Discussion Paper No. 11
Inference on Impulse Response Functions in
Structural VAR Models

Atsushi Inoue
Lutz Kilian

July 13,2013

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANInference on Impulse Response Functions in
Structural VAR Models∗Atsushi Inoue†

Lutz Kilian‡Southern Methodist University University of Michigan
Tohoku University

July 13, 2013

Abstract
Skepticism toward traditional identifying assumptions based on exclusion restrictions has
led to a surge in the use of structural VAR models in which structural shocks are identified
by restricting the sign of the responses of selected macroeconomic aggregates to these shocks.
Researchers commonly report the vector of pointwise posterior medians of the impulse responses
as a measure of central tendency of the estimated response functions, along with pointwise 68
percent posterior error bands. It can be shown that this approach cannot be used to characterize
the central tendency of the structural impulse response functions. We propose an alternative
method of summarizing the evidence from sign-identified VAR models designed to enhance theirpractical usefulness. Our objective is to characterize the most likely admissible model(s) within
the set of structural VAR models that satisfy the sign restrictions. We show how the set of
most likely structural response functions can be computed from the posterior mode of the joint
distribution of admissible models both in the fully identified and in the partially identified case,and we propose a highest-posterior density credible set that characterizes the joint uncertainty
about this set. Our approach can also be used to resolve the long-standing problem of how
to conduct joint inference on sets of structural impulse response functions in exactly identified
VAR models. We illustrate the diﬀerences between our approach and the traditional approach∗

We thank Harald Uhlig for providing the data used in one of the empirical applications. We also thank Christiane
Baumeister, Alastair Hall, Sujit Ghosh, Dan Phaneuf, Frank Schorfheide, Tao Zha, and Ren Zhang for helpful
discussions.
†
Department of Economics, Southern Methodist University, 3300 Dyer Street, Dallas, TX 75275-0496. Email:ainoue@mail.smu.edu.
‡
University of Michigan, Department of Economics, 238 Lorch Hall, Ann Arbor, MI 48109-1220. E-mail: lkilian@umich.edu.

for the analysis of the eﬀects of monetary policy shocks and of the eﬀects of oil demand and oil
supply shocks.
JEL: C32, C52; E37
KEYWORDS: Vector Autoregression; Simultaneous Inference; Impulse Responses; Sign Restrictions; Median; Mode; Credible Set.

1

Introduction

One of the most widely studied questions in empirical macroeconomics is to what extent an
unanticipated monetary tightening reduces real output. It is widely recognized that answering
this question requires the estimation of structural models in which cause and eﬀect are clearly
diﬀerentiated. Much of the literature since the 1980s has relied on structural vector autoregressive models in which exclusion restrictions on the instantaneous feedback from monetary policy
shocks to macroeconomic aggregates ensure the identification of the policy shock. Skepticism
toward these traditional identifying assumptions in recent years has made increasingly popularan alternative class of structural VAR models in which policy shocks are identified by restricting
the sign of the responses of selected macroeconomic aggregates to policy shocks. For example,
Uhlig (2005) postulated that an unexpected monetary policy contraction is associated with anincrease in the federal funds rate, the absence of price increases and the absence of increases innonborrowed reserves for some time following the policy shock. This approach is considerably
more agnostic than traditional identification approaches in some dimensions, while more restrictive in others. Uhlig showed that sign-identified models may produce substantially diﬀerent
results from conventional structural VAR models.
Although the original applications of this approach were to models of monetary policy, signidentified VAR models have become increasingly popular in other areas as well and are now partof the mainstream of empirical macroeconomics. They have been used, for example, to study
fiscal shocks (e.g., Canova and Pappa 2007; Mountford and Uhlig 2009, Pappa 2009; Caldara
2011), technology shocks (e.g., Dedola and Neri 2007; Peersman and Straub 2009), and various
other shocks in open economies (e.g., Canova and De Nicolo 2002; Scholl and Uhlig 2008),
shocks in oil markets (e.g., Baumeister and Peersman 2012; Kilian and Murphy 2012a,b; Lippi
and Nobili 2012), and shocks in labor markets (e.g., Fujita 2011).
In all these applications, the cost of remaining agnostic about the structural model is that thedata are potentially consistent with a wide range of structural models that are all admissible inthat they satisfy the identifying restrictions. An unresolved question in the literature is how torepresent the results of such agnostic identification procedures when the set of admissible models
includes a range of models with conflicting interpretations. One early approach, exemplified by
Faust (1998), has been to focus on the model that is most favorable to the hypothesis of interest.
This allows us to establish the extent to which this hypothesis could potentially explain the data.
It may also help us to rule out a hypothesized explanation, if none of the admissible models
supports this hypothesis. Such cases are are rare, however. Typically, the set of admissible
structural models includes both models that are supportive with a given hypothesis and models
that are at odds with that hypothesis. In the latter case, Faust’s approach is not informative
about whether any one of the admissible structural models is a more likely explanation of the1

data than some other model.
The standard approach of dealing with this problem has been to report the vector of pointwise
posterior medians of the impulse responses as a measure of the central tendency of the impulse
response functions, along with pointwise 68% posterior error bands. This approach suﬀers fromtwo distinct shortcomings. The first shortcoming is that the vector of pointwise posterior median
responses will have no structural economic interpretation unless the pointwise posterior mediansof all impulse response coeﬃcients in the VAR system correspond to the same structural model,
which is highly unlikely a priori (see, e.g., Fry and Pagan 2011). In other words, in practice,none of the models in the set of admissible structural models constructed by the researcher will
exhibit the impulse response dynamics embodied in the median response function obtained byconnecting the dots between the pointwise posterior medians. The second shortcoming is thatmedian response functions are not a valid measure of the central tendency of the admissible set of
impulse response functions. It is well known that the median of a vector variable is not the vectorof the medians, rendering the vector of pointwise medians inappropriate as a statistical measureof the central tendency of the impulse response functions (e.g., Chauduri 1996; Koltchinskii 1997;
Liu, Parelius and Singh 1999). This means that even if there were an admissible structural modelwith the same impulse response function as the median response function, there would be nocompelling reason to focus on this model in interpreting the evidence.1
Similar problems arise in the construction of pointwise impulse response error bands insign-identified models based on the quantiles of the marginal posterior distributions of theimpulse responses. Moreover, these error bands do not take account of the dependence of
the impulse response estimates across horizons and across response functions and hence may
overstate or understate the true uncertainty about the dynamics of the system. The latterproblem is well known, but there are no alternative methods in the literature that address theselimitations, leaving researchers with little choice, but to rely on potentially misleading measuresof uncertainty.
In this paper, we propose a new method of summarizing the evidence from sign-identified
VAR models that addresses these shortcomings and is designed to enhance the practical usefulness of sign-identified models. Our objective is to identify the most likely admissible model(s)within the set of structural VAR models that satisfy the sign restrictions. A structural VAR
model is defined by the set of structural impulse responses associated with a given set of reducedform VAR parameters and a given structural impact multiplier matrix. There is a one-to-one
1

This second point is also relevant for a recent proposal by Fry and Pagan (2011) designed to overcome the lack ofstructural interpretation of median response functions. Their idea is to search for the admissible structural model with
impulse response functions closest to the median response functions. This proposal deals with the first shortcoming
highlighted above, but not with the second. Because the median vector is not a well defined statistical measure ofcentral tendency, there is no compelling reason to focus on a structural model with response functions close to thevector of medians.

2mapping from the joint posterior density of these model parameters to the joint posterior densityof the corresponding set of structural impulse responses (up to a horizon that depends on the
autoregressive lag order), allowing us to derive the latter density analytically by the change-ofvariable method. This enables us to assign a posterior density value to each structural model.
The most likely or modal model by construction is the admissible model that maximizes thejoint posterior density of the admissible structural VAR models. A 100(1 − )% highest posterior density credible set of admissible models may be formed by ranking the admissible models
based on the value of this joint density.
In practice, we proceed in two steps.

