Testing for stationarity-ergodicity and for comovements between nonlinear discrete time Markov processes

In this paper we introduce a class of nonlinear data generating processes (DGPs) that are first order Markov and can be represented as the sum of a linear plus a bounded nonlinear component. We use the concepts of geometric ergodicity and of linear stochastic comovement, which correspond to the linear concepts of integratedness and cointegratedness, to characterize the DGPs. We show that the stationarity test due to Kwiatowski et al. (1992, Journal of Econometrics, 54, 159–178) and the cointegration test of Shin (1994, Econometric Theory, 10, 91–115) are applicable in the current context, although the Shin test has a different limiting distribution. We also propose a consistent test which has a null of linear cointegration (comovement), and an alternative of `non-linear cointegration'. Monte Carlo evidence is presented which suggests that the test has useful finite sample power against a variety of nonlinear alternatives. An empirical illustration is also provided.