# Identification and Inference Under Narrative Restrictions††thanks: We thank
Isaiah Andrews, Sophocles Mavroeidis, José Luis Montiel-Olea, Mikkel Plagborg-
Møller, Morten Ravn, Christian Wolf and seminar participants at several venues
for helpful comments. We gratefully acknowledge financial support from ERC
grants (numbers 536284 and 715940) and the ESRC Centre for Microdata Methods
and Practice (CeMMAP) (grant number RES-589-28-0001).

Raffaella Giacomini, Toru Kitagawa and Matthew Read University College London,
Department of Economics/Cemmap. Email<EMAIL_ADDRESS>College
London, Department of Economics/Cemmap. Email<EMAIL_ADDRESS>College London, Department of Economics. Email<EMAIL_ADDRESS>
###### Abstract

We consider structural vector autoregressions subject to ‘narrative
restrictions’, which are inequality restrictions on functions of the
structural shocks in specific periods. These restrictions raise novel problems
related to identification and inference, and there is currently no frequentist
procedure for conducting inference in these models. We propose a solution that
is valid from both Bayesian and frequentist perspectives by: 1) formalizing
the identification problem under narrative restrictions; 2) correcting a
feature of the existing (single-prior) Bayesian approach that can distort
inference; 3) proposing a robust (multiple-prior) Bayesian approach that is
useful for assessing and eliminating the posterior sensitivity that arises in
these models due to the likelihood having flat regions; and 4) showing that
the robust Bayesian approach has asymptotic frequentist validity. We
illustrate our methods by estimating the effects of US monetary policy under a
variety of narrative restrictions.

JEL classification: C32, E52

Keywords: Frequentist coverage, global identification, identified set,
multiple priors

##  1 Introduction

Estimating the dynamic causal effects of structural shocks is a key challenge
in macroeconomics. A common approach to this problem is to use a structural
vector autoregression (SVAR) with sign or zero restrictions on the model’s
structural parameters. Recently, a number of papers have augmented these
restrictions with restrictions that involve the values of the structural
shocks in specific periods. For example, Antolín-Díaz and Rubio-Ramírez (2018)
(AR18) propose restricting the signs of structural shocks and their
contributions to the change in particular variables in certain historical
episodes. Ludvigson, Ma and Ng (2018) independently propose restricting the
sign or magnitude of the structural shocks in specific periods. A burgeoning
empirical literature has adopted similar restrictions, including Ben Zeev
(2018), Furlanetto and Robstad (2019), Cheng and Yang (2020), Inoue and Kilian
(2020), Kilian and Zhou (2020a, 2020b), Laumer (2020), Redl (2020), Zhou
(2020) and Ludvigson, Ma and Ng (2020). The fact that these restrictions are
placed on the shocks rather than the parameters raises novel problems related
to identification, estimation and inference. This paper clarifies the nature
of these problems and proposes a solution that is valid from both Bayesian and
frequentist perspectives.

Henceforth, we refer to any restrictions that can be written as inequalities
involving structural shocks in particular periods as ‘narrative restrictions’
(NR). An example of NR are ‘shock-sign restrictions’, such as the restriction
in AR18 that the US economy was hit by a positive monetary policy shock in
October 1979. This is when the Federal Reserve markedly increased the federal
funds rate following Paul Volcker becoming chairman, and is widely considered
an example of a positive monetary policy shock (e.g., Romer and Romer (1989)).
AR18 also consider ‘historical-decomposition restrictions’, such as the
restriction that the change in the federal funds rate in October 1979 was
overwhelmingly due to a monetary policy shock. This is an inequality
restriction that simultaneously constrains the historical decomposition of the
federal funds rate with respect to all structural shocks in the SVAR. Other
restrictions on the structural shocks also fit into this framework. For
example, we additionally consider ‘shock-rank restrictions’, such as the
restriction that the monetary policy shock in October 1979 was the largest
positive realization of this shock in the sample period.

From a frequentist perspective, NR are fundamentally different from
traditional identifying restrictions, such as sign restrictions on impulse
responses (e.g., Uhlig (2005)). Under normally distributed structural shocks,
traditional sign restrictions induce set-identification, because they generate
a set-valued mapping from the SVAR’s reduced-form parameters to its structural
parameters that represents observational equivalence (i.e., an identified
set). This set-valued mapping corresponds to the flat region of the
structural-parameter likelihood and, by the definition of observational
equivalence (e.g., Rothenberg (1971)), does not depend on the realization of
the data. NR also result in the structural-parameter likelihood possessing
flat regions and hence generate a set-valued mapping from the reduced-form
parameters to the structural parameters. Crucially, this mapping depends not
only on the reduced-form parameters, but also on the realization of the data.
The data-dependence of this mapping implies that the standard concept of an
identified set does not apply. In turn, this means that: 1) it is unclear
whether NR are point- or set-identifying restrictions; and 2) there is no
known valid frequentist procedure to conduct inference in these
models.111Ludvigson et al. (2018, in press) conduct inference using a
bootstrap procedure, but its frequentist validity is unknown.

From a Bayesian perspective, AR18 and the empirical papers that adopt their
approach conduct standard (single-prior) Bayesian inference under NR in much
the same way as under traditional sign restrictions. However, we highlight two
features of this approach that can spuriously affect inference. First, the
conditional likelihood used by AR18 to construct the posterior (distribution)
implies that, for some types of NR, a component of the prior (distribution) is
updated only in the direction that makes the NR unlikely to hold ex ante. This
occurs because the numerator of the conditional likelihood – the likelihood of
the reduced-form VAR – is flat with respect to the orthonormal matrix that
maps reduced-form VAR innovations into structural shocks, whereas the
denominator – the ex ante probability that the NR hold – depends on this
matrix. Second, standard Bayesian inference under NR may be sensitive to the
choice of prior when the NR yield a likelihood with flat regions. A flat
likelihood implies that the conditional posterior of the orthonormal matrix is
proportional to its conditional prior whenever the likelihood is nonzero.
Posterior inference may therefore be sensitive to the choice of conditional
prior for the orthonormal matrix. This is a problem that also occurs in set-
identified models under traditional restrictions (e.g., Poirier (1998)).

To address the above issues, we study identification under NR and propose a
framework for conducting estimation and inference that is potentially
appealing to both Bayesians and frequentists. We proceed in four main steps.
First, we formalize the identification problem under NR. Second, we propose a
simple modification of the existing Bayesian approach that eliminates the
source of posterior distortion arising under NR. The modification is to use
the unconditional likelihood, rather than the conditional likelihood, to
construct the posterior. Third, as a tool for assessing and/or eliminating
posterior sensitivity occurring due to the likelihood having flat regions, we
propose a robust (multiple-prior) Bayesian approach to estimation and
inference. Finally, we show that the robust Bayesian approach has frequentist
validity in large samples.

To the best of our knowledge, this is the first paper to formally study
identification under general NR. Plagborg-Møller and Wolf (in press[b])
suggest that shock-sign restrictions, in particular, could in principle be
recast as an external instrument (or ‘proxy’) and used to point-identify
impulse responses in a proxy SVAR or local projection framework. We explore
this idea in Appendix F and highlight the potential sensitivity of this
approach to the realization of the unrestricted shocks in the time periods
that enter the NR. Petterson, Seim and Shapiro (2020) derive bounds for a
slope parameter in a single equation given restrictions on the plausible
magnitude of the residuals, but the restrictions are over the entire sample
and the setting is non-probabilistic.

We make two main contributions to the study of identification under NR. First,
we provide a necessary and sufficient condition for global identification of
an SVAR under NR and show that this condition is satisfied in a simple
bivariate example with a single shock-sign restriction. That is, in contrast
with traditional sign restrictions, NR may be formally point-identifying
despite generating a set-valued mapping from reduced-form to structural
parameters in any particular sample. However, this point-identification result
does not deliver a point estimator, because the observed likelihood is almost
always flat at the maximum. Second, to develop a frequentist-valid procedure
for inference, we introduce the notion of a ‘conditional identified set’. The
conditional identified set extends the standard notion of an identified set to
a setting where identification is defined in a repeated sampling experiment
conditional on the set of observations entering the NR. This provides an
interpretation for the set-valued mapping induced by the NR as the set of
observationally equivalent structural parameters in such a conditional
frequentist experiment.

In terms of inference under NR, this paper makes contributions from both a
Bayesian and a frequentist point of view.

The paper’s contribution to Bayesian inference is to address the issues
associated with the current approach to standard Bayesian inference under NR.
First, we advocate using the unconditional likelihood – the joint probability
of observing the data and the NR being satisfied – when constructing the
posterior, rather than the conditional likelihood. Regardless of the type of
NR imposed, the unconditional likelihood is flat with respect to the
orthonormal matrix that maps reduced-form VAR innovations into structural
shocks. This removes the source of posterior distortion that arises due to
conditioning on the NR holding. Standard Bayesian inference under the
unconditional likelihood requires a simple change to existing computational
algorithms. Second, to address posterior sensitivity to the choice of prior,
we adapt the robust Bayesian approach of Giacomini and Kitagawa (in press[a])
(GK) to a setting with NR.

In the context of an SVAR under traditional identifying restrictions, the
robust Bayesian approach of GK involves decomposing the prior for the
structural parameters into a prior for the reduced-form parameters, which is
revised by the data, and a conditional prior for the orthonormal matrix given
the reduced-form parameters, which is unrevisable. Considering the class of
all conditional priors for the orthonormal matrix that are consistent with the
identifying restrictions generates a class of posteriors, which can be
summarized by a set of posterior means (an estimator of the identified set)
and a robust credible region. This removes the source of posterior
sensitivity.222Giacomini, Kitagawa and Read (2019) extend this approach to
proxy SVARs where the parameters of interest are set-identified using external
instruments.

We show that this approach can also be used to summarize posterior sensitivity
under NR, since the unconditional likelihood at the realized data possesses
flat regions and the posterior can therefore be sensitive to the choice of
prior, as in standard set-identified models. There are, however, some
modifications needed to account for the novel features of the NR. In
particular, one cannot use a conditional prior for the orthonormal matrix to
impose the NR due to the data-dependent mapping between reduced-form and
structural parameters. However, by considering the class of all conditional
priors consistent with any traditional identifying restrictions (if present),
one can trace out all possible posteriors that are consistent with the
traditional restrictions and the NR. This is because traditional restrictions
truncate the support of the conditional prior, while NR truncate the support
of the likelihood. Consequently, the posterior given any particular
conditional prior is only supported on the common support of the conditional
prior and the likelihood.

If the researcher has a credible conditional prior, we recommend reporting the
standard Bayesian posterior under the unconditional likelihood together with
the robust Bayesian output. This allows other researchers to assess the extent
to which posterior inference may be driven by prior choice. In the absence of
a credible conditional prior, the robust Bayesian output should be reported as
an alternative to the standard Bayesian posterior.

The paper’s contribution to frequentist inference is to provide an
asymptotically valid approach to inference under NR, which, to the best of our
knowledge, was not previously available. To explore the asymptotic frequentist
properties of our robust Bayesian procedure, we assume a fixed number of NR.
This assumption is empirically relevant given that applications typically
impose no more than a handful of NR. We provide conditions under which the
robust credible region provides asymptotically valid frequentist coverage of
the conditional identified set for the impulse response. Since the conditional
identified set is guaranteed to include the true impulse response, the robust
credible region also provides valid coverage of the true impulse response. Our
robust Bayesian approach should therefore appeal to Bayesians as well as
frequentists.

We illustrate our methods by estimating the effects of monetary policy shocks
in the United States. We find that posterior inferences about the response of
output obtained under restrictions based on the October 1979 episode may be
sensitive to the choice of conditional prior for the orthonormal matrix. In
contrast, under an extended set of restrictions constructed by AR18 based on
multiple historical episodes, output falls with high posterior probability
following a positive monetary policy shock regardless of the choice of
conditional prior. We also estimate the set of output responses that are
consistent with the restriction that the monetary policy shock in October 1979
was the largest positive realization of the shock in the sample period.
Compared with the extended set of restrictions, this shock-rank restriction
results in broadly similar robust posterior inferences about the output
response.

Outline. The remainder of the paper is structured as follows. Section 2
highlights the econometric issues that arise when imposing NR using a simple
bivariate example. Section 3 describes the general SVAR($p$) framework.
Section 4 formally analyzes identification under NR and introduces the concept
of a conditional identified set. Section 5 discusses how to conduct standard
and robust Bayesian inference under NR. Section 6 explores the frequentist
properties of the robust Bayesian approach. Section 7 contains the empirical
application and Section 8 concludes. The appendices contain proofs and other
supplemental material.

Generic notation: For the matrix $\mathbf{X}$, $\mathrm{vec}(\mathbf{X})$ is
the vectorization of $\mathbf{X}$ and $\mathrm{vech}(\mathbf{X})$ is the half-
vectorization of $\mathbf{X}$ (when $\mathbf{X}$ is symmetric).
$\mathbf{e}_{i,n}$ is the $i$th column of the $n\times n$ identity matrix,
$\mathbf{I}_{n}$. $\mathbf{0}_{n\times m}$ is a $n\times m$ matrix of zeros.
$1(.)$ is the indicator function. $\lVert.\rVert$ is the Euclidean norm.

##  2 Bivariate example

This section sets out the econometric issues that arise when imposing NR using
the simplest possible SVAR as an example. Consider the SVAR($0$)
$\mathbf{A}_{0}\mathbf{y}_{t}=\bm{\varepsilon}_{t}$, for $t=1,\ldots,T$, where
$\mathbf{y}_{t}=(y_{1t},y_{2t})^{\prime}$ and
$\bm{\varepsilon}_{t}=(\varepsilon_{1t},\varepsilon_{2t})^{\prime}$ with
$\bm{\varepsilon}_{t}\overset{iid}{\sim}N(\mathbf{0}_{2\times
1},\mathbf{I}_{2})$. We abstract from dynamics for ease of exposition, but
this is without loss of generality. The orthogonal reduced form of the model
reparameterizes $\mathbf{A}_{0}$ as
$\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1}$, where $\bm{\Sigma}_{tr}$ is the
lower-triangular Cholesky factor (with positive diagonal elements) of
$\bm{\Sigma}=\mathbb{E}(\mathbf{y}_{t}\mathbf{y}_{t}^{\prime})=\mathbf{A}_{0}^{-1}\left(\mathbf{A}_{0}^{-1}\right)^{\prime}$.
We parameterize $\bm{\Sigma}_{tr}$ directly as

$\bm{\Sigma}_{tr}=\begin{bmatrix}\sigma_{11}&0\\\
\sigma_{21}&\sigma_{22}\end{bmatrix}\quad(\sigma_{11},\sigma_{22}>0),$ (1)

and denote the vector of reduced-form parameters as
$\bm{\phi}=\mathrm{vech}(\bm{\Sigma}_{tr})$. $\mathbf{Q}$ is an orthonormal
matrix in the space of $2\times 2$ orthonormal matrices, $\mathcal{O}(2)$:

$\mathbf{Q}\in\mathcal{O}(2)=\left\\{\begin{bmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{bmatrix}:\theta\in[-\pi,\pi]\right\\}\cup\left\\{\begin{bmatrix}\cos\theta&\sin\theta\\\
\sin\theta&-\cos\theta\end{bmatrix}:\theta\in[-\pi,\pi]\right\\},$ (2)

where the first set is the set of ‘rotation’ matrices and the second set is
the set of ‘reflection’ matrices.

Given the ‘sign normalization’
$\mathrm{diag}(\mathbf{A}_{0})\geq\mathbf{0}_{2\times 1}$, the set of values
for $\mathbf{A}_{0}$ that are consistent with the reduced-form parameters in
the absence of additional restrictions is

$\mathbf{A}_{0}\in\left\\{\frac{1}{\sigma_{11}\sigma_{22}}\begin{bmatrix}\sigma_{22}\cos\theta-\sigma_{21}\sin\theta&\sigma_{11}\sin\theta\\\
-\sigma_{21}\cos\theta-\sigma_{22}\sin\theta&\sigma_{11}\cos\theta\end{bmatrix}:\sigma_{22}\cos\theta\geq\sigma_{21}\sin\theta,\cos\theta\geq
0,\theta\in[-\pi,\pi]\right\\}\\\
\cup\left\\{\frac{1}{\sigma_{11}\sigma_{22}}\begin{bmatrix}\sigma_{22}\cos\theta-\sigma_{21}\sin\theta&\sigma_{11}\sin\theta\\\
\sigma_{22}\sin\theta+\sigma_{21}\cos\theta&-\sigma_{11}\cos\theta\end{bmatrix}:\sigma_{22}\cos\theta\geq\sigma_{21}\sin\theta,\cos\theta\leq
0,\theta\in[-\pi,\pi]\right\\}.$ (3)

###  2.1 Shock-sign restrictions

Consider the ‘shock-sign restriction’ that $\varepsilon_{1k}$ is nonnegative
for some $k\in\left\\{1,\ldots,T\right\\}$:

$\varepsilon_{1k}=\mathbf{e}_{1,2}^{\prime}\mathbf{A}_{0}\mathbf{y}_{k}=(\sigma_{11}\sigma_{22})^{-1}\left(\sigma_{22}y_{1k}\cos\theta+(\sigma_{11}y_{2k}-\sigma_{21}y_{1k})\sin\theta\right)\geq
0.$ (4)

Given the realization of the data in period $k$, Equation (4) implies that the
restricted structural shock can be written as a function
$\varepsilon_{1k}(\theta,\bm{\phi},\mathbf{y}_{k})$. Under the sign
normalization and the shock-sign restriction, $\theta$ is restricted to the
set

$\theta\in\left\\{\theta:\sigma_{21}\sin\theta\leq\sigma_{22}\cos\theta,\cos\theta\geq
0,\sigma_{22}y_{1k}\cos\theta\geq(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\sin\theta,-\pi\leq\theta\leq\pi\right\\}\\\
\cup\left\\{\theta:\sigma_{21}\sin\theta\leq\sigma_{22}\cos\theta,\cos\theta\leq
0,\sigma_{22}y_{1k}\cos\theta\geq(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\sin\theta,-\pi\leq\theta\leq\pi\right\\}.$
(5)

Since $y_{1k}$ and $y_{2k}$ enter the inequalities characterising this set,
the shock-sign restriction induces a set-valued mapping from $\bm{\phi}$ to
$\theta$ that depends on the realization of $\mathbf{y}_{k}$. For example, if
$\sigma_{21}<0$, $\sigma_{21}y_{1k}-\sigma_{11}y_{2k}>0$ and $y_{1k}>0$,

$\theta\in\left[\arctan\left(\frac{\sigma_{22}}{\sigma_{21}}\right),\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right)\right].$
(6)

The direct dependence of this mapping on the realization of the data implies
that the standard notion of an identified set – the set of observationally
equivalent structural parameter values given the reduced-form parameters –
does not apply. Consequently, it is not obvious whether existing frequentist
procedures for conducting inference in set-identified models are valid under
NR. Moreover, it is unclear whether the restrictions are, in fact, set-
identifying in a formal frequentist sense. We formally analyze identification
under NR in Section 4.

