Geometric methods in econometrics and statistics

Abstract

Econometrics and statistics rely on asymptotic approximations to construct hypothesis tests and confidence regions. Asymptotic approximations can also be used more abstractly to study the quality (efficiency) of estimators and tests. These approximations are closely related to local (differential) properties of the functionals of the statistical model whose values are being estimated and tested. I consider statistical models and estimands motivated by economic theory and applications and study their local and also global properties: I study the local properties of functionals to characterize the efficiency bounds of their estimators and the directions of most rapid (gradient) change with respect to different metrics of distance on the model. I use gradient flows to describe global evolutions on the statistical model governed by changes in a scalar functional. These flows can be used to describe economic policy and to study structural estimators motivated by economic theory.