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I have a question regarding the confidence bands of impulse response functions. I see that some empirical studies report confidence bands around impulse response functions either as 95% confidence bands or as +/- 2 standard errors (SEs) confidence bands.

As I am not well educated in statistics, I want to ask whether both ways lead to the same results or if there is a difference between the two procedures?

Thanks for your help!

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  • $\begingroup$ Others may contradict me, but I prefer CIs over SE, because 2*SE are close to CIs but not quite the same (e.g. z*=1.96). The CI is more intuitive to interpret for most people. $\endgroup$
    – Carol Eisen
    Commented Apr 19, 2017 at 17:54
  • $\begingroup$ It is a property of the Normal distribution that the interval from -1.96 SE + mean to 1.96 SE + mean covers 95% of the distribution. People often round this off to 2. The confidence intervals for the mean are constructed using the sample estimates in place of the population parameters. But this does not apply in general to other distributions. $\endgroup$
    – Michael R. Chernick
    Commented Apr 19, 2017 at 17:54

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Caveat: I'm not an expert in this field.

If the theoretical distribution of sample impulse response function (IRF) is Gaussian (that is, at every time point the distribution of errors is Gaussian) then 1.96 standard errors covers about 95.0% of the distribution of the sample IRF, and 2 standard errors covers about 95.4% of the distribution of the sample IRF. As Michael Chernick pointed out in the comments, this is "close enough" that people use ±2 standard errors to mean "95% theoretical coverage". Note that the IRF for a typical Vector Autoregression model is asymptotically Gaussian [1].

If you imagine testing, at every time point, the hypothesis that the IRF is zero against the alternative that the IRF is nonzero, and if you assume that the test statistic has a Gaussian distribution, then a theoretical 95% confidence interval is equivalent to the 95% theoretical coverage interval described above.

So under a typical set of assumptions, they're more or less the same. But there are several ways to compute a confidence interval for an impulse response function [2], including simulation-based methods. These could in principle lead to a 95% confidence interval that does not line up with the theoretical Gaussian 95% coverage interval.

[1] Cf. lecture notes by Luca Gambetti, http://pareto.uab.es/lgambetti/VAR_Forecasting.pdf, citing: Hamilton, James D., 1994, Time Series Analysis, Princeton University Press.

[2] Griffiths, William and Lütkepohl, Helmut. (1990). Confidence intervals for impulse response functions: a comparison of asymptotic theory and simulation approaches. https://www.une.edu.au/__data/assets/pdf_file/0018/20484/emetwp42.pdf.

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