Research Discussion Paper – RDP 2023-07 Identification and Inference under Narrative Restrictions

Understanding the dynamic causal effects of structural shocks is one of the central problems in macroeconometrics, and there is increasing empirical demand for methods that require minimal identifying assumptions. Replacing point-identifying restrictions in structural vector autoregressions (SVARs) with set-identifying sign restrictions is an early example of this search for robustness (e.g. Uhlig 2005). A more recent example is the idea of substituting or augmenting traditional restrictions on structural parameters with ‘narrative restrictions’ (henceforth NR), which are inequalities involving structural shocks in given time periods (Antolín-Díaz and Rubio-Ramírez (2018) (henceforth AR18); Ludvigson, Ma and Ng (2018)). These restrictions force the SVAR's predictions to be consistent with narratives about the nature of structural shocks driving macroeconomic variation in particular historical episodes. The promise of these restrictions is that they may deliver informative inferences about the effects of structural shocks under weak or uncontroversial restrictions on the structural parameters.

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 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. Other restrictions also fit within this framework, including restrictions on shock magnitudes (e.g. Ludvigson et al 2018) or rankings (e.g. Ben Zeev 2018).

A burgeoning literature imposes NR in a broad range of empirical applications.^[1] However, the non-standard nature of these restrictions raises econometric challenges. Under these restrictions, there are no formal results on identification or the validity of frequentist approaches to inference.^[2] Moreover, as we show in this paper, the Bayesian procedure of AR18, which is used by the majority of the literature, may be sensitive to prior choice. This paper contributes to the literature by formally analysing identification and inference in models with NR, and by providing an approach to inference that eliminates prior sensitivity. Importantly, this approach is valid from both Bayesian and frequentist perspectives.

From a frequentist perspective, NR are fundamentally different from traditional restrictions. 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 – an identified set – that represents observational equivalence. The identified set corresponds to the flat region of the likelihood and, by the definition of observational equivalence, does not depend on the realisation of the data (e.g. Rothenberg 1971). NR also result in the likelihood possessing flat regions and hence generate a set-valued mapping from the reduced-form parameters to the structural parameters. Crucially, this mapping additionally depends on the realisation 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) existing results on identification in SVARs are inapplicable; and 2) there is no known valid frequentist procedure for inference.

From a Bayesian perspective, the bulk of the empirical literature conducts Bayesian inference under NR in a similar way as under traditional restrictions by following a procedure in AR18. We highlight two issues with the existing approach: the potentially spurious effects on inference of using a conditional likelihood to construct the posterior; and the sensitivity of inference to prior choice due to the likelihood possessing flat regions. The prior sensitivity of the existing Bayesian approach makes it difficult to know whether apparently informative inference obtained in empirical studies (e.g. narrow credible intervals) reflects the informativeness of NR or the choice of prior. Removing the effect of the prior allows us to understand if NR deliver on their promise of offering informative inference under minimal assumptions, in contrast with traditional sign restrictions, which have been shown to provide little information in some settings (e.g. Baumeister and Hamilton 2015; Wolf 2020; Read 2022b).

The paper proceeds in four main steps. First, we formalise the identification problem under NR. Second, we propose using the unconditional likelihood, rather than the conditional likelihood, to construct the posterior. Third, we consider a robust (multiple-prior) Bayesian approach to assess and/or eliminate the posterior sensitivity that remains when using the unconditional likelihood. 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 study identification under general NR. Plagborg-Møller and Wolf (2021b) note that shock-sign restrictions could in principle be cast as an external instrument (or ‘narrative proxy’) and used to point identify impulse responses in a local projection. Plagborg-Møller (2022) argues that such an approach possesses several appealing robustness properties relative to the likelihood-based approach of AR18 that we consider here, including that it allows for imperfect narrative information and non-invertibility.^[3] Petterson, Seim and Shapiro (2023) derive bounds for a slope parameter in a single equation given restrictions on the magnitude of the residuals, but the setting is non-probabilistic.

We make two main contributions to the understanding of identification under NR. First, we provide a necessary and sufficient condition for global identification of a SVAR under NR and as an example show that this condition is satisfied in a bivariate SVAR with a single shock-sign restriction. This means that, 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. This result does not, however, deliver a point estimator, because the observed likelihood is almost always flat at the maximum. Second, we introduce the notion of a ‘conditional identified set’, which extends the standard notion of an identified set to a setting where identification is defined in a repeated sampling experiment conditional on the 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. We make use of the conditional identified set when analysing the frequentist properties of our procedure.

The fact that NR deliver a set of maximum likelihood estimators is reminiscent of maximum score estimation, where the objective function yields a set of maximisers (Manski 1975, 1985). Our conditional identified set, which fixes the flat regions of the likelihood in the conditional frequentist experiment, shares geometric properties with the finite sample identified set introduced by Rosen and Ura (2020) in the maximum score context; however, their finite sample inference procedure does not apply here.

In terms of inference under NR, our contribution can be viewed from both a Bayesian and a frequentist point of view. Our first message to Bayesian researchers is to base analysis on the unconditional likelihood, rather than the conditional likelihood used by AR18. Conditioning can be problematic because, for some types of NR, a component of the prior is updated only in the direction that makes the NR unlikely to hold ex ante. This is due to conditioning on a non-ancillary event, which results in loss of information.

