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

Abstract

Abstract. 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 Kwiatkowski, Phillips, Schmidt and Shin (1992), and the cointegration test of Shin (1994) 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 "nonlinear 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.JEL Classification: C12, C22.

Nonlinear Equilibrium Correction in U.S. Real Money Balances, 1869-1997

“…7 Balke and Fomby (1997) show that the Johansen method of estimating the cointegrating vector may work reasonably well in the presence of nonlinear threshold cointegration, and conjecture that these results may also hold for smooth transition nonlinearities in adjustment -see also Corradi, Swanson and White (2000).…”

Likelihood-based inference for cointegration with nonlinear error-correction

“…See for example the term-structure studies in Anderson (1997), Balke and Fomby (1997), Bec and Rahbek (2004), Corradi et al (2000), Hansen and Seo (2002) and Seo (2003) where the (speed of) adjustment is parameterized as a function of interest rate spreads, and similarly, for example, Kapetanios et al (2006) and Psaradakis et al (2004) The asymptotic properties of the QMLE of the cointegrating vectors β depend on whether the aforementioned linear trend is present or not. When there is a linear trend in data,β is super-consistent in all directions but one in which it is T 3/2 -consistent.…”

“…With the focus of deriving tests for 'stationarity-ergodicity', that is cointegration in Markovian nonlinear error correction models, Corradi et al (2000) finds asymptotic properties of the linear OLS estimator in the case of a single cointegration vector, when the data generating process is a nonlinear error correction model. Likewise, Kapetanios et al (2006) and Seo (2006) study test statistics for cointegration in single equation nonlinear error correction models.…”

The Performance of Variance Ratio Unit Root Tests Under Nonlinear Stationary TAR and STAR Processes: Evidence from Monte Carlo Simulations and Applications

“…As shown by Corradi et al (2000), the Kwiatkowski et al (KPSS) test has a well-defined limiting distribution under the null hypothesis of general nonlinear stationary processes. Accordingly, it is possible that the test has a good size performance under nonlinear stationary TAR and STAR processes.…”