Heterogeneous Treatment Effects in the Presence of Self-Selection: A Propensity Score Perspective

The MTE approach builds on the generalized Roy model for discrete choices (Heckman and Vytlacil 2007a; Roy 1951). Consider two potential outcomes, $Y_1$ and $Y_0$, a binary indicator $D$ for treatment status, and a vector of pretreatment covariates $X$. $Y_1$ denotes the potential outcome if the individual were treated ($D=1$), and $Y_0$ denotes the potential outcome if the individual were not treated ($D=0$). We specify the outcome equations as

where ${\mathrm{\mu }}_0(X)=\mathbb{E}[Y_0|X]$, ${\mathrm{\mu }}_1(X)=\mathbb{E}[Y_1|X]$, the error term $\mathrm{ϵ}$ captures all unobserved factors that affect the baseline outcome ($Y_0$), and the error term $\mathrm{\eta }$ captures all unobserved factors that affect the treatment effect ($Y_1-Y_0$). In general, the error terms $\mathrm{ϵ}$ and $\mathrm{\eta }$ need not be statistically independent of $X$, although they have zero conditional means by construction. The observed outcome $Y$ can be linked to the potential outcomes through the switching regression model (Quandt 1958, 1972):

Treatment assignment is represented by a latent index model. Let $I_D$ be a latent tendency for treatment, which depends on both observed ($Z$) and unobserved ($V$) factors:

where ${\mathrm{\mu }}_D(Z)$ is an unspecified function and $V$ is a latent random variable representing unobserved, individual-specific resistance to treatment, assumed to be continuous with a strictly increasing distribution function. The $Z$ vector includes all the components of $X$, but it also includes some instrumental variables (IV) that affect only the treatment status $D$. The key assumptions associated with Equations 1, 2, 4, and 5 are

• Assumption 1. $(\mathrm{ϵ},\mathrm{\eta },V)$ are statistically independent of $Z$ given $X$ (independence).

• Assumption 2. ${\mathrm{\mu }}_D(Z)$ is a nontrivial function of $Z$ given $X$ (rank condition).

The latent index model characterized by Equations 4 and 5 combined with Assumptions 1 and 2 is equivalent to the Imbens-Angrist (Imbens and Angrist 1994) assumptions of independence and monotonicity for the interpretation of IV estimands as local average treatment effects (LATE; Vytlacil 2002). Given Assumptions 1 and 2, the latent resistance $V$ is allowed to be correlated with $\mathrm{ϵ}$ and $\mathrm{\eta }$ in a general way. For example, research considering heterogeneous returns to schooling has argued that individuals may self-select into college on the basis of their anticipated gains. In this case, $V$ will be negatively correlated with $\mathrm{\eta }$ because individuals with higher values of $\mathrm{\eta }$ tend to have lower levels of unobserved resistance $U$.²

To define the MTE, it is best to rewrite the treatment assignment Equations 4 and 5 as

where $F_{V|X}(·)$ is the cumulative distribution function of $V$ given $X$ and $P(Z)=\mathrm{Pr}(D=1|Z)=F_{V|X}({\mathrm{\mu }}_D(Z))$ denotes the propensity score given $Z$. $U=F_{V|X}(V)$ is the quantile of $V$ given $X$, which by definition follows a standard uniform distribution. From Equation 6, we can see that $Z$ affects treatment status only through the propensity score $P(Z)$.³

The MTE is defined as the expected treatment effect conditional on pretreatment covariates $X=x$ and the normalized latent variable $U=u$:

Because $U$ is the quantile of $V$, the variation of $\mathrm{MTE}(x,u)$ over values of $u$ reflects how treatment effect varies with different quantiles of the unobserved resistance to treatment. Alternatively, $\mathrm{MTE}(x,u)$ can be interpreted as the average treatment effect among individuals who are indifferent between treatment or not with covariates $X=x$ and the propensity score $P(Z)=u$.

