Efficient counterfactual estimation in semiparametric discrete choice models: a note on Chiong, Hsieh, and Shum (2017)

TL;DR

This paper improves the computational efficiency of estimating bounds on counterfactual demand in semiparametric discrete choice models by reducing cycle enumeration to shortest path problems, enabling faster calculations even with many markets.

Contribution

It demonstrates that cycle enumeration in the existing algorithm can be replaced with shortest path computations, significantly reducing computational complexity.

Findings

Shortest path approach reduces computation time

Using cycles of all lengths yields small but positive efficiency gains

Algorithm scales well to thousands of markets

Abstract

I suggest an enhancement of the procedure of Chiong, Hsieh, and Shum (2017) for calculating bounds on counterfactual demand in semiparametric discrete choice models. Their algorithm relies on a system of inequalities indexed by cycles of a large number $M$ of observed markets and hence seems to require computationally infeasible enumeration of all such cycles. I show that such enumeration is unnecessary because solving the "fully efficient" inequality system exploiting cycles of all possible lengths $K=1,\dots,M$ can be reduced to finding the length of the shortest path between every pair of vertices in a complete bidirected weighted graph on $M$ vertices. The latter problem can be solved using the Floyd--Warshall algorithm with computational complexity $O\left(M^3\right)$, which takes only seconds to run even for thousands of markets. Monte Carlo simulations illustrate the efficiency gain from using cycles of all lengths, which turns out to be positive, but small.