Forecast Error Variance Decompositions with Local Projections

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. KEYWORDS: Forecast error variance decomposition; Local projections.

ACKNOWLEDGMENTS

We thank our editor Todd Clark, an anonymous referee, and an associate editor for helpful comments and suggestions. We also are grateful to Oscar Jordà, Mikkel Plagborg-Møller, and Christian Wolf for comments on an earlier version of the paper.

Notes

Notes

1 For example, Jordà (2005) suggested an estimator close in spirit to LP-A and LP-B estimators that we cover in Section 3.1 and Appendix B. Our baseline estimator of FEVDs performs better than these estimators for empirically relevant sample sizes. Another method is to compute FEVDs by using VARs that directly include a structural shock (Plagborg-Møller and Wolf 2018). While this method identifies the same population FEVDs, it requires a large number of lags (Baek and Lee 2019), a feature that may be too costly in practice given the curse of dimensionality in VARs and the noise generated by many estimated parameters.

2 Coibion et al. (2017) is among the very few papers reporting FEVDs based on the local projection method.

3 One may use alternative implementations of bootstrap to refine asymptotic inference. We tried the block bootstrap for local projections following Kilian and Kim (2011). However, this block bootstrap method performs worse than the VAR-based bootstrap in simulations. Results are in Appendix E1.

4 Our bootstrap procedure implicitly assumes homoscedasticity of shocks. If a researcher suspects important heteroskedasticity in shocks, one should use alternative bootstrap methods (e.g., Gonçalves and Kilian 2004). An extensive discussion of practical considerations for various bootstrap methods is in Kilian and Lütkepohl (2017, Ch. 12).

5 As the number of variables in C_t increases, the number of parameters in the VAR increases rapidly. When C_t is a large vector, or when a VAR is not a good representation of the DGP for control variables, VAR-based bootstrap might not be an appealing option. In this case, one may consider other forms of bootstrap (e.g., block bootstrap). Alternatively, one may correct for biases by simulating asymptotic distributions of primitive quantities in Equations (3)(3) $$s_h=\frac{\text{var}\left({\psi }_{z,0}{z}_{t+h}+\cdots +{\psi }_{z,h}{z}_t\right)}{\text{var}\left(f_{t+h|t-1}\right)}.$$(3) , (7)(7) $$s_h=\frac{\left(\sum _{i=0}^h{\psi }_{z,i}^2\right){\sigma }_z^2}{\text{var}\left(f_{t+h|t-1}\right)}$$(7) , and (7’) such as ${\widehat{\psi }}_{z,i},{\widehat{\sigma }}_z^2,$ and $\widehat{\text{var}}\left({\widehat{v}}_{t+h|t-1}\right)$. By considering s_h as a non-linear function of those parameters, such simulations would detect biases due to the non-linearity. See Appendices A and B for implementation and F and G for the results.

6 We also considered percentile-t bootstrap and found similar results.

7 This value and pattern are motivated by a 3 percent response of real GDP to a 100bp monetary policy shock estimated in Coibion (2012).

8 The bias can be further reduced by using higher values of L_z and L_y by reducing errors in ${\widehat{f}}_{t+h|t-1}$ due to the truncation.

9 Given the parameter values in Table 1, $\Delta y_t=g_y+\left(1-L\right){\left(1-0.9L\right)}^{-1}{z}_t+{\left(1-0.9L\right)}^{-1}{e}_t^p.$ By pre-multiplying $\left(1-0.9L\right)$, we have $\Delta y_t=0.1g_y+0.9\Delta y_{t-1}-z_{t-1}+z_t+e_t^p.$

10 For this information set, we construct the true FEVD using a stationary Kalman filter similar to the method in Appendix C. We also tried various combinations of shocks and endogenous variables in the information set and found similar results. Figures for inflation and results with large samples are in Appendix G. Note that monetary policy shocks are nearly invertible in the Smets-Wouters model (see Wolf 2017 for more details). While this may be a problem if we use shocks identified and recovered from a DSGE model, the spirit of our exercise is to assume that we have access to other information (as in e.g. Romer and Romer 2004) so that we can observe monetary policy shocks directly.

11 Appendix H presents results for military spending shocks constructed in Ramey and Zubairy (2018).

12 The ordering of variables in the VAR is TFP measure (from Fernald 2014), output growth rate, inflation, monetary policy innovations (from Coibion et al. 2017), and fed funds rate. For the VAR-based analysis, we follow the practice and compute FEVDs using shocks in these variables where shocks are identified recursively from reduced-form residuals.