Impulse Response Estimation by Smooth Local Projections

Regis Barnichon, Christian Brownlees

Abstract

Local Projections (LP) is a popular methodology for the estimation of Impulse Responses (IR). Compared to the traditional VAR approach, LP allow for more flexible IR estimation by imposing weaker assumptions on the dynamics of the data. The nonparametric nature of LP comes at an efficiency cost and in practice the LP estimator may suffer from excessive variability. In this work we propose an IR estimation methodology based on B-spline smoothing called Smooth Local Projections (SLP). The SLP approach preserves the flexibility of standard LP, can substantially increase precision and is straightforward to implement. A simulation study shows that SLP can deliver substantial gains in IR estimation over LP. We illustrate our technique by studying the effects of monetary shocks where we highlight how SLP can easily incorporate commonly employed structural identification strategies.

One way to think about the large, single regression that is done in this paper is to analogize it to panel regression

• You stack the variables for each of the units. Here, the “units” are the different horizons. So you end up with roughly N * H data points in a single regression rather than N data points in H different regressions. You can think about the outcome variable as $y_{it}$, where i is the unit and t is the time point, except here you effectively have $y_{th}$.

Steps of smooth local projections

• Transform the impulse variable X from a Nx1 dimensional vector to a NxK dimensional matrix, via the transformation X @ B, where B is a vector of K values coming from K basis functions
• Residualize the outcome variable and the transformed impulse variable by the set of controls for each impulse horizon, appropriately lagged
• Stack the residualized outcome variable and the NxK impulse variable by the horizon, yielding (NxH)x1 and (NxH)xK dimensional matrices for each
• Run a penalized regression, where the r-th difference between subsequent splined Xs is penalized by $\lambda$. Note you can do this in one-shot via standard least squares by stacking the r-th difference matrix multiplied by $\lambda$ on to the X matrix. This is effectively finds the weights for the basis functions that work best across all horizons
• Get the final IRF by multiplying the matrix of basis functions (splines) by the weights vector

References

• Estimation and Inference of Impulse Responses by Local Projections
• Estimation of Smooth Impulse Response Functions

online