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.

翻译：尽管学界普遍认识到线性（结构）向量自回归可能遗漏经济时间序列的重要特征，但由于对非线性VAR模型中如何纳入共同随机趋势缺乏理解，非线性SVAR的应用迄今几乎完全局限于平稳时间序列的建模。这一局限不当地缩小了此类模型的适用序列范围——或要求这些序列先转化为平稳形式，从而潜在地导致设定偏误——并阻碍了长期识别约束在这类模型中的运用。为解决上述问题，我们构建了一类灵活的可加时间可分非线性SVAR模型，涵盖具有阈值型内生机制转换的模型（包括分段线性与平滑转换两类）。我们将Granger-Johansen表示定理推广至该类模型，获得在线性模型情形下精确退化为常规条件的判定准则。进一步证明，作为推论，这类模型能够支持线性协整SVAR中可用的同类型长期识别约束。