When conducting Bayesian inference, AR18 construct the posterior using the
conditional likelihood, which is the likelihood of observing the data
conditional on the NR holding. Letting
$\mathbf{y}^{T}=(\mathbf{y}_{1}^{\prime},\ldots,\mathbf{y}_{T}^{\prime})^{\prime}$
represent a realization of the random variable $\mathbf{Y}^{T}$, the
conditional likelihood is

$p\left(\mathbf{y}^{T}|\theta,\bm{\phi},\varepsilon_{1k}(\theta,\bm{\phi},\mathbf{y}_{k})\geq
0\right)=\frac{\prod_{t=1}^{T}(2\pi)^{-1}|\bm{\Sigma}|^{-\frac{1}{2}}\exp\left(-\frac{1}{2}\mathbf{y}_{t}^{\prime}\bm{\Sigma}^{-1}\mathbf{y}_{t}\right)}{\mathrm{Pr}(\varepsilon_{1k}\geq
0|\theta,\bm{\phi})}1\left(\varepsilon_{1k}(\theta,\bm{\phi},\mathbf{y}_{k})\geq
0\right).$ (7)

The numerator in the first term is a function of $\bm{\phi}$ and
$\mathbf{y}^{T}$, while the denominator is equal to 1/2, because the marginal
distribution of $\varepsilon_{1k}$ is standard normal. The conditional
likelihood therefore depends on $\theta$ only through the indicator function
$1\left(\varepsilon_{1k}(\theta,\bm{\phi},\mathbf{y}_{k})\geq 0\right)$. This
indicator function truncates the likelihood, with the truncation points
depending on $\mathbf{y}_{k}$. To illustrate, the left panel of Figure 1 plots
the likelihood given different realizations of the data drawn from a data-
generating process with $\sigma_{21}<0$ and assuming for simplicity that the
econometrician knows $\bm{\phi}$.444The data-generating process assumes
$\mathbf{A}_{0}=\begin{bmatrix}1&0.5\\\ 0.2&1.2\end{bmatrix}$, which implies
that $\theta=\arcsin(0.5\sigma_{22})$ with $\mathbf{Q}$ equal to the rotation
matrix. We assume the time series is of length $T=3$ and draw sequences of
structural shocks such that $\varepsilon_{1,1}\geq 0$. $T$ is a small number
to control Monte Carlo sampling error in the exercises below. The analysis
with known $\bm{\phi}$ replicates the situation with a large sample, where the
likelihood for $\bm{\phi}$ concentrates at the truth. The assumption that
$\bm{\phi}$ is known also facilitates visualizing the likelihood, which
otherwise is a function of four parameters. The conditional likelihood is flat
over the region for $\theta$ satisfying the shock-sign restriction and is zero
outside this region. The support of the nonzero region depends on the
realization of $\mathbf{y}_{k}$.

Figure 1: Shock-sign Restriction

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Notes:

$T=3$, $\bm{\phi}$ is known and
$\varepsilon_{1k}(\theta,\bm{\phi},\mathbf{y}_{k})\geq 0$ is the narrative
sign restriction; likelihood in top-left panel is zero outside of plotted
intervals; posterior density of $\eta=\sigma_{11}\cos\theta$ is approximated
using 1,000,000 draws of $\theta$ from its uniform posterior.

The flat likelihood function implies that the posterior will be proportional
to the prior in the region where the likelihood function is nonzero, and it
will be zero outside this region. The standard approach to Bayesian inference
in SVARs identified via sign restrictions assumes a uniform (or Haar) prior
over $\mathbf{Q}$, as does the approach in AR18.555See, for example, Uhlig
(2005), Rubio-Ramírez, Waggoner and Zha (2010), Baumeister and Hamilton (2015)
and Arias, Rubio-Ramírez and Waggoner (2018). In the bivariate example, this
is equivalent to a prior for $\theta$ that is uniform over the interval
$[-\pi,\pi]$. This prior implies that the posterior for $\theta$ is also
uniform over the interval for $\theta$ where the likelihood function is
nonzero.

The impact impulse response of $y_{1t}$ to a positive standard-deviation shock
$\varepsilon_{1t}$ is $\eta\equiv\sigma_{11}\cos\theta$. The right panel of
Figure 1 plots the posterior for $\eta$ induced by a uniform prior over
$\theta$ given the same realizations of the data for which the likelihood was
plotted in the left panel. The uniform posterior for $\theta$ induces a
posterior for $\eta$ that assigns more probability mass to more-extreme values
of $\eta$. This highlights that even a ‘uniform’ prior may be informative for
parameters of interest, which is also the case under traditional sign
restrictions (Baumeister and Hamilton (2015)). One difference is that the
conditional prior under sign restrictions is never updated by the data,
whereas the support and shape of the posterior for $\eta$ under NR may depend
on the realization of $\mathbf{y}_{k}$ through its effect on the truncation
points of the likelihood, so there may be some updating of the conditional
prior by the data. For example, when $\sigma_{21}<0$,
$\sigma_{21}y_{1k}-\sigma_{11}y_{2k}>0$ and $y_{1k}>0$,

$\eta\in\left[\sigma_{11}\cos\left(\arctan\left(\max\left\\{-\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right)\right),\sigma_{11}\right].$
(8)

However, the conditional prior is not updated at values of $\theta$
corresponding to the flat region of the likelihood. Posterior inference about
$\eta$ may therefore still be sensitive to the choice of prior, as in standard
set-identified SVARs.

###  2.2 Historical-decomposition restrictions

The historical decomposition is the contribution of a particular structural
shock to the observed unexpected change in a particular variable over some
horizon. The contribution of the first shock to the change in the first
variable in the $k$th period is

$H_{1,1,k}(\theta,\bm{\phi},\mathbf{y}_{k})=\sigma_{22}^{-1}\left(\sigma_{22}y_{1k}\cos^{2}\theta+(\sigma_{11}y_{2k}-\sigma_{21}y_{1k})\cos\theta\sin\theta\right),$
(9)

while the contribution of the second shock is

$H_{1,2,k}(\theta,\bm{\phi},\mathbf{y}_{k})=\sigma_{22}^{-1}\left(\sigma_{22}y_{1k}\sin^{2}\theta+(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\cos\theta\sin\theta\right).$
(10)

Consider the restriction that the first structural shock in period $k$ was
positive and (in the language of AR18) the ‘most important contributor’ to the
change in the first variable, which requires that
$|H_{1,1,k}(\theta,\bm{\phi},\mathbf{y}_{k})|\geq|H_{1,2,k}(\theta,\bm{\phi},\mathbf{y}_{k})|$.
Under these restrictions and the sign normalization, $\theta$ must satisfy a
set of inequalities that depends on $\bm{\phi}$ and $\mathbf{y}_{k}$. As in
the case of the shock-sign restriction, this set of restrictions generates a
set-valued mapping from $\bm{\phi}$ to $\theta$ that depends on
$\mathbf{y}_{k}$.666See Appendix A for this set of inequalities. It is more
difficult to analytically characterize the induced mapping than in the shock-
sign example, so we do not pursue this.

Let
$\mathcal{D}(\theta,\bm{\phi},\mathbf{y}_{k})=1\\{\varepsilon_{1k}(\theta,\bm{\phi},\mathbf{y}_{k})\geq
0,|H_{1,1,k}(\theta,\bm{\phi},\mathbf{y}_{k})|\geq|H_{1,2,k}(\theta,\bm{\phi},\mathbf{y}_{k})|\\}$
represent the indicator function equal to one when the NR are satisfied and
equal to zero otherwise, and let
$\tilde{\mathcal{D}}(\theta,\bm{\phi},\bm{\varepsilon}_{k})=1\\{\varepsilon_{1k}\geq
0,|\tilde{H}_{1,1,k}(\theta,\bm{\phi},\varepsilon_{1k})|\geq|\tilde{H}_{1,2,k}(\theta,\bm{\phi},\varepsilon_{2k})|\\}$
represent the indicator function for the same event in terms of the structural
shocks rather than the data. The conditional likelihood function given the
restrictions is then

$p\left(\mathbf{y}^{T}|\theta,\bm{\phi},\mathcal{D}(\theta,\bm{\phi},\mathbf{y}_{k})=1\right)=\frac{\prod_{t=1}^{T}(2\pi)^{-\frac{n}{2}}|\bm{\Sigma}|^{-\frac{1}{2}}\exp\left(-\frac{1}{2}\mathbf{y}_{t}^{\prime}\bm{\Sigma}^{-1}\mathbf{y}_{t}\right)}{\mathrm{Pr}(\tilde{\mathcal{D}}(\theta,\bm{\phi},\bm{\varepsilon}_{k})=1|\theta,\bm{\phi})}\mathcal{D}(\theta,\bm{\phi},\mathbf{y}_{k}).$
(11)

As in the case of the shock-sign restriction, the numerator of the first term
does not depend on $\theta$. In contrast, the probability in the denominator
now depends on $\theta$ through the historical decomposition. Intuitively,
changing $\theta$ changes the impulse responses of $y_{1t}$ to the two shocks
and thus changes the ex ante probability that
$|\tilde{H}_{1,1,k}(\theta,\bm{\phi},\varepsilon_{1k})|\geq|\tilde{H}_{1,2,k}(\theta,\bm{\phi},\varepsilon_{2k})|$.
The conditional likelihood therefore depends on $\theta$ both through this
probability and through the indicator function determining the truncation
points of the likelihood. Consequently, the likelihood function is not
necessarily flat when it is nonzero.

To illustrate, the left panel of Figure 2 plots the conditional likelihood
evaluated at a random realization of the data satisfying the restrictions
using the same data-generating process as above and assuming that $\bm{\phi}$
is known. The probability in the denominator of the conditional likelihood is
approximated by drawing 1,000,000 realizations of $\bm{\varepsilon}_{k}$ and
computing the proportion of draws satisfying the restrictions at each value of
$\theta$. This probability is plotted in the right panel of Figure 2. The
likelihood is again truncated according to a set-valued mapping from
$\bm{\phi}$ and $\mathbf{y}_{k}$ to $\theta$, but an important difference from
the case with the shock-sign restriction is that the likelihood is no longer
flat within the region where it is nonzero. In particular, the conditional
likelihood has a maximum at the value of $\theta$ that minimizes the ex ante
probability that the NR are satisfied (within the set of values of $\theta$
that are consistent with the restrictions). The posterior for $\theta$ induced
by a uniform prior will therefore assign greater posterior probability to
values of $\theta$ that yield a lower ex ante probability of satisfying the
NR.

Figure 2: Historical-decomposition Restriction

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Notes:

$T=3$ and $\bm{\phi}$ is known;
$\varepsilon_{1,1}(\bm{\phi},\theta,\mathbf{y}_{k})\geq 0$ and
$|H_{1,1,1}(\bm{\phi},\theta,\mathbf{y}_{k})|\geq|H_{2,1,1}(\bm{\phi},\theta,\mathbf{y}_{k})|$
are the narrative sign restrictions;
$\mathrm{Pr}(\tilde{\mathcal{D}}(\theta,\bm{\phi},\bm{\varepsilon}_{k})=1|\theta,\bm{\phi})$
is approximated using 1,000,000 Monte Carlo draws.

If we view the narrative event as a part of the observables and its
probability of occurring depends on the parameter of interest, conditioning on
the narrative event implies that we are conditioning on a non-ancillary
statistic. When conducting likelihood-based inference, conditioning on a non-
ancillary statistic is undesirable, because it represents a loss of
information about the parameter of interest. The probability that the shock-
sign restriction is satisfied is independent of the parameters, so the event
that the restriction is satisfied is ancillary. In the case where there is
also a restriction on the historical decomposition, the probability that the
NR are satisfied depends on $\theta$, so the event that the NR are satisfied
is not ancillary. Conditioning on this non-ancillary event results in the
likelihood no longer being flat, but the shape of the likelihood is fully
driven by the inverse probability of the conditioning event. That is, the loss
of information for $\theta$ can be viewed as distorting the shape of the
posterior in the sense that the prior is updated toward values of $\theta$
that make the event that the NR are satisfied less likely ex ante. We
therefore advocate forming the likelihood without conditioning on the
restrictions holding.

The joint (or unconditional) likelihood of observing the data and the NR
holding is obtained by multiplying the conditional likelihood by the
probability that the NR are satisfied:

$p\left(\mathbf{y}^{T},\tilde{\mathcal{D}}(\theta,\bm{\phi},\bm{\varepsilon}_{k})=1|\theta,\bm{\phi}\right)=\prod_{t=1}^{T}(2\pi)^{-\frac{n}{2}}|\bm{\Sigma}|^{-\frac{1}{2}}\exp\left(-\frac{1}{2}\left(\mathbf{y}_{t}^{\prime}\bm{\Sigma}^{-1}\mathbf{y}_{t}\right)\right)\mathcal{D}(\theta,\bm{\phi},\mathbf{y}_{k}).$
(12)

Conditional on being nonzero, the unconditional likelihood is flat with
respect to $\theta$. The unconditional likelihood depends on $\theta$ only
through the points of truncation. To illustrate, Figure 2 plots the
unconditional likelihood given the same realization of the data used to plot
the conditional likelihood. As in the case of the shock-sign restriction, the
flat unconditional likelihood implies that posterior inference may be
sensitive to the choice of prior. We describe our approach to addressing this
posterior sensitivity in Section 5.2.

##  3 General framework

This section describes the general SVAR($p$) and outlines the restrictions
that we consider.

###  3.1 SVAR($p$)

Let $\mathbf{y}_{t}$ be an $n\times 1$ vector of endogenous variables
following the SVAR($p$) process:

$\mathbf{A}_{0}\mathbf{y}_{t}=\sum_{l=1}^{p}\mathbf{A}_{l}\mathbf{y}_{t-l}+\bm{\varepsilon}_{t},\quad
t=1,...,T,$ (13)

where $\mathbf{A}_{0}$ is invertible and
$\bm{\varepsilon}_{t}\overset{iid}{\sim}N(\mathbf{0}_{n\times
1},\mathbf{I}_{n})$ are structural shocks. The initial conditions
$(\mathbf{y}_{1-p},...,\mathbf{y}_{0})$ are given. We omit exogenous
regressors (such as a constant) for simplicity of exposition, but these are
straightforward to include. Letting
$\mathbf{x}_{t}=(\mathbf{y}_{t-1}^{\prime},\ldots,\mathbf{y}_{t-p}^{\prime})^{\prime}$
and $\mathbf{A}_{+}=(\mathbf{A}_{1},\ldots,\mathbf{A}_{p})$, rewrite the
SVAR($p$) as

$\mathbf{A}_{0}\mathbf{y}_{t}=\mathbf{A}_{+}\mathbf{x}_{t}+\bm{\varepsilon}_{t},\quad
t=1,...,T.$ (14)

$(\mathbf{A}_{0},\mathbf{A}_{+})$ are the structural parameters. The reduced-
form VAR($p$) representation is

$\mathbf{y}_{t}=\mathbf{B}\mathbf{x}_{t}+\mathbf{u}_{t},\quad t=1,...,T,$ (15)

where $\mathbf{B}=(\mathbf{B}_{1},\ldots,\mathbf{B}_{p})$,
$\mathbf{B}_{l}=\mathbf{A}_{0}^{-1}\mathbf{A}_{l}$ for $l=1,\ldots,p$, and
$\mathbf{u}_{t}=\mathbf{A}_{0}^{-1}\bm{\varepsilon}_{t}\overset{iid}{\sim}N(\mathbf{0}_{n\times
1},\bm{\Sigma})$ with
$\bm{\Sigma}=\mathbf{A}_{0}^{-1}(\mathbf{A}_{0}^{-1})^{\prime}$.
$\bm{\phi}=(\mathrm{vec}(\mathbf{B})^{\prime},\mathrm{vech}(\bm{\Sigma})^{\prime})^{\prime}\in\bm{\Phi}$
are the reduced-form parameters. We assume that $\mathbf{B}$ is such that the
VAR($p$) can be inverted into an infinite-order vector moving average
(VMA($\infty$)) representation.777The VAR($p$) is invertible into a
VMA($\infty$) process when the eigenvalues of the companion matrix lie inside
the unit circle. See Hamilton (1994) or Kilian and Lütkepohl (2017).

