How Well Are State-Dependent Local Projections Capturing Nonlinearities?

We evaluate how well state-dependent local projections recover true impulse responses in nonlinear environments. Using quadratic vector autoregressions as a laboratory, we show that linear local projections fail to capture any nonlinearities when shocks are symmetrically distributed. Popular state-dependent local projections specifications capture distinct aspects of nonlinearity: those interacting shocks with their signs capture higher-order effects, while those interacting shocks with lagged states capture state dependence. However, their gains over linear specifications are concentrated in tail shocks or tail states; and, for lag-based specifications, hinge on how well the chosen observable proxies the latent state. Our proposed specification-which augments the linear specification with a squared shock term and an interaction between the shock and lagged observables-best approximates the true responses across the entire joint distribution of shocks and states. An application to monetary policy reveals economically meaningful state dependence, whereas higher-order effects, though statistically significant, prove economically modest.