A Note on Identification of Match Fixed Effects as Interpretable Unobserved Match Affinity

TL;DR

This paper clarifies the interpretability issues of match fixed effects in models and proposes normalization conditions to make these effects comparable and meaningful, demonstrated with TIMSS data.

Contribution

It introduces normalization conditions that identify interpretable match fixed effects, enabling meaningful comparison of unobserved match affinity.

Findings

Normalized match fixed effects are interpretable as unobserved match affinity.

Reported fixed effects without normalization are not directly interpretable.

Application to TIMSS data illustrates distribution of match effects within a school.

Abstract

We highlight that match fixed effects, represented by the coefficients of interaction terms involving dummy variables for two elements, lack identification without specific restrictions on parameters. Consequently, the coefficients typically reported as relative match fixed effects by statistical software are not interpretable. To address this, we establish normalization conditions that enable identification of match fixed effect parameters as interpretable indicators of unobserved match affinity, facilitating comparisons among observed matches. Using data from middle school students in the 2007 Trends in International Mathematics and Science Study (TIMSS), we highlight the distribution of comparable match fixed effects within a specific school.