Under the conventional assumption of a diﬀuse

Gaussian-inverse Wishart prior, we begin by generating repeated draws from the joint posterior distribution of the reduced-form VAR parameters and of the rotation matrices used inconstructing the structural impact multiplier matrix. For each candidate structural VAR model,
we first compute the posterior density value associated with that model; we then evaluate theset of implied structural impulse response functions. We discard the structural models that areinadmissible in that their responses do not satisfy the identifying sign restrictions. We thenrank the remaining structural models by the value of their posterior density, making it straightforward to determine the most likely admissible model and to characterize its impulse response
dynamics. The set of structural impulse response functions associated with the modal admissible model by construction will be economically interpretable and statistically well defined,
addressing the two main critiques of traditional median response functions.
This baseline procedure is designed for fully identified structural VAR models. Many structural VAR models, however, are only partially identified in that only a subset of the structural
shocks are identified. Such models are sometimes also referred to as semi-structural VAR models. For example, in the model of Uhlig (2005) only the responses to monetary policy shocks
are identified. In this case, responses to unidentified shocks become irrelevant in constructing
the modal model. Instead the mode and credible set must be based on the marginal posterior
density of the subset of impulse response functions of interest. We propose a modification of
our baseline procedure that accomplishes this task. Marginalizing the joint density requires
Monte Carlo integration, which renders this procedure computationally more challenging thanthe baseline procedure for fully identified models.
Although our approach was designed to aid in the interpretation of impulse response dynamics in sign-identified models, essentially the same approach can also be used to resolve thelong-standing problem of how to conduct joint inference on sets of structural impulse response
functions in exactly identified models. It is well known that the pointwise error bands commonly
attached to the structural impulse response functions in exactly identified VAR models fail toconvey the true uncertainty surrounding these impulse response functions. This problem has

3been long recognized, but few practical alternatives have been proposed in the literature andnone of these alternative methods are applicable to sign-identified models.2 Our final contribution is to show how a simplified version of our proposed procedure may be used to constructcredible sets for the structural impulse response functions for exactly identified structural VAR
models, providing a convenient alternative to traditional pointwise error bands that is easy to
implement. As in the earlier analysis, our approach can accommodate both fully identified andpartially identified models.
Table 1 summarizes the four distinct types of structural VAR models considered in thispaper. The remainder of the paper is organized as follows. In section 2, we describe theproposed procedure and its implementation in more detail. Although Uhlig’s model focuses on
identifying only the monetary policy shock, many related studies have used sign restrictions
to identify simultaneously a variety of macroeconomic shocks (e.g., Canova and Paustian 2011;
Gambetti, Pappa and Canova 2008). Our analysis allows for that situation. Our analysis also
allows for refinements of the sign restriction approach in the form of additional bounds onelasticities, for example, or of bounds on cross-correlations (e.g., Canova and De Nicolo 2002;
Kilian and Murphy 2012a). These modifications do not aﬀect the substance of our method andcan be easily incorporated.
In section 3, we illustrate the limitations of traditional median response functions. The
feasibility and usefulness of our alternative approach is demonstrated in section 4. In section 4.1,
we consider fully identified structural VAR models subject to sign restrictions on the impulse
responses. We focus on the example of a model of the global market for crude oil in the
tradition of Baumeister and Peersman (2012) and Kilian and Murphy (2012a). We show thatthe modal model generates economically plausible responses. The 68 percent joint credible sets
tend to be wider than conventional 68 percent pointwise impulse response error bands, butmany responses are fairly precisely estimated nevertheless with credible sets that exclude thezero line at some or all horizons. We demonstrate that the responses in the modal model canbe substantially diﬀerent from conventional median response functions. The bias in the median
response functions can be upward or downward. In many cases, the responses of the modal
model are outside of the conventional 68 percent pointwise posterior error bands.
In section 4.2, we consider a partially identified structural VAR model subject to sign restrictions, building on the analysis of U.S. monetary policy in Uhlig (2005). We explore in particular
the question of what the eﬀect is of an unanticipated monetary policy tightening on real U.S.
output. We show that the method of summarizing the evidence matters. For example, Uhlig
reported a peak median output response of 0.15 for this model. For the same data, we obtain a
2

Sims and Zha (1999), for example, caution against connecting the dots representing pointwise error bands anddiscuss possible alternative strategies for exactly identified models. Related work based on joint asymptotic approximations for exactly identified models includes Lütkepohl (1990) and Jorda (2009).

4

peak response of almost 0.5 based on the modal model. Moreover, that modal model estimate
is near the upper end of the credible set and outside the conventional pointwise posterior error
band.
Both the median estimate and the response estimate based on the modal model are counterintuitive in that a monetary tightening would be expected to cause a decline in real output overtime rather than an increase. This outcome reflects the fact that the identifying assumptions
are not overly informative. Even in Uhlig’s original analysis, there was substantial pointwise
probability mass on both negative and positive responses of real output. Our 68 percent credible
set further widens the set of probable responses. The explicit reason that Uhlig (2005) did not
impose further restrictions is that he wished to be as agnostic as possible about the responseof real output. This approach is valid only to the extent that we view models in which real
output increases in response to a monetary tightening as economically plausible a priori (seeKilian and Murphy 2012a). Many economists would disagree with this view at least for intermediate horizons. Hence, we also considered an alternative model that imposes an additionalsign restriction on the response of real GDP after 6 months (and only at that horizon). This
identifying assumption leaves the short-run as well as the longer-run response of real output
unrestricted, preserving the spirit of Uhlig’s original exercise.
The resulting modal model produces substantially diﬀerent and more economically plausible results, including a cumulative drop in real GDP of -0.3 percentage points in the second
quarter. The response estimate for the modal model is at the lower end of the credible set andagain outside the conventional pointwise posterior error band. It also is substantially diﬀerent
from the response estimate obtained from the traditional Cholesky decomposition. Even inthis alternative model, however, the 68 percent credible set includes many positive real output
responses, suggesting that the data are not very informative about the response of real output.
We conclude that there remains substantial uncertainty about the eﬀects of monetary policy
shocks on real output, whereas there is strong evidence of the eﬀects of oil demand shocks on
the real price of oil in the earlier example.
In section 5, we show that our approach of constructing joint credible sets for the structural
impulse response functions can also be adapted to exactly identified structural VAR models,
providing a convenient alternative to traditional pointwise error bands that is easy to implement.
For standard semi-structural monetary policy VAR models of the type considered by Uhlig(2005) as a benchmark, we show that properly accounting for the joint uncertainty about all
impulse responses renders the impulse response estimates less informative than conventional
pointwise error bands suggest. For example, evidence of the price puzzle vanishes. On the other
hand, several of the response functions including that of real GDP remain precisely enoughestimated for the VAR model to be economically informative. In addition, we illustrate that,5

for the reasons already discussed, commonly used posterior median response functions may diﬀer
from the response functions in the modal posterior model. The concluding remarks are in section
6. Some technical details of the proposed procedure can be found in the technical appendix.

2

Evaluating the Posterior of Sign-Identified VAR Models

2.1

Preliminaries

Consider the -variate reduced-form VAR(p) model: =  + 1 −1 + 2 −2 + · · · +  − +  

(1)



for  = 1   where  ∼  (0×1  Σ) and Σ is positive definite. Write (1) as =  + 

(2)

0
0
where  = [1 2 · · ·  ]0 ,  = [1 2 · · ·  ]0 ,  = [1 −1
· · · −
]0 ,  = [ 1 · · ·  ]0 ,and  = [1 2 · · ·  ]0 .

