Many macroeconomic models let shocks move the economy in different ways depending on the state of the system. This paper studies a class of structural vector autoregressions (SVARs) in which the endogenous variables enter nonlinearly on the left-hand side. The authors call this feature endogenous nonlinearity. They show that, under weak regularity conditions, the model’s parameters and the structural shocks can be recovered from the data in much the same way as in a linear SVAR.

What the researchers study is a very general model written as f0(z_t) = f1(z_{t-1}) + ε_t, where f0 and f1 are possibly nonlinear functions and ε_t are the structural shocks. This formulation covers many familiar ideas, including models with endogenous regime switching and piecewise affine rules. A concrete example is a censored-and-kinked SVAR, which can capture situations like a zero lower bound on interest rates, where the regime depends on the current value of an endogenous variable.

Their main theoretical result (Theorem 2.2) is a nonparametric identification statement. It says that data on the observed variables identify the functions f0 and f1 and the shocks ε_t up to an orthogonal transformation. In plain terms, the data determine the shocks and parameters except for a rotation that mixes the shock labels but preserves their uncorrelatedness. That is the same fundamental limit that appears in linear SVARs.

The practical implication is important. Because the only remaining ambiguity is this orthogonal rotation, the usual ways researchers pin down linear SVARs carry over. In particular, the number of additional restrictions needed to achieve exact identification remains p(p−1)/2 for a p-variable system. In models with endogenous regime switching built from piecewise affine functions, the authors show it is enough to impose identifying restrictions in just one regime (or distribute restrictions across regimes). This avoids having to replicate a full set of restrictions for every regime.