As is standard in the literature that considers set-identified SVARs, we
reparameterize the model into its orthogonal reduced form (e.g., Arias et al.
(2018)):

$\mathbf{y}_{t}=\mathbf{B}\mathbf{x}_{t}+\bm{\Sigma}_{tr}\mathbf{Q}\bm{\varepsilon}_{t},\quad
t=1,...,T,$ (16)

where $\bm{\Sigma}_{tr}$ is the lower-triangular Cholesky factor of
$\bm{\Sigma}$ (i.e. $\bm{\Sigma}_{tr}\bm{\Sigma}_{tr}^{\prime}=\bm{\Sigma}$)
with diagonal elements normalized to be non-negative, $\mathbf{Q}$ is an
$n\times n$ orthonormal matrix and $\mathcal{O}(n)$ is the set of all such
matrices. The structural and orthogonal reduced-form parameterizations are
related through the mapping $\mathbf{B}=\mathbf{A}_{0}^{-1}\mathbf{A}_{+}$,
$\bm{\Sigma}=\mathbf{A}_{0}^{-1}(\mathbf{A}_{0}^{-1})^{\prime}$ and
$\mathbf{Q}=\bm{\Sigma}_{tr}^{-1}\mathbf{A}_{0}^{-1}$ with inverse mapping
$\mathbf{A}_{0}=\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1}$ and
$\mathbf{A}_{+}=\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1}\mathbf{B}$.

The VMA($\infty$) representation of the model is

$\mathbf{y}_{t}=\sum_{h=0}^{\infty}\mathbf{C}_{h}\mathbf{u}_{t-h}=\sum_{h=0}^{\infty}\mathbf{C}_{h}\bm{\Sigma}_{tr}\mathbf{Q}\bm{\varepsilon}_{t},\quad
t=1,...,T,$ (17)

where $\mathbf{C}_{h}$ is the $h$th term in
$(\mathbf{I}_{n}-\sum_{l=1}^{p}\mathbf{B}_{l}L^{l})^{-1}$ and $L$ is the lag
operator. $\mathbf{C}_{h}$ is defined recursively by
$\mathbf{C}_{h}=\sum_{l=1}^{\min\\{k,p\\}}\mathbf{B}_{l}\mathbf{C}_{h-l}$ for
$h\geq 1$ with $\mathbf{C}_{0}=\mathbf{I}_{n}$. The $(i,j)$th element of the
matrix $\mathbf{C}_{h}\bm{\Sigma}_{tr}\mathbf{Q}$, which we denote by
$\eta_{i,j,h}(\bm{\phi},\mathbf{Q})$, is the horizon-$h$ impulse response of
the $i$th variable to the $j$th structural shock:

$\eta_{i,j,h}(\bm{\phi},\mathbf{Q})=\mathbf{e}_{i,n}^{\prime}\mathbf{C}_{h}\bm{\Sigma}_{tr}\mathbf{Q}\mathbf{e}_{j,n}=\mathbf{c}_{i,h}^{\prime}(\bm{\phi})\mathbf{q}_{j},$
(18)

where
$\mathbf{c}_{i,h}^{\prime}(\bm{\phi})=\mathbf{e}_{i,n}^{\prime}\mathbf{C}_{h}\bm{\Sigma}_{tr}$
is the $i$th row of $\mathbf{C}_{h}\bm{\Sigma}_{tr}$ and
$\mathbf{q}_{j}=\mathbf{Q}\mathbf{e}_{j,n}$ is the $j$th column of
$\mathbf{Q}$.

###  3.2 Narrative restrictions

In the absence of any identifying restrictions, it is well-known that
$\mathbf{Q}$ is set-identified. Consequently, functions of $\mathbf{Q}$, such
as the impulse responses, are also set-identified. Imposing traditional
identifying restrictions on the SVAR is equivalent to restricting $\mathbf{Q}$
to lie in a subspace of $\mathcal{O}(n)$. It is conventional to impose a ‘sign
normalization’ on the structural shocks. We normalize the diagonal elements of
$\mathbf{A}_{0}$ to be non-negative, so a positive value of $\varepsilon_{it}$
is a positive shock to the $i$th equation in the SVAR at time $t$. The sign
normalization implies that
$\mathrm{diag}(\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1})\geq\mathbf{0}_{n\times
1}$.

It is common to impose sign restrictions on the impulse responses (e.g., Uhlig
(2005)) or on the structural parameters themselves. For example, the
restriction that the horizon-$h$ impulse response of the $i$th variable to the
$j$th shock is nonnegative is $c_{i,h}^{\prime}(\bm{\phi})\mathbf{q}_{j}\geq
0$, which is a linear inequality restriction on a single column of
$\mathbf{Q}$ that depends only on the reduced-form parameter $\bm{\phi}$.
Restrictions on elements of $\mathbf{A}_{0}$ take a similar form.

In contrast, NR constrain the values of the structural shocks in particular
periods. The structural shocks are

$\bm{\varepsilon}_{t}=\mathbf{A}_{0}\mathbf{u}_{t}=\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t}.$
(19)

The shock-sign restriction that the $i$th structural shock at time $k$ is
positive is

$\varepsilon_{ik}(\bm{\phi},\mathbf{Q},\mathbf{u}_{k})=\mathbf{e}_{i,n}^{\prime}\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{k}=(\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{k})^{\prime}\mathbf{q}_{i}\geq
0.$ (20)

We can treat $\mathbf{u}_{t}$ as observable given $\bm{\phi}$ and the data, so
we suppress the dependence of $\mathbf{u}_{t}$ on $\bm{\phi}$ and
$(\mathbf{y}_{t}^{\prime},\mathbf{x}_{t}^{\prime})^{\prime}$ for notational
convenience. The restriction in (20) is a linear inequality restriction on a
single column of $\mathbf{Q}$. In contrast with traditional sign restrictions,
the shock-sign restriction depends directly on the data through the reduced-
form VAR innovations.

In addition to shock-sign restrictions, AR18 consider restrictions on the
historical decomposition, which is the cumulative contribution of the $j$th
shock to the observed unexpected change in the $i$th variable between periods
$k$ and $k+h$:

$H_{i,j,k,k+h}\left(\bm{\phi},\mathbf{Q},\left\\{\mathbf{u}_{t}\right\\}_{t=k}^{k+h}\right)=\sum_{l=0}^{h}\mathbf{e}_{i,n}^{\prime}\mathbf{C}_{l}\bm{\Sigma}_{tr}\mathbf{Q}\mathbf{e}_{j,n}\mathbf{e}_{j,n}^{\prime}\bm{\varepsilon}_{k+h-l}=\sum_{l=0}^{h}\mathbf{c}_{i,l}^{\prime}(\bm{\phi})\mathbf{q}_{j}\mathbf{q}_{j}^{\prime}\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{k+h-l}.$
(21)

One example of a restriction on the historical decomposition is that the $j$th
structural shock was the ‘most important contributor’ to the change in the
$i$th variable between periods $k$ and $k+h$, which requires that
$|H_{i,j,k,k+h}|\geq\max_{l\neq j}|H_{i,l,k,k+h}|$. Another example is that
the $j$th structural shock was the ‘overwhelming contributor’ to the change in
the $i$th variable between periods $k$ and $k+h$, which requires that
$|H_{i,j,k,k+h}|\geq\sum_{l\neq j}|H_{i,l,k,k+h}|$. From Equation (21), it is
clear that these restrictions are nonlinear inequality constraints that
simultaneously constrain every column of $\mathbf{Q}$ and that depend on the
realizations of the data in particular periods in addition to the reduced-form
parameters.

Other restrictions also naturally fit into this framework. For instance, we
can consider restrictions on the relative magnitudes of a particular
structural shock in different periods. We refer to these restrictions as
‘shock-rank restrictions’, since they imply a (possibly partial) ordering of
the shocks. As an example, one could impose that the $i$th shock in period $k$
was the largest positive realization of this shock in the observed sample.
This requires that
$\varepsilon_{ik}(\bm{\phi},\mathbf{Q},\mathbf{u}_{k})\geq\max_{t\neq
k}\\{\varepsilon_{it}(\bm{\phi},\mathbf{Q},\mathbf{u}_{t})\\}$, which can be
expressed as a system of $T-1$ linear inequality restrictions on a single
column of $\mathbf{Q}$:
$(\bm{\Sigma}_{tr}^{-1}(\mathbf{u}_{k}-\mathbf{u}_{t}))^{\prime}\mathbf{q}_{i}\geq
0$ for $t\neq k$. Alternatively, one could impose that the $i$th shock in
period $k$ was the largest-magnitude realization of that shock, or
$|\varepsilon_{ik}(\bm{\phi},\mathbf{Q},\mathbf{u}_{k})|\geq\max_{t\neq
k}\left\\{|\varepsilon_{it}(\bm{\phi},\mathbf{Q},\mathbf{u}_{t})|\right\\}$.
If $\varepsilon_{ik}(\bm{\phi},\mathbf{Q},\mathbf{u}_{k})\geq 0$, this would
require that
$(\bm{\Sigma}_{tr}^{-1}(\mathbf{u}_{k}-\mathbf{u}_{t}))^{\prime}\mathbf{q}_{i}\geq
0$ and
$(\bm{\Sigma}_{tr}^{-1}(\mathbf{u}_{k}+\mathbf{u}_{t}))^{\prime}\mathbf{q}_{i}\geq
0$ for $t\neq k$, which is a system of $2(T-1)$ linear inequalities
constraining $\mathbf{q}_{i}$. These restrictions could also be applied to a
subset of the observations rather than the full sample (e.g.,
$\varepsilon_{ik}(\bm{\phi},\mathbf{Q},\mathbf{u}_{k})>\varepsilon_{it}(\bm{\phi},\mathbf{Q},\mathbf{u}_{t})$
for some $t\in\\{1,\ldots,T\\}$).888Similar to the shock-rank restrictions we
describe, Ben Zeev (2018) imposes a restriction on the timing of the maximum
three-year average of a particular shock, as well as restrictions on the sign
and relative magnitudes of this three-year average in specific periods.
Restrictions on averages of shocks can also be implemented in the framework we
consider.

The collection of NR can be represented in the general form
$N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})\geq\mathbf{0}_{s\times 1}$, where $s$
is the number of restrictions. As an illustration, consider the case where
there is a single shock-sign restriction in period $k$,
$\varepsilon_{1k}(\bm{\phi},\mathbf{Q},\mathbf{u}_{k})\geq 0$, as well as the
restriction that the first structural shock was the most important contributor
to the change in the first variable in period $k$. Then,

$N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})=\begin{bmatrix}(\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{k})^{\prime}\mathbf{q}_{1}\\\
|\mathbf{e}_{1,n}^{\prime}\bm{\Sigma}_{tr}\mathbf{q}_{1}\mathbf{q}_{1}^{\prime}\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{k}|-\max_{j\neq
1}|\mathbf{e}_{1,n}^{\prime}\bm{\Sigma}_{tr}\mathbf{q}_{j}\mathbf{q}_{j}^{\prime}\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{k}|\end{bmatrix}\geq\mathbf{0}_{2\times
1}.$ (22)

Traditional sign and zero restrictions can also be applied alongside NR. We
follow AR18 by explicitly allowing for sign restrictions on impulse responses
and on elements of $\mathbf{A}_{0}$. We denote such sign restrictions by
$S(\bm{\phi},\mathbf{Q})\geq\mathbf{0}_{\tilde{s}\times 1}$, where $\tilde{s}$
is the number of traditional sign restrictions. It is straightforward to
additionally allow for zero restrictions, including ‘short-run’ zero
restrictions (as in Sims (1980)), ‘long-run’ zero restrictions (as in
Blanchard and Quah (1989)), or restrictions arising from external instruments
(as in Mertens and Ravn (2013) and Stock and Watson (2018)); for example, see
GK and Giacomini et al. (2019).

###  3.3 Conditional and unconditional likelihoods

When constructing the posterior of the SVAR’s parameters, AR18 use the
likelihood conditional on the NR holding. Define

$\displaystyle D_{N}$
$\displaystyle=D_{N}(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})\equiv
1\\{N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})\geq\mathbf{0}_{s\times 1}\\},$
$\displaystyle r(\bm{\phi},\mathbf{Q})$
$\displaystyle\equiv\Pr(D_{N}(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})=1|\bm{\phi},\mathbf{Q}),$
$\displaystyle f(\mathbf{y}^{T}|\bm{\phi})$
$\displaystyle\equiv\prod_{t=1}^{T}(2\pi)^{-\frac{n}{2}}|\bm{\Sigma}|^{-\frac{1}{2}}\exp\left(-\frac{1}{2}\left(\mathbf{y}_{t}-\mathbf{Bx}_{t}\right)^{\prime}\bm{\Sigma}^{-1}\left(\mathbf{y}_{t}-\mathbf{Bx}_{t}\right)\right).$

The likelihood conditional on $D_{N}=1$ can be written as

$p(\mathbf{y}^{T}|D_{N}=1,\bm{\phi},\mathbf{Q})=\frac{f(\mathbf{y}^{T}|\bm{\phi})}{r(\bm{\phi},\mathbf{Q})}\cdot
D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T}).$ (23)

$f(\mathbf{y}^{T}|\bm{\phi})$ is the joint density of the data given
$\bm{\phi}$ (i.e., the likelihood function of the reduced-form VAR), which
depends only on $\bm{\phi}$ and the data. The indicator function
$D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})$ is equal to one when the NR are
satisfied and is equal to zero otherwise. This determines the truncation
points of the likelihood. $r(\bm{\phi},\mathbf{Q})$ is the ex ante probability
that the NR are satisfied. This will be a constant when there are only shock-
sign or shock-rank restrictions; for example, if there are $s$ shock-sign
restrictions, $r(\bm{\phi},\mathbf{Q})=(1/2)^{s}$. In contrast, when there are
restrictions on the historical decomposition, this probability will depend on
$\bm{\phi}$ and $\mathbf{Q}$.

Consider the case where $\bm{\phi}$ is known, which will be the case
asymptotically because $\bm{\phi}$ is point-identified. When
$r(\bm{\phi},\mathbf{Q})$ depends on $\mathbf{Q}$, the conditional likelihood
will be maximized at the value of $\mathbf{Q}$ that minimizes
$r(\bm{\phi},\mathbf{Q})$ (within the set of values of $\mathbf{Q}$ that
satisfy the restrictions). The posterior based on this likelihood will
therefore place higher posterior probability on values of $\mathbf{Q}$ that
result in a lower ex ante probability that the restrictions are satisfied. As
discussed in Section 2.2, this is an artefact of conditioning on a non-
ancillary event, which represents a loss of information about the parameters.

We therefore advocate constructing the likelihood without conditioning on the
NR holding. The unconditional likelihood (the joint distribution of the data
and $D_{N}$) can be expressed as

$\displaystyle\ p(\mathbf{y}^{T},D_{N}=d|\bm{\phi},\mathbf{Q})$
$\displaystyle=\left[f(\mathbf{y}^{T}|\bm{\phi})D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})\right]^{d}\cdot\left[f(\mathbf{y}^{T}|\bm{\phi})\left(1-D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})\right)\right]^{1-d}$
$\displaystyle=f(\mathbf{y}^{T}|\bm{\phi})\cdot\left[D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})\right]^{d}\cdot\left[1-D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})\right]^{1-d}.$
(24)

For any value of $\bm{\phi}$ such that $\mathbf{y}^{T}$ is compatible with the
NR, there will be a set of values of $\mathbf{Q}$ that satisfy the
restrictions, which depend on the data, but the value of the unconditional
likelihood will be the same for all values of $\mathbf{Q}$ within this set.
The conditional posterior of $\mathbf{Q}|\bm{\phi},\mathbf{y}^{T}$ will
therefore be proportional to the conditional prior for $\mathbf{Q}|\bm{\phi}$
in these regions. Given a fixed number of NR, the likelihood will possess flat
regions even with a time-series of infinite length, so posterior inference may
be sensitive to the choice of conditional prior for $\mathbf{Q}$, even
asymptotically (which is also the case for the conditional likelihood when the
restrictions are ancillary). This motivates considering Bayesian inferential
procedures that are robust to the choice of unrevisable conditional prior for
$\mathbf{Q}$, which we explore in Section 5.2.

###  3.4 Discussion

In this section, we briefly discuss the distributional assumptions for the
structural shocks and the mechanism that generates the NR.

