Reference parameters in Blinder-Oaxaca decomposition: Pooled-sample versus intercept-shift approaches

The Blinder-Oaxaca (BO) decomposition explains two groups’ difference (e.g., in wage between males and females) with regressors. The BO decomposition depends on the chosen counterfactual reference parameter representing no “discrimination”. One popular way to choose the reference parameter is simply using the pooled sample, and another way is allowing different intercepts in the pooled sample. This paper points out two problems in the latter. First, the reference slopes are a matrix-weighted average of the two group slopes where the weights depend only on the variances of the two groups, not on the levels; the levels may represent the competing “forces” to primarily influence the reference parameter, but the role of the variances should be secondary at best. Second, the reference intercept is arbitrary: the same BO decomposition holds with vastly different reference intercepts.