Confidence Bands for Impulse Responses: Bonferroni vs. Wald

Date01 December 2015
Published date01 December 2015
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
doi: 10.1111/obes.12114
Confidence Bands for Impulse Responses: Bonferroni
vs. Wald*
Helmut L ¨
utkepohl, Anna Staszewska-Bystrova‡ and Peter
DIW Berlin and Freie Universit¨at Berlin, Mohrenstr. 58, 10177 Berlin, Germany
University of Lodz, Rewolucji 1905r. 41, 90-214, Lodz, Poland
§University of Giessen, Licher Str. 64 Giessen, Germany
In impulse response analysis estimation uncertainty is typically displayed by constructing
bands around estimated impulse response functions. If they are based on the joint asymp-
totic distribution possibly constructed with bootstrap methods in a frequentist framework,
often individual confidence intervals are simply connected to obtain the bands. Such bands
are known to be too narrow and have a joint coverage probability lower than the desired
one. If instead the Wald statistic is used and the joint bootstrap distribution of the impulse
response coefficient estimators is taken into account and mapped into the band, it is shown
that such a band is typically rather conservative. It is argued that, by using the Bonferroni
method, a band can often be obtained which is smaller than the Wald band.
I. Introduction
The problem of constructing confidence bands around impulse responses of structural
VAR processes is discussed extensively in the literature (see, e.g. Sims and Zha, 1999;
Staszewska, 2007; Jord`a, 2009; L¨utkepohl, Staszewska-Bystrova and Winker, 2015). It is
pointed out in a number of studies that simply connecting individual confidence intervals
with the desired confidence level does not result in a band with a preassigned confidence
JEL Classification numbers: C10, C32
*The paper was presented at the Macromodels International Conference in Warsaw, 21–23 October 2013, at the
meeting of the Econometrics Committee of the Verein f¨ur Socialpolitik, Rauischholzhausen, 27 February–1 March
2014, the European Meeting of the Econometric Society in Toulouse, 25–29 August 2014, the 8th International
Conference on Computational and Financial Econometrics in Pisa, 6–12 December 2014, and at seminars at the New
Economic School in Moscow, 28 May2014, and Macquarie University in Sydney, 6 March 2015. Helpful comments
by the participants and in particular Karim Abadir are gratefully acknowledged.This paper was written while the first
author was a Bundesbank Professor at the Freie Universit¨at Berlin. Financial support was provided bythe Deutsche
Forschungsgemeinschaftthrough the SFB 649 ‘Economic Risk’, the National Science Center, Poland (NCN) through
2013/08/A/HS4/00612, and by an MNiSW/DAAD PPP grant (56268818).
Confidence bands: Bonferroni vs.Wald 801
level but will result in a band that is too narrow and contains the true impulse response
function with probability less than the desired confidence level. Thus, the problem arises
how to construct bands containing the true impulse response function with a preassigned
In a frequentist framework, one could construct an asymptotically valid confidence set
for the estimated VAR parameters for a given confidence level 1 based on the Wald
statistic. One could then consider the band that includes all impulse responses associated
with VAR parameters within the Wald confidence set. Such a strategy leads in fact to
conservative error bands for the impulse responses because the latter are constructed by
considering the area between the minimum and the maximum of the impulse responses
in the confidence set for each propagation horizon. To better understand this procedure,
let us consider just two impulse response coefficients jointly. The confidence band for the
two impulse responses corresponds to a box in the plane. It is defined by the smallest
box containing all impulse responses corresponding to the VAR parameters in the Wald
confidence set. Since impulse responses are nonlinear functions of the VAR parameters,
the image of the VAR parameters will not be a box in the impulse response space but
some other subset in the plane. Hence, the box might contain also other values than those
corresponding to the Wald confidence set for the VAR parameters and, consequently, it
is a conservative set. We show that such a confidence box may even have considerably
more probability content than a set constructed according to the standard Bonferroni prin-
In the following, we first formally compare Wald and Bonfer roni confidence bands
for impulse responses. We demonstrate that in an idealized setting, a confidence band
around impulse responses constructed with Bonferroni’s method may be smaller than the
corresponding set obtained by drawing a box around the confidence region of parameter
estimates obtained via the Wald statistics and projected into the impulse response space.
While both methods are conservative, we show that Bonferroni may result in smaller error
bands than using the Wald statistic.We also point out adjustments to both Bonferroni and
Wald bands that have more precise coverage and at the same time are smaller than the
unadjusted, conservative methods.
Since we have to rely on asymptotic arguments in a frequentist setting, a simulation
experiment is carried out to study the small sample implications of our asymptotic re-
sults. We also illustrate the methods by a structural VAR analysis of the market for crude
oil from Kilian (2009). The example shows that the method for constructing impulse
responses is important in practice because alternative methods may lead to different con-
The structure of this study is as follows. In the next section, the Wald and Bonferroni
bands are first reviewed and compared in a general but idealized setting based on a normal
distribution assumption for the parameter estimators. The advantage of using such an
idealized setting is that analytical results can be obtained. Then the bands are placed
in a more realistic setting where only asymptotic normality is obtained in a frequentist
framework. Impulse response analysis for structural vector autoregressive (VAR) models
is considered as a specific area where the results are relevant. In section III, a Monte
Carlo investigation of error bands for impulse responses is presented and an example is
considered in section IV. Section V concludes and mentions possible extensions.
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd

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