####  3.4.1 Distributional assumptions

Practitioners may be concerned about the robustness of inference with respect
to deviations from the assumption of standard normal shocks. For instance, one
could worry that the periods in which the NR are imposed are ‘unusual’ in the
sense that the structural shocks in these periods were drawn from a
distribution with, say, inflated variance or fat tails. The unconditional
likelihood depends on the normality assumption only through
$f(\mathbf{y}^{T}|\bm{\phi})$. By omitting terms in
$f(\mathbf{y}^{T}|\bm{\phi})$ corresponding to the periods in which the NR are
imposed, one can conduct inference that is robust to the distributional
assumption about the shocks in these particular periods. To illustrate,
consider the case where NR are imposed in period $k$ only and assume the
likelihood function for $\mathbf{y}^{T}$ takes the form

$\tilde{f}(\mathbf{y}^{T}|\bm{\phi})=v(\left\\{\mathbf{y}_{t}-\mathbf{Bx}_{t}\right\\}_{t\neq
k}|\bm{\phi})w(\mathbf{y}_{k}-\mathbf{Bx}_{k}),$ (25)

where

$v(\left\\{\mathbf{y}_{t}-\mathbf{Bx}_{t}\right\\}_{t\neq
k}|\bm{\phi})=\prod_{t\neq
k}(2\pi)^{-\frac{n}{2}}|\bm{\Sigma}|^{-\frac{1}{2}}\exp\left(-\frac{1}{2}\left(\mathbf{y}_{t}-\mathbf{Bx}_{t}\right)^{\prime}\bm{\Sigma}^{-1}\left(\mathbf{y}_{t}-\mathbf{Bx}_{t}\right)\right)$
(26)

and $w(\mathbf{y}_{k}-\mathbf{Bx}_{k})$ is an unknown, potentially non-normal,
density. Replacing $f(\mathbf{y}^{T}|\bm{\phi})$ in Equation (24) with
$v(\left\\{\mathbf{y}_{t}-\mathbf{Bx}_{t}\right\\}_{t\neq k}|\bm{\phi})$
yields an ‘unconditional partial likelihood’ that does not depend on the
distribution of $\bm{\varepsilon}_{k}$, but that is still truncated by the NR.
This would potentially result in a loss of information relative to a
likelihood that correctly specifies the distribution of the shocks in period
$k$. However, when NR are imposed in only a few periods, this loss of
information is likely to be small. In contrast, the conditional likelihood
approach cannot leave fully unspecified the distribution of the restricted
structural shocks, because computing $r(\bm{\phi},\mathbf{Q})$ requires
specifying this distribution.

Concerns about heteroscedasticity or non-normality may also be alleviated by
recognizing that the distributional assumption will become irrelevant
asymptotically. The set of values of $\mathbf{Q}$ with non-zero unconditional
likelihood depends only on $\bm{\phi}$, which summarizes the second moments of
the data, and the realization of the data in the periods in which the NR are
imposed. Under regularity assumptions, the likelihood (and thus the posterior)
of $\bm{\phi}$ will converge to a point at the true value of $\bm{\phi}$
asymptotically regardless of whether the true data-generating process is a VAR
with homoscedastic normal shocks.999See Plagborg-Møller (2019) for a
discussion of this point in the context of a structural VMA model. The set of
values of $\mathbf{Q}$ with non-zero likelihood will therefore converge
asymptotically to the same set regardless of whether the distributional
assumption is correct.

####  3.4.2 Mechanism generating NR

Note that we do not explicitly model the mechanism responsible for revealing
the information underlying the NR (i.e., whether $D_{N}=1$ or $D_{N}=0$) or
the mechanism determining the periods in which this information is revealed
(e.g., the identity of $k$ in examples above), which is consistent with the
papers that impose these restrictions. If the revelation of this information
depends on the data, the likelihood will be misspecified. The exact
implications of this misspecification for estimation or inference will depend
on assumptions about the mechanism revealing the narrative information.
Exploring the consequences of such misspecification may be an interesting area
for further work. In the bivariate example of Section 2, if the identity of
$k$ is randomly determined independently of
$\bm{\varepsilon}_{1},\ldots,\bm{\varepsilon}_{T}$, we can interpret the
current analysis conditional on $k$.

##  4 Identification under NR

This section formally analyzes identification in the SVAR under NR. Section
4.1 considers whether NR are point- or set-identifying in a frequentist sense.
Section 4.2 introduces the notion of a ‘conditional identified set’, which
extends the standard notion of an identified set to the setting where the
mapping from reduced-form to structural parameters depends on the realization
of the data. This provides an interpretation of the mapping induced by the NR.
Additionally, we make use of this object when showing the frequentist validity
of our robust Bayesian procedure in Section 6.

###  4.1 Point-identification under NR

Denoting the true parameter value by $(\bm{\phi}_{0},\textbf{Q}_{0})$, point-
identification for the parametric model (24) requires that there is no other
parameter value $(\bm{\phi},\mathbf{Q})\neq(\bm{\phi}_{0},\mathbf{Q}_{0})$
that is observationally equivalent to
$(\bm{\phi}_{0},\mathbf{Q}_{0})$.101010$(\bm{\phi},\mathbf{Q})\neq(\bm{\phi}_{0},\mathbf{Q}_{0})$
is observationally equivalent to $(\bm{\phi}_{0},\mathbf{Q}_{0})$ if
$p(\mathbf{Y}^{T},D_{N}=d|\bm{\phi},\mathbf{Q})=p(\mathbf{Y}^{T},D_{N}=d|\bm{\phi}_{0},\mathbf{Q}_{0})$
holds for all $\mathbf{Y}^{T}$ and $d\in\\{0,1\\}$.

To assess the existence or non-existence of observationally equivalent
parameter points, we analyze a statistical distance between
$p(\mathbf{y}^{T},D_{N}=d|\bm{\phi},\mathbf{Q})$ and
$p(\mathbf{y}^{T},D_{N}=d|\bm{\phi}_{0},\mathbf{Q}_{0})$ that metrizes
observation equivalence. Specifically, in the current setting where the
support of the distribution of observables can depend on the parameters, it is
convenient to work with the Hellinger distance:

$\displaystyle HD(\bm{\phi},\mathbf{Q})$
$\displaystyle\equiv\sum_{d=0,1}\int_{\mathbf{Y}}\left(p^{1/2}(\mathbf{y}^{T},D_{N}=d|\bm{\phi},\mathbf{Q})-p^{1/2}(\mathbf{y}^{T},D_{N}=d|\bm{\phi}_{0},\mathbf{Q}_{0})\right)^{2}d\mathbf{y}^{T}$
$\displaystyle=2\left(1-\mathcal{H}(\bm{\phi},\mathbf{Q})\right),\mspace{10.0mu}\text{where}$
$\displaystyle\mathcal{H}(\bm{\phi},\mathbf{Q})$
$\displaystyle\equiv\sum_{d=0,1}\int_{\mathbf{Y}}p^{1/2}(\mathbf{y}^{T},D_{N}=d|\bm{\phi},\mathbf{Q})\cdot
p^{1/2}(\mathbf{y}^{T},D_{N}=d|\bm{\phi}_{0},\mathbf{Q}_{0})d\mathbf{y}^{T},$
(27)

and $\mathbf{Y}$ is the sample space for $\mathbf{Y}^{T}$. As is known in the
literature on minimum distance estimation (see, for example, Basu, Shioya and
Park (2011)), $(\bm{\phi},\mathbf{Q})$ and $(\bm{\phi}_{0},\mathbf{Q}_{0})$
are observationally equivalent if and only if $HD(\bm{\phi},\mathbf{Q})=0$ or,
equivalently, $\mathcal{H}(\bm{\phi},\mathbf{Q})=1$.

We similarly define the Hellinger distance for the conditional likelihood as

$\displaystyle HD_{c}(\bm{\phi},\mathbf{Q})$ $\displaystyle\equiv
2\left(1-\mathcal{H}_{c}(\bm{\phi},\mathbf{Q})\right),\mspace{10.0mu}\text{where}$
$\displaystyle\mathcal{H}_{c}(\bm{\phi},\mathbf{Q})$
$\displaystyle\equiv\int_{\mathbf{Y}}p^{1/2}(\mathbf{y}^{T}|D_{N}=1,\bm{\phi},\mathbf{Q})\cdot
p^{1/2}(\mathbf{y}^{T}|D_{N}=1,\bm{\phi}_{0},\mathbf{Q}_{0})d\mathbf{y}^{T}.$
(28)

The next proposition analyzes the conditions for
$\mathcal{H}(\bm{\phi},\mathbf{Q})=1$ and
$\mathcal{H}_{c}(\bm{\phi},\mathbf{Q})=1$, and shows that observational
equivalence of $(\bm{\phi},\mathbf{Q})$ and $(\bm{\phi}_{0},\mathbf{Q}_{0})$
boils down to geometric equivalence of the set of reduced-form VAR innovations
satisfying the NR.

######  Proposition 4.1.

Let $(\bm{\phi}_{0},\mathbf{Q}_{0})$ be the true parameter value and let
$\mathbf{U}\equiv\mathbf{U}(\mathbf{y}^{T};\bm{\phi})=(\mathbf{u}_{1}^{\prime},\ldots,\mathbf{u}_{T}^{\prime})^{\prime}$
collect the reduced-form VAR innovations. Define

$\mathcal{Q}^{\ast}\equiv\left\\{\begin{matrix}\mathbf{Q}\in\mathcal{O}(n):\\{\mathbf{U}:N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})\geq\mathbf{0}_{s\times
1}\\}=\\{\mathbf{U}:N(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{Y}^{T})\geq\mathbf{0}_{s\times
1}\\}\\\ \text{up to $f(\mathbf{Y}^{T}|\bm{\phi}_{0})$-null set},\
\mathrm{diag}(\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1})\geq\mathbf{0}_{n\times
1}\end{matrix}\right\\}.$

The unconditional likelihood model (24) and the conditional likelihood model
(23) are globally identified (i.e., there are no observationally equivalent
parameter points to $(\bm{\phi}_{0},\mathbf{Q}_{0})$) if and only if
$\mathcal{Q}^{\ast}$ is a singleton. If the parameter of interest is an
impulse response to the $j$th structural shock,
$\eta_{i,j,h}(\bm{\phi},\mathbf{Q})$, as defined in (18), then
$\eta_{i,j,h}(\bm{\phi},\mathbf{Q})$ is point-identified if the projection of
$\mathcal{Q}^{\ast}$ onto its $j$th column vector is a singleton.

###### Proof.

See Appendix B. ∎

This proposition provides a necessary and sufficient condition for global
identification of SVARs by NR. As shown in the proof in Appendix B,
$\mathcal{Q}^{\ast}$ defined in this proposition corresponds to the
observationally equivalent $\mathbf{Q}$ matrices given
$\bm{\phi}=\bm{\phi}_{0}$, but, importantly, it does not correspond to any
flat region of the observed likelihood (the conditional identified set in
Definition 4.1 below).

To illustrate this point, consider the simple bivariate example of Section 2
with the NR (4), where $\mathbf{y}_{t}$ itself is the reduced-form error, so
$\mathbf{U}$ in Proposition 4.1 can be set to $\mathbf{y}_{k}$. Given
$\bm{\phi}$, the set of $\mathbf{y}_{k}\in\mathbb{R}^{2}$ satisfying the NR is
the half-space given by

$\left\\{\mathbf{y}_{k}\in\mathbb{R}^{2}:(\sigma_{11}\sigma_{22})^{-1}\begin{pmatrix}\sigma_{22}\cos\theta-\sigma_{21}\sin\theta,&\sigma_{11}\sin\theta\end{pmatrix}\mathbf{y}_{k}\geq
0\right\\}.$ (29)

The condition for point-identification shown in Proposition 4.1 is satisfied
if no $\theta^{\prime}\neq\theta$ can generate the half-space of
$\mathbf{y}_{k}$ identical to (29). Such $\theta^{\prime}$ cannot exist, since
a half-space passing through the origin $(a_{1},a_{2})\mathbf{y}_{k}\geq 0$
can be indexed uniquely by the slope $a_{1}/a_{2}$ and (29) implies the slope
$\sigma_{11}^{-1}(\sigma_{22}(\tan\theta)^{-1}-\sigma_{21})$ is a bijective
map of $\theta$ on a constrained domain due to the sign normalization. Figure
3 plots the Hellinger distances in this bivariate example under the shock-sign
restriction (4) and the historical decomposition restriction. For both the
conditional and unconditional likelihood, the Hellinger distances are
minimized uniquely at the true $\theta$, which is consistent with our point-
identification claim for $\theta$.111111Under the restriction on the
historical decomposition, a notable difference between the conditional and
unconditional likelihood cases is the slope of the Hellinger distance around
the minimum. The Hellinger distance of the unconditional likelihood yields a
steeper slope than the conditional likelihood. This indicates the loss of
information for $\theta$ in the conditional likelihood due to conditioning on
the non-ancillary event.

Figure 3: Hellinger Distance

|  |
---|---|---

Notes:

$T=3$ and $\bm{\phi}$ is known; Hellinger distances are approximated using
Monte Carlo.

Proposition 4.1 also provides conditions under which $(\bm{\phi},\mathbf{Q})$
is not globally identified, but a particular impulse response is. To give an
example of this, consider an SVAR with $n>2$ and with a shock-sign restriction
on the first shock in period $k$. Given $\bm{\phi}$, the set of
$\mathbf{u}_{k}\in\mathbb{R}^{n}$ satisfying the NR is a half-space defined by
$\mathbf{q}_{1}^{\prime}\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{k}\geq 0$. The set of
values of $\mathbf{u}_{k}$ satisfying this inequality is indexed uniquely by
$\mathbf{q}_{1}$ given $\bm{\Sigma}_{tr}$ at its true value, so there are no
values of $\mathbf{Q}$ that are observationally equivalent to $\mathbf{Q}_{0}$
with $\mathbf{q}_{1}\neq\mathbf{Q}_{0}\mathbf{e}_{1,n}$. Any value for the
remaining $n-1$ columns of $\mathbf{Q}$ such that they are orthogonal to
$\mathbf{Q}_{0}\mathbf{e}_{1,n}$ will generate the same half-space for
$\mathbf{u}_{k}$, so $\mathcal{Q}^{\ast}$ is not a singleton and the SVAR is
not globally identified. However, the projection of $\mathcal{Q}^{\ast}$ onto
its first column is a singleton, so $\eta_{i,1,h}(\bm{\phi},\mathbf{Q})$ is
globally identified.

Although a single NR can deliver global identification in the frequentist
sense, the practical implication of this theoretical claim is not obvious. The
observed unconditional likelihood is almost always flat at the maximum, so we
cannot obtain a unique maximum likelihood estimator for the structural
parameter. As a result, the standard asymptotic approximation of the sampling
distribution of the maximum likelihood estimator is not applicable. The SVAR
model with NR possesses features of set-identified models from the Bayesian
standpoint (i.e., flat regions of the likelihood). However, strictly speaking,
it can be classified as a globally identified model in the frequentist sense
when the condition of Proposition 4.1 holds.

###  4.2 Conditional identified set

It is well-known that traditional sign restrictions deliver set-identification
of $\mathbf{Q}$ (or, equivalently, the structural parameters). Given the
reduced-form parameter $\bm{\phi}$ – which is point-identified – there are
multiple observationally equivalent values of $\mathbf{Q}$, in the sense that
there exists $\mathbf{Q}$ and $\tilde{\mathbf{Q}}\neq\mathbf{Q}$ such that
$p(\mathbf{y}^{T}|\bm{\phi},\mathbf{Q})=p(\mathbf{y}^{T}|\bm{\phi},\tilde{\mathbf{Q}})$
for every $\mathbf{y}^{T}$ in the sample space. The identified set for
$\mathbf{Q}$ given $\bm{\phi}$ contains all such observationally equivalent
parameter points, and is defined as

$\mathcal{Q}(\bm{\phi}|S)=\left\\{\mathbf{Q}\in\mathcal{O}(n):S(\bm{\phi},\mathbf{Q})\geq\mathbf{0}_{\tilde{s}\times
1},\mathrm{diag}(\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1})\geq\mathbf{0}_{n\times
1}\right\\}.$ (30)

The identified set is a set-valued map only of $\bm{\phi}$, which carries all
the information about $\mathbf{Q}$ contained in the data.

The complication in applying this definition of the identified set in SVARs
when there are NR is that the reduced-form VAR parameters no longer represent
all information about $\mathbf{Q}$ contained in the data; by truncating the
likelihood, the realizations of the data entering the NR contain additional
information about $\mathbf{Q}$. To address this, we introduce a refinement of
the definition of an identified set.

######  Definition 4.1.

Let $N\equiv N(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})\geq\mathbf{0}_{s\times 1}$
represent a set of NR in terms of the parameters and the data.

(i) The conditional identified set for $\mathbf{Q}$ under NR is

$\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)=\\{\mathbf{Q}\in\mathcal{O}(n):N(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})\geq\mathbf{0}_{s\times
1}\\}.$ (31)

The conditional identified set for the impulse response
$\eta=\eta_{i,j,h}(\bm{\phi},\mathbf{Q})$ under NR is defined by projecting
$\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)$ via
$\eta_{i,j,h}(\bm{\phi},\mathbf{Q})$:

$CIS_{\eta}(\bm{\phi}|\mathbf{y}^{T},N)=\\{\eta_{i,j,h}(\bm{\phi},\mathbf{Q}):\mathbf{Q}\in\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)\\}.$
(32)

(ii) Let $\mathbf{s}:\mathbf{Y}\to\mathbb{R}^{S}$ be a statistic. We call
$\mathbf{s}(\mathbf{Y}^{T})$ a sufficient statistic for the conditional
identified set $\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)$ if the conditional
identified set for $\mathbf{Q}$ depends on the sample $\mathbf{y}^{T}$ through
$\mathbf{s}(\mathbf{y}^{T})$; i.e., there exists
$\tilde{\mathcal{Q}}(\bm{\phi}|\cdot,N)$ such that

$\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)=\tilde{\mathcal{Q}}(\bm{\phi}|\mathbf{s}(\mathbf{y}^{T}),N)$
(33)

holds for all $\bm{\phi}\in\bm{\Phi}$ and $\mathbf{y}^{T}\in\mathbf{Y}$.

