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You like econometrics and use "local projections" because they let you do smart tricks, like exploring the non-linear effects of large vs small financial shocks on output? New paper by Kilian and co-authors says you need to correct your estimates!
Nerdy thread #Econtwitter
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Nerdy thread #Econtwitter
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The paper starts from the definition of IRF. The IRF is the difference between the unconditional h steps ahead forecast of a variable y_(t+h), and a conditional forecast y_(t+h)(delta), where delta is the size of a shock that occurs at time t.
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Jorda (2005) noted that these forecasts can be obtained via direct forecasting, therefore a regression of y_(t+h) on a shock (let us call it x_(t), which you need to get somewhere else!) is an estimate of an IRF and called these Local Projections (LP), hugely popular now.
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The problem arises when you add a nonlinear function of x_t, say f(x_t) in the LP. This could be large changes with respect to a threshold, for instance, because you want to know if large shocks matter more than small shocks (spoiler: they typically do)
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In this case the difference between unconditional and conditional forecast also contains an extra term, E[f(x_t+delta)- f(x_t)], and you need to plug this extra term in your LP to get the correct IRFs. You also need to account for lags of x_t and of f(x_t).
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There is a solution though: Sections 5.1 and 5.2 in the paper show you how to get a sample estimate of these extra expectation term. Spoiler: expectations boil down to an in sample average. You thought you were doomed, and instead all you were missing was an average.
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I close with a gut feeling: empirically this correction might not matter much, the extra term E[f(x_t+delta)- f(x_t)] might be very small in practice. But you might want to do this check not to give your referee an easy weapon to throw your paper under the bus.
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Extra tweet: this is what the extra term looks like. Less ugly than you thought
