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

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Abstract

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.

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Notes

  1. Berkowitz and Kilian (2000, p. 30) conjecture that the poor performance of the bootstrap confidence intervals in the presence of unit roots and roots close to unity is due to small sample bias. The evidence presented in this article suggests that the skewness plays an important role.

  2. Wolf and Wunderli (2015) propose their method for prediction regions, but this can also be adapted to confidence regions for IRFs, see Lütkepohl et al. (2015a).

  3. The AIC has a relatively high accuracy in identifying the correct lag order in finite samples (Lütkepohl 1985; Kilian 2001).

  4. Simulations of DGPs with higher lag orders than considered here or by Kilian (1998b) indicate that the AICc might provide to low estimated lag orders and the AIC might be preferable. Research into the optimal information criterion for bootstrap confidence intervals and bands at different sample sizes may be helpful.

  5. Note that the 100% coverage in the initial period in the third quadrant is due to this IRF being restricted to zero by the Cholesky decomposition, which is in accordance with the assumed DGP.

  6. This supposes that we equally dislike positive and negative deviations from the nominal coverage level. Arguably, too low coverages might be considered as a more severe violation of the idea underlying the construction of confidence intervals.

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Authors and Affiliations

  1. Department of Economics, University of Giessen, Licher Str. 64, 35394, Giessen, Germany

    Daniel Grabowski & Peter Winker

  2. University of Lodz, 41 Rewolucji 1905r. Str., 90-214, Lodz, Poland

    Anna Staszewska-Bystrova

Authors
  1. Daniel Grabowski
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  2. Anna Staszewska-Bystrova
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  3. Peter Winker
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Corresponding author

Correspondence to Daniel Grabowski.

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We are indebted to Lutz Kilian and Helmut Lütkepohl for helpful comments on an earlier draft of this paper. We also wish to thank participants of the 2018 European Summer Meeting of the Econometric Society and of the Jahrestagung des Vereins für Socialpolitik 2018 for helpful discussions. The authors are responsible for all remaining shortcomings. Support from the National Science Center, Poland (NCN) through MAESTRO 4: DEC-2013/08/A/HS4/00612 is gratefully acknowledged.

Appendix

Appendix

See Tables 7, 8, 9, 10, 11, 12, 13, 14, and 15

Table 7 Mean coverage frequencies (in %) for nominal 95% confidence intervals over different parameter settings for \(\alpha _{11}\), over periods \(h{=}0,\ldots ,10\) and over the four impulse responses in a two-dimensional VAR with an unknown lag order
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Table 8 Root mean squared coverage errors (RMSCEs) (in percentage points) for nominal 95% confidence intervals over different parameter settings for \(\alpha _{11}\), over periods \(h{=}0,\ldots ,10\) and over the four impulse responses in a two-dimensional VAR with an unknown lag order
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Table 9 Mean widths for nominal 95% confidence intervals over different parameter settings for \(\alpha _{11}\), over periods \(h{=}0,\ldots ,10\) and over the four impulse responses in a two-dimensional VAR with an unknown lag order
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Table 10 Mean coverage frequencies (in percent) for nominal 95% confidence intervals over different parameter settings for \(\alpha _{11}\), over periods \(h{=}0,\ldots ,10\) and over the four impulse responses in a two-dimensional VAR with chi-squared distributed errors
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Table 11 Root mean squared coverage errors (RMSCEs) (in percentage points) for nominal 95% confidence intervals over different parameter settings for \(\alpha _{11}\), over periods \(h{=}0,\ldots ,10\) and over the four impulse responses in a two-dimensional VAR with chi-squared distributed errors
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Table 12 Mean widths for nominal 95% confidence intervals over different parameter settings for \(\alpha _{11}\), over periods \(h{=}0,\ldots ,10\) and over the four impulse responses in a two-dimensional VAR with chi-squared distributed errors
Full size table
Table 13 Mean coverage frequencies (in percent) for nominal 95% joint confidence bands over different parameter settings for \(\alpha _{11}\) and over the four impulse responses in a two-dimensional VAR with an unknown lag order
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Table 14 Root mean squared coverage errors (RMSCEs) (in percentage points) for nominal 95% joint confidence bands over different parameter settings for \(\alpha _{11}\) and over the four impulse responses in a two-dimensional VAR with an unknown lag order
Full size table
Table 15 Mean widths for nominal 95% joint confidence bands over different parameter settings for \(\alpha _{11}\) and over the four impulse responses in a two-dimensional VAR with an unknown lag order
Full size table

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Grabowski, D., Staszewska-Bystrova, A. & Winker, P. Skewness-adjusted bootstrap confidence intervals and confidence bands for impulse response functions. AStA Adv Stat Anal 104, 5–32 (2020). https://doi.org/10.1007/s10182-018-00347-9

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