RDP 2022-04: The Unit-effect Normalisation in Set-identified Structural Vector Autoregressions 4. Checking for Unboundedness in SVARs
October 2022
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Understanding how often the identified set is bounded is crucial for understanding which inferential outputs (e.g. robust credible intervals) are themselves bounded, and thus for gauging the informativeness of the identifying restrictions. This section explains how to check whether identified sets for unit impulse responses may be unbounded in the general setting of an n -dimensional SVAR with dynamics and where there are both sign and zero restrictions on the structural parameters. In this general setting, analytical expressions for identified sets are not usually available and it is necessary to approximate the bounds of the identified set numerically. In practice, I recommend that researchers compute and report the posterior probability that zero is included within the identified set for the normalising impulse response, since this tells us which inferential outputs are guaranteed to be bounded.
In the n -variable SVAR (described in Section 2), assume that the sign restrictions $S\left(\varphi ,Q\right)\ge {0}_{s\times 1}$ include the restriction that the impact response of the first variable to the first shock is non-negative, ${\eta}_{1,1,0}={{e}^{\prime}}_{1,n}{\Sigma}_{tr}{q}_{1}\ge 0.$ For example, in the context of estimating the effects of monetary policy shocks, this restriction would require that a positive monetary policy shock (the first shock) does not decrease the federal funds rate (the first variable) on impact. Such a restriction seems natural in empirical settings. The identified set for ${\tilde{\eta}}_{i,1,h},\text{\hspace{0.17em}}{\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right),$ will be unbounded only if the identified set for ${\eta}_{1,1,0},\text{\hspace{0.17em}}{\eta}_{1,1,0}\left(\varphi |S,F\right),$ includes zero (or, equivalently, ${\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right)$ is guaranteed to be bounded if ${\eta}_{1,1,0}\left(\varphi |S,F\right)$ does not contain zero).^{[13]} This will be the case if there exists Q satisfying the zero restrictions, a ‘binding’ version of the sign restriction on ${\eta}_{1,1,0}\left({{e}^{\prime}}_{1,n}{\Sigma}_{tr}{q}_{1}=0\right)$ and any remaining sign restrictions. The following proposition formalises this claim.
Proposition 4.1. (Necessary condition for unbounded ${\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right).$) Assume interest is in the impulse response to a unit shock in the first variable, ${\tilde{\eta}}_{i,1,h},$ at some fixed and finite horizon h. The identified set for ${\tilde{\eta}}_{i,1,h},{\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right),\text{\hspace{0.17em}}$ is unbounded only if $0\in {\eta}_{1,1,0}\left(\varphi |S,F\right).$
Proposition 4.1 provides a necessary condition for ${\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right)\text{\hspace{0.17em}}$ to be unbounded. Intuitively, if the identified set for ${\eta}_{1,1,0}$ does not contain zero, it is not possible to construct a sequence for Q converging to the point where ${\eta}_{1,1,0}=0$ such that ${\tilde{\eta}}_{i,1,h}$ diverges. If the identified set for ${\eta}_{1,1,0}$ includes zero, it may be possible to construct such a sequence. However, this condition does not guarantee that ${\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right)\text{\hspace{0.17em}}$ is unbounded and hence is not sufficient. To give an example, consider extending the bivariate model of Section 3 to allow for dynamics:
Assume that B_{1} is diagonal with diagonal elements diag(B_{1}) = (b_{11}, b_{22})′. When ${\sigma}_{21}\ge 0,$ the identified set for ${\eta}_{1,1,0}$ includes zero. However, the identified set for ${\tilde{\eta}}_{1,1,h}$ is ${b}_{11}^{h}$, which is finite for any b_{11} and finite h.^{[14]}
In what follows, I discuss how to check whether ${\eta}_{1,1,0}\left(\varphi |S,F\right)$ includes zero, in which case ${\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right)\text{\hspace{0.17em}}$ may be unbounded. First, consider imposing a set of sign and zero restrictions constraining q_{1} only, $S\left(\varphi ,Q\right)=S\left(\varphi \right){q}_{1}\ge {0}_{s\times 1}$ and $F\left(\varphi ,Q\right)=F\left(\varphi \right){q}_{1}={0}_{f\times 1}.$ The following proposition states a sufficient condition for the identified set for ${\eta}_{1,1,0}$ to include zero in this setting.
