The Oaxaca-Blinder decomposition rewrites the difference in mean outcomes between two groups as the sum of a characteristics effect (how much of the gap is explained by group differences in covariates, evaluated at a common price vector) and a coefficients effect (how much is explained by differences in slope coefficients, i.e., the wage structure). For group-specific linear regressions Yˉg=Xˉg′β^g, the decomposition is:
with alternative reference-group choices (group 1's β, group 2's β, or a pooled estimate) giving different allocations between the two terms.
Key Ideas
Oaxaca (1973) and Blinder (1973): Independently proposed the decomposition to study male-female and black-white wage differentials. The "unexplained" coefficients effect is often interpreted as discrimination, though it captures all unobserved group differences in skill prices.
Reference-group ambiguity (Oaxaca-Ransom 1994): The share attributed to characteristics vs. coefficients depends on which group's β is used as the reference; this is a fundamental index-number problem with no universally preferred solution.
Identifying assumption: The decomposition requires a common counterfactual wage structure (typically the non-discriminating group's coefficients or a pooled estimate).
Linear case: In linear models the decomposition is exact, additive, and path-independent. Each covariate's contribution is simply (Xˉ1k−Xˉ2k)β^2k.
How It Works
Linear Case (Oaxaca-Blinder)
Run separate ordinary least squares (OLS) regressions for each group; compute group means; apply the formula above. Aggregate the characteristics effect across covariates: ∑k(Xˉ1k−Xˉ2k)β^2k.
Nonlinear Extensions
When Y=g(X′β,ψ) for a nonlinear link function g (probit, logit, Tobit), the linear formula does not apply exactly.
Fairlie (2005) simulation-based method: Draw matched pairs from the two groups, replace group 1's characteristic with group 2's value, and compute the change in predicted probability. Sensitive to matching order (path dependence) and choice of random subsample; does not fully account for the nonlinear structure.
Yun (2004) linearisation: Applies a first-order Taylor expansion around the pooled mean; yields a decomposition that is approximately additive but leaves a nonlinearity residual; underestimates contribution when the link function is far from linear in the relevant range.
Schwiebert (2015) mean value (MV) decomposition: Uses the mean value theorem to write the contribution of covariate k as ck=E[g′(X12′β,ψ)(X1k−X2k)βk] where X12 is a mean-value intermediate point. This is exact (no residual), unique (no path dependence), and uses all n1×n2 cross-group pairs. Asymptotically normal; bootstrap standard errors are consistent.
Why It Matters
Policy analysis: Decomposes group wage (or employment, health, etc.) gaps into observable and unobservable components, informing anti-discrimination policy.
Benchmark tool: The characteristics/coefficients split is the standard starting point for any group-difference analysis in labour, health, and education economics.
Nonlinear models: Most outcome variables of interest (employment, participation, benefit receipt) are binary or censored; the extension to nonlinear models is empirically essential and methodologically non-trivial.
Open Questions
Coefficient-effect decomposition for nonlinear models (analogous to attributing the coefficients gap to individual regressors) is harder and lacks a clean analogue of the Schwiebert MV approach.
The interpretation of the "unexplained" component as discrimination requires strong assumptions; alternative explanations (unobserved heterogeneity, selection) are observationally equivalent.
With many covariates, the Oaxaca-Blinder decomposition may suffer from multicollinearity in the characteristics-effect components.