Our second message to Bayesian researchers is that posterior inference may be sensitive to the choice of prior, because the unconditional likelihood has flat regions under NR (the conditional likelihood also has flat regions under shock-sign restrictions). This sensitivity is a problem that also occurs in set-identified models under traditional restrictions (e.g. Poirier 1998; Baumeister and Hamilton 2015). As advocated for by Giacomini and Kitagawa (2021a) (henceforth GK21), this problem can be solved by adopting a robust (multiple-prior) Bayesian approach. GK21 consider robust Bayesian inference in SVARs under traditional set-identifying restrictions, a setting where – unlike in our case – frequentist inference is also available (e.g. Gafarov et al 2018; Granziera et al 2018). They decompose the prior for structural parameters into a prior for reduced-form parameters, which is revisable, and a conditional prior for structural parameters given reduced-form parameters, which is unrevisable. Considering the set of all conditional priors satisfying the identifying restrictions generates a set of posteriors. This removes the source of posterior sensitivity and makes robust Bayesian and frequentist approaches asymptotically equivalent, reconciling the disagreement between frequentist and Bayesian methods that arises in set-identified models (Moon and Schorfheide 2012).^[4]

We explain how this robust Bayesian approach can be adapted to NR. Even if a researcher has a credible prior, we recommend reporting the standard Bayesian posterior (under the unconditional likelihood) together with the robust Bayesian output. This allows researchers to assess the extent to which posterior inference may be driven by prior choice. In the absence of a credible prior, we recommend reporting the robust Bayesian output as an alternative to the standard Bayesian posterior.

This paper's contribution to frequentist inference is to provide the first (to our knowledge) asymptotically valid approach to inference under NR. While other frequentist approaches are in principle possible, one appealing feature of the robust Bayesian approach is its numerical tractability. Proving the frequentist asymptotic validity of the approach is challenging, due to the data-dependent mapping induced by the NR that we discussed above. This means that the results in GK21 about the asymptotic equivalence between Bayesian and frequentist inference are not applicable here. We address these challenges by deriving new results on the asymptotics of robust Bayesian analysis under a fixed number of NR, which we argue is the empirically relevant case given the small number of restrictions typically imposed in the literature. We show that, under regularity conditions, the robust credible region provides asymptotically valid frequentist coverage of the conditional identified set for the impulse response, which also implies correct coverage for the true impulse response.

We illustrate our methods by revisiting the monetary SVAR in AR18. We first examine the robustness of conclusions about the output effects of US monetary policy when NR are imposed based only on the Volcker episode. We find that inferences about the output response are sensitive to prior choice, and the restrictions are largely uninformative in the sense that they admit a wide range of positive and negative output responses. Restrictions based on the Volcker episode in isolation are therefore not sufficient to precisely identify the effects of monetary policy. We then impose an extended set of NR related to multiple episodes, and find robust evidence that output falls following a positive monetary policy shock. Disentangling the informativeness of the different restrictions, the shock-sign restrictions on their own are not particularly informative, and drawing robust conclusions about the output response relies on imposing restrictions on the historical decomposition.

The remainder of the paper is structured as follows. Section 2 highlights the econometric issues that arise when imposing NR using a bivariate example. Section 3 describes the general framework. Section 4 analyses global 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.

Notation: For the matrix X, vec(X) is the vectorisation of X and vech(X) is the half-vectorisation. e_i,n is the ith column of the n × n identity matrix, I_n. 0_n×m is a n × m matrix of zeros. 1 (.) is the indicator function.

We illustrate the econometric issues that arise when imposing NR in the context of the following bivariate SVAR(0): $A_0{y}_t={\epsilon }_t$, for t = 1,…,T, where y_t = (y_1t, y_2t)′ and ${\epsilon }_t={\left({\epsilon }_{1t},{\epsilon }_{2t}\right)}^{\prime }$, with ${\epsilon }_t^{{}_{\sim }^{iid}}\mathcal{N}\left(0_{2\times 1},{\text{I}}_2\right)$. We abstract from dynamics for ease of exposition, but this is without loss of generality. The orthogonal reduced form of the model reparameterises A₀ as $Q^{\prime }{\sum }_{tr}^{-1}$, where ${\sum }_{tr}$ is the lower-triangular Cholesky factor (with positive diagonal elements) of $\sum =𝔼\left(y_t{y^{\prime }}_t\right)=A_0^{-1}{\left(A_0^{-1}\right)}^{\prime }$. We parameterise ${\Sigma }_{tr}$ as

and denote the vector of reduced-form parameters as $\varphi =\text{vech}\left({\Sigma }_{tr}\right)$. Q is an orthonormal matrix in the space of 2×2 orthonormal matrices, $𝒪$ (2):

where $\theta \in \left[-\pi ,\pi \right]$. This formulation of the model, which follows Baumeister and Hamilton (2015), means that the structural parameters can be expressed as functions of the reduced-form parameters and $\theta$. Restrictions on the structural parameters and/or functions of the structural shocks can then be interpreted as restricting $\theta$ to some set. In what follows, we discuss properties of this set that are key for analysing identification and inference under NR.

2.1 Shock-sign restrictions

Consider the ‘shock-sign restriction’ that ${\epsilon }_{1k}$ is non-negative for some $k\in \left\{1,\mathrm{...},T\right\}$:

Equation (3) implies that the restricted structural shock can be written as a function ${\epsilon }_{1k}\left(\theta ,\varphi ,y_k\right)$. Along with the ‘sign normalisation’ $\text{diag}\left(A_0\right)\ge 0_{2\times 1}$, the shock-sign restriction implies that $\theta$ is restricted to the set

The restriction induces a set-valued mapping from $\varphi$ to $\theta$ that depends on the realisation of y_k. Giacomini et al (2022a) characterise this mapping in the case where ${\sigma }_{21}<0$. For example, if $h\left(\varphi ,y_k\right)={\sigma }_{21}{y}_{1k}-{\sigma }_{11}{y}_{2k}<0$, then

where $C\left(\varphi ,y_k\right)={\sigma }_{22}{y}_{1k}/h\left(\varphi ,y_k\right)$. The direct dependence of this mapping on the realisation of the data implies that the standard notion of an identified set – the set of observationally equivalent structural parameters given the reduced-form parameters – does not apply. Consequently, it is not obvious whether the restrictions are, in fact, set identifying in a formal frequentist sense, nor whether existing frequentist procedures for conducting inference in set-identified models are valid. We analyse identification under these restrictions in Section 4.