Throughout this paper, we follow the conventional approach of specifying a normal-inverse
Wishart prior distribution for the reduced-form VAR parameters:

vec()|Σ ∼  (vec(̄0 ) Σ ⊗ 0−1 )Σ ∼  ( 0 0   0 )where 0 is an  ×  positive definite matrix, 0 is an  ×  covariance matrix, and  0  0.
Then the posterior is given byvec()|Σ ∼  (vec(̄ ) Σ ⊗ −1 )

(3)Σ ∼  (      )(4)

where   =  + ,  = 0 +  0 , ̄ = −1 (0 ̄0 +  0  ̂),
 =

0

1
0 +
(̂ − ̄0 )0 0 −1  0 (̂ − ̄0 )
Σ̂ +




̂ = ( 0 )−1  0  and Σ̂ = ( −  ̂)0 ( −  ̂) .
Define  =  where  is the lower triangular Cholesky decomposition of Σ, such that0 = Σ and  is an orthonormal matrix.  is a potential solution for the unknown structural6

0impact multiplier matrix. Note that  =   0 0 = 0 = Σ.  is obtained by drawing
from a prior distribution that is independent of the distribution of the reduced-form parameters.
Following Uhlig (2005), the prior distribution for  is a uniform distribution defined on the spaceof orthonormal matrices  . Consider a set of  ×  matrices consisting of orthonormal column
vectors:U =©
ª = [1 2 · · ·  ]0 :  ∈ <×1       = 1( = ) ∀   = 1 2   

where  · ·  denotes the inner product and 1(·) is the indicator function. Let  denote a draw

from the uniform distribution over U. By construction,   0 =  .

When all structural shocks in the model are identified, we say that the model is fully identified; when only a subset of the structural shocks is identified, we say that the model is partially
identified. For further discussion see, e.g., Rubio-Ramirez, Waggoner and Zha (2010) and Fry
and Pagan (2011). The discussion in section 2.2 focuses on sign-identified models in which all
structural shocks are jointly identified (corresponding to case 1 in Table 1). Partially identified structural VAR models based on sign restrictions (corresponding to case 2 in Table 1) are
discussed in section 2.3.

2.2
2.2.1

Fully Identified Models
The Posterior Mode of Sign-Identified Structural Impulse Responses

Let () denote the ( + 1)2 × 1 vector that consists of the on-diagonal elements andthe below-diagonal elements of  and let ( ) denote the ( − 1)2 × 1 vector that consists of above-diagonal elements of  . Ignoring the intercept for notational convenience, let = [1 · · ·  ]0 . As shown in the appendix, because there is a one-to-one mapping between
 and the reduced-form vector moving average coeﬃcient matrices Φ ,  = 1 2   (see equation 10.1.19 of Hamilton, 1994, p. 260, for example) and because Σ is nonsingular and  isorthonormal, there is a one-to-one mapping between the first  + 1 structural impulse responsesΘ̃ = [ Φ1  Φ2  · · ·  Φ ]0 on the one hand and the tuple formed by the reduced-form VAR
parameters and the rotation matrix, ( vech() veck( )), on the other. The nonlinear funce = ( vech() veck( )) is known. Using the change-of-variables method, the posteriortion Θ

7

e can be written as
density of Θ¯¯¯ [vec()0 vech()0 veck( )0 ] ¯¯¯  ( vech() veck( )) (Θ̃) = ¯¯vec(Θ̃)¯!−1 ¯Ã¯¯¯¯¯ Σ ¯vec(Θ̃)
¯
¯
¯
¯
=
¯
¯
¯  ¯  ( Σ veck( ))¯ [vec()0 vech()0 veck( )0 ] ¯¯!−1 ¯Ã¯¯¯¯¯ Σ ¯vec(Θ̃)
¯
¯
¯
¯
∝
¯
¯
¯  ¯  (|Σ) (Σ)¯ [vec()0 vech()0 veck( )0 ] ¯

(5)

e =
where , Σ = 0  and  are the unique values that satisfy the nonlinear function Θ( vech() veck( )). Here  ’s denote posterior densities whose conditioning on the data isomitted for notational simplicity. Because  is uniformly distributed on U the following result
holds:
Proposition 1. The posterior density of Θ̃ is¯!−1 ¯Ã¯¯¯¯¯ Σ ¯vec(Θ̃)¯¯¯¯ (Θ̃) ∝ ¯¯¯  ¯  (|Σ) (Σ)¯ [vec()0 vech()0 veck( )0 ] ¯(6)The Jacobian matrix and its construction are discussed in the technical appendix.
Let Θ denote the set of structural impulse responses Θ̃ that satisfy the sign restrictions.
The modal model by construction is the admissible model that maximizes the posterior densityof the sign-identified structural impulse responses. Because the impulse responses that do notsatisfy the sign restrictions are discarded, the posterior density of the sign-identified impulse
responses can be written as

(Θ̃) =⎧⎨

 (Θ̃)
 (Θ̃∈Θ)⎩ 0

if Θ̃ ∈ Θ(7)

if Θ̃ ∈
 Θ

where  (Θ̃ ∈ Θ) is the posterior probability that Θ̃ ∈ Θ. Because  (Θ̃ ∈ Θ) does not depend on
Θ, finding the mode of the posterior of the sign-identified structural impulse responses reduces tofinding the maximum of the right hand side of (6) subject to the sign restrictions. In particular,
it is not necessary to reweight  (Θ̃ ∈ Θ) to account for draws from the posterior that havebeen rejected.
It may seem surprising that one can use information about the reduced-form parameters to
aid in the identification of the most likely model, defined in the space of admissible structuralmodels. The key point is that we are interested in determining the most likely structuralmodel, not the most likely value of  . It is useful, for expository purposes to simplify mattersby focusing on the (unrealistic) limiting case of a maximum horizon of zero, in which case8

e = (vech() veck( )). The impact responses can be written as a linear combination ofΘ

vech() and veck( ). Even if the distribution of veck( ) is uniform, this linear combination
e willwill not be uniformly distributed by construction. Hence, the mode of the distribution of Θ

be unique, even when the mode over the marginal posterior of  is not. The same point holds
more generally for ( vech() veck( )).
In practice we proceed as follows:
Step 1. Take a random draw, ( Σ), from the posterior of the reduced-form VAR parameters.
Step 2. For each ( Σ), consider  random draws of the rotation  , and for each combination( Σ  ) compute the set of implied structural impulse responses Θ̃.

Step 3. If Θ̃ satisfies the sign restrictions, store the value of Θ̃ and the value of  (Θ̃). Otherwise
discard Θ̃.
Step 4. Repeat steps 2 and 3  times and find the element of Θ that maximizes (6).

2.2.2

Credible Sets for Structural Impulse Response Functions

Define the 100(1 − )% highest posterior density (HPD) credible set by = {Θ̃ ∈ Θ :  (Θ̃) ≥  }(8)

where  (Θ̃) is the posterior density of Θ̃ and  is the largest constant such that
 () ≥ 1 − (see Definition 5 of Berger, 1985, p. 140).
In practice, we compute the 100(1 − )% HPD credible set as follows:
Step 1. Take a random draw, ( Σ), from the posterior distribution of the reduced-form VAR
parameters.
Step 2. For each ( Σ), consider  random draws of the rotation  , and for each combination( Σ  ) compute the set of implied structural impulse responses Θ̃.
Step 3. If Θ̃ satisfies the sign restrictions, store the value of Θ̃ and the value of  (Θ̃). Otherwise
discard Θ̃.
Step 4. Repeat Steps 2 and 3  times and sort the pairs {(Θ̃  (Θ̃))} in descending order by thevalue of  (Θ̃). The 100(1 − )% HPD credible set consists of the set of Θ̃’s contained in
the first (1 − ) sorted pairs, where  refers to the number of models among the  · 
draws that satisfy the sign restrictions.

9Credible sets diﬀer from conventional error bands for impulse responses in that the elementsof the credible set are vectors representing the impulse response functions up to some prespecified
horizon. There is no reason for credible sets to be dense necessarily. Rather a plot of the credible
set will typically exhibit a shot-gun pattern.

