When "Normalization Without Loss of Generality" Loses Generality

TL;DR

This paper develops a framework analyzing how normalization choices affect economic models, showing that normalization can sometimes distort the interpretation and inference of parameters.

Contribution

It formalizes the concept of modeling equivalence and demonstrates how normalization impacts identification, inference, and model interpretation in economics.

Findings

Normalization can create point identification where none exists.

Normalization may distort the topological structure of parameter spaces.

Fidelity, invariance, and regularity cannot all hold at boundary singularities.

Abstract

Normalization is ubiquitous in economics, and a growing literature shows that ``normalizations'' can matter for interpretation, counterfactual analysis, misspecification, and inference. This paper provides a general framework for these issues, based on the formalized notion of modeling equivalence that partitions the space of unknowns into equivalence classes, and defines normalization as a WLOG selection of one representative from each class. A counterfactual parameter is normalization-free if and only if it is constant on equivalence classes; otherwise any point identification is created by the normalization rather than by the model. Applications to discrete choice, demand estimation, and network formation illustrate the insights made explicit through this criterion. We then study two further sources of fragility: an extension trilemma establishes that fidelity, invariance, and regularity cannot simultaneously hold at a boundary singularity, while a normalization can itself introduce a coordinate singularity that distorts the topological and metric structures of the parameter space, with consequences for estimation and inference.