When conducting Bayesian inference, AR18 construct the posterior using the conditional likelihood – the likelihood of observing the data conditional on the NR holding. Letting $y^T={\left({y^{\prime }}_1,\mathrm{...},{y^{\prime }}_T\right)}^{\prime }$, the conditional likelihood is

The denominator in the first term – the ex ante probability that the NR is satisfied – equals ½, because ${\epsilon }_{1k}$ is standard normal. The conditional likelihood therefore depends on $\theta$ only through the indicator function $1\left({\epsilon }_{1k}\left(\theta ,\varphi ,y_k\right)\ge 0\right)$, which truncates the likelihood, with the truncation points depending on y_k. To illustrate, the left panel of Figure 1 plots the conditional likelihood as a function of $\theta$ given two realisations of a data-generating process and fixing $\varphi$ to its true value.^[5] The conditional likelihood is flat over the interval for $\theta$ satisfying the shock-sign restriction and is zero outside this interval. The support of the non-zero region depends on y_k.

The flat likelihood implies that the posterior for $\theta$ is proportional to the prior in the region where the likelihood is non-zero, and is zero outside this region. The standard approach to Bayesian inference in SVARs under sign restrictions assumes a uniform prior over Q, as do AR18.^[6] In the bivariate example, this is equivalent to a prior for $\theta$ that is uniform (Baumeister and Hamilton 2015). This prior implies that the posterior for $\theta$ is also uniform over the interval for $\theta$ where the likelihood is non-zero.

The impact impulse response of y_1t to a positive standard deviation shock ${\epsilon }_{1t}$ 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 realisations of the data for which the likelihood was plotted in the left panel. It can be seen that the posterior for $\eta$ 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 also occurs under traditional sign restrictions (Baumeister and Hamilton 2015). One difference is that the 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 realisation of y_k through its effect on the truncation points of the likelihood, so there may be some updating of the prior. However, the 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 kth period is

while the contribution of the second shock is

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 $\left|H_{1,1,k}\left(\theta ,\varphi ,y_k\right)\right|\ge \left|H_{1,2,k}\left(\theta ,\varphi ,y_k\right)\right|$. Under these restrictions, $\theta$ must satisfy a set of inequalities that depends on $\varphi$ and y_k. As in the case of the shock-sign restriction, this set of inequalities generates a set-valued mapping from $\varphi$ to $\theta$ that depends on y_k.

Let $\mathcal{D}\left(\theta ,\varphi ,y_k\right)=1\left\{{\epsilon }_{1k}\left(\theta ,\varphi ,y_k\right)\ge 0,\left|H_{1,1,k}\left(\theta ,\varphi ,y_k\right)\right|\ge \left|H_{1,2,k}\left(\theta ,\varphi ,y_k\right)\right|\right\}$ represent the indicator function equal to one when the NR are satisfied and equal to zero otherwise, and let $\tilde{\mathcal{D}}\left(\theta ,\varphi ,{\epsilon }_k\right)=1\left\{{\epsilon }_{1k}\ge 0,\left|{\tilde{H}}_{1,1,k}\left(\theta ,\varphi ,{\epsilon }_{1k}\right)\right|\ge \left|{\tilde{H}}_{1,2,k}\left(\theta ,\varphi ,{\epsilon }_{2k}\right)\right|\right\}$ denote the indicator function for the same event in terms of the structural shocks rather than the data. The conditional likelihood given the restrictions is then

In contrast to the case of shock-sign restrictions, 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 $\left|{\tilde{H}}_{1,1,k}\left(\theta ,\varphi ,{\epsilon }_{1k}\right)\right|\ge \left|{\tilde{H}}_{1,2,k}\left(\theta ,\varphi ,{\epsilon }_{2k}\right)\right|$. Consequently, the likelihood is not necessarily flat when it is non-zero.

To illustrate, the top panel of Figure 2 plots the conditional likelihood under the historical decomposition NR using the same data-generating process as in Figure 1. The bottom panel plots the probability in the denominator of the conditional likelihood. The likelihood is again truncated, but it is no longer flat – it has a maximum at the value of $\theta$ that minimises the ex ante probability that the NR are satisfied (within the set of values of $\theta$ that are consistent with the restriction given the realisation of the data). The posterior for $\theta$ induced by the usual uniform prior will therefore assign greater posterior probability to values of $\theta$ that yield a lower ex ante probability of satisfying the NR.

If we view the narrative event ($\text{i}\text{.e}\text{.}\tilde{\mathcal{D}}\left(\theta ,\varphi ,{\epsilon }_k\right)$) as observable and its probability of occurring depends on the parameter of interest, then conditioning on the narrative event implies that we are conditioning on a non-ancillary statistic. This is undesirable when conducting likelihood-based inference, because it represents a loss of information about the parameter of interest. Unlike for the shock-sign restriction, the probability that the historical decomposition restriction is satisfied depends on $\theta$, so the event that the NR are satisfied is not ancillary. Conditioning on this event means that the shape of the likelihood (within the non-zero region) is fully driven by the inverse probability of the conditioning event.

Based on this consideration, we therefore advocate constructing the posterior using the joint (or unconditional) likelihood of observing the data and the NR holding:

For all types of NR, the unconditional likelihood is flat with respect to $\theta$ (when it is non-zero) and depends on $\theta$ only through the points of truncation. Of course, this means that posterior inference based on the unconditional likelihood may be sensitive to the choice of prior, as when using the conditional likelihood under shock-sign restrictions. In Section 5.2, we propose how to deal with this posterior sensitivity.

This section describes the general SVAR, outlines the identifying restrictions we consider, and defines the conditional and unconditional likelihoods in this general setting.

3.1 SVAR

Let y_t be an n × 1 vector of variables following the SVAR(p) process:

where: A₀ is invertible; $x_t={\left({y^{\prime }}_{t-1},\mathrm{...},{y^{\prime }}_{t-p,},{z^{\prime }}_t\right)}^{\prime }$ with z_t containing any exogenous variables (e.g. a constant); $A_{+}=\left(A_1,\mathrm{...},A_p,A_z\right)$; and ${\epsilon }_t{}^{{}_{\sim }^{iid}}\mathcal{N}\left(0_{n\times 1},I_n\right)$ are structural shocks. The initial conditions $\left(y_{1-p},\mathrm{...},y_0\right)$ are given.