Proposition 4.2. (Sufficient condition for ${\eta}_{1,1,0}\left(\varphi |S,F\right)\text{\hspace{0.17em}}$ to include zero.) Assume that any sign and zero restrictions constrain q_{1} only, ${\eta}_{1,1,0}={{e}^{\prime}}_{1,n}{\Sigma}_{tr}{q}_{1}\ge 0$ is contained within the set of sign restrictions $S\left(\varphi \right){q}_{1}\ge {0}_{s\times 1}$ and the number of zero restrictions in $F\left(\varphi \right){q}_{1}={0}_{f\times 1}$ satisfies $0\le f<n-1.$ If $s+f\le n,$ then $0\in {\eta}_{1,1,0}\left(\varphi |S,F\right)\text{\hspace{0.17em}}.$
The sufficient condition in Proposition 4.2 is easily verifiable; it simply requires counting the number of identifying restrictions imposed. When the total number of identifying restrictions is no more than the number of endogenous variables in the VAR, the identifying restrictions cannot rule out the possibility that the first variable does not respond to its own shock on impact. Proposition 4.1 then implies that the identified set for a unit impulse response may potentially always be unbounded, and the identifying restrictions may be extremely uninformative about these impulse responses.
The assumption that $0\le f<n-1$ rules out point identification of q_{1} (and thus any impulse response to the first shock).^{[15]} If the set of sign restrictions were to exclude the restriction ${\eta}_{1,1,0}\ge 0,$ a sufficient condition for $0\in {\eta}_{1,1,0}\left(\varphi |S,F\right)$ would be $s+f\le n-1,$ because one could augment the sign restrictions with ${\eta}_{1,1,0}\ge 0$ and then apply Proposition 4.2. Although Proposition 4.2 only applies when the identifying restrictions constrain a single column of Q, this is the case in many empirical applications; examples include Uhlig (2005) and Arias et al (2019) (see also the references in Gafarov et al (2018)). While the condition $s+f\le n$ is unlikely to hold in applications that impose dynamic sign restrictions (i.e. sign restrictions at multiple horizons), these restrictions are not always imposed. For example, the condition is satisfied in Arias et al (2019), who identify a monetary policy shock by imposing sign and zero restrictions on elements of A_{0} (see Section 5). To identify an unconventional monetary policy shock, Gafarov et al (2018) impose four restrictions (one zero restriction and three signs restrictions) in a four-variable system. Beaudry, Nam and Wang (2011) identify an ‘optimism’ shock by imposing two restrictions (one zero restriction and one sign restriction) in a five-variable system.
When $s+f>n,$ whether it is possible to construct q_{1} satisfying ${{e}^{\prime}}_{1,n}{\Sigma}_{tr}{q}_{1}=0$ and the remaining identifying restrictions depends on the reduced-form parameters. Geometrically, the condition ${{e}^{\prime}}_{1,n}{\Sigma}_{tr}{q}_{1}=0$ and the zero restrictions are jointly satisfied when q_{1} lies in an (n – f – 1)-dimensional hyperplane that is orthogonal to ${{e}^{\prime}}_{1,n}{\Sigma}_{tr}$ and the rows of $F\left(\varphi \right)$, while the remaining sign restrictions require q_{1} to lie within the intersection of s – 1 half-spaces. The identified set for ${\eta}_{1,1,0}$ will include zero if and only if the intersection of this hyperplane and these half-spaces is non-empty. When s + f > n, the hyperplane and half-spaces are not guaranteed to intersect; whether they intersect depends on the values of the reduced-form parameters, which determine the orientations of the hyperplane and half-spaces.
