Modified Delayed Acceptance MCMC for Quasi-Bayesian Inference with Linear Moment Conditions

TL;DR

This paper tackles efficient quasi-Bayesian inference based on linear moment conditions by introducing a modified delayed acceptance MCMC framework that leverages a surrogate kernel and a tailored approximate conditional-posterior proposal. It presents two implementations, DA--MCMC--Exact and DA--MCMC--Approx, balancing exact incorporation of priors with computational stability in high dimensions. Across synthetic heteroskedastic linear models and empirical IV regressions, the proposed methods substantially improve sampling efficiency and throughput relative to standard MCMC and existing DA approaches, with Exact excelling in per-iteration efficiency and Approx delivering superior scalability. The framework is broadly applicable to moment-based and risk-based quasi-Bayesian formulations when first-order conditions are linear, offering a robust, scalable tool for practical quasi-Bayesian analysis in econometrics and statistics.

Abstract

We develop a computationally efficient framework for quasi-Bayesian inference based on linear moment conditions. The approach employs a delayed acceptance Markov chain Monte Carlo (DA-MCMC) algorithm that uses a surrogate target kernel and a proposal distribution derived from an approximate conditional posterior, thereby exploiting the structure of the quasi-likelihood. Two implementations are introduced. DA-MCMC-Exact fully incorporates prior information into the proposal distribution and maximizes per-iteration efficiency, whereas DA-MCMC-Approx omits the prior in the proposal to reduce matrix inversions, improving numerical stability and computational speed in higher dimensions. Simulation studies on heteroskedastic linear regressions show substantial gains over standard MCMC and conventional DA-MCMC baselines, measured by multivariate effective sample size per iteration and per second. The Approx variant yields the best overall throughput, while the Exact variant attains the highest per-iteration efficiency. Applications to two empirical instrumental variable regressions corroborate these findings: the Approx implementation scales to larger designs where other methods become impractical, while still delivering precise inference. Although developed for moment-based quasi-posteriors, the proposed approach also extends to risk-based quasi-Bayesian formulations when first-order conditions are linear and can be transformed analogously. Overall, the proposed algorithms provide a practical and robust tool for quasi-Bayesian analysis in statistical applications.