A Specification Test for Nonlinear Nonstationary Models By

We provide a limit theory for a general class of kernel smoothed U-statistics that may be used for specification testing in time series regression with nonstationary data. The test framework allows for linear and nonlin-ear models with endogenous regressors that have autoregressive unit roots or near unit roots. The limit theory for the specification test depends on the self-intersection local time of a Gaussian process. A new weak convergence result is developed for certain partial sums of functions involving nonstation-ary time series that converges to the intersection local time process. This result is of independent interest and is useful in other applications. Simulations examine the finite sample performance of the test. 1. Introduction. One of the advantages of nonparametric modeling is the opportunity for specification testing of particular parametric models against general alternatives. The past three decades have witnessed many developments in such specification tests involving nonparametric and semiparametric techniques that allow for independent, short memory and long-range dependent data. Recent research on the nonparametric modeling of nonstationary data opens up some new possibilities that seem relevant to applications in many fields, including nonlinear diffusion models in continuous time [Bandi and Phillips (2003, 2007)] and cointe-gration models in economics and finance. Cointegration models were originally developed in a linear parametric framework that has been widely used in econometric applications. That framework was extended in Park and Phillips (1999, 2001) to allow for nonlinear parametric formulations under certain restrictions on the function nonlinearity. While considerably broadening the class of allowable nonstationary models, the potential for parametric misspecification in these models is still present and is important to test in applied work. The hypothesis of linear cointegration is of particular interest in this context, given the vast empirical literature.