RDP 2022-04: The Unit-effect Normalisation in Set-identified Structural Vector Autoregressions Appendix B: Proofs of Propositions
October 2022
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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 ${q}_{1}\in {\mathbb{R}}^{n}$ 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 ${\eta}_{1,1,0}$ $\left(\text{i}\text{.e}\text{.}\text{\hspace{0.17em}}{{e}^{\prime}}_{1,n}{\Sigma}_{tr}{q}_{1}=0\right)$, 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 $\text{rank}\left(G\left(\varphi \right)\right)=n,\text{\hspace{0.17em}}\text{rank}\left(\tilde{F}\left(\varphi \right)\right)=f+1.$ 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 ${\mathbb{R}}^{n-f-1}$ . Let $\stackrel{\u2323}{S}\left(\varphi \right){\stackrel{\u2323}{q}}_{1}\ge {0}_{\left(s-1\right)\times 1}$ represent the transformed sign restrictions, where ${\stackrel{\u2323}{q}}_{1}\in {\mathbb{R}}^{n-f-1}$ and $\stackrel{\u2323}{S}\left(\varphi \right)$ is obtained from $\stackrel{\u2323}{S}\left(\varphi \right)$ using the transformation described in Read. Corollary 4.1 of Read states that the set $\left\{{\tilde{q}}_{1}\in {\mathbb{R}}^{n}:\tilde{F}\left(\varphi \right){\tilde{q}}_{1}={0}_{\left(f+1\right)\times 1},\tilde{S}\left(\varphi \right){\tilde{q}}_{1}\ge {0}_{\left(s-1\right)\times 1}\right\}$ will be non-empty if and only if the set $\left\{{\stackrel{\u2323}{q}}_{1}\in {\mathbb{R}}^{n-f-1}:\stackrel{\u2323}{S}\left(\varphi \right){\stackrel{\u2323}{q}}_{1}\ge {0}_{\left(s-1\right)\times 1}\right\}$ is non-empty, in which case $0\in {\eta}_{1,1,0}\left(\varphi |S,F\right).$ I proceed by showing that the set $\left\{{\stackrel{\u2323}{q}}_{1}\in {\mathbb{R}}^{n-f-1}:\stackrel{\u2323}{S}\left(\varphi \right){\stackrel{\u2323}{q}}_{1}\ge {0}_{\left(s-1\right)\times 1}\right\}$ is always non-empty.
If $\text{rank}\left(\stackrel{\u2323}{S}\left(\varphi \right)\right)<s-1=n-f-1,$ then $N\left(\stackrel{\u2323}{S}\left(\varphi \right)\right)$ has dimension $n-f-1-\text{rank}\left(\stackrel{\u2323}{S}\left(\varphi \right)\right)\ge 1,$ so it is always possible to construct ${\stackrel{\u2323}{q}}_{1}\in {\mathbb{R}}^{n-f-1}$ satisfying $\stackrel{\u2323}{S}\left(\varphi \right){\stackrel{\u2323}{q}}_{1}={0}_{\left(s-1\right)\times 1}$ by taking any column of $N\left(\stackrel{\u2323}{S}\left(\varphi \right)\right)$, and the set $\left\{{\stackrel{\u2323}{q}}_{1}\in {\mathbb{R}}^{n-f-1}:\stackrel{\u2323}{S}\left(\varphi \right){\stackrel{\u2323}{q}}_{1}\ge {0}_{\left(s-1\right)\times 1}\right\}$ is non-empty. If $\text{rank}\left(\stackrel{\u2323}{S}\left(\varphi \right)\right)=s-1=n-f-1,$ there cannot exist $x>{0}_{\left(n-f-1\right)\times 1}$ such that $\stackrel{\u2323}{S}{\left(\varphi \right)}^{\prime}x={0}_{\left(n-f-1\right)\times 1}$ (since $\stackrel{\u2323}{S}\left(\varphi \right)$ has full rank), so by Gordan's Theorem (e.g. Mangasarian 1994; Border 2020) there must exist ${\stackrel{\u2323}{q}}_{1}\in {\mathbb{R}}^{n-f-1}$ such that $\stackrel{\u2323}{S}\left(\varphi \right){\stackrel{\u2323}{q}}_{1}>{0}_{\left(s-1\right)\times 1}$, so the set $\left\{{\stackrel{\u2323}{q}}_{1}\in {\mathbb{R}}^{n-f-1}:\stackrel{\u2323}{S}\left(\varphi \right){\stackrel{\u2323}{q}}_{1}\ge {0}_{\left(s-1\right)\times 1}\right\}$ 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]