Reverse regression. A common method of analyzing statistical data to detect discrimination in the workplace is to fit the regression y = α + x' β + γd + ε, (1) where y is the wage rate and d is a dummy variable indicating either membership (d = 1) or nonmembership (d = 0) in the class toward which it is suggested the discrimination is directed. The regressors x include factors specific to the particular type of job as well as indicators of the qualifications of the individual.The hypothesis of interest is H0: γ ≥ 0 versus H1: γ < 0. The regression seeks to answer the question, “In a given job, are individuals in the class (d = 1) paid less than equally qualified individuals not in the class (d = 0)?” Consider an alternative approach. Do individuals in the class in the same job as others, and receiving the same wage, uniformly have higher qualifications? If so, this might also be viewed as a form of discrimination. To analyze this question, Conway and Roberts (1983) suggested the following procedure:
1. Fit (1) by ordinary least squares. Denote the estimates a, b, and c.
2. Compute the set of qualification indices, q = ai + Xb. (2) Note the omission of cd from the fitted value.
3. Regress q on a constant, y and d. The equation is q = α* + β* y + γ* d + ε*. (3) The analysis suggests that ifγ < 0, γ* > 0
a. Prove that the theory notwithstanding, the least squares estimates c and c* are related by where
y1 = mean of y for observations with d = 1,
y = mean of y for all observations,
P = mean of d,
R2 = coefficient of determination for (1),
r2yd = squared correlation between y and d.
b. Will the sample evidence necessarily be consistent with the theory? Asymposium on the Conwayand Roberts’s paper appeared in the Journal of Business and Economic Statistics in April1983.