Unlike the standard identified set $\mathcal{Q}(\bm{\phi}|S)$, the conditional
identified set $\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)$ depends on the sample
$\mathbf{y}^{T}$ because of the aforementioned data-dependent support of the
likelihood. In terms of the observed likelihood, however, they share the
property that the likelihood is flat on the (conditional) identified set.
Hence, given the sample $\mathbf{y}^{T}$ and the reduced-form parameters
$\bm{\phi}$, all values of $\mathbf{Q}$ in
$\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)$ fit the data equally well and, in
this particular sense, they are observationally equivalent.

When the NR concern shocks in only a subset of the time periods in the data,
the conditional identified set under these NR depends on the sample only
through a few observations entering the NR. The sufficient statistics
$\textbf{s}(\mathbf{y}^{T})$ defined in Definition 4.1(ii) represent such
observations. For instance, in the toy example of Section 2.1, the conditional
identified set depends only on the observations in period $k$, so
$\mathbf{s}(\mathbf{y}^{T})=\mathbf{y}_{k}$. If we extend the example of
Section 2.1 to the SVAR($p$), the shock-sign restriction in Equation (4) can
be expressed as

$\varepsilon_{1k}=\mathbf{e}_{1,2}^{\prime}\mathbf{A}_{0}\mathbf{u}_{k}=\mathbf{e}_{1,2}^{\prime}\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1}(\mathbf{y}_{k}-\mathbf{B}\mathbf{x}_{k})\geq
0.$ (34)

Hence, the conditional identified set
$\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)$ depends on the data only through
$(\mathbf{y}_{k}^{\prime},\mathbf{x}_{k}^{\prime})^{\prime}=(\mathbf{y}_{k}^{\prime},\mathbf{y}_{k-1}^{\prime},\cdots,\mathbf{y}_{k-p}^{\prime})^{\prime}$,
so we can set
$\mathbf{s}(\mathbf{y}^{T})=(\mathbf{y}_{k}^{\prime},\mathbf{y}_{k-1}^{\prime},\cdots,\mathbf{y}_{k-p}^{\prime})^{\prime}$.

If the conditional distribution of $\mathbf{Y}^{T}$ given
$\mathbf{s}(\mathbf{Y}^{T})=\mathbf{s}(\mathbf{y}^{T})$ is nondegenerate, we
can consider a frequentist experiment (repeated sampling of $\mathbf{Y}^{T}$)
conditional on the sufficient statistics set to the observed value. In this
conditional experiment, we can view the conditional identified set
$\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)$ as the standard identified set in
set-identified models, since it no longer depends on the data in the
conditional experiment where $\mathbf{s}(\mathbf{y}^{T})$ is fixed. This is
the reason that we refer to $\mathcal{Q}(\bm{\phi}|\mathbf{y}^{T},N)$ as the
conditional identified set. In Section 6 below, we show the frequentist
validity of the robust-Bayes credible region by establishing conditional
coverage of the conditional identified set for an impulse response.

##  5 Posterior inference under NR

This section presents approaches to conducting posterior inference in SVARs
under NR. Section 5.1 discusses how to modify the standard Bayesian approach
in AR18 to use the unconditional likelihood rather than the conditional
likelihood. Section 5.2 explains how to conduct robust Bayesian inference
under NR, which further addresses the issue of posterior sensitivity due to
the flat unconditional likelihood. Section 5.3 describes how to numerically
implement the robust Bayesian procedure.

###  5.1 Standard Bayesian inference

AR18 propose an algorithm for drawing from the uniform-normal-inverse-Wishart
posterior of $(\bm{\phi},\mathbf{Q})$ given a set of traditional sign
restrictions and NR. This is the posterior induced by a normal-inverse-Wishart
prior over $\bm{\phi}$ and an unconditionally uniform prior over $\mathbf{Q}$.
The algorithm proceeds by drawing $\bm{\phi}$ from a normal-inverse-Wishart
distribution and $\mathbf{Q}$ from a uniform distribution over
$\mathcal{O}(n)$, and checking whether the restrictions are satisfied. If the
restrictions are not satisfied, the joint draw is discarded and another draw
is made. If the restrictions are satisfied, the ex ante probability that the
NR are satisfied at the drawn parameter values is approximated via Monte Carlo
simulation. Once the desired number of draws are obtained satisfying the
restrictions, the draws are resampled with replacement using as importance
weights the inverse of the probability that the NR are satisfied.121212Based
on the results in Arias et al. (2018), AR18 argue that their algorithm draws
from a normal-generalized-normal posterior over the SVAR’s structural
parameters $(\mathbf{A}_{0},\mathbf{A}_{+})$ induced by a conjugate normal-
generalized-normal prior, conditional on the restrictions.

This algorithm essentially draws from the posterior under the unconditional
likelihood and then uses importance sampling to transform these draws into
draws from the posterior given the conditional likelihood. To draw from the
uniform-normal-inverse-Wishart posterior using the unconditional likelihood to
construct the posterior, one therefore simply needs to omit the importance-
sampling step from this algorithm. Approximating the probability used to
construct the importance weights requires Monte Carlo integration, which can
be computationally expensive, particularly when the NR constrain the
structural shocks in multiple periods. Omitting the importance-sampling step
can therefore ease the computational burden of drawing from the posterior.
However, as discussed above, standard Bayesian inference under the
unconditional likelihood may be sensitive to the choice of conditional prior
for $\mathbf{Q}|\bm{\phi}$, because the likelihood possesses flat regions.

By rejecting draws that do not satisfy the restrictions, the algorithm
described above places more weight on draws of $\bm{\phi}$ that are less
likely to satisfy the restrictions under the uniform distribution over
$\mathcal{O}(n)$. As discussed in Uhlig (2017), one may instead prefer to use
a prior that is conditionally uniform over $\mathbf{Q}|\bm{\phi}$. To draw
from the posterior of $(\bm{\phi},\mathbf{Q})$ under the unconditional
likelihood given an arbitrary prior over $\bm{\phi}$ and a conditionally
uniform prior over $\mathbf{Q}|\bm{\phi}$, one can repeat Step 2 of Algorithm
1 in Section 5.3.

###  5.2 Robust Bayesian inference

This section explains how to conduct robust Bayesian inference about a scalar-
valued function of the structural parameters under NR and traditional sign
restrictions. The approach can be viewed as performing global sensitivity
analysis to assess whether posterior conclusions are robust to the choice of
prior on the flat regions of the likelihood. We assume that the object of
interest is a particular impulse response $\eta$, although the discussion in
this section also applies to any other scalar-valued function of the
structural parameters, such as the forecast error variance decomposition or
the historical decomposition.

Let $\pi_{\bm{\phi}}$ be a prior over the reduced-form parameter
$\bm{\phi}\in\bm{\Phi}$, where $\bm{\Phi}$ is the space of reduced-form
parameters such that $\mathcal{Q}(\bm{\phi}|S)$ is non-empty. A joint prior
for $(\bm{\phi},\mathbf{Q})\in\bm{\Phi}\times\mathcal{O}(n)$ can be written as
$\pi_{\bm{\phi},\mathbf{Q}}=\pi_{\mathbf{Q}|\bm{\phi}}\pi_{\bm{\phi}}$, where
$\pi_{\mathbf{Q}|\bm{\phi}}$ is supported only on $\mathcal{Q}(\bm{\phi}|S)$.
When there are only traditional identifying restrictions,
$\pi_{\mathbf{Q}|\bm{\phi}}$ is not updated by the data, because the
likelihood function is not a function of $\mathbf{Q}$. Posterior inference may
therefore be sensitive to the choice of conditional prior, even
asymptotically. As discussed above, a similar issue arises under NR. The
difference under NR is that $\pi_{\mathbf{Q}|\bm{\phi}}$ is updated by the
data through the truncation points of the unconditional likelihood. However,
at each value of $\bm{\phi}$, the unconditional likelihood is flat over the
set of values of $\mathbf{Q}$ satisfying the NR. Consequently, the conditional
posterior for $\mathbf{Q}|\bm{\phi},\mathbf{Y}^{T}$ is proportional to the
conditional prior for $\mathbf{Q}|\bm{\phi}$ at each $\bm{\phi}$ whenever the
conditional identified set for $\mathbf{Q}$ given $(\bm{\phi},\mathbf{Y}^{T})$
is nonempty.

Rather than specifying a single prior for $\mathbf{Q}|\bm{\phi}$, the robust
Bayesian approach of GK considers the class of all priors for
$\mathbf{Q}|\bm{\phi}$ that are consistent with the traditional identifying
restrictions:

$\Pi_{\mathbf{Q}|\bm{\phi}}=\left\\{\pi_{\mathbf{Q}|\bm{\phi}}:\pi_{\mathbf{Q}|\bm{\phi}}(\mathcal{Q}(\bm{\phi}|S))=1\right\\}.$
(35)

Notice that we cannot impose the NR using a particular conditional prior on
$\mathbf{Q}|\bm{\phi}$ due to the data-dependent mapping from $\bm{\phi}$ to
$\mathbf{Q}$. However, by considering all possible conditional priors for
$\mathbf{Q}|\bm{\phi}$ that are consistent with the traditional identifying
restrictions, we trace out all possible conditional posteriors for
$\mathbf{Q}|\bm{\phi},\mathbf{Y}^{T}$ that are consistent with the traditional
identifying restrictions and the NR. This is because the NR truncate the
unconditional likelihood function and the traditional identifying restrictions
truncate the prior for $\mathbf{Q}|\bm{\phi}$, so the posterior for
$\mathbf{Q}|\bm{\phi},\mathbf{Y}^{T}$ is supported only on the values of
$\mathbf{Q}$ that satisfy both sets of restrictions.

Given a particular prior for $(\bm{\phi},\mathbf{Q})$ and using the
unconditional likelihood, the posterior is

$\displaystyle\pi_{\bm{\phi},\mathbf{Q}|\mathbf{Y}^{T},D_{N}=1}$
$\displaystyle\propto
p(\mathbf{Y}^{T},D_{N}=1|\bm{\phi},\mathbf{Q})\pi_{\mathbf{Q}|\bm{\phi}}\pi_{\bm{\phi}}$
$\displaystyle\propto
f(\mathbf{Y}^{T}|\bm{\phi})D_{N}(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})\pi_{\bm{\phi}}\pi_{\mathbf{Q}|\bm{\phi}}$
$\displaystyle\propto\pi_{\bm{\phi}|\mathbf{Y}^{T}}\pi_{\mathbf{Q}|\bm{\phi}}D_{N}(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T}).$
(36)

The final expression for the posterior makes it clear that any prior for
$\mathbf{Q}|\bm{\phi}$ that is consistent with the traditional identifying
restrictions is in effect further truncated by the NR (through the likelihood)
once the data are realized. Generating this posterior using every prior within
the class of priors for $\mathbf{Q}|\bm{\phi}$ generates a class of posteriors
for $(\bm{\phi},\mathbf{Q})$:

$\Pi_{\bm{\phi},\mathbf{Q}|\mathbf{Y}^{T},D_{N}=1}=\left\\{\pi_{\bm{\phi},\mathbf{Q}|\mathbf{Y}^{T},D_{N}=1}=\pi_{\bm{\phi}|\mathbf{Y}^{T}}\pi_{\mathbf{Q}|\bm{\phi}}D_{N}(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T}):\pi_{\mathbf{Q}|\bm{\phi}}\in\Pi_{\mathbf{Q}|\bm{\phi}}\right\\}.$
(37)

Marginalizing each posterior in this class of posteriors induces a class of
posteriors for $\eta$, $\Pi_{\eta|\mathbf{Y}^{T},D_{N}=1}$. Each prior within
the class of priors $\Pi_{\mathbf{Q}|\bm{\phi}}$ therefore induces a posterior
for $\eta$. Associated with each of these posteriors are quantities such as
the posterior mean, median and other quantiles. For example, as we consider
each possible prior within $\Pi_{\mathbf{Q}|\bm{\phi}}$, we can trace out the
set of all possible posterior means for $\eta$. This will always be an
interval, so we can summarize this ‘set of posterior means’ by its endpoints:

$\left[\int_{\bm{\Phi}}l(\bm{\phi},\mathbf{Y}^{T})d\pi_{\bm{\phi}|\mathbf{Y}^{T}},\int_{\bm{\Phi}}u(\bm{\phi},\mathbf{Y}^{T})d\pi_{\bm{\phi}|\mathbf{Y}^{T}}\right],$
(38)

where
$l(\bm{\phi},\mathbf{Y}^{T})=\inf\\{\eta(\bm{\phi},\mathbf{Q}):\mathbf{Q}\in\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)\\}$,
$u(\bm{\phi},\mathbf{Y}^{T})=\sup\\{\eta(\bm{\phi},\mathbf{Q}):\mathbf{Q}\in\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)\\}$
and

$\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)=\left\\{\mathcal{Q}(\bm{\phi}|S)\cap\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N)\right\\}$
(39)

is the set of values of $\mathbf{Q}$ that are consistent with the traditional
identifying restrictions and the NR. In contrast, in GK the set of posterior
means is obtained by finding the infimum and supremum of
$\eta(\bm{\phi},\mathbf{Q})$ over $\mathcal{Q}(\bm{\phi}|S)$ and averaging
these over $\pi_{\bm{\phi}|\mathbf{Y}^{T}}$. The important difference from GK
is that the current set of posterior means depends on the data not only
through the posterior for $\bm{\phi}$ but also through the set of admissible
values of $\mathbf{Q}$ under the NR. As a result, in contrast with GK, we
cannot interpret the set of posterior means (38) as a consistent estimator for
the identified set for $\eta$ (which is not well-defined, as we discussed
above). Nevertheless, the set of posterior means still carries a robust
Bayesian interpretation similar to GK in that it clarifies posterior results
that are robust to the choice of prior on the non-updated part of the
parameter space (i.e., on the flat regions of the likelihood).

As in GK, we can also report a robust credible region with credibility level
$\alpha$, which is the shortest interval estimate for $\eta$ such that the
posterior probability put on the interval is greater than or equal to $\alpha$
uniformly over the posteriors in $\Pi_{\eta|\mathbf{Y}^{T},D_{N}=1}$ (see
Proposition 1 of GK). One may also be interested in posterior lower and upper
probabilities, which are the infimum and supremum, respectively, of the
probability for a hypothesis over all posteriors in the class.

GK provide conditions under which their robust Bayesian approach has a valid
frequentist interpretation, in the sense that the robust credible region is an
asymptotically valid confidence set for the true identified set. For the same
reason as mentioned above, however, frequentist validity of the robust
credible region does not immediately extend to the NR case. We provide
conditions under which the robust credible region has a valid frequentist
interpretation in Section 6.

###  5.3 Numerical implementation of robust Bayesian approach

This section describes a general algorithm to implement our robust Bayesian
procedure under NR. GK propose numerical algorithms for conducting robust
Bayesian inference in SVARs identified using traditional sign and zero
restrictions. Their Algorithm 1 uses a numerical optimization routine to
obtain the lower and upper bounds of the identified set at each draw of
$\bm{\phi}$. Obtaining the bounds via numerical optimization is not generally
applicable under the class of NR considered in AR18, since the constraints on
the historical decomposition are not differentiable everywhere in
$\mathbf{Q}$. We therefore adapt Algorithm 2 of GK, which approximates the
bounds of the identified set at each draw of $\bm{\phi}$ using Monte Carlo
simulation.

Algorithm 1. Let
$N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})\geq\mathbf{0}_{s\times 1}$ be the set
of NR and let $S(\bm{\phi},\mathbf{Q})\geq\mathbf{0}_{\tilde{s}\times 1}$ be
the set of traditional sign restrictions (excluding the sign normalization).
Assume the object of interest is
$\eta_{i,j^{*},h}=c_{i,h}^{\prime}(\bm{\phi})\mathbf{q}_{j^{*}}$.

  * •

Step 1: Specify a prior for $\bm{\phi}$, $\pi_{\bm{\phi}}$, and obtain the
posterior $\pi_{\bm{\phi}|\mathbf{Y}^{T}}$.

  * •

Step 2: Draw $\bm{\phi}$ from $\pi_{\bm{\phi}|\mathbf{Y}^{T}}$ and check
whether $\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$ is empty using the
subroutine below.

    * –

Step 2.1: Draw an $n\times n$ matrix of independent standard normal random
variables, $\mathbf{Z}$, and let $\mathbf{Z}=\tilde{\mathbf{Q}}\mathbf{R}$ be
the QR decomposition of $\mathbf{Z}$.131313This is the algorithm used by
Rubio-Ramírez et al. (2010) to draw from the uniform distribution over
$\mathcal{O}(n)$, except that we do not normalize the diagonal elements of
$\mathbf{R}$ to be positive. This is because we impose a sign normalization
based on the diagonal elements of
$\mathbf{A}_{0}=\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1}$ in Step 2.2.

    * –

Step 2.2: Define

$\mathbf{Q}=\left[\mathrm{sgn}((\bm{\Sigma}_{tr}^{-1}\mathbf{e}_{1,n})^{\prime}\tilde{\mathbf{q}}_{1})\frac{\tilde{\mathbf{q}}_{1}}{\lVert\tilde{\mathbf{q}}_{1}\rVert},\ldots,\mathrm{sgn}((\bm{\Sigma}_{tr}^{-1}\mathbf{e}_{n,n})^{\prime}\tilde{\mathbf{q}}_{n})\frac{\tilde{\mathbf{q}}_{n}}{\lVert\tilde{\mathbf{q}}_{n}\rVert}\right],$

where $\tilde{\mathbf{q}}_{j}$ is the $j$th column of $\tilde{\mathbf{Q}}$.

