Skewness-adjusted bootstrap confidence intervals and confidence bands for impulse response functions

  • Daniel GrabowskiEmail author
  • Anna Staszewska-Bystrova
  • Peter Winker
Original Paper


Inference on impulse response functions from vector autoregressive models is commonly done using bootstrap methods. These methods can be inaccurate in small samples and for persistent processes. This article investigates the construction of skewness-adjusted confidence intervals and joint confidence bands for impulse responses with improved small sample performance. We suggest to adjust the skewness of the bootstrap distribution of the autoregressive coefficients before the impulse response functions are computed. Using extensive Monte Carlo simulations, the approach is shown to improve the coverage accuracy in small- and medium-sized samples and for unit-root processes.


Bootstrap Confidence intervals Joint confidence bands Vector autoregression 

JEL Classifications

C15 C32 



  1. Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control 19(6), 716–723 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Basawa, I.V., Mallik, A.K., Cormick, W.P., Reeves, J.H., Taylor, R.L.: Bootstrapping unstable first-order autoregressive processes. Ann. Stat. 19, 1098–1101 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Benkwitz, A., Neumann, M.H., Lütekpohl, H.: Problems related to confidence intervals for impulse responses of autoregressive processes. Econom. Rev. 19(1), 69–103 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Berkowitz, J., Kilian, L.: Recent developments in bootstrapping time series. Econom. Rev. 19(1), 1–48 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bose, A.: Edgeworth correction by bootstrap in autoregressions. Ann. Stat., 1709–1722 (1988)Google Scholar
  6. Efron, B.: Nonparametric standard errors and confidence intervals. Can. J. Stat. 9, 139–172 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Efron, B., Tibshirani, R.: Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Stat. Sci. 1(1), 54–75 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Grabowski, D., Staszewska-Bystrova, A., Winker, P.: Generating prediction bands for path forecasts from SETAR models. Stud. Nonlinear Dyn. Econom. 21(5) (2017)Google Scholar
  9. Hall, P.: The Bootstrap and Edgeworth Expansion. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  10. Hurvich, C.M., Tsai, C.L.: Regression and time series model selection in small samples. Biometrika 76, 297–307 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hurvich, C.M., Tsai, C.L.: A corrected Akaike information criterion for vector autoregressive model selection. J. Time Ser. Anal. 14, 272–279 (1993)MathSciNetzbMATHGoogle Scholar
  12. Inoue, A., Kilian, L.: Bootstrapping autoregressive processes with possible unit roots. Econometrica 70(1), 377–391 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Inoue, A., Kilian, L.: The continuity of the limit distribution in the parameter of interest is not essential for the validity of the bootstrap. Econom. Theory 19(6), 944–961 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Inoue, A., Kilian, L.: Inference on impulse response functions in structural VAR models. J. Econom. 177(1), 1–13 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Inoue, A., Kilian, L.: Joint confidence sets for structural impulse responses. J. Econom. 192(2), 421–432 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kilian, L.: Small-sample confidence intervals for impulse response functions. Rev. Econ. Stat. 80(2), 218–230 (1998a)CrossRefGoogle Scholar
  17. Kilian, L.: Accounting for lag order uncertainty in autoregressions: the endogenous lag order bootstrap algorithm. J. Time Ser. Anal. 19(5), 531–548 (1998b)CrossRefzbMATHGoogle Scholar
  18. Kilian, L.: Finite-sample properties of percentile and percentile-t bootstrap confidence intervals for impulse responses. Rev. Econ. Stat. 81(4), 652–660 (1999)CrossRefGoogle Scholar
  19. Kilian, L.: Impulse response analysis in vector autoregressions with unknown lag order. J. Forecast. 20, 161–179 (2001)CrossRefGoogle Scholar
  20. Kilian, L., Lütkepohl, H.: Structural Vector Autoregressive Analysis. Cambridge University Press, Cambridge (2017)CrossRefzbMATHGoogle Scholar
  21. Kim, J.H.: Bias-corrected bootstrap prediction regions for vector autoregression. J. Forecast. 23(2), 141–154 (2004)MathSciNetCrossRefGoogle Scholar
  22. Lütkepohl, H.: Comparison of criteria for estimating the order of a vector autoregressive process. J. Time Ser. Anal. 6, 35–52 (1985)MathSciNetCrossRefGoogle Scholar
  23. Lütkepohl, H.: Asymptotic distributions of impulse response functions and forecast error variance decompositions of vector autoregressive models. Rev. Econ. Stat. 72(1), 116–125 (1990)CrossRefGoogle Scholar
  24. Lütkepohl, H.: New Introduction to Multiple Time Series Analysis. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  25. Lütkepohl, H., Staszewska-Bystrova, A., Winker, P.: Comparison of methods for constructing joint confidence bands for impulse response functions. Int. J. Forecast. 31(3), 782–798 (2015)CrossRefGoogle Scholar
  26. Lütkepohl, H., Staszewska-Bystrova, A., Winker, P.: Confidence bands for impulse responses: Bonferroni vs. Wald. Oxf. Bull. Econ. Stat. 77, 800–821 (2015)CrossRefGoogle Scholar
  27. Pope, A.L.: Biases of estimators in multivariate non-Gaussian autoregressions. J. Time Ser. Anal. 11(3), 249–258 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Sims, C.A., Zha, T.: Error bands for impulse responses. Econometrica 67(5), 1113–1155 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Staszewska, A.: Representing uncertainty about response paths: the use of heuristic optimisation methods. Comput. Stat. Data Anal. 52(1), 121–132 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Staszewska-Bystrova, A., Winker, P.: Constructing narrowest pathwise bootstrap prediction bands using Threshold Accepting. Int. J. Forecast. 29(2), 221–233 (2013)CrossRefGoogle Scholar
  31. Staszewska-Bystrova, A., Winker, P.: Measuring forecast uncertainty of corporate bond spreads by Bonferroni-type prediction bands. Cent. Eur. J. Econ. Model. Econom. 2, 89–104 (2014)zbMATHGoogle Scholar
  32. Stine, R.A.: Estimating properties of autoregressive forecasts. J. Am. Stat. Assoc. 82(400), 1072–1078 (1987)MathSciNetCrossRefGoogle Scholar
  33. Sugiura, N.: Further analysis of the data by Akaike’s information criterion and the finite corrections. Commun. Stat. A7, 13–26 (1978)CrossRefzbMATHGoogle Scholar
  34. Wolf, M., Wunderli, D.: Bootstrap joint prediction regions. J. Time Ser. Anal. 36(3), 352–376 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of GiessenGiessenGermany
  2. 2.University of LodzLodzPoland

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