The reduced-form VAR(p) representation is

where: $B=\left(B_1,\mathrm{...},B_p,B_z\right)\text{with}{B}_l=A_0^{-1}{A}_l;$ and $u_t=A_0^{-1}{\epsilon }^{{}_{\sim }^{iid}}\mathcal{N}\left(0_{n\times 1},\Sigma \right)$ with $\Sigma =A_0^{-1}{\left(A_0^{-1}\right)}^{\prime }$. $\varphi ={\left(\text{vec}{\left(B\right)}^{\prime },\text{vech}{\left(\Sigma \right)}^{\prime }\right)}^{\prime }\in \Phi$ are the reduced-form parameters.

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

where: ${\Sigma }_{tr}$ is the lower-triangular Cholesky factor of $\Sigma \left(\text{i}\text{.e}\text{.}{\Sigma }_{tr}{{\Sigma }^{\prime }}_{tr}=\Sigma \right)$ with non-negative diagonal elements; and Q is an n × n orthonormal matrix with $𝒪\left(n\right)$ the set of all such matrices. The structural and orthogonal reduced-form parameterisations are related through the mapping $B=A_0^{-1}{A}_{+},\Sigma =A_0^{-1}{\left(A_0^{-1}\right)}^{\prime }$ and $Q={\Sigma }_{tr}^{-1}{A}_0^{-1}$ with inverse mapping $A_0=Q^{\prime }{\Sigma }_{tr}^{-1}$ and $A_{+}=Q^{\prime }{\Sigma }_{tr}^{-1}B.$

We assume B is such that the VAR(p) can be inverted into an infinite-order vector moving average (VMA$\left(\infty \right)$) representation:^[7]

where C_h is the hth term in ${\left(I_n-{\Sigma }_{l=1}^p{B}_l{L}^l\right)}^{-1}$ and L is the lag operator.^[8] The (i, j)th element of the matrix $C_h{\Sigma }_{tr}Q$, which we denote by ${\eta }_{i,j,h}\left(\varphi ,Q\right)$, is the horizon-h impulse response of the ith variable to the jth structural shock:

with ${c^{\prime }}_{i,h}\left(\varphi \right)={e^{\prime }}_{i,n}{C}_h{\Sigma }_{tr}$ the ith row of $C_h{\Sigma }_{tr}$ and q_j = Qe_j,n the jth column of Q.

3.2 Narrative restrictions

In the absence of identifying restrictions, Q – and functions of Q such as ${\eta }_{i,j,h}\left(\varphi ,Q\right)$ – are set identified, since any $Q\in 𝒪\left(n\right)$ is consistent with the joint distribution of the data, which is summarised by the reduced-form parameters. Imposing identifying restrictions is equivalent to restricting Q to lie in a subspace of $𝒪\left(n\right)$. Throughout, we impose the ‘sign normalisation’ $\text{diag}\left(A_0\right)=\text{diag}\left(Q^{\prime }{\Sigma }_{tr}^{-1}\right)\ge 0_{n\times 1}$, so a positive value of ${\epsilon }_{it}$ is a positive shock to the ith equation in the SVAR at time t.

It is common to impose sign restrictions on the impulse responses (e.g. Uhlig 2005) or on the structural parameters (e.g. Arias, Caldara and Rubio-Ramírez 2019). For example, the restriction ${\eta }_{i,j,h}\left(\varphi ,Q\right)={c^{\prime }}_{i,h}\left(\varphi \right)q_j\ge 0$ is a linear inequality restriction on a single column of Q that depends only on the reduced-form parameters $\varphi$. Restrictions on elements of A₀ take a similar form.

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

The shock-sign restriction that the ith structural shock at time k is positive is

We can treat u_t as observable given $\varphi$ and the data, so we suppress the dependence of u_t on $\varphi$ and ${\left({y^{\prime }}_t,{x^{\prime }}_t\right)}^{\prime }$ for notational convenience. The restriction in Equation (17) is a linear inequality restriction on a single column of 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 jth shock to the observed unexpected change in the ith variable between periods k and k+h (i.e. the contribution to the (h+1)-step-ahead forecast error):

An example is the restriction that the jth structural shock was the ‘most important contributor’ to the change in the ith variable between periods k and k + h, which requires that $\left|H_{i,j,k,k+h}\right|\ge {\max }_{l\ne j}\left|H_{i,l,k,k+h}\right|$. Another is that the jth structural shock was the ‘overwhelming contributor’ to the change in the ith variable between periods k and k + h, which requires that $\left|H_{i,j,k,k+h}\right|\ge {\Sigma }_{l\ne j}\left|H_{i,l,k,k+h}\right|$. From Equation (18), it is clear that these restrictions are nonlinear inequality constraints that simultaneously constrain every column of Q and that depend on the realisations of the data in particular periods in addition to the reduced-form parameters.

Other restrictions also naturally fit within this framework. For instance, Ludvigson et al (2018) restrict the magnitudes of structural shocks in particular periods (e.g. ${\epsilon }_{ik}\left(\varphi ,Q,u_k\right)<\lambda$ for some specified scalar $\lambda$). One could also consider restrictions on the relative magnitudes of a particular shock in different periods (e.g. ${\epsilon }_{ik}\left(\varphi ,Q,u_k\right)\ge {\epsilon }_{ij}\left(\varphi ,Q,u_j\right)$ for $j\ne k$).^[9]

A collection of NR can be represented in the general form $N\left(\varphi ,Q,Y^T\right)\ge 0_{s\times 1}$, where s is the number of restrictions. As an illustration, consider the case where there is a single shock-sign restriction in period $k,{\epsilon }_{1k}\left(\varphi ,Q,u_k\right)\ge 0$, 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,