    * –

Step 2.3: Check whether $\mathbf{Q}$ satisfies
$\mathbf{N}(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})\geq\mathbf{0}_{s\times 1}$
and $S(\bm{\phi},\mathbf{Q})\geq\mathbf{0}_{\tilde{s}\times 1}$. If so, retain
$\mathbf{Q}$ and proceed to Step 3. Otherwise, repeat Steps 2.1 and 2.2 (up to
a maximum of $L$ times) until $\mathbf{Q}$ is obtained satisfying the
restrictions. If no draws of $\mathbf{Q}$ satisfy the restrictions,
approximate $\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$ as being empty and
return to Step 2.

  * •

Step 3: Repeat Steps 2.1–2.3 until $K$ draws of $\mathbf{Q}$ are obtained. Let
$\\{\mathbf{Q}_{k},k=1,...,K\\}$ be the $K$ draws of $\mathbf{Q}$ that satisfy
the restrictions and let $\mathbf{q}_{j^{*},k}$ be the $j^{*}$th column of
$\mathbf{Q}_{k}$. Approximate
$[l(\bm{\phi},\mathbf{Y}^{T}),u(\bm{\phi},\mathbf{Y}^{T})]$ by
$[\min_{k}\mathbf{c}_{i,h}^{\prime}(\bm{\phi})\mathbf{q}_{j^{*},k}$,
$\max_{k}\mathbf{c}_{i,h}^{\prime}(\bm{\phi})\mathbf{q}_{j^{*},k}]$.

  * •

Step 4: Repeat Steps 2–3 $M$ times to obtain
$[l(\bm{\phi_{m}},\mathbf{Y}^{T}),u(\bm{\phi_{m}},\mathbf{Y}^{T})]$ for
$m=1,...,M$. Approximate the set of posterior means using the sample averages
of $l(\bm{\phi_{m}},\mathbf{Y}^{T})$ and $u(\bm{\phi_{m}},\mathbf{Y}^{T})$.

  * •

Step 5: To obtain an approximation of the smallest robust credible region with
credibility $\alpha\in(0,1)$, define
$d(\eta,\bm{\phi},\mathbf{Y}^{T})=\max\\{|\eta-l(\bm{\phi},\mathbf{Y}^{T})|,|\eta-u(\bm{\phi},\mathbf{Y}^{T})|\\}$
and let $\hat{z}_{\alpha}(\eta)$ be the sample $\alpha$-th quantile of
$\\{d(\eta,\bm{\phi_{m}},\mathbf{Y}^{T}),m=1,...,M\\}$. An approximated
smallest robust credible interval for $\eta_{i,j^{*},h}$ is an interval
centered at $\arg\min_{\eta}\hat{z}_{\alpha}(\eta)$ with radius
$\min_{\eta}\hat{z}_{\alpha}(\eta)$.

Algorithm 1 approximates
$[l(\bm{\phi},\mathbf{Y}^{T}),u(\bm{\phi},\mathbf{Y}^{T})]$ at each draw of
$\bm{\phi}$ via Monte Carlo simulation. The approximated set will be too
narrow given a finite number of draws of $\mathbf{Q}$, but the approximation
error will vanish as the number of draws goes to infinity. The algorithm may
be computationally demanding when the restrictions substantially truncate
$\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$, because many draws of
$\mathbf{Q}$ from $\mathcal{O}(n)$ may be rejected at each draw of
$\bm{\phi}$. However, the same draws of $\mathbf{Q}$ can be used to compute
$l(\bm{\phi},\mathbf{Y}^{T})$ and $u(\bm{\phi},\mathbf{Y}^{T})$ for different
objects of interest, which cuts down on computation time. For example, the
same draws of $\mathbf{Q}$ can be used to compute the impulse responses of all
variables to all shocks at all horizons of interest. They can also be used to
compute other parameters by replacing $\eta_{i,j^{*},h}$ with some other
function, such as the forecast error variance decomposition, an element of
$\mathbf{A}_{0}$, the historical decomposition or the structural shocks
themselves in particular periods.141414Impulse responses to a unit shock –
rather than a standard-deviation shock – can be computed as in Algorithm 3 of
Giacomini et al. (2019). Step 3 is parallelizable, so reductions in computing
time are possible by distributing computation across multiple processors.
Other algorithms may be computationally more efficient than Algorithm 1 in
particular cases. We discuss these in Appendix D.

##  6 Frequentist coverage under a few NR

In this section, we show that the robust Bayes credible region attains
asymptotically valid frequentist coverage in a setting where the number of NR
is small relative to the length of the sampled periods in a sense that we make
precise in the next assumption. This assumption is empirically relevant given
that applications typically impose these restrictions in at most a handful of
periods.

######  Assumption 6.1.

(fixed-dimensional $\mathbf{s}(\mathbf{Y}^{T})$): The conditional identified
set under NR has sufficient statistics $\mathbf{s}(\mathbf{Y}^{T})$, as
defined in Definition 4.1(ii), and the dimension of
$\mathbf{s}(\mathbf{Y}^{T})$ does not depend on $T$.

Let $(\bm{\phi}_{0},\mathbf{Q}_{0})$ be the true parameter values. We view the
sample $\mathbf{Y}^{T}$ as being drawn from $p(\mathbf{Y}^{T}|\bm{\phi}_{0})$.
Let $p(\mathbf{Y}^{T}|\bm{\phi}_{0},\mathbf{s})$ be the conditional
distribution of the sample $\mathbf{Y}^{T}$ given the sufficient statistics
for the conditional identified set $\mathbf{s}=\mathbf{s}(\mathbf{Y}^{T})$ at
$\bm{\phi}=\bm{\phi}_{0}$. We denote by $p(\mathbf{s}|\bm{\phi}_{0})$ the
distribution of the sufficient statistics $\mathbf{s}(\mathbf{Y}^{T})$ at
$\bm{\phi}=\bm{\phi}_{0}$. The next assumption assumes that in the conditional
experiment given $\mathbf{s}(\mathbf{Y}^{T})$, the sampling distribution for
the maximum likelihood estimator
$\hat{\bm{\phi}}\equiv\arg\max_{\bm{\phi}}p(\mathbf{Y}^{T}|\bm{\phi})$
centered at $\bm{\phi}_{0}$ and the posterior for $\bm{\phi}$ centered at
$\hat{\bm{\phi}}$ asymptotically coincide.

######  Assumption 6.2.

(Conditional Bernstein-von Mises property for $\bm{\phi}$): For
$p(\mathbf{s}|\bm{\phi}_{0})$-almost every $\mathbf{s}$ and
$p(\mathbf{Y}^{T}|\bm{\phi}_{0},\mathbf{s})$-almost every sampling sequence
$\mathbf{Y}^{T}$, the posterior for $\sqrt{T}(\bm{\phi}-\hat{\bm{\phi}})$
asymptotically coincides with the sampling distribution of
$\sqrt{T}(\hat{\bm{\phi}}-\bm{\phi}_{0})$ with respect to
$p(\mathbf{Y}^{T}|\bm{\phi}_{0},\mathbf{s})$, as $T\to\infty$, in the sense
stated in Assumption 5(i) in GK.

This is a key assumption for establishing the asymptotic frequentist validity
of the robust credible region under NR. It holds, for instance, when
$\mathbf{s}(\mathbf{y}^{T})$ corresponds to one or a few observations in the
whole sample, as we had in the toy example of Section 2.1. In this case, the
influence of $\mathbf{s}(\mathbf{y}^{T})$ vanishes in the conditional sampling
distribution of $\sqrt{T}(\hat{\bm{\phi}}-\bm{\phi}_{0})$ as $T\to\infty$, as
the latter asymptotically agrees with the asymptotically normal sampling
distribution for the maximum likelihood estimator with variance-covariance
matrix given by the inverse of the Fisher information matrix. By the well-
known Bernstein-von Mises theorem for regular parametric models, the posterior
for $\sqrt{T}(\bm{\phi}-\hat{\bm{\phi}})$ asymptotically agrees with this
sampling distribution.

The last assumption requires convexity and smoothness of the conditional
identified set, and is analogous to Assumption 5(ii) of GK for standard set-
identified models.

######  Assumption 6.3.

(Almost-sure convexity and smoothness of the impulse response identified set):
Let $\widetilde{CIS}_{\eta}(\bm{\phi}|\mathbf{s}(\mathbf{Y}^{T}),N)$ be the
conditional identified set for $\eta$ with the sufficient statistics
$\mathbf{s}(\mathbf{Y}^{T})$. For $p(\mathbf{Y}^{T}|\bm{\phi}_{0})$-almost
every $\mathbf{Y}^{T}$,
$\widetilde{CIS}_{\eta}(\bm{\phi}|\mathbf{s}(\mathbf{y}^{T}),N)$ is closed and
convex,
$\widetilde{CIS}_{\eta}(\bm{\phi}|\mathbf{s}(\mathbf{y}^{T}),N)=[\tilde{\bm{\ell}}(\bm{\phi},\mathbf{s}(\mathbf{Y}^{T})),\tilde{\mathbf{u}}(\bm{\phi},\mathbf{s}(\mathbf{Y}^{T}))]$,
and its lower and upper bounds are differentiable in $\bm{\phi}$ at
$\bm{\phi}=\bm{\phi}_{0}$ with nonzero derivatives.

Propositions B.1–B.3 in Appendix B provide primitive conditions for Assumption
6.3 to hold in the case where there are shock-sign restrictions. Imposing
Assumptions 6.1, 6.2 and 6.3, we obtain the following theorem.

######  Theorem 6.4.

For $\gamma\in(0,1)$, let $\widehat{C}_{\alpha}^{\ast}$ be the volume-
minimizing robust credible region for $\eta$ with credibility
$\alpha$,151515The volume-minimizing robust credible region
$\widehat{C}_{\alpha}^{\ast}$ is defined as a shortest interval among the
connected intervals $C_{\alpha}$ satisfying
$P_{\mathbf{Y}^{T}|\mathbf{s},\bm{\phi}}(\widetilde{CIS}_{\eta}(\bm{\phi}_{0}|\mathbf{s}(\mathbf{Y}^{T}),N)\subset
C_{\alpha}|\mathbf{s}(\mathbf{Y}^{T}),\bm{\phi}_{0})\geq\alpha.$ See
Proposition 1 in GK for a procedure to compute the volume-minimizing credible
region. which satisfies

$\inf_{\pi\in\Pi_{\bm{\phi},\mathbf{Q}|\mathbf{Y}^{T},D_{N}=1}}\pi(\widehat{C}_{\alpha}^{\ast})=\pi_{\bm{\phi}|\mathbf{Y}^{T},D_{N}=1}(CIS_{\eta}(\bm{\phi}|\mathbf{Y}^{T},N)\subset\hat{C}_{\alpha}^{\ast}|\mathbf{Y}^{T},D_{N}=1)=\alpha.$
(40)

Under Assumptions 6.1, 6.2, and 6.3, $\widehat{C}_{\alpha}^{\ast}$ attains
asymptotically valid coverage for the true impulse response, $\eta_{0}$,
conditional on $\mathbf{s}(\mathbf{Y}^{T})$.

$\liminf_{T\to\infty}P_{\mathbf{Y}^{T}|\mathbf{s},\bm{\phi}}(\eta_{0}\in\widehat{C}_{\alpha}^{\ast}|\mathbf{s}(\mathbf{Y}^{T}),\bm{\phi}_{0})\geq\\\
\lim_{T\to\infty}P_{\mathbf{Y}^{T}|\mathbf{s},\bm{\phi}}(\widetilde{CIS}_{\eta}(\bm{\phi}_{0}|\mathbf{s}(\mathbf{Y}^{T}),N)\subset\widehat{C}_{\alpha}^{\ast}|\mathbf{s}(\mathbf{Y}^{T}),\bm{\phi}_{0})=\alpha.$
(41)

Accordingly, $\widehat{C}_{\alpha}^{\ast}$ attains asymptotically valid
coverage for $\eta_{0}$ unconditionally,

$\liminf_{T\to\infty}P_{\mathbf{Y}^{T}|\bm{\phi}}(\eta_{0}\in\widehat{C}_{\alpha}^{\ast}|\bm{\phi}_{0})\geq\lim_{T\to\infty}P_{\mathbf{Y}^{T}|\bm{\phi}}(\widetilde{CIS}_{\eta}(\bm{\phi}_{0}|\mathbf{s}(\mathbf{Y}^{T}),N)\subset\widehat{C}_{\alpha}^{\ast}|\bm{\phi}_{0})=\alpha.$
(42)

###### Proof.

See Appendix B. ∎

This theorem shows that the robust credible region of GK applied to the SVAR
model with NR attains asymptotically valid frequentist coverage for the true
impulse response as well as the conditional impulse-response identified set.
Even if the point-identification condition of Proposition 4.1 holds for the
impulse response, it is not obvious if the standard Bayesian credible region
can attain frequentist coverage. This is because the Bernstein-von Mises
theorem does not seem to hold for the impulse response due to the non-standard
features of models with NR.

One could also consider asymptotics under an increasing number of
restrictions. We conjecture that, under certain assumptions about how the NR
are generated, the class of posteriors for $\eta$ will converge to a point
mass at its true value. An implication is that the posterior mean under any
conditional prior for $\mathbf{Q}$ that places probability one on the
identified set would be consistent for the true value. This result would be an
interesting contrast to the case under traditional set-identifying
restrictions, where the location of the posterior mean within the identified
set is determined purely by the conditional prior and the posterior quantiles
lie strictly within the identified set (e.g., Moon and Schorfheide (2012)). We
do not analyze this case here, since the assumption that there is a fixed
number of restrictions seems to be of primary interest. However, Appendix C
provides numerical evidence in support of our conjecture. We leave formal
investigation of this conjecture for future work.

##  7 Empirical application: the dynamic effects of a monetary policy shock

AR18 estimate the effects of monetary policy shocks on the US economy using a
combination of sign restrictions on impulse responses and NR. The reduced-form
VAR is the same as that used in Uhlig (2005). The model’s endogenous variables
are real GDP, the GDP deflator, a commodity price index, total reserves, non-
borrowed reserves (all in natural logarithms) and the federal funds rate; see
Arias, Caldara and Rubio-Ramírez (2019) for details on the variables. The data
are monthly and run from January 1965 to November 2007. The VAR includes 12
lags and we include a constant.

As NR, AR18 impose that the monetary policy shock in October 1979 was positive
and that it was the overwhelming contributor to the unexpected change in the
federal funds rate in that month. This was the month in which the Federal
Reserve markedly and unexpectedly increased the federal funds rate following
the appointment of Paul Volcker as chairman of the Federal Reserve, and is
widely considered to be an example of a positive monetary policy shock (e.g.,
Romer and Romer (1989)). The traditional sign restrictions considered in Uhlig
(2005) are also imposed. Specifically, the response of the federal funds rate
is restricted to be non-negative for $h=0,1,\ldots,5$ and the responses of the
GDP deflator, the commodity price index and nonborrowed reserves are
restricted to be nonpositive for $h=0,1,\ldots,5$.

We assume a Jeffreys’ (improper) prior over the reduced-form parameters,
$\pi_{\bm{\phi}}=\pi_{\mathbf{B},\bm{\Sigma}}\propto|\bm{\Sigma}|^{-\frac{n+1}{2}}$,
which is truncated so that the VAR is stable. The posterior for the reduced-
form parameters, $\pi_{\bm{\phi}|\mathbf{Y}^{T}}$, is then a normal-inverse-
Wishart distribution, from which it is straightforward to obtain independent
draws (for example, see Del Negro and Schorfheide (2011)). We obtain 1,000
draws from the posterior of $\bm{\phi}$ such that the VAR is stable and
$\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$ is non-empty. We use Algorithm 1
with $K=10,000$ draws of $\mathbf{Q}$ at each draw of $\bm{\phi}$ to
approximate $l(\bm{\phi},\mathbf{Y}^{T})$ and $u(\bm{\phi},\mathbf{Y}^{T})$.
If we cannot obtain a draw of $\mathbf{Q}$ satisfying the restrictions after
100,000 draws of $\mathbf{Q}$, we approximate
$\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$ as being empty at that draw of
$\bm{\phi}$.

We explore the sensitivity of posterior inference to the choice of prior for
$\mathbf{Q}|\bm{\phi}$ when the unconditional likelihood is used to construct
the posterior. For brevity, we report only the impulse responses of the
federal funds rate and real GDP to a positive standard-deviation monetary
policy shock (Figure 4). As a point of comparison, we report results obtained
using a conditionally uniform prior for $\mathbf{Q}|\bm{\phi}$. Under this
prior, the 68 per cent highest posterior density credible intervals for the
response of real GDP exclude zero at horizons greater than a year or
so.161616The results are not directly comparable to those presented in Figure
6 of AR18. First, we present responses to a standard-deviation shock, whereas
AR18 describe their responses as being to a 25 basis point shock (although,
from close inspection of their Figure 6, it is evident that this normalization
is not imposed correctly, because the impact response of the federal funds
rate fans out around zero). Second, we use a prior for $\mathbf{Q}$ that is
conditionally uniform given $\bm{\phi}$, whereas AR18 use a prior that is
unconditionally uniform. In contrast, the 68 per cent robust credible
intervals include zero at all horizons. Under the single prior, the posterior
probability that the output response is negative two years after the shock is
95 per cent. In contrast, the posterior lower probability of this event – the
smallest probability over the class of posteriors generated by the class of
priors – is only 54 per cent. The results suggest that posterior inference
about the effect of monetary policy on output can be sensitive to the choice
of (unrevisable) prior for $\mathbf{Q}|\bm{\phi}$.

