Reverse regression. A common method of analyzing statistical data to detect discrimination in the workplace is to fit the regression y = + x'

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 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

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.

(-)(1-R) (1-P)(1-rd) C,