  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 nonlinear 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
nonstationary 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.
