-stationary regressors and $\{ϵ_t\}$ is a stationary sequence. We propose to estimate both $θ_0$ and $g(⋅)$ by a semi-parametric least-squares (SLS) estimation method. Under certain conditions, we then show that the proposed SLS estimator of $θ_0$ is still asymptotically normal with the same rate as for the case of stationary time series. In addition, we also establish an asymptotic distribution for the nonparametric estimator of the function $g(⋅)$. Some numerical examples are provided to show that our theory and estimation method work well in practice.