  While it is widely recognised that linear (structural) VARs may omit
important features of economic time series, the use of nonlinear SVARs has to
date been almost entirely confined to the modelling of stationary time series,
because of a lack of understanding as to how common stochastic trends may be
accommodated within nonlinear VAR models. This has unfortunately circumscribed
the range of series to which such models can be applied -- and/or required that
these series be first transformed to stationarity, a potential source of
misspecification -- and prevented the use of long-run identifying restrictions
in these models. To address these problems, we develop a flexible class of
additively time-separable nonlinear SVARs, which subsume models with
threshold-type endogenous regime switching, both of the piecewise linear and
smooth transition varieties. We extend the Granger-Johansen representation
theorem to this class of models, obtaining conditions that specialise exactly
to the usual ones when the model is linear. We further show that, as a
corollary, these models are capable of supporting the same kinds of long-run
identifying restrictions as are available in linear cointegrated SVARs.
