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Joint confidence sets for structural impulse responses

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  • Inoue, Atsushi
  • Kilian, Lutz

Abstract

Many questions of economic interest in structural VAR analysis involve estimates of multiple impulse response functions. Other questions relate to the shape of a given impulse response function. Answering these questions requires joint inference about sets of structural impulse responses, allowing for dependencies across time as well as across response functions. Such joint inference is complicated by the fact that the joint distribution of the structural impulse response estimators becomes degenerate when the number of structural impulse responses of interest exceeds the number of model parameters, as is often the case in applied work. This degeneracy may be overcome by transforming the estimator appropriately. We show that the joint Wald test is invariant to this transformation and converges to a nonstandard distribution, which can be approximated by the bootstrap, allowing the construction of asymptotically valid joint confidence sets for any subset of structural impulse responses, regardless of whether the joint distribution of the structural impulse responses is degenerate or not. We propose to represent the joint confidence sets in the form of “shotgun plots” rather than joint confidence bands for impulse response functions. Several empirical examples demonstrate that this approach not only conveys the same information as confidence bands about the statistical significance of response functions, but may be used to provide economically relevant additional information about the shape of and comovement across response functions that is lost when reducing the joint confidence set to two-dimensional bands.

Suggested Citation

  • Inoue, Atsushi & Kilian, Lutz, 2016. "Joint confidence sets for structural impulse responses," Journal of Econometrics, Elsevier, vol. 192(2), pages 421-432.
  • Handle: RePEc:eee:econom:v:192:y:2016:i:2:p:421-432
    DOI: 10.1016/j.jeconom.2016.02.008
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    3. Anna Staszewska-Bystrova & Peter Winker, 2014. "Measuring Forecast Uncertainty of Corporate Bond Spreads by Bonferroni-Type Prediction Bands," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 6(2), pages 89-104, June.
    4. Guerron-Quintana, Pablo & Inoue, Atsushi & Kilian, Lutz, 2017. "Impulse response matching estimators for DSGE models," Journal of Econometrics, Elsevier, vol. 196(1), pages 144-155.
    5. Neil Kellard & Denise Osborn & Jerry Coakley & Simone D. Grose & Gael M. Martin & Donald S. Poskitt, 2015. "Bias Correction of Persistence Measures in Fractionally Integrated Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(5), pages 721-740, September.
    6. Daniel Grabowski & Anna Staszewska-Bystrova & Peter Winker, 2020. "Skewness-adjusted bootstrap confidence intervals and confidence bands for impulse response functions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(1), pages 5-32, March.
    7. Blazsek, Szabolcs & Escribano, Álvaro & Licht, Adrian, 2019. "Co-integration and common trends analysis with score-driven models : an application to the federal funds effective rate and US inflation rate," UC3M Working papers. Economics 28451, Universidad Carlos III de Madrid. Departamento de Economía.
    8. Inoue, Atsushi & Kilian, Lutz, 2022. "Joint Bayesian inference about impulse responses in VAR models," Journal of Econometrics, Elsevier, vol. 231(2), pages 457-476.
    9. Lütkepohl, Helmut & Staszewska-Bystrova, Anna & Winker, Peter, 2020. "Constructing joint confidence bands for impulse response functions of VAR models – A review," Econometrics and Statistics, Elsevier, vol. 13(C), pages 69-83.
    10. Montiel Olea, José Luis & Nesbit, James, 2021. "(Machine) learning parameter regions," Journal of Econometrics, Elsevier, vol. 222(1), pages 716-744.
    11. Paul Beaudry & Fabrice Collard & Patrick Feve & Alain Guay & Franck Portier, 2022. "Dynamic Identification in VARs," Working Papers 22-08, Chair in macroeconomics and forecasting, University of Quebec in Montreal's School of Management.
    12. Lynda Khalaf & Beatriz Peraza López, 2020. "Simultaneous Indirect Inference, Impulse Responses and ARMA Models," Econometrics, MDPI, vol. 8(2), pages 1-26, April.
    13. Carsten Trenkler & Enzo Weber, 2020. "Identifying shocks to business cycles with asynchronous propagation," Empirical Economics, Springer, vol. 58(4), pages 1815-1836, April.
    14. Atsushi Inoue & Òscar Jordà & Guido M. Kuersteiner, 2023. "Significance Bands for Local Projections," Working Paper Series 2023-15, Federal Reserve Bank of San Francisco.
    15. José Luis Montiel Olea & Mikkel Plagborg‐Møller, 2021. "Local Projection Inference Is Simpler and More Robust Than You Think," Econometrica, Econometric Society, vol. 89(4), pages 1789-1823, July.
    16. Bojaj, Martin M. & Muhadinovic, Milica & Bracanovic, Andrej & Mihailovic, Andrej & Radulovic, Mladen & Jolicic, Ivan & Milosevic, Igor & Milacic, Veselin, 2022. "Forecasting macroeconomic effects of stablecoin adoption: A Bayesian approach," Economic Modelling, Elsevier, vol. 109(C).
    17. Inoue, Atsushi & Kilian, Lutz, 2020. "The uniform validity of impulse response inference in autoregressions," Journal of Econometrics, Elsevier, vol. 215(2), pages 450-472.
    18. Konstantin A. Kholodilin & Aleksei Netsunajev, 2016. "Crimea and Punishment: The Impact of Sanctions on Russian and European Economies," Discussion Papers of DIW Berlin 1569, DIW Berlin, German Institute for Economic Research.
    19. Helmut Lütkepohl & Anna Staszewska-Bystrova & Peter Winker, 2018. "Calculating joint confidence bands for impulse response functions using highest density regions," Empirical Economics, Springer, vol. 55(4), pages 1389-1411, December.
    20. Jonas E. Arias & Juan F. Rubio-Ramirez & Daniel F. Waggoner, 2020. "Uniform Priors for Impulse Responses," Working Papers 22-30, Federal Reserve Bank of Philadelphia.
    21. Hafner, Christian M. & Herwartz, Helmut & Wang, Shu, 2023. "Causal inference with (partially) independent shocks and structural signals on the global crude oil market," LIDAM Discussion Papers ISBA 2023004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    22. Paul Carrillo‐Maldonado, 2023. "Partial identification for growth regimes: The case of Latin American countries," Metroeconomica, Wiley Blackwell, vol. 74(3), pages 557-583, July.
    23. Lenard Lieb & Stephan Smeekes, 2017. "Inference for Impulse Responses under Model Uncertainty," Papers 1709.09583, arXiv.org, revised Oct 2019.
    24. Bojaj, Martin M. & Djurovic, Gordana & Fabris, Nikola & Milovic, Nikola, 2023. "Top 1% and inequality connectedness in the EMU and WB," International Review of Economics & Finance, Elsevier, vol. 83(C), pages 139-155.

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    More about this item

    Keywords

    Joint inference; Shotgun plots; Confidence bands; Impulse response shapes; Bootstrap; Degenerate limiting distribution;
    All these keywords.

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods

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