Cointegration in Non-linear time series

#1
I have the following problem:
In*non-linear time series regression yt = g(xt)+et,
if xt is non-stationary (i.e. I(1)) and
yt is stationary (i.e. I(0)),
how to test that this relationship is cointegrated and not spurious?
Would it be enough to use KPSS test for unit root on residuals et (this seems to be the only test applicable for the non-linear case)?
*
Note: both yt and xt are stationary long-term but the regression is only fit on the data available short-term where xt becomes non-stationary.

Any advice (possibly with references) can help.