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Generalized Bayes in Conditional Moment Restriction Models

Sid Kankanala

TL;DR

The paper develops a generalized Bayes framework for nonparametric conditional moment restriction models with endogenous variables, using Gaussian process priors to regularize the infinite-dimensional structural function $h_0$. A quasi-likelihood based on $m(W,h)=\mathbb{E}[\rho(Y,h(X))|W]$ yields a quasi-Bayes posterior, for which the authors establish posterior contraction and a nonparametric Bernstein--von Mises theorem, guaranteeing exact frequentist coverage for optimally weighted credible sets. The methodology is demonstrated through a Chilean production-function application, where a fully nonparametric $F(\cdot)$ exhibits economically meaningful patterns such as inverted-U marginal product of capital and input interactions; simulations further show favorable performance relative to NPIV/NPQIV and random forests, especially in ill-posed, multivariate settings. The results provide a broadly applicable, data-driven framework for regularized inference in nonlinear conditional moment models with infinite-dimensional parameters.

Abstract

This paper develops a generalized (quasi-) Bayes framework for conditional moment restriction models, where the parameter of interest is a nonparametric structural function of endogenous variables. We establish contraction rates for a class of Gaussian process priors and provide conditions under which a Bernstein-von Mises theorem holds for the quasi-Bayes posterior. Consequently, we show that optimally weighted quasi-Bayes credible sets achieve exact asymptotic frequentist coverage, extending classical results for parametric GMM models. As an application, we estimate firm-level production functions using Chilean plant-level data. Simulations illustrate the favorable performance of generalized Bayes estimators relative to common alternatives.

Generalized Bayes in Conditional Moment Restriction Models

TL;DR

The paper develops a generalized Bayes framework for nonparametric conditional moment restriction models with endogenous variables, using Gaussian process priors to regularize the infinite-dimensional structural function $h_0$. A quasi-likelihood based on $m(W,h)=\mathbb{E}[\rho(Y,h(X))|W]$ yields a quasi-Bayes posterior, for which the authors establish posterior contraction and a nonparametric Bernstein--von Mises theorem, guaranteeing exact frequentist coverage for optimally weighted credible sets. The methodology is demonstrated through a Chilean production-function application, where a fully nonparametric $F(\cdot)$ exhibits economically meaningful patterns such as inverted-U marginal product of capital and input interactions; simulations further show favorable performance relative to NPIV/NPQIV and random forests, especially in ill-posed, multivariate settings. The results provide a broadly applicable, data-driven framework for regularized inference in nonlinear conditional moment models with infinite-dimensional parameters.

Abstract

This paper develops a generalized (quasi-) Bayes framework for conditional moment restriction models, where the parameter of interest is a nonparametric structural function of endogenous variables. We establish contraction rates for a class of Gaussian process priors and provide conditions under which a Bernstein-von Mises theorem holds for the quasi-Bayes posterior. Consequently, we show that optimally weighted quasi-Bayes credible sets achieve exact asymptotic frequentist coverage, extending classical results for parametric GMM models. As an application, we estimate firm-level production functions using Chilean plant-level data. Simulations illustrate the favorable performance of generalized Bayes estimators relative to common alternatives.

Paper Structure

This paper contains 33 sections, 15 theorems, 193 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature
  3. Models and Procedures
  4. Framework
  5. Gaussian process priors
  6. First stage estimation
  7. Motivation
  8. Simulation Evidence
  9. Theory
  10. Assumptions on the Generalized Residual
  11. Consistency
  12. Contraction Rates
  13. Inference
  14. Simulations
  15. Results
  16. ...and 18 more sections

Key Result

Theorem 1

Suppose Conditions residuals-fsbasis hold and $h_0 \in L^2(\mathbb{P})$ is the unique structural function that satisfies $\mathbb{E}( \| m(W,h_0) \|_{\ell^2}^2) = 0$. Let $K = K_n \rightarrow \infty$ denote any sequence that satisfies $n^{d/2(\alpha + d)} \lessapprox K_n$ and $\log(n) K_n = o(n)$.

Figures (7)

  • Figure 1: Sample size $n = 1000$. NPIV posterior for the design in santos2012inference. The red dashed line shows the true function, the dark blue solid line is the posterior mean, and the light blue lines are posterior draws.
  • Figure 2: Scatter plot of true vs. predicted values for the multivariate NP design. Quasi-Bayes (NPIV) predictions. The red 45$^\circ$ line denotes perfect prediction ($\text{True} = \text{Predicted}$).
  • Figure 3: Scatter plot of true vs. predicted values for the multivariate NP design. Random forest (OLS) predictions. The red 45$^\circ$ line denotes perfect prediction ($\text{True} = \text{Predicted}$).
  • Figure 4: Estimated production function $\widehat{F}(k,l)$ at selected labor quantiles.
  • Figure 5: Estimated marginal product $\partial_k \widehat{F}(k,l)$ at the 0.75 labor quantile, as a function of log capitak $k$, illustrating the classical inverted-U pattern.
  • ...and 2 more figures

Theorems & Definitions (44)

  • Example 1: Nonparametric Instrumental Variables
  • Example 2: Nonparametric Quantile IV
  • Example 3: Production functions
  • Remark 1: Weighting
  • Example : Matérn Gaussian Priors
  • Remark 2: Centering
  • Remark 3: Scaling
  • Remark 4: Frequentist estimation
  • Theorem 1: Consistency
  • Theorem 2: Identified Set Consistency
  • ...and 34 more