  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 $g(\cdot)$ 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 $g(\cdot)$. Some numerical examples are
provided to show that our theory and estimation method work well in practice.