A wide range of causal estimands, such as ATE and TT, can be expressed as weighted averages of $\mathrm{MTE}(x,u)$ (Heckman et al. 2006). To obtain population-level causal effects, $\mathrm{MTE}(x,u)$ needs to be integrated twice, first over $u$ given $X=x$ and then over $x$. The weights for integrating over $u$ are shown in Table 1. Note that these weights are conditional on $X=x$. To estimate overall ATE, TT, and treatment effect of the untreated (TUT), we need to further integrate estimates of $\mathrm{ATE}(x)$, $\mathrm{TT}(x)$, and $\mathrm{TUT}(x)$ against appropriate marginal distributions of $X$.

Table 1. Weights for Constructing $\mathrm{ATE}(x)$, $\mathrm{TT}(x)$, and $\mathrm{TUT}(x)$ from $\mathrm{MTE}(x,u)$

Table 1. Weights for Constructing $\mathrm{ATE}(x)$, $\mathrm{TT}(x)$, and $\mathrm{TUT}(x)$ from $\mathrm{MTE}(x,u)$

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The estimation of $\mathrm{MTE}(x,u)$, however, is not straightforward because neither the counterfactual outcome nor the latent variable $U$ is observed. Moreover, the estimation of weights can be practically challenging (except for the ATE case) because it involves estimating the conditional density of $P(Z)$ given $X$ and the latter is usually a high-dimensional vector. We turn to these estimation issues now.

Given Assumptions 1 and 2, $\mathrm{MTE}(x,u)$ can be nonparametrically identified using the method of local instrumental variables (LIV).⁴ To see how it works, let us first write out the expectation of the observed outcome $Y$ given the covariates $X=x$ and the propensity score $P(Z)=p$. According to Equation 3, we have

Taking the partial derivative of Equation 8 with respect to $p$, we have

Because $\mathbb{E}(Y|X=x,P(Z)=p)$ is a function of observed (or estimable) quantities, the previous equation means $\mathrm{MTE}(x,u)$ is identified as long as $u$ falls within $\mathrm{supp}(P(Z)|X$), the conditional support of $P(Z)$ given $X=x$. In other words, $\mathrm{MTE}(x,u)$ is nonparametrically identified over $\mathrm{supp}(X,P(Z))$, the support of the joint distribution of $X$ and $P(Z)$.

In practice, however, it is difficult to condition on $X$ nonparametrically, especially when $X$ is high-dimensional. Therefore, in most empirical work using LIV, it is assumed that $(X,Z)$ is jointly independent of $(\mathrm{ϵ},\mathrm{\eta },V)$ (e.g., Carneiro et al. 2011; Carneiro and Lee 2009; Maestas, Mullen, and Strand 2013). Under this assumption, the MTE is additively separable in $x$ and $u$:

The additive separability not only simplifies estimation, but it allows $\mathrm{MTE}(x,u)$ to be identified over $\mathrm{supp}(X)\times \mathrm{supp}(P(Z))$ (instead of $\mathrm{supp}(X,P(Z)$)). The previous equation also suggests a necessary and sufficient condition for the MTE to be additively separable:

• Assumption 3. $\mathbb{E}[\mathrm{\eta }|X=x,U=u]$ does not depend on x (additive separability).

This assumption is implied by (but does not imply) the full independence between $(X,Z)$ and $(\mathrm{ϵ},\mathrm{\eta },V)$ (for a similar discussion, see Brinch, Mogstad, and Wiswall 2017).

In most applied work, the conditional means of $Y_0$ and $Y_1$ given $X$ are further specified as linear in parameters: ${\mathrm{\mu }}_0(X)={\mathrm{\beta }}_0^{T}X$ and ${\mathrm{\mu }}_1(X)={\mathrm{\beta }}_1^{T}X$. Given the linear specification and Assumptions 1, 2, and 3, $\mathbb{E}[Y|X=x,P(Z)=p]$ can be written as

where $K(p)$ is an unknown function that can be estimated either parametrically or nonparametrically.⁵

