Nonparametric Estimation in a Nonlinear Cointegration Type Model

We derive an asymptotic theory of nonparametric estimation for a time series regression model $Z_{t}=f(X_{t})+W_{t}$, where $\{X_{t}\}$ and $\{Z_{t}\}$ are observed nonstationary processes and $\{W_{t}\}$ is an unobserved stationary process. In econometrics, this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results are of wider interest. The class of nonstationary processes allowed for $\{X_{t}\}$ is a subclass of the class of null recurrent Markov chains. This subclass contains random walk, unit root processes and nonlinear processes. We derive the asymptotics of a nonparametric estimate of f(x) under the assumption that $\{W_{t}\}$ is a Markov chain satisfying some mixing conditions. The finite-sample properties of $\hat{f}(x)$ are studied by means of simulation experiments.