# Forecast Error Variance Decompositions with Local Projections

*Yuriy Gorodnichenko, Byoungchan Lee*
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## Abstract

We propose and study properties of an estimator of the forecast error variance decomposition in the local projections framework. We find for empirically relevant sample sizes that, after being bias-corrected with bootstrap, our estimator performs well in simulations. We also illustrate the workings of our estimator empirically for monetary policy and productivity shocks.

A simple method for estimating forecast error decompositions for local projections. :

- Calculate forecast errors for the endogenous variable of interest at each horizon $h$ based on the information set at $t−1$ as the residuals of a regression of the change in the endogenous variable from $t−1→h$ on lags of all variables
- For each horizon $h$, regress the forecast errors at that horizon onto the shock innovations up through that horizon
- The $R_{2}$ of those regressions is the forecast error variance due to the shock

This quantity can be understood as an R2 of the population projection of ft+h|t−1 on Zh t , or the probability limit of sample R2’s. This observation suggests a natural estimator of sh. First, the forecast errors for each horizon h are estimated using local projections. Second, the estimated forecast errors for the horizon h at time t are regressed on shocks that happen between t and t + h.The R2 in this regression is an estimate of sh.

The forecast errors can be interpreted as the change in the endogenous variable that couldn’t have be forecasted based on available information. The size (variance) of these errors will of course tend to grow with the horizon. Some portion of this variance is due to the shock variable, while the rest could be due to any number of factors (including direct shocks to the endogenous variable itself).

It might be easiest to simply include the future values of the shock of interest when running the forecast error regression above and see how much of a difference they make to the R-squared, in other words, identify the partial R^2 of those shocks.

Note that one may implement this estimator by augmenting Equation (5) with shocks $z_{t}$, …, $z_{t+h}$ and calculating the partial $R_{2}$.