Estimation in semi-parametric regression with non-stationary regressors

In this paper, we consider a partially linear model of the form Y-t = X-t(tau)theta(0) + g(V-t) + epsilon(t), t = 1,...,n, where {V-t} is a beta null recurrent Markov chain, {X-t} is a sequence of either strictly stationary or non-stationary regressors and {epsilon(t)} is a stationary sequence. We propose to estimate both theta(0) and g(.) by a semi-parametric least-squares (SLS) estimation method. Under certain conditions, we then show that the proposed SLS estimator of theta(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.