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Projection Inference for set-identified SVARs

Bulat Gafarov, Matthias Meier, José Luis Montiel Olea

Abstract

We study the properties of the classical \emph{projection} method to conduct simultaneous inference about the coefficients of the structural impulse-response function and their identified set in Structural Vector Autoregressions. We show that -- as the sample size grows large -- projection inference produces regions for the structural parameters and their identified set with both frequentist coverage and robust Bayesian credibility of at least $1-α$. We then calibrate the radius of the Wald ellipsoid to guarantee that -- for a given posterior on the reduced-form parameters -- the robust Bayesian credibility of the projection method is exactly $1-α$. We illustrate the main results of the paper using a demand/supply model of the U.S.~labor market.

Projection Inference for set-identified SVARs

Abstract

We study the properties of the classical \emph{projection} method to conduct simultaneous inference about the coefficients of the structural impulse-response function and their identified set in Structural Vector Autoregressions. We show that -- as the sample size grows large -- projection inference produces regions for the structural parameters and their identified set with both frequentist coverage and robust Bayesian credibility of at least . We then calibrate the radius of the Wald ellipsoid to guarantee that -- for a given posterior on the reduced-form parameters -- the robust Bayesian credibility of the projection method is exactly . We illustrate the main results of the paper using a demand/supply model of the U.S.~labor market.

Paper Structure

This paper contains 20 sections, 4 theorems, 65 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Overview and Related Literature
  3. Overview
  4. Related Literature
  5. Basic Model, Main Assumptions, and Frequentist Results
  6. Model
  7. Assumptions for frequentist inference
  8. Main result concerning frequentist inference
  9. Robust Bayesian Credibility
  10. Calibrated Projection for a Robust Bayesian
  11. Implementation of Baseline and Calibrated Projection
  12. Projection as a mathematical optimization problem
  13. Implementing baseline projection in an example
  14. Results of the implementation of baseline projection
  15. Implementing calibrated projection in our example
  16. ...and 5 more sections

Key Result

Theorem 3.1

Consider the projection region for the collection of structural coefficients $\lambda^H \equiv \{\lambda_{k_h,i_h,j_h}\}_{h=1}^{H}$ given by: where and $CS_{T}(1-\alpha; \mu)$ is the $1-\alpha$ Wald confidence ellipsoid for $\mu$. If the class of data generating processes $\mathcal{P}$ satisfies Assumption ass:A1, then: That is, the projected confidence interval in (equation:ProjectionCSJoint)

Figures (5)

  • Figure 1: 68% Projection Region and 68% Credible Set.
  • Figure 2: 68% Projection Region and 68% Calibrated Projection.
  • Figure 3: Accuracy of SQP/IP for a demand shock
  • Figure 4: Simulation error in Projection region.
  • Figure 5: 68% Projection Region and 68% Credible Set.

Theorems & Definitions (11)

  • Definition 1: Coefficients of the Structural IRF
  • Definition 2: Identified Set and its bounds
  • Theorem 3.1: Frequentist Coverage of Projection Inference for $\lambda^{H}$
  • proof
  • Theorem 4.2: Asymptotic Robust Bayesian Credibility of Projection
  • proof
  • Theorem 5.3: Calibration of Robust Credibility
  • proof
  • Definition 3
  • Lemma 2.1
  • ...and 1 more