RDP 2022-04: The Unit-effect Normalisation in Set-identified Structural Vector Autoregressions Appendix B: Proofs of Propositions

Proof of Proposition 4.1. Assume $0\notin {\eta }_{1,1,0}\left(\varphi |S,F\right),$ so there exists $\delta >0$ such that ${\eta }_{1,1,0}\left(\varphi ,Q\right)>\delta$ for all $Q\in Q\left(\varphi |S,F\right).$ Given that the impulse response horizon h is fixed and finite, $\left|{\eta }_{i,1,h}\left(\varphi ,Q\right)\right|<\infty$ for all $Q\in Q\left(\varphi |S,F\right)$.^[26] There thus exists $\kappa <\infty$ such that $\left|{\eta }_{i,1,h}\left(\varphi ,Q\right)\right|<\kappa$ for all $Q\in Q\left(\varphi |S,F\right).$ It follows that $\left|{\tilde{\eta }}_{i,1,h}\left(\varphi ,Q\right)\right|<\frac{\kappa }{\delta }<\infty$ for all $Q\in Q\left(\varphi |S,F\right)$, so ${\tilde{\eta }}_{i,1,h}\left(\varphi |S,F\right)$ is bounded.

Proof of Proposition 4.2. Assume that the sign restrictions are ordered such that the first row of $S\left(\varphi \right)$ corresponds to the sign restriction ${\eta }_{1,1,0}={e^{\prime }}_{1,n}{\Sigma }_{tr}{q}_1\ge 0.$ Let $G\left(\varphi \right)={\left(S{\left(\varphi \right)}^{\prime },F{\left(\varphi \right)}^{\prime }\right)}^{\prime }$ collect the coefficient vectors of the sign and zero restrictions.

First, consider the case where $\text{rank}\left(G\left(\varphi \right)\right)<n$ and let $N\left(G\left(\varphi \right)\right)$ represent an orthonormal basis for the null space of $G\left(\varphi \right)$. By the rank-nullity theorem, $N\left(G\left(\varphi \right)\right)$ has dimension $n-\text{rank}\left(G\left(\varphi \right)\right)\ge 1.$ Thus, it is always possible to construct $q_1\in {ℝ}^n$ satisfying $G\left(\varphi \right)q_1=0_{\left(s+f\right)\times 1}$ by taking any column of $N\left(G\left(\varphi \right)\right)$. Such a vector clearly satisfies the identifying restrictions with ${\eta }_{1,1,0}=0$, so $0\in {\eta }_{1,1,0}\left(\varphi |S,F\right)$.

