Differences from Differencing : Should Local Projections with Observed Shocks be Estimated in Levels or Differences?

We show there are substantial small sample gains from estimating local projections (LPs) in a cumulated differences specification vs. a specification in levels when the impulse response of interest is to an externally identified ("observed") shock. The cumulated differences specification reduces estimation bias and improves confidence interval coverage over LPs specified in levels, with the improvement increasing for more persistent processes, at longer horizons, and for smaller sample sizes. For processes with persistence ranging from zero to moderately high, LPs estimated in cumulated differences display consistently negligible bias, while LPs estimated in levels display increasing bias as persistence rises. For near unit root and unit root processes, LPs estimated in cumulated differences are biased, but the improvement in bias over the levels specification is largest in these cases. We demonstrate these results using simulation evidence as well as analytic results for the example case of an autoregression. Overall, the cumulated differences specification appears to be an effective approach to reduce bias and improve the accuracy of confidence intervals in LPs estimated with observed shocks, regardless of the true integration properties of the data