Traditional sign and zero restrictions can also be imposed alongside NR. We follow AR18 by explicitly allowing for sign restrictions on impulse responses and on elements of A₀. We denote such sign restrictions by $S\left(\varphi ,Q\right)\ge 0_{\tilde{s}\times 1}$, where $\tilde{s}$ is the number of traditional sign restrictions. It is straightforward to additionally allow for zero restrictions, so long as these are not over-identifying. These include ‘short-run’ zero restrictions (Sims 1980), ‘long-run’ zero restrictions (Blanchard and Quah 1989), or restrictions arising from external instruments (Mertens and Ravn 2013; Stock and Watson 2018; Aria, Rubio-Ramírez and Waggoner 2021).^[10]

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

The likelihood conditional on D_N = 1 can be written as

$f\left(y^T|\varphi \right)$ is the joint density of the data given $\varphi$ (i.e. the likelihood of the reduced-form VAR), which depends only on $\varphi$ and the data. The indicator function $D_N\left(\varphi ,Q,Y^T\right)$ equals one when the NR are satisfied and is zero otherwise. This determines the truncation points of the likelihood. $r\left(\varphi ,Q\right)$ is the ex anteprobability that the NR are satisfied. This is constant when there are only shock-sign restrictions; for example, if there are s shock-sign restrictions, $r\left(\varphi ,Q\right)={\left(1/2\right)}^s$. When there are restrictions on the historical decomposition, this probability depends on $\varphi$ and Q.

Consider the case where $\varphi$ is known, which will be the case asymptotically because $\varphi$ is point identified. When $r\left(\varphi ,Q\right)$ depends on Q, the conditional likelihood is maximised at the value of Q that minimises $r\left(\varphi ,Q\right)$ (within the set of values of Q satisfying the restrictions). The posterior based on this likelihood therefore places higher posterior probability on values of 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.

We therefore advocate constructing the likelihood without conditioning on the NR holding. The unconditional likelihood (the joint distribution of the data and D_N) is

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

3.4 Discussion of assumptions

3.4.1 Distributional assumptions

Researchers may be concerned about misspecification with regards to 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, different variance or fat tails. The unconditional likelihood depends on the normality assumption only through the reduced-form VAR likelihood, $f\left(y^T|\varphi \right)$. By omitting terms in $f\left(y^T|\varphi \right)$ corresponding to the periods in which the NR are imposed, one can thus 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 for y^T takes the form

where

and $w\left(y_t-B{x}_k\right)$ is an unknown, potentially non-normal, density. Replacing $f\left(y^T|\varphi \right)$ in Equation (24) with $v\left({\left\{y_t-B{x}_t\right\}}_{t\ne k}|\varphi \right)$ yields an ‘unconditional partial likelihood’ that does not depend on the distribution of ${\epsilon }_k$, but 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 is likely to be small. In contrast, when using the conditional likelihood, the distribution of the structural shocks must be specified in all periods to be able to compute $r\left(\varphi ,Q\right)$.

Concerns about misspecification may also be alleviated by recognising that the distributional assumption is irrelevant asymptotically. The set of values of Q with non-zero unconditional likelihood depends only on $\varphi$ and the realisation of the data in the periods in which the NR are imposed. Under regularity assumptions, the likelihood (and thus the posterior) of $\varphi$ will converge to a point at the true value of $\varphi$ asymptotically regardless of whether the true data-generating process is a VAR with homoskedastic normal shocks.^[11] The set of values of 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

In line with the existing literature, 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). If the revelation of this information depends on the data, the likelihood will be misspecified. The exact implications of this misspecification for identification 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 model of Section 2, if the identity of k is randomly determined independently of ${\epsilon }_1,\mathrm{...},{\epsilon }_T$, we can interpret the current analysis conditional on k.

This section formally analyses 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 realisation of the data. This provides an interpretation of the set-valued mapping induced by the NR. Additionally, we make use of the conditional identified set when investigating the frequentist properties of our robust Bayesian procedure in Section 6.

4.1 Point identification under NR

Denoting the true parameter value by $\left({\varphi }_0,Q_0\right)$, point identification for the parametric model (Equation (24)), which is based on the unconditional likelihood, requires that there is no other parameter value $\left({\varphi }_0,Q_0\right)\ne \left({\varphi }_0,Q_0\right)$ that is observationally equivalent to $\left({\varphi }_0,Q_0\right)$.^[12]

To assess the existence of observationally equivalent parameters, we analyse a statistical distance between $p\left(y^T,D_N=d|\varphi ,Q\right)$ and $p\left(y^T,D_N=d|{\varphi }_0,Q_0\right)$ that metrises observational equivalence. Since the support of the distribution of observables can depend on the parameters, it is convenient to work with the Hellinger distance:

and Y is the sample space for Y^T. As is known in the literature on minimum distance estimation, $\left(\varphi ,Q\right)$ and $\left({\varphi }_0,Q_0\right)$ are observationally equivalent if and only if $HD\left(\varphi ,Q\right)=0$ or, equivalently, $\mathscr{H}\left(\varphi ,Q\right)=1$ (e.g. Basu, Shioya and Park 2011).

We similarly define the Hellinger distance for the conditional likelihood as

The next proposition analyses the conditions for $\mathscr{H}\left(\varphi ,Q\right)=1$ and ${\mathscr{H}}_c\left(\varphi ,Q\right)=1$, and shows that observational equivalence of $\left(\varphi ,Q\right)$ and $\left({\varphi }_0,Q_0\right)$ boils down to geometric equivalence of the set of reduced-form VAR innovations satisfying the NR.

Proposition 4.1. Let $\left({\varphi }_0,Q_0\right)$ be the true parameter value and let $U\equiv U\left(y^T;\varphi \right)={\left({u^{\prime }}_1,\mathrm{...},{u^{\prime }}_T\right)}^{\prime }$ collect the reduced-form VAR innovations. Define

The unconditional likelihood model (Equation (24)) and the conditional likelihood model (Equation (23)) are globally identified (i.e. there are no observationally equivalent parameter points to $\left({\varphi }_0,Q_0\right)$) if and only if ${𝒬}^{*}$ is a singleton. If the parameter of interest is an impulse response to the jth structural shock, ${\eta }_{i,j,h}\left(\varphi ,Q\right)$ as defined in Equation (15), then ${\eta }_{i,j,h}\left(\varphi ,Q\right)$ is point identified if the projection of ${𝒬}^{*}$ onto its jth column vector is a singleton.

