Nonparametric estimation in a nonlinear cointegration type model

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We derive an asymptotic theory of nonparametric estimation for a time series regression model Zₜ=f(Xₜ)+Wₜ, where {Xₜ} and {Zₜ} are observed nonstationary processes and {Wₜ} 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 {Xₜ} 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 {Wₜ} is a Markov chain satisfying some mixing conditions. The finite-sample properties of f(x) are studied by means of simulation experiments.