Stratified Sample Design for Fair Lending Binary Logit Models

Logistic regressions are commonly used to assess for fair lending across groups of loan applicants. This paper considers estimation of the disparate treatment parameter when the sample is stratified jointly by loan outcome and race covariate. We use Monte Carlo analysis to investigate the finite-sample properties of two estimators of the disparate treatment parameter under six stratified sampling designs and three data generating processes; one estimator is consistent irrespective of sample design while the other is not. Unfortunately the inconsistent estimator is employed inadvertently in fair lending studies. We demonstrate the gains in using the consistent estimator as well as providing recommendations on sample design. We also study the effect of sample design on the empirical power of a test for statistical significance of the disparate treatment parameter. We recommend adopting a sample design that approximately balances by outcome and racial group, when using the estimator that adjusts for the stratification scheme. However, if the standard logit estimator is employed, then our results suggest a sample design that balances by outcome and allocates across racial groups proportionally to the population. Though our study is framed in terms of fair lending applications, our results apply generally to the estimation of logistic regressions that use stratified or choice-based sample designs.