As a simple example of applying Proposition 4.2, consider the bivariate model of Section 3. Here, s = 3 > 2 = n, so the condition in Proposition 4.2 is not satisfied and zero is not necessarily included within the identified set for ${\eta}_{1,1,0}$; in particular, zero is excluded when ${\sigma}_{21}<0.$ Removing the restriction ${\eta}_{2,1,0}\le 0$ means that s = n = 2, so the condition in Proposition 4.2 is satisfied and the identified set for ${\eta}_{1,1,0}$ includes zero regardless of the sign of ${\sigma}_{21}<0.$ Geometrically, when s = n, the intersection of the half-spaces generated by the sign restrictions always includes the boundary of the half-space representing the restriction ${\eta}_{1,1,0}\ge 0$ (i.e. the hyperplane ${\eta}_{1,1,0}={{e}^{\prime}}_{1,n}{\Sigma}_{tr}{q}_{1}=0$). Graphically, one can see this by deleting the line ‘SR2’ in Figure 1.
When the conditions in Proposition 4.2 do not hold (i.e. when s + f > n or the identifying restrictions constrain multiple columns of Q), it is necessary to check whether ${\eta}_{1,1,0}\left(\varphi |S,F\right)$ includes zero to determine whether ${\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right)\text{\hspace{0.17em}}$ may be unbounded. The following proposition formulates a necessary and sufficient condition for ${\eta}_{1,1,0}\left(\varphi |S,F\right)$ to include zero. I subsequently discuss how to check this condition in practice.
Proposition 4.3. (Necessary and sufficient condition for $0\in {\eta}_{1,1,0}\left(\varphi |S,F\right)$.) Let $\tilde{F}\left(\varphi ,Q\right)={0}_{\left(f+1\right)\times 1}$ represent an augmented set of zero restrictions that includes a ‘binding’ version of the sign restriction ${\eta}_{1,1,0}\ge 0(i.e.\text{\hspace{0.17em}}{{e}^{\prime}}_{1,n}{\Sigma}_{tr}{q}_{\text{1}}=0)$ with $0\le f<n-1$ and let $\tilde{S}\left(\varphi ,Q\right)\ge {0}_{\left(s-1\right)\times 1}$ collect the remaining sign restrictions. The identified set for ${\eta}_{1,1,0}$ includes zero if and only if the identified set for Q given the augmented set of restrictions, $Q\left(\varphi |\tilde{S},\tilde{F}\right),$ is non-empty.
The rationale underlying Proposition 4.3 is straightforward, so I omit a formal proof. If $Q\left(\varphi |\tilde{S},\tilde{F}\right)$ is non-empty, there exists a value of Q satisfying the identifying restrictions (i.e. within $Q\left(\varphi |S,F\right)$ ) such that ${\eta}_{1,1,0}=0.$ If $Q\left(\varphi |\tilde{S},\tilde{F}\right)$ is empty, there cannot be a value of Q within $Q\left(\varphi |S,F\right)$ such that ${\eta}_{1,1,0}=0$. An implication of the proposition is that one can check whether ${\eta}_{1,1,0}\left(\varphi |S,F\right)$ contains zero by applying existing numerical algorithms to check whether $Q\left(\varphi |\tilde{S},\tilde{F}\right)$ is non-empty. For example, in the case where there are sign and zero restrictions constraining q_{1} only, Algorithm 4.1 in Read (2022) can be applied.^{[16]} In the general case where the identifying restrictions (potentially nonlinearly) constrain multiple columns of Q, one can check whether $Q\left(\varphi |\tilde{S},\tilde{F}\right)$ is non-empty by drawing from a uniform distribution over $Q\left(\varphi |\tilde{F}\right)$ (e.g. using the algorithms in Arias et al (2018) or Giacomini and Kitagawa (2021)) until a draw is obtained satisfying the remaining sign restrictions. If no such draw can be obtained after a large number of draws, this suggests that $Q\left(\varphi |\tilde{S},\tilde{F}\right)$ is empty, in which case ${\tilde{\eta}}_{i,1,h}\left(\varphi |S,F\right)$ must be bounded.
Given draws of $\varphi $ from its posterior distribution and having checked whether ${\eta}_{1,1,0}\left(\varphi |S,F\right)$ contains zero at each draw, one can determine whether different inferential outputs are guaranteed to be bounded. Remark 4.1 relates the posterior probability that ${\eta}_{1,1,0}\left(\varphi |S,F\right)$ includes zero to the boundedness of different summaries of the robust Bayesian class of posteriors for ${\tilde{\eta}}_{i,1,h}$.
