Identification and feasible inference for marginal treatment effects in the returns-to-education selection framework under weaker assumptions

The returns-to-education selection framework of Carneiro, Heckman, and Vytlacil (2011) carries several maintained limitations — identification of global treatment parameters confined to the empirical support of the propensity score, a strong full-independence assumption adopted for empirical feasibility, binary schooling, computationally infeasible inference for a semiparametric single-index implementation, reliance on instrument validity, and the divergent tails of the normal selection model — some of which later work has partially overtaken: shape-constrained partial identification now supplies principled extrapolation beyond common support, and nonparametric tests for instrument validity in heterogeneous-effect models exist. The objective is to establish nonparametric identification, feasible estimation, and valid inference for marginal treatment effects (MTEs) and their policy-relevant functionals in the selection framework under strictly weaker independence assumptions, robustness to invalid instruments, and multi-valued schooling treatments.

Background

Estimating the causal return to education is complicated by the fact that people choose their own schooling. In the potential-outcomes notation standard in this literature, each person has earnings $Y_1$ with college and $Y_0$ without it; the causal effect $Y_1-Y_0$ is never fully observed. The Roy model — a self-selection model in which agents choose whichever option gives them the higher personal payoff — explains why observed college/non-college wage gaps need not equal any average causal effect: those who enroll are, in part, those who expect to gain. The version used by Carneiro, Heckman, and Vytlacil (2011) (CHV) is a generalized Roy selection model,

$$Y = \mu_0(\mathbf X) + U_0 + S\,[\mu_1(\mathbf X) - \mu_0(\mathbf X) + (U_1 - U_0)],$$ $$S = \mathbf 1\{\mu_S(\mathbf Z) \ge U_S\},$$

where $\mathbf X$ are observed individual characteristics, $\mathbf Z$ are observed variables containing instruments (variables that shift schooling but are excluded from the wage equation), $\mu_0,\mu_1,\mu_S$ are unspecified functions, and $U_0,U_1,U_S$ are unobservables: idiosyncratic wage components and the person-specific component of schooling preferences. Under independence of $(\mathbf X,\mathbf Z)$ from $(U_0,U_1,U_S)$, $U_S$ can be normalized to be uniform on $[0,1]$, so $U_S$ is a person's quantile of latent resistance to college, and the propensity score $P(\mathbf Z)=\Pr(S=1\mid \mathbf Z)$ equals the choice index $\mu_S(\mathbf Z)$.

The marginal treatment effect (MTE) is $\mathrm{MTE}(x,u)=E[Y_1-Y_0\mid \mathbf X=x,\,U_S=u]$ — the return for a person with characteristics $x$ who is indifferent about college at resistance level $u$. The local instrumental variables (LIV) result identifies it nonparametrically: under the independence assumption, $\mathrm{MTE}(x,p)$ equals the derivative of the conditional mean $E[Y\mid \mathbf X=x,\,P(\mathbf Z)=p]$ with respect to $p$, so the MTE can be traced out by nonparametric regression over the observed values of the propensity score. Every standard treatment parameter is then a weighted integral of the MTE: the average treatment effect $\mathrm{ATE}=E[Y_1-Y_0]$ for the whole population, the effect of treatment on the treated $\mathrm{TT}=E[Y_1-Y_0\mid S=1]$, the effect of treatment on the untreated $\mathrm{TUT}=E[Y_1-Y_0\mid S=0]$, and conventional instrumental-variables (IV) estimands. This weighting view explains why different instruments and methods deliver different numbers: each answers a different question. To speak to policy one wants the policy-relevant treatment effect (PRTE) — the average effect of an explicit policy that shifts the choice probability — and its marginal version, the MPRTE, defined by Carneiro, Heckman, and Vytlacil (2010) for infinitesimal policy shifts; the MPRTE weights the MTE exactly at the margins the data reach, so unlike the $\mathrm{ATE}$ it requires no extrapolation beyond observed support.