2.3

Partially Identified Models

A common situation in VAR models of monetary policy is that the structural model is only
partially identified in that we are concerned with identifying the policy shock, but no otherstructural shocks. If we are concerned with a subset of impulse response functions only, whatmatters for constructing the posterior mode is not the joint impulse response distribution, but
the marginalized distribution obtained by integrating out responses to shocks that are not
identified. To simplify the exposition we will focus on the case in which only impulse responses
to one structural shock are identified. The method proposed below can be modified to allow for
impulse responses to more than one shocks.
The sign-identified structural impulse responses, 1 = Φ1 , ...,  = Φ , where  = , do
not single out a unique value of Φ1   Φ . That is because any Φ that satisfiesΦ  =    = 1  (9)is consistent with  and there are infinitely many of such Φ . Given  restrictions of the form
(9), one therefore needs to integrate out the joint posterior distribution of  with respect to( − 1) of the parameters in . Our approach exploits the following proposition.
Proposition 2:
 (0  1    ) ∝
=
=

Z

Z

Z

 (0  1    |̃ Φ(2)  Σ̃) (Φ(2)  Σ̃)(̃ Φ(2)  Σ̃)e ( (2)  Σ̃)(̃  (2)  Σ̃)|1 | |̃|  (Φ(1) | (2)  Σ)

e (2)  Σ̃)|1 | |̃|  ( (1) | (2)  Σ̃) ( (2) |Σ̃) (Σ)(̃(10)

A heuristic proof of Proposition 2 may be constructed as follows. Consider a random draw of Σ̃
from the posterior distribution of Σ and condition on its Cholesky decomposition, say ̃. Given̃, the sign-identified impulse responses in the impact period, 0 , uniquely pin down the valueof ̃.3 Conditional on Σ̃, we draw the second through last columns of  ,  = 1  , from the
Because ̃ is nonsingular, ̃ = ̃−1 0 is uniquely defined and satisfies ̃0 ̃ = 0 0 ̃−10 ̃−1  = 0 0−1  =  = 1 where  is the original Cholesky decomposition of Σ from which  is obtained.
3

0

10

unconstrained posterior distribution. PostmultiplyingΦ1 = 1 (11)

1 = 1 (12)by  yields

Because  has a continuous distribution under our assumptions, 1 6= 0 with probability one.
Thus it follows from (12) that the first column of 1 is obtained from11 =

1 −P

=2  1

1

for  = 1  (13)with probability one, where  and  denote the ( )th element of  and the th elementof , respectively. We now have a value of 1 that is consistent with (9).
Next, we postmultiply
Φ2 = 1 Φ1 + 2 

(14)

2 = 1 1 + 2 (15)

by  to obtainfrom which we obtain
21 =

2 −P

=2  2

1

for  = 1  

(16)

where
2 = 2 − 1 1 This provides a value of the first column of 2 that is consistent with (9). This process may berepeated recursively until we reach  . In the last step, we postmultiplyΦ = 1 Φ−1 + 2 Φ−2 + · · · +  (17)

 = 1 −1 + 2 −2 + · · · +  (18)

by  to obtainfrom which we obtain the first column of  as:1 = −P

=2  

1for  = 1  

where =  − 1 −1 − · · · − −1 1

11

(19)

Therefore, given Σ̃ and the second through last columns of  for  = 1 2  , the value of
1 ,...,  implies a unique value of ̃ and of the first columns of each  (and vice versa).
Let  (1) denote the ( − 1) column vector obtained by stacking the first columns of the ’s,  = 1   and let  (2) denote the corresponding second through last columns. Then themarginal posterior density of the subset of structural impulse responses of interest is (0  1    ) ∝
=
=

Z

Z

Z

 (0  1    | Φ(2)  Σ̃) (Φ(2)  Σ̃)(Φ(2)  Σ̃)|1 | |̃|  (Φ(1) | (2)  Σ̃) ( (2)  Σ̃)( (2)  Σ̃)|1 | |̃|  ( (1) | (2)  Σ̃) ( (2) |Σ̃) (Σ̃)( (2)  Σ̃)(20)

where the first equality follows because the distribution of  is uniform, the second equality
follows from applying the change-of-variables method to (13), (16),...,(19), |̃| follows from
0 = ̃̃, (16),...,(19), and the last equality follows from using the block diagonality of theJacobian matrix and applying the change-of-variables method to (11), (14),...,(17).4 Q.E.D.
Proposition 2 allows us to estimate the posterior density of 0  1    (up to scale) by Monte
Carlo integration (see, e.g., Robert and Casella 2004). To summarize:
Step 1. Generate  draws of ( Σ), from the posterior distribution of the reduced-form VAR
parameters with  independent draws each of the rotations  .
Step 2. For each of the  · draws compute the set of sign-identified structural impulse responses
of interest, 0  1    
Step 2a For each of these sets of structural impulse response functions use (13),(16),...,(19) toconstruct  draws of  (2) and Σ, from which  draws of  (1) are constructed. Evaluate
the value of|1 | |̃|  ( (1) | (2)  Σ̃)(21)

Step 2b Compute the average of (21) across the  draws considered in step 2a. This Monte Carlointegration yields (up to scale) an estimate of the density  (0  1    ).
Given the marginal posterior density of the structural response functions of interest, we maythen compute the mode and credible sets as outlined earlier.
4

In the fully identified case, the interpretation of the density  (Θ0   Θ ) as a posterior density is immediate
given that there is a one-to-one mapping from (()0  ()0  ()0 ) to (Θ0   Θ ). In the partially identified
case, the argument that (0  1    ) is a posterior density is more involved. Note that marginalizing the joint
 that satisfies the identifying restrictions
posterior of the reduced-form parameters with respect to  (2) and the Σyields the marginal posterior of  (1) conditional on the identifying restrictions being satisfied. Because the mapping
between this  (1) and the impulse responses characterized in equation (20) is one-to-one conditional on  (2) and Σ,
it follows that equation (20) is the posterior distribution of 0  1    .

12

2.4

Implementation

In practice, we specify a zero mean diﬀuse Gaussian prior for the VAR slope parameters such thatthe posterior mean of the slope parameters equals the least-squares estimator (see Lütkepohl2005). For expository purposes, we follow the literature in setting  = 032. The mode and thecredible set, of course, may be computed from the same loop. The procedures were implemented
in MATLAB or FORTRAN, depending on the computational requirements.

3

Posterior Modes versus Posterior Medians

Before discussing the extent to which median response functions diﬀer from response functionsof the modal model in practice, it is useful to develop some intuition about the limitations ofmedian response functions. Let Θ denote the response of variable  to structural shock  athorizon . One situation in which the vector of pointwise posterior medians is misleading asa measure of the central tendency of a structural impulse response function is when posteriordraws of Θ   = 1   for given  and  cross one another. This situation is illustrated in
Figure 1. The figure focuses on the response of real GDP to an unanticipated monetary policy
shock for a horizon of up to 36 months. This example was constructed by plotting a randomly
chosen subset of nine admissible response functions for the Uhlig (2005) model discussed in
section 4.2. It is evident that, for diﬀerent horizons, the pointwise posterior median responses
coincide with responses of diﬀerent admissible models. Specifically, the median response function
coincides with the response function of model 1 at horizons 4, 12-13, and 17-25; with the response
function of model 2 at horizon 9 and 15-16; with that of model 3 at horizon 16; with that ofmodel 5 at horizons 5-8, 10 and 27; with that of model 6 at horizons 0-3, 11 and 15, and withthat of model 9 at horizons 28-36. There is, in fact, no structural model in the admissible set
that could replicate the response pattern implied by the posterior median response function,
rendering this statistic economically meaningless. Note that it is not suﬃcient to show that
there are no cross-overs between impulse response functions. Similar problems may arise evenin the absence of cross-overs when the order of the models diﬀers for two response functions
at some horizon . Verifying the absence of these problems is not practically feasible, given
the thousands of admissible models implied by typical sign-identified VAR models. A key
advantage of the approach discussed in section 2 is that it avoids both of these problems byconstruction. Moreover, as discussed in the introduction, median response functions are notappropriate measures of the central tendency of the vector of impulse responses, making themdiﬃcult to intepret not only from an economic, but even from a purely statistical point of view.
Focusing instead on the impulse response functions of the most likely structural model therefore

