Bayesian Estimation and Comparison of Conditional Moment Models

TL;DR

This paper develops a Bayesian framework for analyzing models with conditional moment restrictions, providing theoretical results, model comparison tools, and practical algorithms, with applications in finance and causal inference.

Contribution

It introduces a Bayesian approach using nonparametric empirical likelihood for conditional moment models, including model comparison and high-dimensional variable selection.

Findings

Bernstein-von Mises theorems established for the posterior distribution

Model selection via marginal likelihood favors less misspecified models

Efficient MCMC algorithms developed for high-dimensional parameters

Abstract

We consider the Bayesian analysis of models in which the unknown distribution of the outcomes is specified up to a set of conditional moment restrictions. The nonparametric exponentially tilted empirical likelihood function is constructed to satisfy a sequence of unconditional moments based on an increasing (in sample size) vector of approximating functions (such as tensor splines based on the splines of each conditioning variable). For any given sample size, results are robust to the number of expanded moments. We derive Bernstein-von Mises theorems for the behavior of the posterior distribution under both correct and incorrect specification of the conditional moments, subject to growth rate conditions (slower under misspecification) on the number of approximating functions. A large-sample theory for comparing different conditional moment models is also developed. The central result is that the marginal likelihood criterion selects the model that is less misspecified. We also introduce sparsity-based model search for high-dimensional conditioning variables, and provide efficient MCMC computations for high-dimensional parameters. Along with clarifying examples, the framework is illustrated with real-data applications to risk-factor determination in finance, and causal inference under conditional ignorability.