True and Pseudo-True Parameters

TL;DR

This paper examines the relevance of pseudo-true parameters in misspecified models for decision-making, showing their importance is limited and providing methods for constructing reliable confidence intervals.

Contribution

It characterizes conditions under which Bayesian posteriors concentrate on pseudo-true values and develops confidence intervals that are robust to misspecification.

Findings

Posteriors concentrate on pseudo-true values only under specific priors.

Pseudo-true values are generally not relevant for decision-making.

Constructed confidence intervals guarantee correct coverage regardless of misspecification.

Abstract

Parameter estimates in misspecified models converge to pseudo-true parameter values, which minimize a population objective function. Pseudo-true values often differ from quantities of economic interest, raising questions of how, if at all, they are relevant for decision-making. To study this question we consider Bayesian decision-makers facing a linear population minimum distance problem. Within a class of priors motivated by the minimum distance objective, we characterize prior sequences under which posteriors concentrate on the pseudo-true value. This convergence is fragile to small changes in priors, implying that pseudo-true values are relevant for decision-making only in special cases. Constructive results are nevertheless possible in this setting, and we derive simple confidence intervals that guarantee correct average coverage for the true parameter under every prior in the class we study, with no bound on the magnitude of misspecification.