Identification of Unobservables in Observations

TL;DR

This paper demonstrates how to identify unobserved variables in economic data by establishing a unique mapping from observable data to unobservables, ensuring the latent variables can be recovered at the population level.

Contribution

It introduces a method for identifying unobserved variables using the joint distribution of observables and unobservables, ensuring uniqueness of the latent values in each observation.

Findings

Existence of a unique mapping from observables to unobservables when observables are distinct.

Identification of the joint distribution of observables and unobservables from observable data.

Application of the method to three illustrative examples.

Abstract

In empirical studies, the data usually don't include all the variables of interest in an economic model. This paper shows the identification of unobserved variables in observations at the population level. When the observables are distinct in each observation, there exists a function mapping from the observables to the unobservables. Such a function guarantees the uniqueness of the latent value in each observation. The key lies in the identification of the joint distribution of observables and unobservables from the distribution of observables. The joint distribution of observables and unobservables then reveal the latent value in each observation. Three examples of this result are discussed.