Skip to main content
SpringerLink
Log in

Bayesian Inference of Local Projections with Roughness Penalty Priors

  • Published:
Computational Economics Aims and scope Submit manuscript

Abstract

A local projection is a statistical framework that accounts for the relationship between an exogenous variable and an endogenous variable, measured at different time points. Local projections are often applied in impulse response analyses and direct forecasting. While local projections are becoming increasingly popular because of their robustness to misspecification and their flexibility, they are less statistically efficient than standard methods, such as vector autoregression. In this study, we seek to improve the statistical efficiency of local projections by developing a fully Bayesian approach that can be used to estimate local projections using roughness penalty priors. By incorporating such prior-induced smoothness, we can use information contained in successive observations to enhance the statistical efficiency of an inference. We apply the proposed approach to an analysis of monetary policy in the United States, showing that the roughness penalty priors successfully estimate the impulse response functions and improve the predictive accuracy of local projections.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. See, e.g., Geweke (2005) for a general introduction to Bayesian analysis.

  2. Bayesian inference using the Jeffreys prior for \(\varvec{\Sigma }\) almost always fails for the synthetic and real data used in the subsequent section.

  3. See, for example, De Boor (1978) and Eilers and Marx (1996) for a detailed description of B-splines.

  4. All programs were written in Matlab 2016a (64 bit) and executed on an Ubuntu Desktop 16.04 LTS (64 bit), running on Intel Xeon E5-2607 v3 processors (2.6 GHz).

  5. From this point of view, local projections are similar to functional data models such as Guo (2002) and Morris and Carroll (2006).

  6. When the degree of the B-spline bases is increased to 5, the weight set becomes {1/120, 13/60, 33/60, 13/60, 1/120}. The added weights are too small to affect the estimate \(\left( 1/120\approx 0.0083\right) \).

  7. The time series of monetary policy shocks is from Yuriy Gorodnichenko’s website (https://eml.berkeley.edu//~ygorodni/index.htm).

  8. In the “Online Appendix”, Figs. A.1, A.2, and A.3 display credible intervals for all the specifications.

  9. Both the DIC and WAIC are asymptotically related to the AIC. Thus, one might consider evaluating the statistical significance of the difference in the values of the criteria of two models by applying a rule of thumb that is originally proposed to Bayes factor. As Burnham and Anderson (2004) describe, the AIC can be interpreted as an approximation of the log marginal likelihood of a model under a “savvy” prior that is a function of sample size and the number of model parameters. According to Jeffreys’s (1961) rule of thumb, the statistical significance of the difference between two models is “weak” if the difference in the AIC/DIC/WAIC is 0-2, “positive” if 2–6, “strong” if 6–10, or “very strong” if >10 (see also Raftery 1995). When this rule of thumb is directly applied to Table 3, one might be able to interpret the results as follows: the statistical significance of the differences related to the prior choice is “very strong,” and the significance of the differences attributable to the use of B-splines is “weak” or “positive” when the Normal prior is used while it is “very strong” when the N-RP prior is used.

  10. We minimize the information criteria using the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm with bounds (Byrd et al. 1995), using a Matlab routine minConf_TMP.m written by Mark Schmidt. (https://www.cs.ubc.ca/~schmidtm/Software/minConf.html).

  11. We also considered Jeffreys prior for \(\sigma _{\left( h_{i}\right) }^{2}\) and obtained almost the same result for the half-t prior (thus, it is not reported).

  12. As T increases (e.g., \(T=500\)), the relative performance of the FGLS approach with unrestricted covariance becomes comparable with that with diagonal covariance (not reported).

References

  • Aikman, D., Bush, O., & Taylor, A. M. (2016). Monetary versus macroprudential policies: Causal impacts of interest rates and credit controls in the era of the UK radcliffe report. NBER working paper 22380.

  • Alvarez, I., Niemi, J., & Simpson, M. (2014). Bayesian inference for a covariance matrix. In 26th Annual conference on applied statistics in agriculture.

  • Auerbach, A. J., & Gorodnichenko, Y. (2013). Fiscal multipliers in recession and expansion. In A. Alesina & F. Giavazzi (Eds.), Fiscal policy after the financial crisis (pp. 63–98). Chicago: University of Chicago Press.