Now, consider the case where $\text{rank}\left(G\left(\varphi \right)\right)=n.$ Let $\tilde{F}\left(\varphi \right)={\left({\left({e^{\prime }}_{1,n}{\Sigma }_{tr}\right)}^{\prime },F{\left(\varphi \right)}^{\prime }\right)}^{\prime }$ represent the coefficients of the zero restrictions augmented with a ‘binding’ version of the sign restriction on ${\eta }_{1,1,0}$ $\left(\text{i}\text{.e}\text{.}{e^{\prime }}_{1,n}{\Sigma }_{tr}{q}_1=0\right)$, and let $\tilde{S}\left(\varphi \right)$ represent the coefficients of the remaining sign restrictions (i.e. the last s – 1 rows of $S\left(\varphi \right)$). Since $\text{rank}\left(G\left(\varphi \right)\right)=n,\text{rank}\left(\tilde{F}\left(\varphi \right)\right)=f+1.$ Proposition 4.1 in Read (2022) states that the system of zero and sign restrictions, $\tilde{F}\left(\varphi \right){\tilde{q}}_1=0_{\left(f+1\right)\times 1}$ and $\tilde{S}\left(\varphi \right){\tilde{q}}_1=0_{\left(s-1\right)\times 1}$, can be transformed into a set of sign restrictions in ${ℝ}^{n-f-1}$ . Let $\stackrel{⌣}{S}\left(\varphi \right){\stackrel{⌣}{q}}_1\ge 0_{\left(s-1\right)\times 1}$ represent the transformed sign restrictions, where ${\stackrel{⌣}{q}}_1\in {ℝ}^{n-f-1}$ and $\stackrel{⌣}{S}\left(\varphi \right)$ is obtained from $\stackrel{⌣}{S}\left(\varphi \right)$ using the transformation described in Read. Corollary 4.1 of Read states that the set $\left\{{\tilde{q}}_1\in {ℝ}^n:\tilde{F}\left(\varphi \right){\tilde{q}}_1=0_{\left(f+1\right)\times 1},\tilde{S}\left(\varphi \right){\tilde{q}}_1\ge 0_{\left(s-1\right)\times 1}\right\}$ will be non-empty if and only if the set $\left\{{\stackrel{⌣}{q}}_1\in {ℝ}^{n-f-1}:\stackrel{⌣}{S}\left(\varphi \right){\stackrel{⌣}{q}}_1\ge 0_{\left(s-1\right)\times 1}\right\}$ is non-empty, in which case $0\in {\eta }_{1,1,0}\left(\varphi |S,F\right).$ I proceed by showing that the set $\left\{{\stackrel{⌣}{q}}_1\in {ℝ}^{n-f-1}:\stackrel{⌣}{S}\left(\varphi \right){\stackrel{⌣}{q}}_1\ge 0_{\left(s-1\right)\times 1}\right\}$ is always non-empty.

If $\text{rank}\left(\stackrel{⌣}{S}\left(\varphi \right)\right)<s-1=n-f-1,$ then $N\left(\stackrel{⌣}{S}\left(\varphi \right)\right)$ has dimension $n-f-1-\text{rank}\left(\stackrel{⌣}{S}\left(\varphi \right)\right)\ge 1,$ so it is always possible to construct ${\stackrel{⌣}{q}}_1\in {ℝ}^{n-f-1}$ satisfying $\stackrel{⌣}{S}\left(\varphi \right){\stackrel{⌣}{q}}_1=0_{\left(s-1\right)\times 1}$ by taking any column of $N\left(\stackrel{⌣}{S}\left(\varphi \right)\right)$, and the set $\left\{{\stackrel{⌣}{q}}_1\in {ℝ}^{n-f-1}:\stackrel{⌣}{S}\left(\varphi \right){\stackrel{⌣}{q}}_1\ge 0_{\left(s-1\right)\times 1}\right\}$ is non-empty. If $\text{rank}\left(\stackrel{⌣}{S}\left(\varphi \right)\right)=s-1=n-f-1,$ there cannot exist $x>0_{\left(n-f-1\right)\times 1}$ such that $\stackrel{⌣}{S}{\left(\varphi \right)}^{\prime }x=0_{\left(n-f-1\right)\times 1}$ (since $\stackrel{⌣}{S}\left(\varphi \right)$ has full rank), so by Gordan's Theorem (e.g. Mangasarian 1994; Border 2020) there must exist ${\stackrel{⌣}{q}}_1\in {ℝ}^{n-f-1}$ such that $\stackrel{⌣}{S}\left(\varphi \right){\stackrel{⌣}{q}}_1>0_{\left(s-1\right)\times 1}$, so the set $\left\{{\stackrel{⌣}{q}}_1\in {ℝ}^{n-f-1}:\stackrel{⌣}{S}\left(\varphi \right){\stackrel{⌣}{q}}_1\ge 0_{\left(s-1\right)\times 1}\right\}$ is non-empty.

Footnote

Allowing for arbitrarily large impulse response horizons h would require restricting the support of the reduced-form parameter space $\Phi$ such that the infinite-order vector moving average representation of the VAR exists; this will be the case if the eigenvalues of the companion matrix lie inside the unit circle (e.g. Hamilton 1994; Kilian and Lütkepohl 2017). By avoiding this assumption I allow for mildly explosive processes. [26]