This proposition provides a necessary and sufficient condition for global identification of SVARs by NR. As shown in the proof in Appendix B, ${𝒬}^{*}$ defined in this proposition corresponds to the set of observationally equivalent values of Q given $\varphi ={\varphi }_0$, 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 bivariate model of Section 2 with the shock-sign restriction (Equation (3)), where y_t itself is the reduced-form error, so U in Proposition 4.1 can be set to y_k. Given $\varphi$, the set of $y_k\in {ℝ}^2$ satisfying the NR is the half-space

The condition for point identification shown in Proposition 4.1 is satisfied if no ${\theta }^{\prime }\ne \theta$ can generate a half-space identical to Equation (29). Such ${\theta }^{\prime }$ cannot exist, since a half-space passing through the origin $\left(a_1,a_2\right)y_k\ge 0$ can be indexed uniquely by the slope a₁/a₂ and Equation (29) implies the slope ${\sigma }_{11}^{-1}\left({\sigma }_{22}{\left(\tan \theta \right)}^{-1}-{\sigma }_{21}\right)$ is a bijective map of $\theta$ on a constrained domain due to the sign normalisation. Figure 3 plots the squared Hellinger distances in the bivariate model under the shock-sign restriction (top panel) and the historical decomposition restriction (bottom panel). For both the conditional and unconditional likelihood, the squared Hellinger distances are minimised uniquely at the true $\theta$, which is consistent with our point-identification claim for $\theta$.^[13]

Proposition 4.1 also provides conditions under which $\left(\varphi ,Q\right)$ is not globally identified, but a particular impulse response is. To give an example, consider an SVAR with n > 2 and with a shock-sign restriction on the first shock in period k. Given $\varphi$, the set of $u_k\in {ℝ}^n$ satisfying the NR is a half-space defined by ${q^{\prime }}_1{\Sigma }_{tr}^{-1}{u}_k\ge 0$. The set of values of u_k satisfying this inequality is indexed uniquely by q₁ given ${\Sigma }_{tr}$ at its true value, so there are no values of Q that are observationally equivalent to Q₀ with $q_1\ne Q_0{e}_{1,n}$. Any value for the remaining n – 1 columns of Q such that they are orthogonal to Q₀e_1,n will generate the same half-space for u_k, so ${𝒬}^{*}$ is not a singleton and the SVAR is not globally identified. However, the projection of ${𝒬}^{*}$ onto its first column is a singleton, so ${\eta }_{i,j,h}\left(\varphi ,Q\right)$ is globally identified for all i and h.

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 $S\left(\varphi ,Q\right)\ge 0_{\tilde{s}\times 1}$ set identify Q or, equivalently, the structural parameters. Given the reduced-form parameters $\varphi$ – which are point identified – there are multiple observationally equivalent values of Q, in the sense that there exists Q and $\tilde{Q}\ne Q$ such that $p\left(y^T|\varphi ,Q\right)=p\left(y^T|\varphi ,\tilde{Q}\right)$ for every y^T in the sample space. The identified set for Q given $\varphi$ contains all such observationally equivalent parameter points, and is defined as

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

The complication in applying this definition of the identified set in SVARs when there are NR is that $\varphi$ no longer represents all information about Q contained in the data; by truncating the likelihood, the realisations of the data entering the NR contain additional information about Q. To address this, we introduce a refinement of the definition of an identified set.

Definition 4.1. Let $N\equiv N\left(\varphi ,Q,y^T\right)\ge 0_{s\times 1}$ represent a set of NR in terms of the parameters and the data.

(i) The conditional identified set for Q under NR is

The conditional identified set for the impulse response $\eta ={\eta }_{i,j,h}\left(\varphi ,Q\right)$ under NR is defined by projecting $𝒬\left(\varphi |y^T,N\right)via{\eta }_{i,j,h}\left(\varphi ,Q\right)$:

(ii) Let $s:Y\to {ℝ}^S$ be a statistic. We call s(y^T) a sufficient statistic for the conditional identified set $𝒬\left(\varphi |y^T,N\right)$ if the conditional identified set for Q depends on the sample y^T through s(y^T); that is, there exists $\widetilde{𝒬}\left(\varphi |y^T,N\right)$ such that

holds for all $\varphi \in \Phi$ and $y^T\in Y$.

Unlike the standard identified set $𝒬\left(\varphi |S\right)$, the conditional identified set $𝒬\left(\varphi |y^T,N\right)$ depends on the sample 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 y^T and the reduced-form parameters $\varphi$, all values of Q in $𝒬\left(\varphi |y^T,N\right)$ fit the data equally well and, in this particular sense, they are observationally equivalent.

When the NR involve shocks in only a subset of time periods (as is typically the case), the conditional identified set depends on the sample only through the observations entering the NR, which are represented by the sufficient statistic s(y^T) in Definition 4.1(ii). For instance, in the example of Section 2.1 s(y^T) = y_k. If we extend the example to the SVAR(p), the shock-sign restriction in Equation (3) is

Hence, the conditional identified set $𝒬\left(\varphi |y^T,N\right)$ depends on the data only through ${\left({y^{\prime }}_k^{},{x^{\prime }}_k\right)}^{\prime }={\left({y^{\prime }}_k,{y^{\prime }}_{k-1},\mathrm{...},{y^{\prime }}_{k-p}\right)}^{\prime }$, so we can set $s\left(y^T\right)={\left({y^{\prime }}_k,{y^{\prime }}_{k-1},\mathrm{...},{y^{\prime }}_{k-p}\right)}^{\prime }$.

