In the literature on nonlinear cointegration, a long-standing open problem relates to how a (nonlinear) vector autoregression, which provides a unified description of the short- and long-run dynamics of a collection of time series, can generate 'nonlinear cointegration' in the profound sense of those series sharing common nonlinear stochastic trends. We consider this problem in the setting of the censored and kinked structural VAR (CKSVAR), which provides a flexible yet tractable framework within which to model time series that are subject to threshold-type nonlinearities, such as those arising due to occasionally binding constraints, of which the zero lower bound (ZLB) on short-term nominal interest rates provides a leading example. We provide a complete characterisation of how common linear and {nonlinear stochastic trends may be generated in this model, via unit roots and appropriate generalisations of the usual rank conditions, providing the first extension to date of the Granger-Johansen representation theorem to a nonlinearly cointegrated setting, and thereby giving the first successful treatment of the open problem. The limiting common trend processes include regulated, censored and kinked Brownian motions, none of which have previously appeared in the literature on cointegrated VARs. Our results and running examples illustrate that the CKSVAR is capable of supporting a far richer variety of long-run behaviour than is a linear VAR, in ways that may be particularly useful for the identification of structural parameters.

翻译：在非线性协整的文献中，一个长期存在的开放性问题涉及如何让（非线性）向量自回归模型——该模型统一描述了时间序列集合的短期与长期动态——在深层意义上生成“非线性协整”，即这些序列共享共同的非线性随机趋势。我们在线段删截与拐点结构VAR（CKSVAR）的设置中研究此问题，该框架灵活且易于处理，适用于受阈值型非线性影响的时间序列建模，例如由偶尔约束（短期名义利率的零下限（ZLB）是其主要实例）所产生的情况。我们完整刻画了该模型如何通过单位根及通常秩条件的适当推广生成共同线性和非线性随机趋势，首次将Granger-Johansen表示定理扩展至非线性协整场景，从而成功处理了这一开放性问题。极限共同趋势过程包括受调控、删截和拐点的布朗运动，这些过程此前均未在协整VAR文献中出现。我们的结论与运行实例表明，CKSVAR能够支持比线性VAR更为丰富的长期行为多样性，这可能对结构参数的识别特别有用。