Local projections
Good example papers
- Longer-Run Economic Consequences of Pandemics
- Government Spending Multipliers in Good Times and in Bad: Evidence from U.S. Historical Data
- Macroeconomic Shocks and Their Propagation
I’m generally finding that despite the work of Local Projection Inference is Simpler and More Robust Than You Think that for local projections you’re much better off simply using a shock that is already stationary rather than including lags as controls in order to get to effective stationarity. Relying on lags of the independent variables just hasn’t worked very well for me. A quick survey of papers shows that most of the time the shock is already stationary or is made stationary via first differencing rather than via extra lags. They do include lags, but they are lags of an already stationary variable, and I think the intent there is mainly to control for autocorrelation rather than force stationarity.
One thing I’ve grappled with is whether to use lags of the first difference of non-stationary variables as controls or lags of the variables in levels. The vast majority of the literature is either silent on the question or uses first differences. But similar to this discussion in the context of VARs (VARs in levels vs differences), everything should go through just fine in levels as long as you include lags. In fact, there should hypothetically be even less concern in the context of local projections than in VARs because with LP you are only concerned with the coefficient on a single variable. Unlike a VAR, where the impulse response is a complicated function of multiple parameters, in an LP it’s just a single parameter. To be fair it’s the same parameter of a series of separate regressions, but it’s effectively just one parameter. Further, first differencing is inappropriate if the variables do not in fact have unit roots.
Spurious local projections
Have to be careful when running local projections to ensure that the outcome and impulse variable are stationary. If they aren’t you can get some very weird results. The issue is that the typical transformations people do are not necessarily foolproof in a wide array of situations. With macro data taking first differences or long differences is generally enough to make data stationary, so that’s what people do. But in some cases even the first differences data can trend, in which case this won’t be enough. In those situations you must detrend the data or equivalently include a time trend among the regressors
Panel LP
Nickell bias is a pervasive issue in panel local projections and one often not accounted for, leading to results that shade toward zero. The bias is especially problematic when the cross-sectional dimension is large. Some papers that cover this problem and provide a solution include
The solution is an easy modification of the fixed effects estimator that involves running the same specification on two subsets of the data, averaging them, and the subtracting this average from two times the uncorrected fixed effects estimator.