If the conditional distribution of Y^T given s(Y^T) = s(y^T) is non-degenerate, we can consider a frequentist sampling experiment (repeated sampling of Y^T) conditional on the sufficient statistics set to their observed values. We can then view the conditional identified set $𝒬\left(\varphi |y^T,N\right)$ as the standard identified set in set-identified models, since it no longer depends on the data in the conditional experiment where s(y^T) is fixed. This motivates referring to $𝒬\left(\varphi |y^T,N\right)$ as the conditional identified set.

The conditional identified set resembles the finite-sample identified set introduced by Rosen and Ura (2020) in the context of maximum score estimation (Manski 1975, 1985). Their set corresponds to the plateau of the population objective function in the conditional frequentist sampling experiment given the regressors. If we impose only the shock-sign restrictions, and given knowledge of the true data-generating processes, the construction of the conditional identified set coincides with the construction of the finite-sample identified set for the scale-normalised coefficients, as they both solve the system of inequalities in Equations (3) or (34).^[14] Despite these common geometric features, there are several differences between the SVAR under NR and maximum score estimation. First, the SVAR under NR is a likelihood-based parametric model, while maximum score estimation is a semi-parametric binary regression without a likelihood. Second, NR directly trim the support of the sample objective function (the likelihood) by the intersection of inequalities, while the maximum score objective function counts the number of inequalities satisfied in the sample. Third, the number of NR depends on the researcher's choice, while the number of inequalities in maximum score estimation is driven by the support points of the regressors observed in the sample.

This section presents approaches to conducting Bayesian 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 a flat likelihood.

5.1 Standard Bayesian inference

AR18 propose an algorithm for drawing from the uniform normal-inverse-Wishart posterior of $\left(\varphi ,Q\right)$ given traditional sign restrictions and NR. This is the posterior induced by a normal-inverse-Wishart prior for $\varphi$ and a uniform prior for Q. The algorithm draws $\varphi$ from a normal-inverse-Wishart distribution and Q from a uniform distribution over $𝒪\left(n\right)$, and checks whether the restrictions are satisfied. If not, 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 sufficient 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.^[15]

This algorithm can be interpreted as drawing from the posterior based on the unconditional likelihood and then using importance sampling to transform into draws from the posterior based on the conditional likelihood. Drawing from the posterior based on the unconditional likelihood therefore simply requires omitting the importance-sampling step. Constructing 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 computational burden.

The algorithm described above places more weight on values of $\varphi$ (relative to the notional normal-inverse-Wishart prior) that are more likely to satisfy the restrictions under the uniform distribution over $𝒪\left(n\right)$ (i.e. values with ‘larger’ conditional identified sets). As discussed in Uhlig (2017), it may instead be preferable to use a prior that is conditionally uniform over the identified set for Q. To draw from the posterior of $\left(\varphi ,Q\right)$ under the unconditional likelihood given a conditionally uniform prior for Q simply requires obtaining a fixed number of draws of Q at each draw of $\varphi$.

5.2 Robust Bayesian inference

Standard Bayesian inference based on the unconditional likelihood (or based on the conditional likelihood under shock-sign restrictions) is potentially sensitive to the choice of conditional prior for Q given $\varphi$, because the likelihood possesses flat regions. 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 an impulse response $\eta$, but the discussion applies to any other scalar-valued function of the structural parameters.

Let ${\pi }_{\varphi }$ be a prior over the reduced-form parameters $\varphi \in \Phi$, where $\Phi$ is the space of reduced-form parameters such that $𝒬\left(\varphi |S\right)$ is non-empty. A joint prior for $\left(\varphi ,Q\right)\in \Phi \times 𝒪\left(n\right)$ can be written as ${\pi }_{\varphi ,Q}={\pi }_{Q|\varphi }{\pi }_{\varphi }$, where ${\pi }_{Q|\varphi }$ is supported only on $𝒬\left(\varphi |S\right)$. When there are only traditional identifying restrictions, ${\pi }_{Q|\varphi }$ is not updated by the data, because the likelihood is not a function of 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 ${\pi }_{Q|\varphi }$ is updated by the data through the truncation points of the unconditional likelihood. However, at each value of $\varphi$, the unconditional likelihood is flat over the set of values of Q satisfying the NR. Consequently, the conditional posterior for $Q|\varphi ,Y^T$ is proportional to the conditional prior for $Q|\varphi$ at each $\varphi$ whenever the conditional identified set for Q given $\left(\varphi ,Y^T\right)$ is non-empty.

Rather than specifying a single conditional prior for Q, the robust Bayesian approach of GK21 considers the set of all conditional priors for Q that are consistent with the identifying restrictions:

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

Given a particular prior for $\left(\varphi ,Q\right)$ and using the unconditional likelihood, the posterior is

The final expression for the posterior makes it clear that any prior for $Q|\varphi$ that is consistent with the traditional identifying restrictions is in effect further truncated by the NR (through the likelihood) once the data are realised. Generating this posterior using every prior in the set of conditional priors yields a set of posteriors for $\left(\varphi ,Q\right)$:

Marginalising each posterior in this set induces a set of posteriors for $\eta ,{\Pi }_{\eta |Y^T,D_{N=1}}$. 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 ${\Pi }_{Q|\varphi }$, we can trace out the set of all possible posterior means for $\eta$. This will always be an interval, so we can summarise this ‘set of posterior means’ by its end points:

where $\ell \left(\varphi ,Y^T\right)=\inf \left\{\eta \left(\varphi ,Q\right):Q\in 𝒬\left(\varphi |Y^T,N,S\right)\right\},u\left(\varphi ,Y^T\right)$ $=\sup \left\{\eta \left(\varphi ,Q\right):Q\in 𝒬\left(\varphi |Y^T,N,S\right)\right\}$ and $𝒬\left(\varphi |Y^T,N,S\right)=\left\{\theta \left(\varphi |S\right)\cap \theta \left(\varphi |Y^T,N\right)\right\}$ is the set of values of Q that are consistent with the traditional identifying restrictions and the NR (i.e. the conditional identified set). In contrast, in GK21 the set of posterior means is obtained by finding the infimum and supremum of $\eta \left(\varphi ,Q\right)$ over $𝒬\left(\varphi |S\right)$ and averaging these over ${\pi }_{\varphi |Y^T}$. The important difference from GK21 is that the current set of posterior means depends on the data not only through the posterior for $\varphi$ but also through the conditional identified set generated by the NR. As a result, in contrast with GK21, we cannot interpret the set of posterior means (Equation (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 GK21 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 GK21, we can also report a robust credible region with credibility level $\alpha$. This 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 ${\Pi }_{\eta |Y^T,D_{N=1}}$ (see Proposition 1 of GK21). We can also report posterior lower and upper probabilities. These are the infimum and supremum, respectively, of the probability for a hypothesis over all posteriors in the set.

