Asymptotic properties of Bayesian inference in linear regression with a structural break

TL;DR

This paper establishes that Bayesian inference for slope parameters in linear regression models with a structural break is asymptotically equivalent to traditional methods, providing more reliable credible intervals especially in finite samples.

Contribution

It proves a Bernstein-von Mises theorem for Bayesian inference in models with structural breaks, bridging Bayesian and frequentist approaches.

Findings

Bayesian credible intervals outperform confidence intervals in finite samples.

As sample size grows, Bayesian and frequentist inferences converge.

Simulation and empirical analysis support the theoretical results.

Abstract

This paper studies large sample properties of a Bayesian approach to inference about slope parameters $\gamma$ in linear regression models with a structural break. In contrast to the conventional approach to inference about $\gamma$ that does not take into account the uncertainty of the unknown break location $\tau$, the Bayesian approach that we consider incorporates such uncertainty. Our main theoretical contribution is a Bernstein-von Mises type theorem (Bayesian asymptotic normality) for $\gamma$ under a wide class of priors, which essentially indicates an asymptotic equivalence between the conventional frequentist and Bayesian inference. Consequently, a frequentist researcher could look at credible intervals of $\gamma$ to check robustness with respect to the uncertainty of $\tau$. Simulation studies show that the conventional confidence intervals of $\gamma$ tend to undercover in finite samples whereas the credible intervals offer more reasonable coverages in general. As the sample size increases, the two methods coincide, as predicted from our theoretical conclusion. Using data from Paye and Timmermann (2006) on stock return prediction, we illustrate that the traditional confidence intervals on $\gamma$ might underrepresent the true sampling uncertainty.