productivity considerations. That is, α can be zero even though R is systematically related to the difference between the index Pf and the unbiased productivity index E(P | X1,X2). We define this as adverse impact discrimination. The legal requirement that firms validate hiring criteria having an adverse impact on protected classes of workers is designed to prevent this form of discrimination.
In general, it will be difficult to detect that the firm is behaving in accordance with equation (A7.1) without information on P. Suppose, however, that the researcher has an unbiased indicator P* of P as well as data on X1 but not X2. Then the researcher can estimate the coefficients θ1 of the conditional expectation
If firms are hiring on the basis of expected productivity given X1 and X2, then E(y | X1) = E(y | X1θ1). Consequently, one can test the null hypothesis that firms are hiring on the basis of expected productivity given X1 and X2 by testing the restriction that
One can test this restriction by regressing y on X1θ1 and X1 (with one element of X1 excluded because of collinearity) and testing the null hypothesis that the elements of X1 have no effect on y, holding X1θ1 constant. From a regression of E(y | X1) on R and X1θ1, one can estimate the race gap for workers with a given value of X1 that is due to the firm’s policy. Without special assumptions, however, one cannot estimate the effect on group R of the firm’s misuse of X1 and X2 without having data on both variables. Unfortunately, even a noisy indicator of productivity is unavailable in most of the data sets used to study racial differences.