Figure 4: Impulse Responses to a Monetary Policy Shock

|  |
---|---|---

Notes:

Circles and dashed lines are, respectively, posterior means and 68 per cent
(pointwise) highest posterior density intervals under the uniform prior for
$\mathbf{Q}|\bm{\phi}$; vertical bars are sets of posterior means and solid
lines are 68 per cent (pointwise) robust credible regions obtained using
Algorithm 1 with 10,000 draws from
$\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$; results are based on 1,000 draws
from the posterior of $\bm{\phi}$ with nonempty
$\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$; impulse responses are to a
standard-deviation shock.

AR18 also consider an alternative set of restrictions. Specifically, they
impose that the monetary policy shock was: positive in April 1974, October
1979, December 1988 and February 1994; negative in December 1990, October
1998, April 2001 and November 2002; and the most important contributor to the
observed unexpected change in the federal funds rate in these months. The
choice of these dates is based on a synthesis of information from different
sources, including the chronology of monetary policy actions from Romer and
Romer (1989), an updated series of the monetary policy shocks constructed
using Greenbook forecasts in Romer and Romer (2004), the high-frequency
monetary policy surprises from Gürkaynak, Sack and Swanson (2005), and minutes
from Federal Open Markets Committee meetings. Under this extended set of
restrictions, the set of posterior means and the robust credible interval are
tightened noticeably, particularly at shorter horizons (Figure 5). The
posterior lower probability of a negative output response two years after the
shock is now 80 per cent, compared with 54 per cent under the October 1979
restrictions.

Finally, we investigate how posterior inference about the output response is
affected by replacing AR18’s extended set of restrictions with a shock-rank
restriction. Specifically, we estimate the set of output responses that are
consistent with the restriction that the monetary policy shock in October 1979
was the largest positive realization of the monetary policy shock in the
sample period.171717The large number of inequality constraints and tight
conditional identified set induced by the shock-rank restriction poses
computational challenges when using Algorithm 1. Accordingly, we use an
alternative algorithm to obtain the results. The algorithm adapts an algorithm
in Amir-Ahmadi and Drautzburg (2021) and is described in Appendix D. This
restriction appears plausible given that the change in the federal funds rate
in October 1979 was more positive than the change in the federal funds rate in
the other periods identified by AR18 as containing notable monetary policy
shocks (Table 1). The shock-rank restriction somewhat shrinks the set of
posterior means and robust credible regions relative to those obtained under
the restrictions on the historical decomposition. Nevertheless, the two sets
of restrictions lead to similar (robust) posterior inferences about the output
response. The 68 per cent robust credible intervals include zero at all
horizons under both sets of restrictions. The posterior lower probability that
output falls two years after the shock is 73 per cent under the shock-rank
restriction, compared with 80 per cent under the restriction on the historical
decomposition.

Table 1: Monthly Change in Federal Funds Rate (ppt)

Oct 79 | Apr 74 | Dec 88 | Feb 94 | Dec 90 | Oct 98 | Apr 01 | Nov 02
---|---|---|---|---|---|---|---
2.34 | 1.16 | 0.41 | 0.20 | –0.50 | –0.44 | –0.51 | –0.41

  * •

Source: FRED

Figure 5: Impulse Responses to a Monetary Policy Shock – Extended Restrictions
vs Shock-rank Restriction

|  |
---|---|---

Notes:

Solid lines represent set of posterior means and dashed lines represent 68 per
cent (pointwise) robust credible regions; results are based on 1,000 draws
from the posterior of $\bm{\phi}$ with nonempty
$\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$; results under shock-rank
restriction are obtained using Algorithm D.1; results under restrictions on
the historical decomposition are obtained using Algorithm 1 with 1,000 draws
from $\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$; impulse responses are to a
standard-deviation shock.

In general, $\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$ may be empty at
particular values of $\bm{\phi}$. The proportion of draws of $\bm{\phi}$ where
$\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)$ is empty can therefore be used to
assess the plausibility of the restrictions (see GK). Under the October 1979
restrictions, the posterior plausibility of the restrictions is one (i.e.,
every draw of $\bm{\phi}$ has a nonempty conditional identified set). In
contrast, the posterior plausibility under AR18’s extended set of restrictions
is 53 per cent, while it is only 17 per cent under the shock-rank restriction.

##  8 Conclusion

Directly restricting the values of structural shocks to be consistent with
historical narratives offers a potentially useful approach to disciplining
SVARs, but raises novel issues related to identification and inference. These
restrictions generate a set-valued mapping from the model’s reduced-form
parameters to its structural parameters that depends on the realization of the
data entering the restrictions. This means that these restrictions do not fit
neatly into the existing framework for analyzing identification in SVARs. In
particular, we show that these restrictions may be point-identifying in a
frequentist sense. We also highlight issues associated with existing standard
Bayesian approaches to estimation and inference. Conditioning on the
restrictions holding may result in the posterior placing more weight on
parameters that yield a lower ex ante probability that the restrictions are
satisfied. We therefore advocate using the unconditional likelihood when
constructing the posterior. However, the observed unconditional likelihood
will almost always possess flat regions, which implies that a component of the
prior will not be updated by the data. Posterior inference may therefore be
sensitive to the choice of prior. To address this, we provide robust Bayesian
tools to assess or eliminate the sensitivity of posterior inference to the
choice of prior. We also provide conditions under which these tools have a
valid frequentist interpretation, so our approach should appeal to both
Bayesians and frequentists.

While we focus on SVARs in the paper, our analysis could be extended to other
settings. For example, Plagborg-Møller and Wolf (in press[a]) explain how to
impose traditional SVAR identifying restrictions in the local projection
framework under the assumption that the structural shocks are invertible. In
Appendix E we briefly discuss how NR could also be imposed within the local
projection framework, but we leave a formal analysis of this problem to future
research.

##  Appendix A Bivariate example derivations

Set of values of $\theta$ under shock-sign restriction. This section derives
analytical expressions for the set of values of $\theta$ consistent with the
shock-sign restriction in the bivariate example of Section 2. Throughout, we
assume that $\theta\in[-\pi,\pi]$.

Under the shock-sign restriction $\varepsilon_{1k}\geq 0$ and the sign
normalization $\mathrm{diag}(\mathbf{A}_{0})\geq\mathbf{0}_{2\times 1}$,
$\theta$ is restricted to lie in the set

$\theta\in\left\\{\theta:\sigma_{21}\sin\theta\leq\sigma_{22}\cos\theta,\cos\theta\geq
0,\sigma_{22}y_{1k}\cos\theta\geq(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\sin\theta\right\\}\\\
\cup\left\\{\theta:\sigma_{21}\sin\theta\leq\sigma_{22}\cos\theta,\cos\theta\leq
0,\sigma_{22}y_{1k}\cos\theta\geq(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\sin\theta\right\\}.$
(A.1)

Consider the case where $\sigma_{21}<0$ and
$\sigma_{21}y_{1k}-\sigma_{11}y_{2k}<0$. Then $\theta$ is restricted to the
set

$\theta\in\left\\{\theta:\tan\theta\geq\frac{\sigma_{22}}{\sigma_{21}},\cos\theta>0,\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\leq\tan\theta\right\\}\cup\left\\{\frac{\pi}{2}\right\\}\\\
\cup\left\\{\theta:\tan\theta\leq\frac{\sigma_{22}}{\sigma_{21}},\cos\theta<0,\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\geq\tan\theta\right\\}.$
(A.2)

The inequalities in the first set hold if and only if
$\tan\theta\geq\max\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}$
and $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$, which implies that

$\arctan\left(\max\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right)\leq\theta<\frac{\pi}{2}.$
(A.3)

The inequalities on the second line hold if and only if
$\tan\theta\leq\min\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}$
and $\theta\in[-\pi,-\frac{\pi}{2})\cup(\frac{\pi}{2},\pi]$. Since
$\sigma_{21}<0$, $\tan\theta$ must be negative, which implies that
$\theta\in(\frac{\pi}{2},\pi]$. It follows that

$\frac{\pi}{2}<\theta\leq\pi+\arctan\left(\min\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right).$
(A.4)

Taking the union of (A.3), (A.4) and $\left\\{\frac{\pi}{2}\right\\}$ implies
that

$\theta\in\left[\arctan\left(\max\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right),\pi+\arctan\left(\min\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right)\right].$
(A.5)

Next, consider the case where $\sigma_{21}<0$ and
$\sigma_{21}y_{1k}-\sigma_{11}y_{2k}>0$. Then $\theta$ is restricted to the
set

$\theta\in\left\\{\theta:\tan\theta\geq\frac{\sigma_{22}}{\sigma_{21}},\cos\theta>0,\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\geq\tan\theta\right\\}\\\
\cup\left\\{\theta:\tan\theta\leq\frac{\sigma_{22}}{\sigma_{21}},\cos\theta<0,\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\leq\tan\theta\right\\}.$
(A.6)

If $y_{1k}>0$ or if $y_{1k}<0$ and
$\frac{\sigma_{22}}{\sigma_{21}}<\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}$,
the second set of inequalities is not satisfied for any $\theta$, while the
first set of inequalities is satisfied for

$\theta\in\left[\arctan\left(\frac{\sigma_{22}}{\sigma_{21}}\right),\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right)\right].$
(A.7)

If $y_{1k}<0$ and
$\frac{\sigma_{22}}{\sigma_{21}}>\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}$,
the first set of inequalities has no solution and the second set is satisfied
for

$\theta\in\left[\pi+\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right),\pi+\arctan\left(\frac{\sigma_{22}}{\sigma_{21}}\right)\right].$
(A.8)

In the case where $\sigma_{21}>0$ and $\sigma_{21}y_{1k}-\sigma_{11}y_{2k}<0$,
$\theta$ is restricted to the set

$\theta\in\left\\{\theta:\tan\theta\leq\frac{\sigma_{22}}{\sigma_{21}},\cos\theta>0,\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\leq\tan\theta\right\\}\\\
\cup\left\\{\theta:\tan\theta\geq\frac{\sigma_{22}}{\sigma_{21}},\cos\theta<0,\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\geq\tan\theta\right\\}.$
(A.9)

If $y_{1k}>0$ or if $y_{1k}<0$ and
$\frac{\sigma_{22}}{\sigma_{21}}>\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}$,
the second set of inequalities has no solution, while the first is satisfied
for

$\theta\in\left[\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right),\arctan\left(\frac{\sigma_{22}}{\sigma_{21}}\right)\right].$
(A.10)

If $y_{1k}<0$ and
$\frac{\sigma_{22}}{\sigma_{21}}<\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}$,
the first set of inequalities has no solution and the second set is satisfied
for

$\theta\in\left[-\pi+\arctan\left(\frac{\sigma_{22}}{\sigma_{21}}\right),-\pi+\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right)\right].$
(A.11)

Finally, in the case where $\sigma_{21}>0$ and
$\sigma_{21}y_{1k}-\sigma_{11}y_{2k}>0$, $\theta$ is restricted to the set

$\theta\in\left\\{\theta:\tan\theta\leq\frac{\sigma_{22}}{\sigma_{21}},\cos\theta>0,\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\geq\tan\theta\right\\}\cup\left\\{-\frac{\pi}{2}\right\\}\\\
\cup\left\\{\theta:\tan\theta\geq\frac{\sigma_{22}}{\sigma_{21}},\cos\theta<0,\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\leq\tan\theta\right\\}.$
(A.12)

The first set of inequalities holds if and only if
$\tan\theta\leq\min\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}$
and $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$, which implies that

$-\frac{\pi}{2}<\theta\leq\arctan\left(\min\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right).$
(A.13)

The second set of inequalities holds if and only if
$\tan\theta\geq\max\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}$
and $\theta\in[-\pi,-\frac{\pi}{2})\cup(\frac{\pi}{2},\pi]$. Since
$\sigma_{21}>0$, $\tan\theta$ must be positive, which implies that
$\theta\in[-\pi,-\frac{\pi}{2})$. It follows that

$-\pi+\arctan\left(\max\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right)\leq\theta<-\frac{\pi}{2}.$
(A.14)

Taking the union of (A.13), (A.14) and $\left\\{-\frac{\pi}{2}\right\\}$
implies that

$\theta\in\left[-\pi+\arctan\left(\max\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right),\arctan\left(\min\left\\{\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right)\right].$
(A.15)

Set of values of $\eta$ under shock-sign restriction. Here we derive the
expression for the set of impulse responses $\eta\equiv\sigma_{11}\cos\theta$
consistent with the shock-sign restriction (i.e., (8) in Section 2).

In the absence of restrictions, the set of admissible values for the matrix of
contemporaneous impulse responses is

$\mathbf{A}_{0}^{-1}\in\left\\{\begin{bmatrix}\sigma_{11}\cos\theta&-\sigma_{11}\sin\theta\\\
\sigma_{21}\cos\theta+\sigma_{22}\sin\theta&\sigma_{22}\cos\theta-\sigma_{21}\sin\theta\end{bmatrix}:\theta\in[-\pi,\pi]\right\\}\\\
\cup\left\\{\begin{bmatrix}\sigma_{11}\cos\theta&\sigma_{11}\sin\theta\\\
\sigma_{21}\cos\theta+\sigma_{22}\sin\theta&\sigma_{21}\sin\theta-\sigma_{22}\cos\theta\end{bmatrix}:\theta\in[-\pi,\pi]\right\\}.$
(A.16)

Assume that $\sigma_{21}<0$, $\sigma_{21}y_{1k}-\sigma_{11}y_{2k}>0$ and
$y_{1k}>0$. Within the interval for $\theta$ defined in (A.7), $\eta$ is
maximized at $\theta=0$, so $\eta_{ub}=\sigma_{11}$. The lower bound
$\eta_{lb}$ occurs at one of the endpoints of the interval for $\theta$, so it
satisfies

$\displaystyle\eta_{lb}$
$\displaystyle=\min\left\\{\sigma_{11}\cos\left(\arctan\left(\frac{\sigma_{22}}{\sigma_{21}}\right)\right),\sigma_{11}\cos\left(\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right)\right)\right\\}$
$\displaystyle=\min\left\\{\sigma_{11}\cos\left(-\arctan\left(\frac{\sigma_{22}}{\sigma_{21}}\right)\right),\sigma_{11}\cos\left(\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right)\right)\right\\}$
$\displaystyle=\min\left\\{\sigma_{11}\cos\left(\arctan\left(-\frac{\sigma_{22}}{\sigma_{21}}\right)\right),\sigma_{11}\cos\left(\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right)\right)\right\\}$
$\displaystyle=\sigma_{11}\cos\left(\max\left\\{\arctan\left(-\frac{\sigma_{22}}{\sigma_{21}}\right),\arctan\left(\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right)\right\\}\right)$
$\displaystyle=\sigma_{11}\cos\left(\arctan\left(\max\left\\{-\frac{\sigma_{22}}{\sigma_{21}},\frac{\sigma_{22}y_{1k}}{\sigma_{21}y_{1k}-\sigma_{11}y_{2k}}\right\\}\right)\right).$
(A.17)

The second line follows from the fact that $\cos(.)$ is an even function and
the third line follows from the fact that $\arctan(.)$ is an odd function. The
arguments entering the $\cos(.)$ functions on the third line are both in the
interval $[0,\frac{\pi}{2})$, so the fourth line follows from the fact that
$\cos(.)$ is a decreasing function over this domain. The final line follow
from the fact that $\arctan(.)$ is an increasing function.

Restriction on the historical decomposition. Under the restrictions that the
first structural shock is positive in period $k$ and was the most important
(or overwhelming) contributor to the change in the first variable, $\theta$ is
restricted to lie in the set

$\theta\in\Big{\\{}\theta:\sigma_{21}\sin\theta\leq\sigma_{22}\cos\theta,\cos\theta\geq
0,\sigma_{22}y_{1k}\cos\theta\geq(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\sin\theta,\\\
|\sigma_{22}y_{1k}\cos^{2}\theta+(\sigma_{11}y_{2k}-\sigma_{21}y_{1k})\cos\theta\sin\theta|\geq|\sigma_{22}y_{1k}\sin^{2}\theta+(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\cos\theta\sin\theta|\Big{\\}}\\\
\cup\Big{\\{}\theta:\sigma_{21}\sin\theta\leq\sigma_{22}\cos\theta,\cos\theta\leq
0,\sigma_{22}y_{1k}\cos\theta\geq(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\sin\theta,\\\
|\sigma_{22}y_{1k}\cos^{2}\theta+(\sigma_{11}y_{2k}-\sigma_{21}y_{1k})\cos\theta\sin\theta|\geq|\sigma_{22}y_{1k}\sin^{2}\theta+(\sigma_{21}y_{1k}-\sigma_{11}y_{2k})\cos\theta\sin\theta|\Big{\\}}.$
(A.18)

As in the case of the shock-sign restriction, this set also depends on the
data $\mathbf{y}_{k}$ independently of the reduced-form parameters.

##  Appendix B Omitted proofs

Proof of Proposition 4.1.

