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Semiparametric Bayesian Inference for a Conditional Moment Equality Model

Christopher D. Walker

Abstract

I propose a semiparametric Bayesian inference framework for conditional moment equalities. The core idea is that these models deterministically map a conditional distribution of data to a structural parameter via the restriction that a conditional expectation equals zero. Consequently, a posterior for the conditional distribution leads to a posterior for the structural parameter by minimizing the distance of the conditional moments to zero. The method has similar flexibility to frequentist semiparametric estimators and does not require converting the conditional moments into unconditional moments. I also establish frequentist asymptotic optimality of my proposal via a semiparametric Bernsteinvon Mises theorem (BvM), which establishes that the posterior for the structural parameter is asymptotically normal and matches the Chamberlain (1987) semiparametric efficiency bound. The BvM conditions are verified for Gaussian process priors and complement the numerical aspects of the paper in which these priors are used to estimate welfare effects.

Semiparametric Bayesian Inference for a Conditional Moment Equality Model

Abstract

I propose a semiparametric Bayesian inference framework for conditional moment equalities. The core idea is that these models deterministically map a conditional distribution of data to a structural parameter via the restriction that a conditional expectation equals zero. Consequently, a posterior for the conditional distribution leads to a posterior for the structural parameter by minimizing the distance of the conditional moments to zero. The method has similar flexibility to frequentist semiparametric estimators and does not require converting the conditional moments into unconditional moments. I also establish frequentist asymptotic optimality of my proposal via a semiparametric Bernsteinvon Mises theorem (BvM), which establishes that the posterior for the structural parameter is asymptotically normal and matches the Chamberlain (1987) semiparametric efficiency bound. The BvM conditions are verified for Gaussian process priors and complement the numerical aspects of the paper in which these priors are used to estimate welfare effects.

Paper Structure

This paper contains 41 sections, 16 theorems, 53 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Literature
  4. Outline
  5. Bayesian Inference Framework
  6. Semiparametric Bayesian Inference
  7. A Flexible Class of Priors
  8. Empirical Illustration: Estimating Welfare Effects
  9. Dataset and Target Parameter
  10. Semiparametric Bayesian Inference
  11. Semiparametric Bernstein-von Mises Theorem
  12. Assumptions about the Data-Generating Process (DGP)
  13. Assumptions about the Prior
  14. Bernstein-von Mises Theorem
  15. Main Result
  16. ...and 26 more sections

Key Result

Theorem 1

If Assumptions as:uemfp, as:dgp2, as:correctspec, as:moments, as:covariance, as:criterion, as:localident, and as:concentration.1--as:concentration.5 hold, then, for $P_{0,Z}^{\infty}$-almost every fixed realization $\{z_{i}\}_{i \geq 1}$ of $\{Z_{i}\}_{i \geq 1}$, as $n\rightarrow \infty$, where $\{\tilde{\mathcal{P}}_{n,W|Z}\}_{n \geq 1}$ are the sets defined in Assumption as:concentration.

Figures (1)

  • Figure 1: Deadweight Loss Posterior Densities

Theorems & Definitions (31)

  • Example 1
  • Remark 1: Other Minimum Distance Estimands
  • Remark 2: Prior on the Structural Parameter
  • Remark 3: Extensions
  • Definition 1
  • Example 2
  • Remark 4: Other Priors
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • ...and 21 more