Finite-Sample NonParametric Bounds with an Application to the Causal Effect of Workforce Gender Diversity on Firm Performance

Classical Manski bounds identify average treatment effects under minimal assumptions but, in finite samples, they rely on latent outcome expectations being bounded by the sample's own extrema or known population bounds, an assumption often violated in firm-level data with heavy-tailed outcomes. We develop a finite-sample, concentration-driven confidence band (concATE) that replaces this requirement with a Dvoretzky-Kiefer-Wolfowitz tail bound, combines it with delta-method variance, and allocates size via a Bonferroni correction. The band extends to a group-sequential design that controls the family-wise error rate when the first "significant" diversity threshold is data-chosen. Applied to data on 901 listed firms (2015 Q2-2022 Q1), concATE shows that senior-level gender diversity has a significant positive effect on firm value (Tobin's Q) only after crossing substantial representation thresholds: in Growth & Innovation sectors the effect becomes statistically significant at the 5% level once women hold roughly 55% of senior leadership roles, whereas in Defensive sectors a significant impact appears only once female leadership reaches about 60%.