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.

Transcribed Image Text:

(-ỹ)(1-R²) (1-P)(1-rd) C,