CHV (2011) implement this program on the NLSY79 (a US longitudinal youth cohort), with $S$ equal to college enrollment by 1991, $Y$ the 1991 log wage, $\mathbf X$ containing ability (AFQT test scores), family background and local labor-market variables, and instruments: presence of a four-year college in the county of residence at age 14, local wage levels and state unemployment around age 17, and public college tuition. They estimate the MTE semiparametrically (kernel-based LIV, with a parametric normal selection model as a benchmark), find strong evidence of selection on gains — returns are higher for those whose unobservables make them more likely to attend — and show that the implied marginal returns differ from conventional IV estimates. The paper is explicit about what the analysis could not deliver. (i) The estimated propensity score covers only $[0.0324,0.9775]$ after trimming, so global parameters such as $\mathrm{ATE}$, $\mathrm{TT}$, $\mathrm{TUT}$ and PRTEs for non-marginal policies are not nonparametrically identified. (ii) To identify the MTE over the marginal (rather than covariate-conditional) support of $P$, the empirical work invokes the stronger assumption that $(\mathbf X,\mathbf Z)$ is fully independent of $(U_0,U_1,U_S)$. (iii) Schooling is reduced to two levels; although methods for multiple schooling levels existed, applying them requires instruments for each schooling transition. (iv) A variant using semiparametric least squares (SLS, a single-index method estimating the direction of $\mathbf X$ nonparametrically) to form the covariate index is computationally very demanding and no standard errors were feasible. (v) All tests and estimators, including the tests for selection on gains, assume the validity of the instruments; if an instrument is correlated with the wage or schooling unobservables, the conclusions may reflect that invalidity. (vi) The normal selection model implies $\mathrm{MTE}(x,0)=+\infty$ and $\mathrm{MTE}(x,1)=-\infty$ at the extremes, so its extrapolated tails are not credible.

Later work has overtaken part of this list. The support and tail items now have a standard answer: the marginal treatment response (MTR) framework — functions describing how the mean of each potential outcome varies with latent resistance, whose difference is the MTE — combined with shape restrictions such as monotonicity yields point or partial identification of policy-relevant parameters with valid inference, replacing the divergent parametric tails with transparent, assumption-indexed extrapolation (Brinch, Mogstad, and Wiswall 2017; Mogstad, Santos, and Torgovitsky 2018). Nonparametric tests for instrument validity in heterogeneous-effect models, including multivalued treatments, now exist (Sun 2023); identification of MTEs for multivalued treatments has been studied (Lee and Salanié 2018; Tsuda 2022), with an empirical demonstration for the multi-margin college choice problem (Mountjoy 2022). The 2024 Handbook of Labor Economics survey by Mogstad and Torgovitsky judges the multivalued-treatment approaches relatively tailored, requiring case-specific work.

Problem Statement

Later literature has overtaken two of these limitations (extrapolation beyond common support and control of parametric tail behavior, via shape-constrained partial identification) and has produced detection tools for a third (instrument-validity testing). The surviving open problem is the consolidated methodological step:

Within the generalized Roy selection framework used to estimate marginal returns to education, establish nonparametric identification (point identification or sharp partial identification over an explicitly stated support) and feasible estimation with valid inference for the marginal treatment effect $\mathrm{MTE}(x,u)$ and its policy-relevant functionals ($\mathrm{MPRTE}$, $\mathrm{ATE}$, $\mathrm{TT}$, $\mathrm{TUT}$, and PRTEs), within a single unified framework that jointly removes all three of the maintained assumption classes that CHV (2011) had to invoke:

(a) Weaker independence. Replace the maintained full independence $(\mathbf X,\mathbf Z)\perp(U_0,U_1,U_S)$ — invoked so that the MTE is identified over the marginal support of $P(\mathbf Z)$ rather than the support conditional on $\mathbf X$ — by strictly weaker dependence conditions (for example limited or conditional independence, or endogenous covariates), while preserving nonparametric recovery of the MTE and the interpretability of $U_S$ as latent resistance.

(b) Instrument invalidity. Estimate the MTE and test for selection on gains when one or more instruments may be correlated with $(U_0,U_1,U_S)$ beyond conditioning on $\mathbf X$ — robustness within the selection framework.

(c) Multi-valued treatments. Extend the same machinery to ordered, multi-level schooling transitions (for example high-school graduation, two-year college, four-year college), so that transition-specific MTEs are identified and estimated with feasible precision, replacing the case-specific constructions that the 2024 Handbook survey documents.

The common objective uniting (a)-(c) is removing the assumptions that currently stand between the selection framework and credible nonparametric recovery of marginal and policy-relevant returns to education.

References

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