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Fast Posterior Sampling in Tightly Identified SVARs Using 'Soft' Sign Restrictions

Matthew Read, Dan Zhu

Abstract

We propose algorithms for conducting Bayesian inference in structural vector autoregressions identified using sign restrictions. The key feature of our approach is a sampling step based on 'soft' sign restrictions. This step draws from a target density that smoothly penalises parameter values that violate the restrictions, facilitating the use of computationally efficient Markov chain Monte Carlo sampling algorithms. An importance-sampling step yields draws conditional on the 'hard' sign restrictions. Relative to standard accept-reject sampling, the method substantially speeds up sampling when identification is tight. It also facilitates implementing prior-robust Bayesian methods. We illustrate the broad applicability of the approach in an oil-market model identified using a rich set of sign, elasticity and narrative restrictions.

Fast Posterior Sampling in Tightly Identified SVARs Using 'Soft' Sign Restrictions

Abstract

We propose algorithms for conducting Bayesian inference in structural vector autoregressions identified using sign restrictions. The key feature of our approach is a sampling step based on 'soft' sign restrictions. This step draws from a target density that smoothly penalises parameter values that violate the restrictions, facilitating the use of computationally efficient Markov chain Monte Carlo sampling algorithms. An importance-sampling step yields draws conditional on the 'hard' sign restrictions. Relative to standard accept-reject sampling, the method substantially speeds up sampling when identification is tight. It also facilitates implementing prior-robust Bayesian methods. We illustrate the broad applicability of the approach in an oil-market model identified using a rich set of sign, elasticity and narrative restrictions.

Paper Structure

This paper contains 32 sections, 2 theorems, 43 equations, 9 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Framework
  3. SVAR and orthogonal reduced form
  4. Identifying restrictions
  5. Sampling problem
  6. Algorithms
  7. Accept-reject sampling
  8. Soft sign restrictions
  9. Slice sampling
  10. Advantages of 'softening'
  11. Empty identified sets
  12. Numerical Illustrations
  13. Connected identified set
  14. Disconnected identified set
  15. Empirical Application: Shocks in the Oil Market
  16. ...and 17 more sections

Key Result

Proposition 1

Assume the conditions in Assumption ass1 hold and let $T:\mathbb{R}^d\rightarrow \mathbb{R}$ be such that $\int_{\mathbb{R}^d}|T(\mathbf{Z})|f_Z(\mathbf{Z})d\mathbf{Z}<\infty$. Then, where $\mathbb{E}_f(.)$ and $\mathbb{E}_\Delta(.)$ are expectations taken under $f$ and $f_\Delta$, respectively.

Figures (9)

  • Figure 1: Illustration of Sampling Using Soft Sign Restrictions
  • Figure 2: Illustration of Sampling Using Soft Sign Restrictions -- Disconnected Identified Set
  • Figure 3: Impulse Responses to Oil Market Shocks -- Standard Bayesian Inference
  • Figure 4: Comparison of Conditional Posterior Approximations Across Samplers
  • Figure 5: Impulse Responses to Oil Market Shocks -- Comparison of Standard and Robust Bayesian Inference
  • ...and 4 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • proof
  • proof