To numerically implement this robust Bayesian procedure, we extend the numerical algorithms in GK21 to handle NR. We approximate the bounds of the conditional identified set at each value of $\varphi$ using a simulation-based approach based on Algorithm 2 of GK21. See Appendix A for details.

This section analyses the frequentist properties of the robust Bayesian approach under NR. GK21 provide conditions under which 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 assume that the number of NR is fixed when the sample size grows, representing situations where the number of NR is ‘small’ relative to the sample size. This setting is empirically relevant given that the literature typically imposes no more than a handful of NR. The sense in which the number of NR is ‘small’ is made precise in the following assumption.

Assumption 1. (Fixed-dimensional s(Y^T)): The conditional identified set under NR has sufficient statistics s(Y^T), as defined in Definition 4.1(ii), and the dimension of s(Y^T) does not depend on T.

Let $\left({\varphi }_0,Q_0\right)$ be the true parameter values. We view the sample Y^T as being drawn from $p\left(Y^T|{\varphi }_0\right)$. Let $p\left(Y^T|s,{\varphi }_0\right)$ be the conditional distribution of the sample Y^T given the sufficient statistics for the conditional identified set s = s(Y^T) at $\varphi ={\varphi }_0$. We denote by $p\left(s|{\varphi }_0\right)$ the distribution of the sufficient statistics s(Y^T) at $\varphi ={\varphi }_0$. The next assumption assumes that in the conditional sampling experiment given s(Y^T), the sampling distribution for the maximum likelihood estimator $\widehat{\varphi }\equiv \arg {\max }_{\varphi }p\left(Y^T|\varphi \right)$ centered at ${\varphi }_0$ and the posterior for $\varphi$ centered at $\widehat{\varphi }$ asymptotically coincide. To characterise the asymptotic properties of our inference proposals, let $Y^{\infty }$ be a sequence of endogenous variables of infinite length, (y_t : t = 1,2,...), generated according to the SVAR(p) model of Equation (11). We denote its true probability law as P₀, whose marginal distribution for the first T realisations, Y^T, corresponds to $p\left(Y^T|{\varphi }_0\right)$.

Assumption 2. (Conditional Bernstein-von Mises property for $\varphi$): For $p\left(s|{\varphi }_0\right)$-almost every s and $p\left(Y^{\infty }|s,{\varphi }_0\right)$-almost every sampling sequence $Y^{\infty }$, the posterior for $\sqrt{T}\left(\varphi -\widehat{\varphi }\right)$ asymptotically coincides with the sampling distribution of $\sqrt{T}\left(\widehat{\varphi }-{\varphi }_0\right)$ under $p\left(Y^T|s,{\varphi }_0\right)$ as $T\to \infty$, in the sense stated in Assumption 5(i) in GK21.

This is a key assumption for establishing the asymptotic frequentist validity of the robust credible region under NR. It holds, for instance, when s(y^T) corresponds to one or a few observations in the whole sample, as we had in the example of Section 2.1. In this case, the influence of s(y^T) vanishes in the conditional sampling distribution of $\sqrt{T}\left(\widehat{\varphi }-{\varphi }_0\right)$ 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 $\sqrt{T}\left(\varphi -\widehat{\varphi }\right)$ 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 GK21 for set-identified models.

Assumption 3. (Almost-sure convexity and smoothness of the impulse response identified set): Let ${\widetilde{CIS}}_{\eta }\left(\varphi |s\left(Y^T\right),N\right)$ be the conditional identified set for $\eta$ with the sufficient statistics s(Y^T). For any T and $p\left(Y^T|{\varphi }_0\right)$-almost every $Y^T,{\widetilde{CIS}}_{\eta }\left(\varphi |s\left(Y^T\right),N\right)$ is closed and convex, ${\widetilde{CIS}}_{\eta }\left(\varphi |s\left(Y^T\right),N\right)=\left[\tilde{\ell }\left(\varphi ,s\left(Y^T\right)\right),\tilde{u}\left(\varphi ,s\left(Y^T\right)\right)\right]$, and its lower and upper bounds are differentiable in $\varphi$ at $\varphi ={\varphi }_0$ with non-zero derivatives.

Propositions B.1-B.3 provide primitive conditions for Assumption 3 to hold in the case where there are shock-sign restrictions. Imposing Assumptions 1, 2 and 3, we obtain the following theorem characterising the asymptotic frequentist properties of the robust credible interval under NR.

Theorem 6.1. For $\alpha \in \left(0,1\right)$, let ${\widehat{C}}_{\alpha }^{*}$ be the volume-minimising robust credible region for $\eta$ with credibility $\alpha$, ^[16] which satisfies

Under Assumptions 1, 2, and 3, ${\widehat{C}}_{\alpha }^{*}$ attains asymptotically valid coverage for the true impulse response, ${\eta }_0$, conditional on s(Y^T):

Accordingly, ${\widehat{C}}_{\alpha }^{*}$ attains asymptotically valid coverage for ${\eta }_0$ unconditionally,

This theorem shows that the robust credible region applied to the SVAR model with NR attains asymptotically valid frequentist coverage for the impulse response conditional identified set and consequently for the true impulse response. Even if the point-identification condition of Proposition 4.1 holds for the impulse response, it is not obvious that the standard (single prior) 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.