Remark 4.1. Let $\alpha ={\pi}_{\varphi |Y}\left(\left\{\varphi :{\eta}_{1,1,0}\left(\varphi |S,F\right)\cap \left\{0\right\}=\varnothing \right\}\right)$ be the posterior probability that ${\eta}_{1,1,0}\left(\varphi |S,F\right)$ excludes zero (in which case the identified set for ${\tilde{\eta}}_{i,1,h}$ is guaranteed to be bounded by Proposition 4.1) and assume the parameter of interest is ${\tilde{\eta}}_{i,1,h}$ Let $\tau \in \left(0,1\right).$ Then:
- if $\alpha =1,$ the set of posterior means is bounded;
- if $\alpha \ge \tau ,$ the set of posterior $\tau $ -quantiles is bounded; and
- if $\alpha \ge 1-\tau /2,$ the robust credible interval with credibility $1-\tau $, constructed by taking the $\tau /2$ quantile of $\ell \left(\varphi \right)$ and the $1-\tau /2$ quantile of $u\left(\varphi \right)$, is bounded.
In practice, I recommend that researchers report $\alpha ,$ since doing so makes it clear which inferential outputs are guaranteed to be bounded, and thus is important for understanding the informativeness of the identifying restrictions. For example, if $\alpha =0.1,$ the 68 per cent robust credible interval is guaranteed to be bounded, whereas the 90 per cent robust credible is not necessarily bounded. In this example, reporting a 68 per cent (robust) credible interval (as is common in the literature using set-identified SVARs) may be misleading about the informativeness of the restrictions; presumably, some researchers would be interested in knowing that there is potentially non-trivial posterior probability assigned to infinitely large responses.
Footnotes
I abstract from the possibility of imposing sign restrictions with strict inequality (i.e. $S\left(\varphi \right)>{0}_{s\times 1}$). In that case, identified sets will be open intervals (as noted in Section 3.1). Consequently, the identified set for ${\tilde{\eta}}_{i,1,h}$ could be unbounded without the identified set for ${\eta}_{1,1,0}$ including zero. When allowing for strict inequalities, the identified set for ${\tilde{\eta}}_{i,1,h}$ will be unbounded only if the closure of the identified set for ${\eta}_{1,1,0}$ includes zero. [13]
I thank Thorsten Drautzburg for suggesting this example. [14]
When rank $\left(F\left(\varphi \right)\right)=n-1,$ the identified set for ${\eta}_{1,1,0}$ (which, given the sign normalisation, is a singleton when non-empty) excludes zero if rank $\left(\tilde{F}\left(\varphi \right)\right)=n.$ This condition would be violated in the (unrealistic) instance where the zero restrictions in $F\left(\varphi \right)$ include the restriction that ${\eta}_{1,1,0}=0.$ Note that the condition $0\le f<n-1$ implicitly rules out the possibility that n = 1, in which case the impulse responses would be trivially point identified. [15]
If f = n – 2 and rank $\left(\tilde{F}\left(\varphi \right)\right)=n-1,$ the unit-length vector ${\tilde{q}}_{1}$ satisfying $\tilde{F}\left(\varphi \right){\tilde{q}}_{1}={0}_{\left(f+1\right)\times 1}$ is pinned down up to sign; such a vector can be found by computing an orthonormal basis for the null space of $\tilde{F}\left(\varphi \right),\text{\hspace{0.17em}}N\left(\tilde{F}\left(\varphi \right)\right).$ If either $\tilde{S}\left(\varphi \right)N\left(\tilde{F}\left(\varphi \right)\right)\ge {0}_{\left(s-1\right)\times 1}$ or $\text{\hspace{0.17em}}-\tilde{S}\left(\varphi \right)N\left(\tilde{F}\left(\varphi \right)\right)\ge {0}_{\left(s-1\right)\times 1},$ then $Q\left(\varphi |\tilde{S},\tilde{F}\right)$ is non-empty. For $0\le f<n-2,$ Algorithm 4.1 in Read (2022) can be applied. [16]