When Do State-Dependent Local Projections Work?

Many empirical studies estimate impulse response functions that depend on the state of the economy. Most of these studies rely on a variant of the local projection (LP) approach to estimate the state-dependent impulse response functions. Despite its widespread application, the asymptotic validity of the LP approach to estimating state-dependent impulse responses has not been established to date. We formally derive this result for a structural state-dependent vector autoregressive process. The model only requires the structural shock of interest to be identified. A sufficient condition for the consistency of the state-dependent LP estimator of the response function is that the first- and second-order conditional moments of the structural shocks are independent of current and future states, given the information available at the time the shock is realized. This rules out models in which the state of the economy is a function of current or future realizations of the outcome variable of interest, as is often the case in applied work. Even when the state is a function of past values of this variable only, consistency may hold only at short horizons. These results show that the state-dependent LP regression (7) recovers the conditional IRF obtained in Proposition 3.1 with h = 0 under Assumption 1. No further assumptions are required (provided a law of large numbers can be applied to ^ Q 11 : 2 and ^ Q 1 y: 2 ; 0 ). In particular, conditional homoskedasticity of " t is not required. Nor do we need to impose further restrictions on the process driving state dependence.