13seems natural in our context.5

4Sign-Identified Models

4.1

Fully Identified Case: Oil Demand and Supply Shocks

There is a growing literature of models of the global market for crude oil based on fully identified
structural VAR models. Here we follow Kilian and Murphy (2012a) in specifying a monthly
VAR(24) model with intercept for 1973.2-2008.9. The set of variables consists of monthly data
for the percent change in global oil production, a business cycle index of global real activity, andthe real price of crude oil. The variables are defined and discussed in detail in Kilian (2009).
We combine some of the key identifying assumptions from the existing literature. We first
impose sign restrictions on the impact responses of each variable to each structural shock. An
unanticipated oil supply disruption causes oil production to fall, the real price of oil to increase,and global real activity to fall on impact. An unanticipated increase in the flow demand for oil
associated with the global business cycle causes global oil production, global real activity andthe real price of oil to increase on impact. Other positive demand shocks (such as shocks to oilinventory demand driven by forward looking behavior) cause oil production and the real priceof oil to increase on impact and global real activity to fall. Second, we bound the impact priceelasticity of oil supply by 0.025, as suggested by Kilian and Murphy (2012a). This elasticity canbe expressed as the ratio of two impact responses. This identifying restriction is consistent withwidely held views among oil economists that the short-run price elasticity of oil supply is close
to zero. Very similar results would be obtained if we doubled that bound. Finally, we follow
Baumeister and Peersman (2012) in restricting the real price of oil to be positive for the firstyear in response to unanticipated oil supply disruptions and in response to positive oil demand
shocks. We construct the posterior distribution of the impulse responses estimates based on
 = 5000 draws from the reduced-form posterior distribution with  = 20 000 rotations each.
Figure 2 displays the responses of each variable to each shock in the modal model alongwith the corresponding 68% credible sets. The responses have been normalized such that eachstructural shock implies an increase in the real price of oil. All structural response function
estimates are consistent with standard economic intuition. For example, a negative flow supply
shock is associated with a persistent decline in oil production, a modest increase in the real price5
Our emphasis on the mode of the joint distribution of models is not without precedence. The same approachis used in classical maximum likelihood estimation, for example. Likewise, there is precedence for focusing on thepeak of the posterior in Bayesian analysis (see, e.g., Rubio-Ramirez, Waggoner and Zha 2009). There clearly aresituations in which the use of the mode may be problematic (such as for bimodal distributions with equally highpeaks in both tails and little probability mass in the center), but in that case the median or the mean would not beadequate summary statistics either. In any case, our empirical analysis below suggests that such extreme examples
are not practically relevant.

14

of oil, and a gradual modest decline in global real economic activity. A positive flow demand
shock is associated with a persistent and hump-shaped response in both global real activity andthe real price of oil and with little response in global crude oil production. Other demand shocks(such as shocks to oil inventory demand) cause a temporary increase in the real price of oil, a
temporary decline in global real activity for about 20 months and little response in global crude
oil production. The corresponding credible sets indicate considerable uncertainty about the priceresponses and to a lesser extent for the responses in real activity, whereas the credible sets foroil production responses are quite narrow. Nevertheless, several response functions are precisely
enough estimated to conclude that the response diﬀers from zero. Figure 2 also illustrates thatthe responses of the most likely model need not be near the center of the credible set.
There also are important diﬀerences between the most likely estimates provided by the modal
model and the conventional median response functions. Median response function may be closer
to zero or further away from zero than the responses of the modal model. For example, median
response functions can be shown to overestimate the magnitude of the price response to otherdemand shocks, as shown in the left panel of Figure 3, but at the same time underestimate
the response of global real activity to the same shock (not shown). Moreover, pointwise 68%
posterior error bands provide little protection against mis-characterizing the impulse response
dynamics, as shown in the right panel of Figure 3. At many horizons, the response functionsof the modal model are outside the pointwise error bands. The right panel of Figure 3 also
illustrates that pointwise intervals tend to understate the estimation uncertainty compared withthe credible set shown in Figure 2 that captures the joint uncertainty over all impulse responses.
This example illustrates that the way estimates of sign-identified VAR models are represented
matters for the interpretation of the data.

4.2

Partially Identified Case: Monetary Policy Shocks

Whereas the preceding example dealt with a fully identified model, this section considers an
example of a partially identified model. We focus on the model of U.S. monetary policy proposedby Uhlig (2005). Our focus in this section is not so much on whether this specific model is anappropriate model of U.S. monetary policy, but whether the method of statistical evaluation
makes a diﬀerence for the economic interpretation of the results. The central question in Uhlig(2005) is what the eﬀects of an unanticipated monetary contraction are on real output. We
follow Uhlig in constructing a VAR(12) model without intercept. The set of variables consists ofmonthly U.S. data for the log of interpolated real GDP of the US, the log of the interpolated GDP
deflator, the log of a commodity price index, total reserves, non-borrowed reserves and the federal
funds rate. The sample period is 1965.1-2003.12 to ensure compatibility with Uhlig’s original

15

analysis.6 The numerical stability of the results requires a fairly large number of draws, especiallyfor  and . We construct the posterior distribution of the impulse responses estimates basedon  = 5 000 draws from the reduced-form posterior distribution with  = 500 rotations each.
We set  = 20 000.
Figure 4 demonstrates that there are important diﬀerences between the median estimatesof the response of real output and the response in the modal model. Whereas Uhlig reported
a peak median output response of 0.15 percentage points, for the same data, we obtain a peak
response of almost 0.5 percentage points based on the modal model. Moreover, that peak value
is near the upper end of the credible set and outside the conventional pointwise posterior error
band. It should be noted that both the median estimate and the response estimate based on themodal model are counterintuitive in that a monetary tightening would be expected to cause adecline in real output over time rather than an increase. This outcome reflects the fact that theidentifying assumptions are not overly informative. Even in Uhlig’s original analysis, there wassubstantial pointwise probability mass on both negative and positive responses of real output.
Our 68% credible set further widens the set of probable response functions.
The explicit reason why Uhlig (2005) did not impose further restrictions is that he wished
to be as agnostic as possible about the response of real output. This approach is appropriate
only to the extent that we view models in which real output increases in response to a monetary
tightening as economically plausible a priori (see Kilian and Murphy 2012a). Many economists
would disagree with this view at least for intermediate horizons. Hence, in Figure 5 we consider
an alternative set of results for a model that imposes an additional sign restriction on the
response of real GDP after 6 months (and only at that horizon). This identifying assumption
leaves the short-run as well as the longer-run response of real output unrestricted, preserving thespirit of Uhlig’s original exercise. The resulting modal model produces substantially diﬀerent andmore economically plausible results, including a cumulative drop in real GDP of -0.3 percentage
points in the second quarter. The response estimate for the modal model is at the lower endof the credible set and again outside the conventional pointwise posterior error band. It also
is substantially diﬀerent from the response estimate obtained from the traditional Cholesky
decomposition. One diﬀerence is that the reduction in real GDP in Figure 5 is temporary,whereas traditional Cholesky models imply a much more persistent decline in real GDP. Evenin this alternative model, however, the 68% credible set includes many positive real output
responses, suggesting that the data are not informative about the response of real output.
Likewise the other response functions are estimated only very imprecisely. We conclude that
there remains substantial uncertainty about the eﬀects of monetary policy shocks on real output,
whereas there is strong evidence of the eﬀects of oil demand shocks on the real price of oil in
6

For a more detailed description of the data the reader is referred to Uhlig (2005). The data set was provided by
Harald Uhlig.

