From sequential binary models to first-best multinomial choice: A Stata implementation

This presentation introduces a Stata implementation of a Gentzkow-style framework for studying complementarity between two binary decisions. Building on Gentzkow (2007) and Li and Mora (2022), the approach models the four possible bundles jointly and treats the multinomial choice as the first-best benchmark. Standard bivariate probit models, even with correlated errors, do not capture Hicksian complementarity. Sequential binary models can be informative because they estimate whether one choice affects the other; we establish the bridge between these sequential representations and the first-best multinomial model. This bridge identifies three types of decision-makers: those for whom no sequence is compatible with the first best, those for whom only one ordering is compatible, and those for whom both orderings are compatible. The accompanying Stata module, gentzkow, currently estimates the model using a mixed logit specification and reports complementarity patterns and type probabilities implied by the estimated utilities. We illustrate the usefulness of the approach with household-level data on three-generation families, focusing on whether families live together and whether they share childcare.