The ABC of Simulation Estimation with Auxiliary Statistics

TL;DR

This paper bridges the gap between frequentist simulated minimum distance and Bayesian Approximate Bayesian Computation methods using a Reverse Sampler, providing new insights into their relationship and properties.

Contribution

It introduces a hybrid Reverse Sampler that combines optimization and importance weighting, connecting ABC estimates with SMD modes and analyzing their differences.

Findings

ABC estimates can be expressed as weighted averages of SMD modes

The Reverse Sampler links Bayesian and frequentist likelihood-free methods

Simulation studies illustrate the theoretical relationships

Abstract

The frequentist method of simulated minimum distance (SMD) is widely used in economics to estimate complex models with an intractable likelihood. In other disciplines, a Bayesian approach known as Approximate Bayesian Computation (ABC) is far more popular. This paper connects these two seemingly related approaches to likelihood-free estimation by means of a Reverse Sampler that uses both optimization and importance weighting to target the posterior distribution. Its hybrid features enable us to analyze an ABC estimate from the perspective of SMD. We show that an ideal ABC estimate can be obtained as a weighted average of a sequence of SMD modes, each being the minimizer of the deviations between the data and the model. This contrasts with the SMD, which is the mode of the average deviations. Using stochastic expansions, we provide a general characterization of frequentist estimators and those based on Bayesian computations including Laplace-type estimators. Their differences are illustrated using analytical examples and a simulation study of the dynamic panel model.