Normalizations and misspecification in skill formation models

TL;DR

This paper examines how scale and location restrictions affect the identification and interpretation of parameters in skill formation models, revealing potential misspecification issues with common assumptions and proposing solutions.

Contribution

It provides new identification results, characterizes the identified set without restrictions, and highlights the impact of standard restrictions on policy-relevant parameters.

Findings

Common scale restrictions can lead to misspecification with CES functions.

Weaker assumptions can achieve point identification of key parameters.

Existing estimators can be adapted to address these issues.

Abstract

An important class of structural models studies the determinants of skill formation and the optimal timing of interventions. In this paper, I provide new identification results for these models and investigate the effects of seemingly innocuous scale and location restrictions on parameters of interest. To do so, I first characterize the identified set of all parameters without these additional restrictions and show that important policy-relevant parameters are point identified under weaker assumptions than commonly used in the literature. The implications of imposing standard scale and location restrictions depend on how the model is specified, but they generally impact the interpretation of parameters and may affect counterfactuals. Importantly, with the popular CES production function, commonly used scale restrictions fix identified parameters and lead to misspecification. Consequently, simply changing the units of measurements of observed variables might yield ineffective investment strategies and misleading policy recommendations. I show how existing estimators can easily be adapted to solve these issues. As a byproduct, this paper also presents a general and formal definition of when restrictions are truly normalizations.