RDP 2025-03: Fast Posterior Sampling in Tightly Identified SVARs Using ‘Soft’ Sign Restrictions Appendix A: Proofs

Proof of Proposition 1. The assumption on T ensures that ${𝔼}_f\left[T\left(Z\right)\right]$ and ${𝔼}_{\text{Δ}}\left[T\left(Z\right)\right]$ exist. We can write

Denote the normalising constants for $f\left(Z|Q\left(Z\right)\in 𝒬\left(\varphi |S\right)\right)$ and $f_{\Delta }\left(Z\right)$ by C_f and $C_{\text{Δ}}$, respectively. Without loss of generality, assume s = 1 so there is a single sign restriction $S\left(\varphi ,Q\right)\ge 0$. The right-hand side of Equation (A1) can be written as

Under Assumption 1,

The first term in Equation (A2) therefore goes to zero as $\text{Δ}\to 0$ by the monotone convergence theorem. In the second term of Equation (A2),

which similarly goes to zero.