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Joint Confidence Sets for Structural Impulse Responses


  • Inoue, Atsushi
  • Kilian, Lutz


Many users of structural VAR models are primarily interested in learning about the shape of structural impulse response functions. This 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 structural impulse response 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 demonstrate by simulation the coverage accuracy of these sets in finite samples under realistic conditions. We make the case for representing these 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 provides economically relevant additional information about the shape of response functions that is lost when reducing the joint confidence set to two-dimensional bands.

Suggested Citation

  • Inoue, Atsushi & Kilian, Lutz, 2014. "Joint Confidence Sets for Structural Impulse Responses," CEPR Discussion Papers 9892, C.E.P.R. Discussion Papers.
  • Handle: RePEc:cpr:ceprdp:9892

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    References listed on IDEAS

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    Cited by:

    1. Nikola Kutin & Zakaria Moussa & Thomas Vallée, 2018. "Factors behind the Freight Rates in the Liner Shipping Industry," Working Papers halshs-01828633, HAL.
    2. 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, CEJEME, vol. 6(2), pages 89-104, June.
    3. 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.
    4. 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.
    5. Skrobotov, Anton & Turuntseva, Marina, 2015. "Theoretical Foundations of SVAR Modeling," Published Papers mak8, Russian Presidential Academy of National Economy and Public Administration.
    6. Lenard Lieb & Stephan Smeekes, 2017. "Inference for Impulse Responses under Model Uncertainty," Papers 1709.09583,, revised May 2018.

    More about this item


    Bootstrap; Confidence regions; Degenerate limiting distribution; Impulse response shapes; Joint inference; Shotgun plots;

    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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