    Chapter  Google Scholar 

  • Barnichon, R., & Matthes, C. (2019). Functional approximations of impulse responses. Journal of Monetary Economics, 99(1), 41–55.

    Google Scholar 

  • Barnichon, R., & Brownlees, C. (2019). Impulse response estimation by smooth local projections. Review of Economics and Statistics, 101(3), 522–230.

    Article  Google Scholar 

  • Byrd, R. H., Peihuang, L., Nocedal, J., & Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16, 1190–1208.

    Article  Google Scholar 

  • Coibion, O., Gorodnichenko, Y., Kueng, L., & Silvia, J. (2017). Innocent bystanders? Monetary policy and inequality. Journal of Monetary Economics, 88, 70–89.

    Article  Google Scholar 

  • De Boor, C. (1978). A practical guide to splines (Vol. 27). New York: Springer.

    Book  Google Scholar 

  • Eilers, P. H. C., & Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11, 89–121.

    Article  Google Scholar 

  • El-Shagi, M. (2019). A simple estimator for smooth local projections. Applied Economics Letters, 26, 830–834.

    Article  Google Scholar 

  • Geweke, J. (2005). Contemporary Bayesian econometrics and statistics (Vol. 537). New York: Wiley.

    Book  Google Scholar 

  • Guo, W. (2002). Functional mixed effects models. Biometrics, 58, 121–128.

    Article  Google Scholar 

  • Huang, A., & Wand, M. P. (2013). Simple marginally noninformative prior distributions for covariance matrices. Bayesian Analysis, 8, 439–452.

    Article  Google Scholar 

  • Hurvich, C. M., Simonoff, J. S., & Tsai, C.-L. (1998). Smoothing parameter selection in nonparametric regression using an improved akaike information criterion. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60, 271–293.

    Article  Google Scholar 

  • Jordà, Ò. (2005). Estimation and inference of impulse responses local projections. American Economic Review, 95, 161–182.

    Article  Google Scholar 

  • Lang, S., & Brezger, A. (2004). Bayesian P-splines. Journal of Computational and Graphical Statistics, 13, 183–212.

    Article  Google Scholar 

  • Miranda-Agrippino, S., & Ricco, G. (2017). The transmission of monetary policy shocks. Staff working paper 657, Bank of England.

  • Morris, J. S., & Carroll, R. J. (2006). Wavelet-based functional mixed models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68, 179–199.

    Article  Google Scholar 

  • Ramey, V. A. (2016). Macroeconomic shocks and their propagation, Chap. 2. In J. B. Taylor & H. Uhlig (Eds.), Handbook of macroeconomics (Vol. 2A, pp. 71–162). Amsterdam: Elsevier.

    Google Scholar 

  • Ramey, V. A., & Zubuairy, S. (2018). Government spending multipliers in good times and in bad: Evidence from U.S. historical data. Journal of Political Economy, 126(2), 850–901.

  • Riera-Crichton, D., Vegh, C. A., & Vuletin, G. (2015). Procyclical and countercyclical fiscal multipliers: Evidence from OECD countries. Journal of International Money and Finance, 52, 15–31.

    Article  Google Scholar 

  • Romer, C. D., & Romer, D. H. (2004). A new measure of monetary shocks: Derivation and implications. American Economic Review, 94, 1055–1084.

    Article  Google Scholar 

  • Rue, H. (2001). Fast sampling of Gaussian Markov random fields. Journal of the Royal Statistical Society, Series B, 63, 325–338.

    Article  Google Scholar 

  • Rue, H., & Held, L. (2005). Gaussian Markov random fields: Theory and applications. Boca Raton: CRC Press.

    Book  Google Scholar 

  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B, 64, 583–639.

    Article  Google Scholar 

  • Stock, J. H., & Watson, M. W. (2007). Why has US inflation become harder to forecast? Journal of Money, Credit and Banking, 39, 3–33.

    Article  Google Scholar 

  • Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research, 11, 3571–3594.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Economics, Waseda University, 1-6-1, Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan

    Masahiro Tanaka

Authors
  1. Masahiro Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masahiro Tanaka.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 210 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tanaka, M. Bayesian Inference of Local Projections with Roughness Penalty Priors. Comput Econ 55, 629–651 (2020). https://doi.org/10.1007/s10614-019-09905-y

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10614-019-09905-y

Keywords

JEL Classification

Search

Navigation