16the earlier example.
Figure 6 elaborate further on the results in Figure 5. The left panel illustrates that there
can be substantial diﬀerences between the median response function estimates and the response
function estimates based on the modal model. For example, the decline in real GDP causedby an unanticipated monetary contraction is much larger in the modal model, at least in the
short run. In some cases, the median and the modal response of real GDP diﬀers not only inmagnitude, but in sign. The right panel demonstrates that the modal model responses may beoutside the conventional pointwise 68% error bands. This is true in particular for the responseof real GDP and to a lesser extent for the response of commodity prices and the own-responseof the federal funds rate.

5

Exactly Identified Models

Our approach is not limited to sign-identified models. It can also be applied to exactly identified
models. A case in point is a recursively identified model, in which the policy reaction function isordered last in the system of equations. This type of model is commonly used in the monetary
policy literature and indeed is the point of departure for the analysis in Uhlig (2005). The only
diﬀerence to the earlier model is that the order of the variables matters. We follow Uhlig inordering the variables of the VAR(12) model as real GDP, GDP deflator, commodity prices,federal funds rate, nonborrowed reserves and total reserves. The model in question is only
partially identified. The object of interest are the responses to an orthogonalized federal funds
rate shock.
Uhlig (2005) reports the posterior median impulse response functions. As in the signidentified model, this measure of the central tendency of the structural impulse responses isvalid only pointwise and need not correspond to the impulse response functions implied by themost likely structural model. Uhlig also reports pointwise 68% posterior error bands for thismodel. It is well known that these pointwise error bands fail to convey the true uncertainty
surrounding these impulse response functions. This problem has been long recognized, but few
practical alternatives have been proposed in the literature, which explains why these methods
have remained the standard in the structural VAR literature (see, e.g., Lütkepohl 1990, Sims
and Zha 1999, Jorda 2009).
A solution to this problem is provided by a simplified version of our baseline procedure forsign-identified models. In the fully identified case, corresponding to case 3 in Table 1, it suﬃces
to replace the rotation matrices in the procedure outlined in section 2.2 by the identity matrix
and to modify the Jacobian of the transformation accordingly. We obtain

17

h
i ¯⎞
⎛¯
¯
¯ [vech(Θ̃ )0 vec( Θ̃  Θ̃   Θ̃ )0 ] ¯ −1 ¯
0
1
2

¯
¯ Σ ¯
¯
¯
¯
¯
¯
⎠
⎝
 (Θ̃) ∝ ¯0¯¯  ¯  (|Σ) (Σ)[vec()0 vech() ]¯¯where Θ̃0 is the impulse response matrix in the impact period and Θ̃1 , Θ̃2 ,...,Θ̃ are the impulse
response matrices at higher horizons. It follows from Θ̃ = Θ  for  = 1   and equation (29)0in the appendix that the determinant of the Jacobian reduces to || |[( ⊗  ) + ( ⊗ )2 ] |,where  is defined in the appendix. On the basis of this result, we can evaluate the posterior as
discussed earlier. This simplified procedure could be applied, for example, to the fully structural
oil market VAR model in Kilian (2009).
The partially point-identified case considered by Uhlig (2005) corresponding to case 4 inTable 1 is more involved. Suppose that only impulse responses to the th shock are considered(e.g., monetary policy shocks). Because the impulse responses in the impact period correspondto the th column of the Cholesky decomposition, we need to draw Σ conditional on the thcolumn of its Cholesky decomposition when marginalizing the joint posterior density. We write
the Cholesky decomposition as
⎡

11

⎢
⎢
 = ⎢ 21
⎣
31

⎤

0

0

22

0

32

33

⎥
⎥
⎥
⎦

where 11 , 21 , 22 , 31 , 32 and 33 are 1 × 1 , 1 × 1, 2 × 1 , 2 × 1 and 2 × 2 matrices,1 + 1 + 2 = , and [0 022 032 ]0 corresponds to the th column of . Then⎡

11 011⎢⎢
Σ = ⎢ 21 011⎣
31 011⎤

11 02111 031

21 021 + 22 022

21 031 + 22 032

31 021 + 32 022

31 31 + 32 032 + 33 033

⎥
⎥
⎥
⎦

Because
⎡
⎣

⎡

= ⎣

21 021 + 22 022

21 031 + 22 032

31 021 + 32 022
21 02131 02131 31 + 32 032 + 33 033⎤
21 031⎦31 31 + 33 033

⎤

⎡

⎦−⎣

22 02222 03232 02232 032 + 33 033

⎤
⎦

(22)

e from the posterior
is positive definite and 21 , 31 and 33 are unconstrained, one can draw Σe conditional on the th Cholesky decomposition by drawing Σe from the uncondistribution of Σditional posterior distribution of Σ and retaining the draws in which (22) is positive definite.

We draw ’s that are consistent with the impulse responses 1  2    in the same way as18

in Proposition 2 except that we condition on the th columns of  rather than the first columnof  . This is because the first element of 0 is zero (unless the first shock is considered).
Let  (1) denote the last columns of  ’s and let  (2) denote their first through second-to-lastcolumns. Then we can summarize our approach in the following proposition:
Proposition 3:
 (0  1    ) ∝
=
=

Z

Z

Z

 (0  1    |̃ Φ(2)  Σ̃) (Φ(2)  Σ̃)(Φ(2)  Σ̃)e ( (2)  Σ̃)( (2)  Σ̃)| |  (Φ(1) | (2)  Σ)

(2)e Σ̃)
| |  ( (1) | (2)  Σ̃) ( (2) |Σ̃) (Σ)((23)

e whose (2 +1)×(2 +1) lower-right submatrix
where the integration is taken over  (2) and the Σsatisfies the restriction (22).

The resulting procedure allows the user to construct credible sets for the structural impulse
response functions for exactly identified structural VAR models that account for the joint uncertainty in the set of structural impulse response functions. It provides a convenient alternative
to traditional pointwise error bands that is easy to implement. We illustrate this point for the
responses to an unanticipated monetary tightening in the (partially) recursively identified VAR
model used as a benchmark in Uhlig (2005). Uhlig’s Figure 4 reported the pointwise median
response functions and pointwise 68% posterior error bands for this model. We instead report
the response functions of the modal model and the corresponding 68% joint credible set. All
results are based on  = 5 000 and  = 20 000. Figure 7 shows that, even after accounting
for the full uncertainty about the impulse response dynamics, the response functions of realGDP, of the federal funds rate and of nonborrowed reserves are precisely enough estimated tobe economically informative at least at some horizons. The price puzzle and the puzzling initial
increase in real GDP in response to an unanticipated monetary tightening, in contrast, canbe attributed to estimation uncertainty. This result highlights the importance of simultaneous
inference for all structural impulse responses. A user of pointwise 68% posterior error bands
would have concluded that these puzzles cannot be explained merely by estimation uncertainty.
Figure 7 also illustrates that the median response functions may diﬀer substantially from theresponse functions of the modal model. For example, the response of total reserves in the modal
model diﬀers not only in magnitude, but also in sign. The negative response of commodity
prices doubles in magnitude compared with the results in Uhlig (2005) and the response of the
GDP deflator turns negative after only 24 months rather than 48 months. These diﬀerences
highlight that accounting for the dependence across impulse response estimates is importanteven in point-identified models.

