Posterior Inference in Curved Exponential Families under Increasing Dimensions

TL;DR

This paper investigates the asymptotic behavior of posterior inference in curved exponential families with increasing dimensions, providing conditions for normality and applying results to high-dimensional econometric models.

Contribution

It establishes new conditions for posterior normality in high-dimensional curved exponential families and improves previous results using concentration of measure techniques.

Findings

Posterior distribution is approximately normal under certain conditions.

Enhanced understanding of high-dimensional econometric models with increasing parameters.

Applicability to models like multinomial, seemingly unrelated regressions, and structural equations.

Abstract

This work studies the large sample properties of the posterior-based inference in the curved exponential family under increasing dimension. The curved structure arises from the imposition of various restrictions on the model, such as moment restrictions, and plays a fundamental role in econometrics and others branches of data analysis. We establish conditions under which the posterior distribution is approximately normal, which in turn implies various good properties of estimation and inference procedures based on the posterior. In the process we also revisit and improve upon previous results for the exponential family under increasing dimension by making use of concentration of measure. We also discuss a variety of applications to high-dimensional versions of the classical econometric models including the multinomial model with moment restrictions, seemingly unrelated regression equations, and single structural equation models. In our analysis, both the parameter dimension and the number of moments are increasing with the sample size.