This is very useful since we can then use results from the treatment effect literature.
11/5/20137The wage structure effect (ΔS) can be interpreted as a treatment effectThe conditional independence assumption (E(ε|X)=0) usually invoked in Oaxaca decompositions can be replaced by the weaker ignorability assumption (Dg╨ε|X) to compute the aggregate decompositionFor example, ability (ε) can be correlated with education (X) as long as the correlation is the same in groups A and B.This is the standard assumption used in “selection on observables” models where matching methods are typically used to estimate the treatment effect.Main result: If we have YG=mG(X, ε) and ignorability, then:ΔSsolely reflects changes in the m(.) functions (ATET)ΔXsolely reflects changes in the distribution of X and ε(ignorability key for this last result).The wage structure effect (ΔS) can be interpreted as a treatment effectA number of estimators for ATET= ΔShave been proposed in the treatment effect literatureInverse probability weighting (IPW), matching, etc.Formal results exist, e.g. IPW is efficient forATET (Hirano, Imbens, and Ridder, 2003)Quantile treatment effects (Firpo, 2007)This has been widely used in the decomposition literature since DiNardo, Fortin, and Lemieux (1996).Formal derivation of the identification result (Handbook chapter)Formal derivation (2)
11/5/20138Formal derivation (3)Some intuition about Proposition 1The wage setting model we use, Yg=mg (X,ε) is very generalIncludes the linear model yg= xβg+ εas a special case There are three reasons why wages can be different between groups g=A and g=B:Differences in the wage setting equations ma (.) and mb(.) Differences in the distribution of X for the two groupsDifferences in the distribution of εfor the two groupsThe ignorability assumption states that the distribution of εgiven X is the same for the two groups, though this does not mean that E(ε|X)=0.So once we control for differences between the X’s in the two groups, we also implicitly control for differences in the ε’s.Only source of difference left is, thus, differences in the wage structures ma (.) and mb(.).A few caveats discussed in the chapterThis general result (Proposition 1) only works for the aggregate decomposition. More assumptions have to be imposed to get at the detailed decomposition.We are implicitly ruling out general equilibrium effects under assumption (simple counterfactual treatment). For instance, in the absence of unions, wages in the nonunionsector may change as firms are no longer confronted with the threat of unionizationThe wage structure observed among nonunion workers, ma (.), is no longer a valid counterfactual for union workers.