# RDP 2023-07: Identification and Inference under Narrative Restrictions 6. Frequentist Coverage

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(YT)): The conditional identified set under NR has sufficient statistics s(YT), as defined in Definition 4.1(ii), and the dimension of s(YT) does not depend on T.

Let $\left({\varphi }_{0},{Q}_{0}\right)$ be the true parameter values. We view the sample YT 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 YT given the sufficient statistics for the conditional identified set s = s(YT) at $\varphi ={\varphi }_{0}$. We denote by $p\left(s|{\varphi }_{0}\right)$ the distribution of the sufficient statistics s(YT) at $\varphi ={\varphi }_{0}$. The next assumption assumes that in the conditional sampling experiment given s(YT), the sampling distribution for the maximum likelihood estimator $\stackrel{^}{\varphi }\equiv \mathrm{arg}{\mathrm{max}}_{\varphi }p\left({Y}^{T}|\varphi \right)$ centered at ${\varphi }_{0}$ and the posterior for $\varphi$ centered at $\stackrel{^}{\varphi }$ asymptotically coincide. To characterise the asymptotic properties of our inference proposals, let ${Y}^{\infty }$ be a sequence of endogenous variables of infinite length, (yt : 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, YT, 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 $\sqrt{T}\left(\varphi -\stackrel{^}{\varphi }\right)$ asymptotically coincides with the sampling distribution of $\sqrt{T}\left(\stackrel{^}{\varphi }-{\varphi }_{0}\right)$ 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(yT) 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(yT) vanishes in the conditional sampling distribution of $\sqrt{T}\left(\stackrel{^}{\varphi }-{\varphi }_{0}\right)$ 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 $\sqrt{T}\left(\varphi -\stackrel{^}{\varphi }\right)$ 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 ${\stackrel{˜}{CIS}}_{\eta }\left(\varphi |s\left({Y}^{T}\right),N\right)$ be the conditional identified set for $\eta$ with the sufficient statistics s(YT). For any T and $p\left({Y}^{T}|{\varphi }_{0}\right)$-almost every ${Y}^{T},{\stackrel{˜}{CIS}}_{\eta }\left(\varphi |s\left({Y}^{T}\right),N\right)$ is closed and convex, ${\stackrel{˜}{CIS}}_{\eta }\left(\varphi |s\left({Y}^{T}\right),N\right)=\left[\stackrel{˜}{\ell }\left(\varphi ,s\left({Y}^{T}\right)\right),\stackrel{˜}{u}\left(\varphi ,s\left({Y}^{T}\right)\right)\right]$, 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 ${\stackrel{^}{C}}_{\alpha }^{*}$ be the volume-minimising robust credible region for $\eta$ with credibility $\alpha$, [16] which satisfies

(39) $π∈ Π ϕ,Q| Y T , D N =1 inf π( C ^ α * )= π ϕ| Y T , D N =1 ( CIS η ( ϕ| Y T ,N )⊂ C ^ α * | Y T , D N =1 )=α$

Under Assumptions 1, 2, and 3, ${\stackrel{^}{C}}_{\alpha }^{*}$ attains asymptotically valid coverage for the true impulse response, ${\eta }_{0}$, conditional on s(YT):

(40) $liminf T→∞ P Y T | s,ϕ ( η 0 ∈ C ^ α * | s( Y T ), ϕ 0 )≥ lim T→∞ P Y T | s,ϕ ( CIS ˜ η ( ϕ 0 | s( Y T ),N )⊂ C ^ α * | s( Y T ), ϕ 0 )=α$

Accordingly, ${\stackrel{^}{C}}_{\alpha }^{*}$ attains asymptotically valid coverage for ${\eta }_{0}$ unconditionally,

(35) $liminf T→∞ P Y T |ϕ ( η 0 ∈ C ^ α * | ϕ 0 )≥ lim T→∞ P Y T |ϕ ( CIS ˜ η ( ϕ 0 | s( Y T ),N )⊂ C ^ α * | ϕ 0 )=α$

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.

## Footnote

${\stackrel{^}{C}}_{\alpha }^{*}$is defined as a shortest interval among the connected intervals ${C}_{\alpha }$ satisfying ${P}_{{Y}^{T}|s,\varphi }\left({\stackrel{˜}{CIS}}_{\eta }\left({\varphi }_{0}|s\left({Y}^{T}\right),N\right)\subset {C}_{\alpha }|s\left({Y}^{T}\right),{\varphi }_{0}\right)\ge \alpha .$ See Proposition 1 in GK21 for a procedure to compute the volume-minimising credible region. [16]