Do lenders still discriminate? A robust approach for assessing differences in menus

We use a new methodology to assess mortgage pricing discrimination faced by minority borrowers. We identify a "menu problem" that comes from the multidimensional nature of mortgage pricing: When getting a mortgage, borrowers can choose to avoid closing costs and pay a high interest rate or contribute to closing costs to get a lower rate. While data on both dimensions of mortgage pricing are by now often available, intuitively attractive metrics of lender pricing discrimination used in the literature can lead to both false and contradictory results. For example, it is sometimes observed that conditional on rate, minority borrowers pay the same closing costs as white borrowers, but conditional on closing costs, minority borrowers pay a higher rate. Though generally underappreciated, the menu problem is broadly relevant in economic assessments of differences in opportunity given data on outcomes. We develop a solution to the menu problem by defining (1) a test statistic for equality in menus and (2) a difference in menus (DIM) metric for assessing whether one group of borrowers would prefer to switch to another group's menus, both based on pairwise dominance relationships in the data. Our proposed solution is robust to arbitrary heterogeneity in borrower preferences across racial groups. We show how our metrics can be computed using methods from optimal transport and also devise a new procedure for hypothesis testing in this class of problems based on directional differentiation. Finally, we implement our methodology on a data set linking 2018-2019 Home Mortgage Disclosure Act (HMDA) data to Optimal Blue rate locks, and present novel results on mortgage pricing discrimination.