Schwiebert (2015) A Detailed Decomposition for Nonlinear Econometric Models

Summary

Schwiebert (2015) proposes a mean-value-theorem (MV) decomposition that extends the Oaxaca-Blinder framework to nonlinear models (probit, logit, Tobit, etc.). The decomposition is unique — unlike Fairlie's simulation-based method, it has no path dependence — and it explicitly absorbs the nonlinear link function rather than treating it as an additive residual. The procedure uses all $n_1 \times n_2$ between-group observation pairs and yields asymptotically normal component estimates with bootstrap-consistent standard errors.

Key Claims

• MV decomposition formula: The contribution of characteristic $k$ to the group-mean outcome differential is $c_k = E[g'(X_{12}'\beta, \psi)(X_{1k} - X_{2k})\beta_k]$, where $g$ is the link function and $X_{12}$ is an intermediate (mean-value) point between a group-1 and a group-2 observation.
• Estimator: $\hat{c}_k = \frac{1}{n_1 n_2}\sum_{i \in I_1}\sum_{j \in I_2} g'(X_{ij}'\hat{\beta}, \hat{\psi})(X_{ik} - X_{jk})\hat{\beta}_k$, computed by averaging over all $n_1 \times n_2$ cross-group pairs. The derivative $g'$ is recovered via the identity $g'(X_{ij}'\hat{\beta}, \hat{\psi}) = [g(X_i'\hat{\beta}, \hat{\psi}) - g(X_j'\hat{\beta}, \hat{\psi})] / [(X_i - X_j)'\hat{\beta}]$.
• Theorem 1 (asymptotic normality): $\hat{c}_k \xrightarrow{a} \mathcal{N}(c_k, \sigma_k^2/n)$; bootstrap standard errors are consistent.
• Uniqueness: No arbitrary matching or ordering of observations; the decomposition is path-independent by construction.
• Advantage over Fairlie (2005): Fairlie draws random subsamples and is sensitive to the choice of reference group and matching order; the MV method uses all pairs, is symmetric, and gives a clear reference point.
• Advantage over Yun (2004): Yun's linearisation (first-order Taylor expansion) does not account for the nonlinear structure of $g$, so the decomposition components do not add up to the total differential without a residual. The MV decomposition absorbs the nonlinearity exactly.
• Empirical application: LaLonde (1986) training data, probit model of programme participation, black vs. white participants (group means 20% vs. 1%). Decomposes the 19-percentage-point gap into covariate contributions with bootstrap standard errors.

Concepts Introduced or Extended

• Oaxaca-Blinder Decomposition

Entities Mentioned

• Jörg Schwiebert

Quotes

"The decomposition suggested in this paper is based on the mean value theorem and exploits the fact that the mean value theorem provides an exact (rather than approximate) decomposition." (p. 55)

My Take

The MV decomposition is a clean theoretical advance: it turns an approximate linearisation into an exact result via the mean value theorem, and it eliminates path dependence by exhausting all cross-group pairs. The cost is computational — $O(n_1 n_2)$ evaluations of the link function — which is manageable for typical survey samples but could be expensive at large scale. The asymptotic theory is standard; the key assumption is that the same coefficient vector $\beta$ applies to both groups (i.e., the "characteristics effect" interpretation requires that group 2's regression model is the appropriate counterfactual). The paper does not address coefficient-effect decomposition for nonlinear models, which remains a harder problem.