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When "Normalization Without Loss of Generality" Loses Generality

Wayne Gao

arXiv 29 Mar 2026 · Econometrics

arXiv:2603.27762 · PDF · DOI · OpenAlex · Extracted main text

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.

Citation extraction

25
references
53
in-text mentions
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distinct cited
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Most heavily cited references

The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.

ReferenceIntensityMentionsSectionsMain text
1Freyberger, J (2025) Normalizations and Misspecification in Skill Formation Models1.00085100%
2Gao, W. Y (2020) Nonparametric Identification in Index Models of Link Formation self1.00073100%
3Agostinelli, F. and M. Wiswall (2025) Estimating the Technology of Children's Skill Formation1.00055100%
4Hamilton, J. D., D. F. Waggoner, and T. Zha (2007) Normalization in Econometrics0.92844100%
5Chen, J. and J. Roth (2024) Logs with Zeros? Some Problems and Solutions0.87452100%
6Graham, B. S (2017) An econometric model of network formation with degree heterogeneity0.84333100%
7Berry, S., J. Levinsohn, and A. Pakes (1995) Automobile Prices in Market Equilibrium0.64422100%
8Manski, C. F (1975) Maximum score estimation of the stochastic utility model of choice0.51121100%
9Agostinelli, F. and M. Wiswall (2016) Identification of Dynamic Latent Factor Models: The Implications of Re-Normalization in a Model of Child Development, Working Pa…0.40511100%
10Berry, S. T. and P. A. Haile (2014) Identification in Differentiated Products Markets Using Market Level Data0.40511100%

Showing the top 10 of 25 scored citations.