Stock, James H., and Mark W. Watson. "Identification and Estimation of Dynamic Causal Effects in Macroeconomics Using External Instruments." The Economic Journal 128.610 (May 2018): 917-948.
External sources of as-if randomness — that is, external instruments — can be used to identify the dynamic causal effects of macroeconomic shocks. One method is a one-step instrumental variables regression (local projections – IV); a more efficient two-step method involves a vector autoregression. We show that, under a restrictive instrument validity condition, the one-step method is valid even if the vector autoregression is not invertible, so comparing the two estimates provides a test of invertibility. If, however, lagged endogenous variables are needed as control variables in the one-step method, then the conditions for validity of the two methods are the same.