###### Proof.

$\mathcal{H}(\bm{\phi},\mathbf{Q})$ can be written as

$\displaystyle\mathcal{H}(\bm{\phi},\mathbf{Q})=$
$\displaystyle\int_{\mathbf{Y}}f^{1/2}(\mathbf{y}^{T}|\bm{\phi})f^{1/2}(\mathbf{y}^{T}|\bm{\phi}_{0})\cdot
D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})D_{N}(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{y}^{T})d\mathbf{y}^{T}$
$\displaystyle+\int_{\mathbf{Y}}f^{1/2}(\mathbf{y}^{T}|\bm{\phi})f^{1/2}(\mathbf{y}^{T}|\bm{\phi}_{0})\cdot(1-D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T}))(1-D_{N}(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{y}^{T}))d\mathbf{y}^{T}.$

Note that the likelihood for the reduced-form parameters
$f(\mathbf{y}^{T}|\bm{\phi})$ point-identifies $\bm{\phi}$, so
$f(\cdot|\bm{\phi})=f(\cdot|\bm{\phi}_{0})$ holds only at
$\bm{\phi}=\bm{\phi}_{0}$. Hence, we set $\bm{\phi}=\bm{\phi}_{0}$ and
consider $\mathcal{H}(\bm{\phi}_{0},\mathbf{Q})$,

$\mathcal{H}(\bm{\phi}_{0},\mathbf{Q})=\int_{\\{\mathbf{y}^{T}:D_{N}(\bm{\phi}_{0},\mathbf{Q},\mathbf{y}^{T})=D_{N}(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{y}^{T})\\}}f(\mathbf{y}^{T}|\bm{\phi}_{0})d\mathbf{y}^{T}.$

Hence, $\mathcal{H}(\bm{\phi}_{0},\mathbf{Q})=1$ if and only if
$D_{N}(\bm{\phi}_{0},\mathbf{Q},\mathbf{y}^{T})=D_{N}(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{y}^{T})$
holds $f(\mathbf{Y}^{T}|\bm{\phi}_{0})$-a.s. In terms of the reduced-form
residuals entering the NR, the latter condition is equivalent to
$\\{\mathbf{U}:N(\bm{\phi}_{0},\mathbf{Q},\mathbf{Y}^{T})\geq\mathbf{0}_{s\times
1}\\}=\\{\mathbf{U}:N(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{Y}^{T})\geq\mathbf{0}_{s\times
1}\\}$ up to $f(\mathbf{Y}^{T}|\bm{\phi}_{0})$-null set. Hence,
$\mathcal{Q}^{\ast}$ defined in the proposition collects observationally
equivalent values of $\mathbf{Q}$ at $\bm{\phi}=\bm{\phi}_{0}$ in terms of the
unconditional likelihood.

Next, consider the conditional likelihood and consider

$\displaystyle\mathcal{H}_{c}(\bm{\phi}_{0},\mathbf{Q})$
$\displaystyle=\frac{1}{r^{1/2}(\bm{\phi},\mathbf{Q})r^{1/2}(\bm{\phi}_{0},\mathbf{Q}_{0})}\int_{\mathbf{Y}}f(\mathbf{y}^{T}|\bm{\phi}_{0})\cdot
D_{N}(\bm{\phi},\mathbf{Q},\mathbf{y}^{T})D_{N}(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{y}^{T})d\mathbf{y}^{T}$
$\displaystyle=\frac{E_{\mathbf{Y}^{T}|\bm{\phi}_{0}}\left[D_{N}(\bm{\phi}_{0},\mathbf{Q},\mathbf{Y}^{T})D_{N}(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{Y}^{T})\right]}{r^{1/2}(\bm{\phi},\mathbf{Q})r^{1/2}(\bm{\phi}_{0},\mathbf{Q}_{0})}$
$\displaystyle\leq 1,$

where the inequality follows by the Cauchy-Schwartz inequality, and it holds
with equality if and only if
$D_{N}(\bm{\phi}_{0},\mathbf{Q},\mathbf{Y}^{T})=D_{N}(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{Y}^{T})$
holds $f(\mathbf{Y}^{T}|\bm{\phi}_{0})$-a.s. Hence, by repeating the argument
for the unconditional likelihood case, we conclude that $\mathcal{Q}^{\ast}$
consists of observationally equivalent values of $\mathbf{Q}$ at
$\bm{\phi}=\bm{\phi}_{0}$ in terms of the conditional likelihood. ∎

Proof of Theorem 6.4. Since $(\bm{\phi}_{0},\mathbf{Q}_{0})$ satisfies the
imposed NR
$N(\bm{\phi}_{0},\mathbf{Q}_{0},\mathbf{y}^{T})\geq\mathbf{0}_{s\times 1}$ and
the other sign restrictions (if any imposed),
$\eta_{0}\in\widetilde{CIS}_{\eta}(\bm{\phi}_{0}|\mathbf{s}(\mathbf{y}^{T}),N)$
holds for any $\mathbf{y}^{T}$. Hence, for all $T$,

$P_{\mathbf{Y}^{T}|\mathbf{s},\bm{\phi}}(\eta_{0}\in\widehat{C}_{\alpha}^{\ast}|\mathbf{s}(\mathbf{Y}^{T}),\bm{\phi}_{0})\geq
P_{\mathbf{Y}^{T}|\bm{\phi}}(\widetilde{CIS}_{\eta}(\bm{\phi}_{0}|\mathbf{s}(\mathbf{Y}^{T}),N)\subset\widehat{C}_{\alpha}^{\ast}|\mathbf{s}(\mathbf{Y}^{T}),\bm{\phi}_{0}).$
(B.1)

Hence, to prove the claim, it suffices to focus on the asymptotic behavior of
the coverage probability for the conditional identified set shown in the
right-hand side.

Under Assumption 6.2 and 6.3, the asymptotically correct coverage for the
conditional identified set can be obtained by applying Proposition 2 in GK. ∎

Primitive Conditions for Assumption 6.3. In what follows, we present
sufficient conditions for convexity, continuity and differentiability (both in
$\bm{\phi}$) of the conditional impulse-response identified set under the
assumption that there is a fixed number of shock-sign restrictions
constraining the first structural shock only (possibly in multiple periods).

######  Proposition B.1.

Convexity. Let the parameter of interest be $\eta_{i,1,h}$, the impulse
response of the $i$th variable at the $h$th horizon to the first structural
shock. Assume that there are shock-sign restrictions on $\varepsilon_{1,t}$
for $t=t_{1},\ldots,t_{K}$, so
$N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})=(\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{1}},\ldots,\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{K}})^{\prime}\mathbf{q}_{1}\geq\mathbf{0}_{K\times
1}$. Then the set of values of $\eta_{i,1,h}$ satisfying the shock-sign
restrictions and sign normalization,
$\\{\eta_{i,1,h}(\bm{\phi},\mathbf{Q})=\mathbf{c}_{i,h}(\bm{\phi})\mathbf{q}_{1}:N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})\geq\mathbf{0}_{K\times
1},\mathrm{diag}(\mathbf{Q}^{\prime}\bm{\Sigma}_{tr}^{-1})\geq\mathbf{0}_{n\times
1},\mathbf{Q}\in\mathcal{O}(n)\\}$ is convex for all $i$ and $h$ if there
exists a unit-length vector $\mathbf{q}\in\mathbb{R}^{n}$ satisfying

$\begin{bmatrix}(\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{1}},\ldots,\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{K}})^{\prime}\\\
(\bm{\Sigma}_{tr}^{-1}\mathbf{e}_{1,n})^{\prime}\end{bmatrix}\mathbf{q}\geq\mathbf{0}_{(K+1)\times
1}.$ (B.2)

Proof of Proposition B.1. If there exists a unit-length vector $\mathbf{q}$
satisfying the inequality in (B.2), it must lie within the intersection of the
$K$ half-spaces defined by the inequalities
$(\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{k}})^{\prime}\mathbf{q}\geq 0$,
$k=1,\ldots,K$, the half-space defined by the sign normalization,
$(\bm{\Sigma}_{tr}^{-1}\mathbf{e}_{1,n})^{\prime}\mathbf{q}\geq 0$, and the
unit sphere in $\mathbb{R}^{n}$. The intersection of these $K+1$ half-spaces
and the unit sphere is a path-connected set. Since
$\eta_{i,1,h}(\bm{\phi},\mathbf{Q})$ is a continuous function of
$\mathbf{q}_{1}$, the set of values of $\eta_{i,1,h}$ satisfying the
restrictions is an interval and is thus convex, because the set of a
continuous function with a path-connected domain is always an interval. ∎

######  Proposition B.2.

Continuity. Let the parameter of interest and restrictions be as in
Proposition B.1, and assume that the conditions in the proposition are
satisfied. If there exists a unit-length vector $\mathbf{q}\in\mathbb{R}^{n}$
such that, at $\bm{\phi}=\bm{\phi}_{0}$,

$\begin{bmatrix}(\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{1}},\ldots,\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{K}})^{\prime}\\\
(\bm{\Sigma}_{tr}^{-1}\mathbf{e}_{1,n})^{\prime}\end{bmatrix}\mathbf{q}>>\mathbf{0}_{(K+1)\times
1},$ (B.3)

then $u(\bm{\phi},\mathbf{Y}^{T})$ and $l(\bm{\phi},\mathbf{Y}^{T})$ are
continuous at $\bm{\phi}=\bm{\phi}_{0}$ for all $i$ and $h$.181818For a vector
$\mathbf{x}=(x_{1},\ldots,x_{m})^{\prime}$, $\mathbf{x}>>\mathbf{0}_{m\times
1}$ means that $x_{i}>0$ for all $i=1,\ldots,m$.

Proof of Proposition B.2. $\mathbf{Y}^{T}$ enters the NR through the reduced-
form VAR innovations, $\mathbf{u}_{t}$. After noting that the reduced-form VAR
innovations are (implicitly) continuous in $\bm{\phi}$, continuity of
$u(\bm{\phi},\mathbf{Y}^{T})$ and $l(\bm{\phi},\mathbf{Y}^{T})$ follows by the
same logic as in the proof of Proposition B.2 of Giacomini and Kitagawa (in
press[b]). We omit the detail for brevity. ∎

######  Proposition B.3.

Differentiability. Let the parameter of interest and restrictions be as in
Proposition B.1, and assume that the conditions in the proposition are
satisfied. Denote the unit sphere in $\mathbb{R}^{n}$ by $\mathcal{S}^{n-1}$.
If, at $\bm{\phi}=\bm{\phi}_{0}$, the set of solutions to the optimization
problem

$\max_{\mathbf{q}\in\mathcal{S}^{n-1}}\quad\left(\min_{\mathbf{q}\in\mathcal{S}^{n-1}}\right)\quad\mathbf{c}_{i,h}^{\prime}(\bm{\phi})\mathbf{q}\quad\textrm{s.t.}\quad\begin{bmatrix}(\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{1}},\ldots,\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{K}}),&\bm{\Sigma}_{tr}^{-1}\mathbf{e}_{1,n}\end{bmatrix}^{\prime}\mathbf{q}\geq\mathbf{0}_{(K+1)\times
1}$ (B.4)

is singleton, the optimized value $u(\bm{\phi},\mathbf{Y}^{T})$
($l(\bm{\phi},\mathbf{Y}^{T})$) is nonzero, and the number of binding
inequality restrictions at the optimum is at most $n-1$, then
$u(\bm{\phi},\mathbf{Y}^{T})$ ($l(\bm{\phi},\mathbf{Y}^{T})$) is almost-surely
differentiable at $\bm{\phi}=\bm{\phi}_{0}$.

Proof of Proposition B.3. One-to-one differentiable reparameterization of the
optimization problem in Equation (B.4) using
$\mathbf{x}=\bm{\Sigma}_{tr}\mathbf{q}$ yields the optimization problem in
Equation (2.5) of Gafarov, Meier and Montiel-Olea (2018) with a set of
inequality restrictions that are now a function of the data through the
reduced-form VAR innovations entering the NR. Noting that $\mathbf{u}_{t}$ is
(implicitly) differentiable in $\bm{\phi}$, differentiability of
$u(\bm{\phi},\mathbf{Y}^{T})$ at $\bm{\phi}=\bm{\phi}_{0}$ follows from their
Theorem 2 under the assumptions that, at $\bm{\phi}=\bm{\phi}_{0}$, the set of
solutions to the optimization problem is singleton, the optimized value
$u(\bm{\phi},\mathbf{Y}^{T})$ is nonzero, and the number of binding sign
restrictions at the optimum is at most $n-1$. Differentiability of
$l(\bm{\phi},\mathbf{Y}^{T})$ follows similarly. Note that Theorem 2 of
Gafarov et al. (2018) additionally requires that the column vectors of
$\begin{bmatrix}(\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{1}},\ldots,\bm{\Sigma}_{tr}^{-1}\mathbf{u}_{t_{K}}),&\bm{\Sigma}_{tr}^{-1}\mathbf{e}_{1,n}\end{bmatrix}$
are linearly independent, but this occurs almost-surely under the probability
law for $\mathbf{Y}^{T}$. ∎

##  Appendix C Asymptotics with increasing number of NR

What happens if the number of NR increases with the sample size? We conjecture
that, under some assumptions about how the NR are generated, the class of
posteriors for particular parameters will converge to a point mass at the
truth. Intuitively, as the number of restrictions increases, the likelihood
tends to be truncated to an increasing extent until the only point with
positive likelihood is the true value of the parameter.

To provide some numerical evidence for this conjecture, we return to the
bivariate example from Section 2. Assume that $\bm{\phi}=\bm{\phi}_{0}$ is
known (which will be the case asymptotically under regularity conditions), so
$\pi_{\bm{\phi}|\mathbf{Y}^{T}}$ is a point mass at $\bm{\phi}=\bm{\phi}_{0}$.
Consider the case where the econometrician observes
$\mathrm{sgn}(\varepsilon_{1t})$ for all $t$ and imposes the shock-sign
restriction
$\mathrm{sgn}(\varepsilon_{1t})\varepsilon_{1t}(\bm{\phi},\mathbf{Q},\mathbf{y}_{t})\geq
0$ for $t=1,\ldots,T$.191919This assumption can be relaxed, so that the
econometrician observes $\mathrm{sgn}(\varepsilon_{1t})$ each period with some
probability.

Figure C.1 plots the conditional identified set for $\mathbf{q}_{1}$ for
different numbers of restrictions given random realizations of a time series
drawn from the data-generating process. As the sample size increases, the
boundaries of the half-spaces generated by the binding shock-sign restrictions
converge towards the true value of $\mathbf{q}_{1}$, $\mathbf{q}_{1,0}$.
Additionally, the conditional identified set for $\mathbf{q}_{2}$ will
converge to its true value, since $\mathbf{q}_{2}$ is orthogonal to
$\mathbf{q}_{1}$ and satisfies a sign normalization.202020Equivalently, we
could show convergence of the conditional identified set for $\theta$ to
$\theta_{0}$, which pins down all impulse responses. In other words, imposing
a growing number of shock-sign restrictions on a single shock is sufficient
for the posterior of all impulse responses to converge to a point mass at the
true value. Note, however, that this will not be the case in higher-
dimensional VARs, since the collapse of the conditional identified set for
$\mathbf{q}_{1}$ does not pin down values for $\mathbf{q}_{j}$,
$j=2,\ldots,n$.

Figure C.1: Illustration of Posterior Consistency

|  |
---|---|---

Notes:

Purple line is true value of $\mathbf{q}_{1}=(q_{1,1},q_{1,2})^{\prime}$;
orange line is boundary of half-space generated by the sign normalization;
blue lines are boundaries of half-spaces generated by ‘binding’ shock-sign
restrictions intersected with half-space generated by the sign normalisation;
red line is intersection of all half-spaces with unit circle.

##  Appendix D Alternative algorithms for robust Bayesian inference

Assume that the object of interest is an impulse response to the first
structural shock. The upper bound of the conditional identified set for the
horizon-$h$ impulse response of the $i$th variable to this shock given
$\bm{\phi}$ and $\mathbf{Y}^{T}$ is the value function associated with the
optimization problem

$u(\bm{\phi},\mathbf{Y}^{T})=\max_{\mathbf{Q}\in\mathcal{Q}(\bm{\phi}|\mathbf{Y}^{T},N,S)}\mathbf{c}_{i,h}^{\prime}(\bm{\phi})\mathbf{q}_{1}.$
(D.1)

$l(\bm{\phi},\mathbf{Y}^{T})$ is obtained by minimising the same objective
function subject to the same constraints. When
$N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})$ and $S(\bm{\phi},\mathbf{Q})$ only
constrain $\mathbf{q}_{1}$, applying the change of variables
$\mathbf{x}=\bm{\Sigma}_{tr}\mathbf{q}_{1}$ yields the optimization problem in
Gafarov et al. (2018) with additional inequality restrictions that are
functions of $\mathbf{Y}^{T}$.

Given a set of active inequality restrictions, Gafarov et al. (2018) provide
an analytical expression for the value function and solution of this
optimization problem. To find the bounds of the identified set, they compute
these quantities for every possible combination of active restrictions and
check which pair solves the optimization problem. Since the bounds are
computed analytically at each set of active restrictions, this algorithm is
computationally inexpensive as long as there is not a very large number of
inequality restrictions. However, if $N(\bm{\phi},\mathbf{Q},\mathbf{Y}^{T})$
contains restrictions on the historical decomposition, all columns of