A Note on "A survey of preference estimation with unobserved choice set heterogeneity" by Gregory S. Crawford, Rachel Griffith, and Alessandro Iaria

TL;DR

This paper critically examines Crawford et al.'s 2021 framework for estimating discrete choice models with unobserved consideration sets, clarifying the necessary sampling corrections and conditions for ignoring them in practical estimators.

Contribution

It derives the correct sampling correction needed for past-choice-based consideration sets and formalizes conditions under which this correction can be ignored.

Findings

Identifies imprecise usage of McFadden's 1978 result in prior work.

Derives the specific sampling correction for consideration set estimation.

Establishes conditions for ignoring the sampling correction in practical estimators.

Abstract

Crawford's et al. (2021) article on estimation of discrete choice models with unobserved or latent consideration sets, presents a unified framework to address the problem in practice by using "sufficient sets", defined as a combination of past observed choices. The proposed approach is sustained in a re-interpretation of a consistency result by McFadden (1978) for the problem of sampling of alternatives, but the usage of that result in Crawford et al. (2021) is imprecise in an important matter. It is stated that consistency would be attained if any subset of the true consideration set is used for estimation, but McFadden (1978) shows that, in general, one needs to do a sampling correction that depends on the protocol used to draw the choice set. This note derives the sampling correction that is required when the choice set for estimation is built from past choices. Then, it formalizes the conditions under which such correction would fulfill the uniform condition property and can therefore be ignored when building practical estimators, such as the ones analyzed by Crawford et al. (2021).