Impulse responses by smooth local projections

Author and guest post by Eren Ocakverdi

This blog piece intends to introduce a new add-in (i.e. SMLOCALIRFS) that implements the methodology developed by Barnichon and Brownlees (2019).

Table of Contents

1. Introduction
2. Smooth Local Projections
3. Application to U.S. Monetary VAR Model
4. Files
5. References

Introduction

Local projections (LP) are nonparametric approaches, and they tend to be inefficient as their estimates frequently show considerable variability. In their paper, Barnichon and Brownlees (2019) propose a novel methodology for the estimation of impulse responses based on penalized B-splines called smooth local projections (SLP). They find that the SLP approach can deliver substantial improvements over regular LP, while preserving its flexibility.

Smooth Local Projections

Regular LP recovers the dynamic multipliers, $\beta_{(h)}$, by estimating the following set of h-step-ahead predictive regressions via OLS: \begin{align*} y_{t+h} = \alpha_{(h)} + \beta_{(h)} x_t + \sum_{i=1}^{p} \gamma_{i(h)}\,\omega_{it} + u_{(h)t+h} \end{align*} However, overparameterization may lead to excessive variability in the estimates of dynamic multipliers. Authors propose an alternative methodology, where they approximate the coefficients using a linear B-splines basis function expansion in the forecast horizon: \begin{align*} Z_{(h)} \approx \sum_{k=1}^{K} z_k\,B_{k(h)}, \qquad Z \in \{\alpha,\beta,\gamma\}, \qquad z \in \{a,b,c\} \end{align*} Here, B-spline basis function is made up of $q+1$ polynomial pieces of order $q$. By using a particular family of penalty matrices $(P)$, authors shrink the estimated impulse responses toward a polynomial of a given order instead of shrinking toward zero like typical shrinkage estimators. The amount of shrinkage $(λ)$, which determines the bias/variance trade-off of the estimator, can be selected by $k-$fold cross-validation.

Application to U.S. Monetary VAR Model

Example data uses the three-variable monetary VAR model of Stock and Watson (2001), where the inflation, unemployment, and federal funds rate variables used in that study are extended to cover the 1959q1-2007q2 period (see Figure 1).

Figure 1: Macroeconomic variables

Proposed methodology is applied to study the effects of monetary shocks on output. Therefore, it is assumed that the response of GDP growth to a monetary shock can be identified by introducing a shock to Fed funds rate. Here, contemporaneous values of GDP growth and inflation as well as four lags of GDP growth, inflation, and the Fed funds rate are used as control variables. To run the procedure on this data, we can use the add-in (see Figure 2).

Figure 2: GUI of the add-in (see the help file in the add-in’s folder for details)

Figure 3 plots the impulse responses of GDP growth to a 1 standard deviation monetary shock obtained from regular LP and smooth LP. Overall, the responses display an expected pattern where GDP growth declines after a contractionary shock. However, the impulse responses obtained by regular LP are more volatile as sharp fluctuations are observed between consecutive quarters (see Figure 3).

Figure 3a: Accumulated response of GDP growth to a monetary shock

Figure 4 plots the coefficients of impulse response estimation of smooth LP along with 95% confidence intervals. Lambda parameter is also determined by running 5-fold cross-validation on alternative values (i.e. 1, 10, 50, 100, 250, 500, 1000, 5000, 10000).

Figure 4: Smooth LP impulse response of GDP growth to a monetary shock with error bands (optimal lambda = 50)

Volatile nature of impulse responses generated by regular LP method can complicate the interpretation of the output. It may not be easy to filter out the impacts coming from measurement errors. Therefore, smoothing can substantially increase the precision without compromising on the flexibility.

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Files

• SMLOCALIRFS_DATA.WF1
• SMLOCALIRFS_EXAMPLE.PRG

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References

1. Barnichon, R., and Brownlees, C. (2019), "Impulse Response Estimation by Smooth Local Projections", The Review of Economics and Statistics, v. 101(3), pp. 522-530..