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Joint Confidence Sets for Structural Impulse Responses

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  • Atsushi Inoue

    ()
    (Southern Methodist University)

  • Lutz Kilian

    ()
    (University of Michigan)

Abstract

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 Classification-JEL: C32, C52, C53.

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Bibliographic Info

Paper provided by Southern Methodist University, Department of Economics in its series Departmental Working Papers with number 1401.

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Date of creation: Feb 2014
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Handle: RePEc:smu:ecowpa:1401

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Postal: Department of Economics, P.O. Box 750496, Southern Methodist University, Dallas, TX 75275-0496
Phone: 214-768-2715
Fax: 214-768-1821
Web page: http://www.smu.edu/economics

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Keywords: Confidence Bands; Simultaneous Inference; Singular Covariance Matrix.;

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References

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  1. Helmut Lütkepohl & Anna Staszewska-Bystrova & Peter Winker, 2014. "Confidence Bands for Impulse Responses: Bonferroni versus Wald," Discussion Papers of DIW Berlin 1354, DIW Berlin, German Institute for Economic Research.
  2. Andrews, Donald W. K., 1987. "Asymptotic Results for Generalized Wald Tests," Econometric Theory, Cambridge University Press, vol. 3(03), pages 348-358, June.
  3. Christopher A. Sims & Tao Zha, 1999. "Error Bands for Impulse Responses," Econometrica, Econometric Society, Econometric Society, vol. 67(5), pages 1113-1156, September.
  4. Lutz Kilian, 1998. "Small-Sample Confidence Intervals For Impulse Response Functions," The Review of Economics and Statistics, MIT Press, vol. 80(2), pages 218-230, May.
  5. Bao, Yong & Ullah, Aman, 2007. "The second-order bias and mean squared error of estimators in time-series models," Journal of Econometrics, Elsevier, vol. 140(2), pages 650-669, October.
  6. Barsky, Robert & Kilian, Lutz, 2004. "Oil and the Macroeconomy Since the 1970s," CEPR Discussion Papers 4496, C.E.P.R. Discussion Papers.
  7. Helmut Lütkepohl & Anna Staszewska-Bystrova & Peter Winker, 2013. "Comparison of Methods for Constructing Joint Confidence Bands for Impulse Response Functions," MAGKS Papers on Economics, Philipps-Universität Marburg, Faculty of Business Administration and Economics, Department of Economics (Volkswirtschaftliche Abteilung) 201325, Philipps-Universität Marburg, Faculty of Business Administration and Economics, Department of Economics (Volkswirtschaftliche Abteilung).
  8. Blanchard, Olivier Jean, 1989. "A Traditional Interpretation of Macroeconomic Fluctuations," American Economic Review, American Economic Association, vol. 79(5), pages 1146-64, December.
  9. Kilian, Lutz, 2001. "Impulse Response Analysis in Vector Autoregressions with Unknown Lag Order," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 20(3), pages 161-79, April.
  10. Atsushi Inoue & Lutz Kilian, 2013. "Inference on Impulse Response Functions in Structural VAR Models," TERG Discussion Papers, Graduate School of Economics and Management, Tohoku University 307, Graduate School of Economics and Management, Tohoku University.
  11. Lütkepohl, Helmut & Poskitt, D.S., 1991. "Estimating Orthogonal Impulse Responses via Vector Autoregressive Models," Econometric Theory, Cambridge University Press, vol. 7(04), pages 487-496, December.
  12. Lutkepohl, Helmut, 1990. "Asymptotic Distributions of Impulse Response Functions and Forecast Error Variance Decompositions of Vector Autoregressive Models," The Review of Economics and Statistics, MIT Press, vol. 72(1), pages 116-25, February.
  13. Bertille Antoine & Eric Renault, 2012. "Efficient Minimum Distance Estimation with Multiple Rates of Convergence," Discussion Papers dp12-03, Department of Economics, Simon Fraser University.
  14. �scar Jordà, 2009. "Simultaneous Confidence Regions for Impulse Responses," The Review of Economics and Statistics, MIT Press, vol. 91(3), pages 629-647, August.
  15. Lutz Kilian & Yun Jung Kim, 2011. "How Reliable Are Local Projection Estimators of Impulse Responses?," The Review of Economics and Statistics, MIT Press, vol. 93(4), pages 1460-1466, November.
  16. Lutz Kilian, 1999. "Finite-Sample Properties of Percentile and Percentile-t Bootstrap Confidence Intervals for Impulse Responses," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 652-660, November.
  17. Kilian, Lutz & Chang, Pao-Li, 2000. "How accurate are confidence intervals for impulse responses in large VAR models?," Economics Letters, Elsevier, vol. 69(3), pages 299-307, December.
  18. Lutz Kilian, 2008. "A Comparison of the Effects of Exogenous Oil Supply Shocks on Output and Inflation in the G7 Countries," Journal of the European Economic Association, MIT Press, MIT Press, vol. 6(1), pages 78-121, 03.
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Cited by:
  1. 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.

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