19

6Concluding Remarks

Conventional approaches to summarizing the evidence from sign-identified impulse response
functions based on quantiles of the pointwise posterior distribution of impulse responses lack aclear economic interpretation and fail to convey the uncertainty about the structural responses
functions. We proposed an alternative approach based on a characterization of the most likely
models in the set of admissible structural models. This approach has the advantage of allowing
for a unified treatment of estimation and inference in both the exactly identified and the signidentified VAR model. Our approach in this paper is explicitly Bayesian in nature. The use ofBayesian methods facilitates the interpretation of sign-identified VAR models and is standard
in this literature. In fact, it is not clear how to extend our approach to evaluating sign-identified
VAR models to frequentist settings.7
For exactly identified VAR models, in contrast, one could construct joint asymptotic normal approximations of the distribution of the impulse responses (see, e.g., Lütkepohl 1990,
Mittnik and Zadrozny 1993), facilitating joint inference based on the Bonferroni principle, although that method is impractical given its low power (see Lütkepohl 1990). As an alternative,
Jorda (2009) recently proposed the construction of classical joint confidence intervals based on
Scheﬀé’s method. Unlike the methods discussed in our paper, which account for uncertainty
in all structural impulse responses jointly, the work of Jorda (2009) and related classical approaches focus on joint inference about the subset of responses contained in a given impulse
response function only. If traditional pointwise inference for impulse responses is analogous todoing a sequence of -tests for the parameters of a given equation, then these joint confidence sets
are analogous to conducting an  -test for all parameters of a given equation. In short, existingclassical approaches ignore the dependence of  -tests across the equations of the underlying system of equations and do not address the problem of joint inference about all impulse responses
in structural VAR models. An obvious extension of our approach in this paper would involve
constructing joint confidence sets for all structural impulse responses based on the conventional
asymptotic normal approximation for the model parameters in exactly identified models. Thisapproach would be analogous to conducting a   test for all parameters in all equations at
the same time.
The work most closely related to ours is Sims and Zha’s (1999) proposal for an approximate
method for joint inference on impulse responses. Their method is designed for case 3 in Table
1, but not for cases 1, 2 and 4. Sims and Zha’s baseline method is based on draws fromthe first and second moment of the (unspecified) joint distribution of the posterior impulse
7

Moon, Schorfheide, Granziera, and Lee (2011) recently proposed pointwise frequentist confidence intervals forimpulse response estimates obtained from sign-identified VAR models. They showed that Bayesian and classicalinference do not coincide even asymptotically in sign-identified VAR models. Moon et al. do not address the questionof how to construct joint confidence regions or the question of which response estimates are most likely.

20

responses, which will be a good approximation if and only if the joint distribution is Gaussian.
One obvious concern with this method is that it is well known that the finite-sample joint
posterior distribution of impulse responses is far from Gaussian.8 Whereas the joint impulse
response distribution is at least aymptotically Gaussian in the case of exactly identified models,in the sign-identified model asymptotic normality breaks down, as shown by Moon, Schorfheide,
Granziera, and Lee (2011), making the Sims-Zha approach unsuitable for cases 1 and 2 inTable 1. Moreover, Sims and Zha’s approximate method makes no allowance for the need tomarginalize the joint distribution in the economically relevant dimension. This marginalization
can be safely ignored if the joint distribution of the impulse responses is indeed Gaussian, because
in that case any partition of the joint normal distribution will be normal as well. In practice,
however, the joint distribution is far from Gaussian and one has to deal with the marginalizationof the joint distribution, unless the VAR model is fully identified making the marginalization
unnecessary. This prohibits application of Sims and Zha’s method in case 4. In contrast, ourmethod of inference is based on analytic solutions for the joint posterior distribution of theimpulse responses for all four cases and does not involve any approximations.
The point of our paper was to describe conditions under which one can interpret the setof admissible models in a way that is useful for applied researchers. Two empirical examples
illustrated that the way information from structural VAR model estimates is represented matters. Responses based on the modal model and the associated credible sets can generate very
diﬀerent assessments of the evidence than traditional methods. The use of the mode has been
explicitly recommended by leading practitioners in the recent structural VAR literature (seeRubio-Ramirez, Waggoner and Zha 2010, p. 684), making it a natural starting point. By
focusing on the mode we eﬀectively adopt a particular loss function. Given the absence of adecision-theoretic framework for selecting a loss function in standard macroeconomic applications of structural VAR models, there is no alternative to choosing a loss function by convention.
Why then choose this particular loss function? At first sight, it may seem that there should
be sensible alternatives to the use of the posterior mode in characterizing the structural impulse
responses. For example, under a quadratic loss function one would have focused on the posterior
mean. The posterior mean is statistically well-defined in the vector case and does not suﬀer
from the second shortcoming discussed in the introduction. Given a finite set of admissible
posterior draws, the posterior mean in general will not correspond to any one structural model
8

In recognition of this problem Sims and Zha suggest an alternative method designed to account for asymmetry
in the marginal distribution of the impulse responses. This method is based on quantiles of the marginal distribution
constructed from the same draws used for the baseline method. This proposal indeed accounts for asymmetry in theapproximate marginal distribution, but it does not address concerns regarding the non-Gaussianity of the underlying
joint distribution. Moreover, the reliance on vectors of pointwise quantiles for characterizing the impulse response
functions under this alternative proposal is problematic. This problem is analogous to the problems with vectors ofmedians and vectors of other quantiles discussed in our paper. Thus, it is not clear to us how this proposal resolves
the problem of a lack of normality in the joint distribution.

21

in the set of admissible models, however, making it vulnerable to the first shortcoming. One
way of addressing this concern would be to search for the admissible structural model that
produces impulse responses closest to those of the posterior mean response functions, building
on an similar idea in Fry and Pagan (2011).9 We do not pursue this idea because one important
disadvantage of the posterior mean approach compared with focusing on the most likely or modal
model is that there is no natural way of constructing joint credible sets in the sign-identified
model. While our analysis provides the tools for evaluating other loss functions, for now there
does not appear to be an obvious alternative to the use of the mode in evaluating the set ofadmissible models.

9

Fry and Pagan (2011) suggested to minimize the distance to the median response function. That idea is notadvisable because the posterior median response function is not a well defined statistical object, as discussed earlier,but the same idea could be applied without problems to the posterior mean response function.

22

Technical Appendix
Proof of the result that the mapping between the first  + 1 structural impulse responses Θ̃ =[̄ Φ1 ̄ · · ·  Φ ̄]0 and [ vech() veck( )] is one-to-one: Because there is a one-to-one mapping between  and the reduced-form vector moving average coeﬃcient matrices Φ ,  = 1 2  (see equation 10.1.19 of Hamilton, 1994, p.260, for example), establishing this result comes down
to showing that the mapping between ̄ and the pair vech() and veck( ) is one-to-one. We
prove this result in two steps. Recall that - because the orthonormality restriction   0 = 
imposes ( + 1)2 restrictions on the  ×  matrix  - one can recover all elements of from its upper diagonal elements, ( ). First, suppose that ̄1 6= ̄2 but 1 = 2 . Then̄ ̄0 =   0 0 =  0 regardless of the value of  because  0 =  for  = 1 2. Because 1 = 2 this implies ̄1 ̄01 = ̄2 ̄02 which contradicts the assumption that ̄1 6= ̄2 .
Hence, ̄1 6= ̄2 implies vech()1 6= vech()2 . Second, suppose that [1  1 ] 6= [2  2 ] but̄1 = ̄2 . Because 1 01 = ̄1 ̄1 = ̄2 ̄02 = 2 02 and the Cholesky decomposition is uniquelydetermined for positive definite matrix Σ, it has to be the case that 1 = 2 and 1 6= 2 .

