🤖 AI Summary
A long-standing open question in nonlinear cointegration concerns how shared nonlinear stochastic trends emerge from nonlinear vector autoregressive (VAR) models. Method: Framing the analysis within the CKSVAR model—driven by occasional constraints such as the zero lower bound on interest rates—we jointly characterize short- and long-run dynamics of time series. Contribution/Results: We extend the Granger–Johansen representation theorem to nonlinear cointegration for the first time, deriving novel unit-root and cointegration-rank criteria tailored to threshold nonlinearity. We further derive new classes of cointegrating trend processes—including controlled, truncated, and kinked Brownian motions. Our theoretical results fully characterize the conditions under which linear and nonlinear common trends arise, demonstrating that CKSVAR supports a substantially richer set of long-run behaviors than linear VAR models. Moreover, the framework establishes a new foundation for structural parameter identification in nonlinear cointegrated systems.
📝 Abstract
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 vector 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.