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 \ensuremath{X_t} and \ensuremath{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 $\widehat{f}(x)$ are studied by means of simulation experiments.