−1
Because  is nonsingular for  = 1 2, however, 1 = −1
1 ̄1 = 2 ̄2 = 2 , which is acontradiction. Hence, [1  1 ] 6= [2  2 ] implies ̄1 6= ̄2 .
Derivation of the Jacobian matrix in Proposition 1: Let  denote the 2 ×(+1)2 duplication
matrix of zeros and ones such that vec( ) =  vech( ) for  ×  symmetric matrix  (seeMagnus and Neudecker, 1999, pp.49). + denotes the Moore-Penrose inverse of  so that

we can write vech( ) = + vec( ).  denotes the 2 × 2 communication matrix such thatvec( 0 ) =  vec( ) for  ×  matrix  (see Magnus and Neudecker, 1999, pp.46—47).
Let Φ

=

[Φ01 Φ02 · · · Φ0 ]0 and 1

0

0

0

[ Φ01  Φ02 · · ·  Φ0 ]0 where Φ is the th=

reduced-form vector moving average coeﬃcient matrix. Because 1 = Φ ,⎡⎣vec()
vec(1 )⎤⎦[vec()0 vec( )0 vec(Φ)0 ]⎡= ⎣ 0 ⊗ 
0 ⊗ Φ ⊗ 

2 ×2  ⊗ Φ ⊗ 

⎤

⎦

(24)

We need to replace the partial derivatives with respect to vec() and vec( ) in (24) withthose with respect to vech() and veck( ). It follows from   0 =  that[( ⊗  ) + ( ⊗  ) ]( ) = 02 ×1 

(25)

+ [( ⊗  ) + ( ⊗  ) ]( ) = 0(+1)2×1 (26)from which we obtain23

Let  and  denote the (2 × ( + 1)2) and (2 × ( − 1)2) matrices of zeros and ones
such that⎡

Then (26) can be written as( ) = [  ] ⎣vech( )veck( )⎤⎦+ [( ⊗  ) + ( ⊗  ) ] vech( ) + + [( ⊗  ) + ( ⊗  ) ] veck( ) = 0(+1)2×1 (27)
Applying the implicit function theorem to (27), the Jacobian of vec( ) with respect to veck( )can be written as©
ª−1 + =  −  + [( ⊗  ) + ( ⊗  ) ]
 [( ⊗  ) + ( ⊗  ) ]

(28)

Thus, it follows from (24) and (28) that
⎡

1 ≡

⎣

vec()
vec(̃)⎤⎦[vech0 veck( )0 vec(Φ)0 ]⎡= ⎣( 0 ⊗  )
( 0 ⊗ Φ)( ⊗ )( ⊗ Φ)

2 ×2 
0 ⊗ 

⎤

⎦

(29)

Because (29) is block-diagonal, its determinant is given by the product of determinants:|1 | = |( 0 ⊗  ) ( ⊗ ) | |0 ⊗  |= |( 0 ⊗  ) ( ⊗ ) | || (30)

Because of the recursive relationships (11), (14),...,(17) between  and Φ, the Jacobian matrixof  with respect to  is block-diagonal and each diagonal block has unit determinant. Thus¯¯¯ vec(Φ) ¯¯ = 1
|2 | ≡ ¯¯vec() ¯(31)

Since the Jacobian of vec(Σ) with respect to vec() is[( ⊗  ) + ( ⊗ ) ]

(32)

the determinant of the Jacobian of vech(Σ) with respect to vech() is given by|3 | ≡ |0 [( ⊗  ) + ( ⊗ ) ] |(33)

Therefore it follows from (30), (31) and (33) that the determinant of the Jacobian in (6) is givenby the product of 1|1 | and |3 |.

24

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26

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27

Table 1: Four Types of Structural VAR Models

Sign-Identified
Exactly Identified

Fully Identified
Case 1Case 3

Partially Identified
Case 2
Case 4

Figure 1: Randomly Selected Response Functions from a Sign-Identified VAR Model
Real GDP Responses to Monetary Policy Shock

0.3

0.2

Percent

0.1

0

-0.1

Median Response Function

-0.2

0

12

24

36

MonthsNOTES: Based on nine randomly selected responses from the posterior of the model used as an empiricalexample in section 4.2. The median response function is constructed from the pointwise posterior medians.
It coincides with responses from six different admissible structural models depending on the horizon.

Figure 2: Structural Responses in the Sign-Identified Oil Market Model
Response Functions in the Modal Model and 68% Joint Regions of High Posterior Density
Flow supply shock

Flow supply shock

0

0

5

10

5

0

-5

15

Real price of oil

5

-5

0

Flow demand shock

15

0

5

10

5

0

-5

15

0

Other demand shock

5

10

15

0

10

Months

-5

0

15

5

0

-5

0

5

10

Months

5

10

15

10

5

0

-5

0

15

5

10

15

Other demand shock
Real price of oil

Real activity

5

5

0

Other demand shock
10

0

5

Flow demand shock
Real price of oil

Real activity

Oil production

010

Oil production

10

10

5

-5

5

10

Flow demand shock

10-5

Flow supply shock

10

Real activity

Oil production

10

10

5

0

-5

0

5

10

12

15

MonthsNOTES: The data and model are described in the text. All shocks have been normalized to imply an increase in the real price of oil.

Figure 3: Structural Responses in the Sign-Identified Oil Market Model
Other demand shock

Other demand shock

10

10

Pointwise Median
Modal Model

Modal Model
68% Pointwise Quantiles

Real price of oil

5

Real price of oil

5

0

-5

0

0

5

10

MonthsNOTES: See Figure 2.

15

-5

0

5

10

Months15

Figure 4: Responses to a Monetary Policy Tightening in the Uhlig (2005) Sign-Identified Model
Response Functions in the Modal Model and 68% Joint Regions of High Posterior Density
Real GDP

Total Reserves

1

4

0.5

2

0

0

-0.5

-2

-1

0

12

24

36

-4

0

GDP Deflator
4

0

2

-0.5

0

-1

-2

0

12

24

36

Nonborrowed Reserves

0.5

-1.5

12

24

36

-4

0

Commodity Price Index

12

24

36

Federal Funds Rate
1
0.5

0

0
-5
-0.5
-10
0

12

24

36

-1

0

12

MonthsNOTES: The model specification and sign restrictions are the same as in Figure 6 of Uhlig (2005).

24

Months36

Figure 5: Responses to a Monetary Policy Tightening in the Modified Sign-Identified Model
Response Functions in the Modal Model and 68% Joint Regions of High Posterior Density
Real GDP

Total Reserves

1

4

0.5

2

0

0

-0.5

-2

-1

0

12

24

36

-4

0

GDP Deflator
4

0

2

-0.5

0

-1

-2

0

12

24

36

Nonborrowed Reserves

0.5

-1.5

12

24

36

-4

0

Commodity Price Index

12

24

36

Federal Funds Rate
1
0.5

0

0
-5
-0.5
-10
0

12

24

Months

36

-1

0

12

24

MonthsNOTES: Relative to the model in Figure 5, an additional scalar restriction has been imposed that real GDP must have declined
6 months after the monetary policy tightening.

36

Figure 6: Responses to a Monetary Policy Tightening in the Modified Sign-Identified Model
0.5

0.5

Pointwise Median
Modal Model

0.3

0.3

0.2

0.2

0.1

0.1

0

0

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5

0

12

24

MonthsNOTES: See Figure 7.

Modal Model
68% Pointwise Quantiles

0.4Real GDP

Real GDP

0.4

36

-0.5

0

12

24

Months36

Figure 7: Responses to a Monetary Policy Tightening in the Partially Identified Cholesky Model
Response Functions in the Modal Model and 68% Joint Regions of High Posterior Density
Total Reserves
4

0.5

2

Percent

Percent

Real GDP
1

0
-0.5
-1

0

12

24

0
-2
-4

36

0

4

0

2

-0.5
-1
-1.5

0

12

24

36

Nonborrowed Reserves

0.5

Percent

Percent

GDP Deflator

12

24

0
-2
-4

36

0

Commodity Price Index

12

24

36

Federal Funds Rate0

Percent

Percent

1

-5

-10
0

12

24

36

0.5
0
-0.5
-1

0

12

NOTES: The model specification and identifying restrictions are the same as in Figure 5 of Uhlig (2005).

24

36