How Well Are State-Dependent Local Projections Capturing Nonlinearities?

TL;DR

This paper evaluates how well local projection methods capture nonlinear dynamics in DSGE models, proposing an augmented specification that improves approximation of true impulse responses.

Contribution

It introduces a new LP specification with squared shocks and state interactions, demonstrating its superior performance in capturing nonlinearities.

Findings

Linear LP fails to capture nonlinear effects for symmetric shocks.

State-dependent LP captures asymmetries and state dependence, especially in tails.

The proposed augmented LP best approximates true responses and is supported by valid inference procedures.

Abstract

We use quadratic vector autoregressions, motivated by pruned second-order perturbation solutions to DSGE models, as a laboratory to evaluate how well popular local projection (LP) specifications recover true impulse responses in nonlinear environments. We derive closed-form population impulse responses under each specification and compare them to truth. Linear LP fails to capture nonlinearities when the shock is symmetrically distributed. State-dependent LP specifications capture distinct aspects of nonlinearity: interacting the shock with its sign captures asymmetric effects, while interacting the shock with observable state proxies captures state dependence. However, their gains over linear LP are concentrated in tail shocks or states, and for the latter depend on proxy quality. Our proposed specification -- augmenting linear LP with a squared shock term and shock-state proxy interactions -- best approximates true responses. We also establish valid estimation and inference procedures for this specification. In a monetary policy application, we find state dependence, while higher-order effects differ across outcomes.