First, in the special case where the error terms $(\mathrm{ϵ},\mathrm{\eta },V)$ are assumed to be jointly normal with zero means and an unknown covariance matrix $\mathrm{\Sigma }$, the generalized Roy model characterized by Equations 1, 2, 4, and 5 is fully parameterized, and the unknown parameters $({\mathrm{\beta }}_1,{\mathrm{\beta }}_0,\mathrm{\gamma },\mathrm{\Sigma })$ can be jointly estimated via maximum likelihood.⁶ This model specification has a long history in econometrics and is usually called the “normal switching regression model” (Heckman 1978; for a review, see Winship and Mare 1992). With the joint normality assumption, Equation 9 reduces to

where ${\mathrm{\sigma }}_{\mathrm{\eta }V}$ is the covariance between $\mathrm{\eta }$ and $V$, ${\mathrm{\sigma }}_V$ is the standard deviation of $V$, and ${\mathrm{\Phi }}^{-1}(·)$ denotes the inverse of the standard normal distribution function.⁷ By plugging in the maximum likelihood estimates (MLE) of $({\mathrm{\beta }}_1,{\mathrm{\beta }}_0,{\mathrm{\sigma }}_{\mathrm{\eta }V},{\mathrm{\sigma }}_V)$, we obtain an estimate of $\mathrm{MTE}(x,u)$ for any combination of $x$ and $u$.

The joint normality of error terms is a strong and restrictive assumption. When errors are not normally distributed, imposition of normality may lead to substantial bias in estimates of the model parameters (Arabmazar and Schmidt 1982). To avoid this problem, Heckman and colleagues (2006) proposed to fit Equation 10 semiparametrically using a double residual procedure (Robinson 1988). In this case, the estimation of $\mathrm{MTE}(x,u)$ can be summarized in four steps:

1. Estimate the propensity scores using a standard logit/probit model, denote them as $\hat{P}$.⁸

2. Fit local linear regressions of $Y$, $X$, and $X\hat{P}$ on $\hat{P}$ and extract their residuals $e_Y$, $e_X$, and $e_{X\hat{P}}$.

3. Fit a simple linear regression of $e_Y$ on $e_X$ and $e_{X\hat{P}}$ (with no intercepts) to estimate the parametric part of Equation 10, that is, $({\mathrm{\beta }}_0,{\mathrm{\beta }}_1-{\mathrm{\beta }}_0)$, and store the remaining variation of $Y$ as $e_Y^{*}=Y-{\hat{\mathrm{\beta }}}_0^{T}X-({\hat{\mathrm{\beta }}}_1-{\hat{\mathrm{\beta }}}_0{)}^{T}X\hat{P}$.

4. Fit a local quadratic regression (Fan and Gijbels 1996) of $e_Y^{*}$ on $\hat{P}$ to estimate $K(p)$ and its derivative $K\prime (p)$.

The MTE is then estimated as

With estimates of $\mathrm{MTE}(x,u)$, we still need appropriate weights to estimate aggregate causal effects such as ATE and TT. As shown in Table 1, most weights involve the conditional density of $P(Z)$ given $X$. Because the latter is often a high-dimensional vector, direct estimation of these weights is challenging. In their empirical application, Carneiro and colleagues (2011) conditioned on an index of $X$, $({\hat{\mathrm{\beta }}}_1-{\hat{\mathrm{\beta }}}_0{)}^{T}X$, instead of $X$. In other words, they used $f[\hat{P}|({\hat{\mathrm{\beta }}}_1-{\hat{\mathrm{\beta }}}_0{)}^{T}X]$ as an approximation to $f[P(Z)|X]$. To estimate the former, we can first estimate the bivariate density $f[\hat{P},({\hat{\mathrm{\beta }}}_1-{\hat{\mathrm{\beta }}}_0{)}^{T}X]$ using kernel methods and then divide the estimated bivariate density by the marginal density $f[({\hat{\mathrm{\beta }}}_1-{\hat{\mathrm{\beta }}}_0{)}^{T}X]$. As we will see, these ad hoc methods for estimating weights are no longer needed with our new approach.

Under the generalized Roy model, a single latent variable $U$ not only summarizes all unobserved determinants of treatment status but also captures all the treatment-effect heterogeneity by unobserved characteristics that may cause selection bias. In fact, the latent index structure implies that all the treatment-effect heterogeneity that is consequential for selection bias exists only along two dimensions: (1) the propensity score $P(Z)$ and (2) the latent variable $U$ representing unobserved resistance to treatment. This is directly reflected in Equation 6: a person is treated if and only if his or her propensity score exceeds his or her (realized) latent resistance $u$. Therefore, given both $P(Z)$ and $U$, treatment status $D$ is fixed (either 0 or 1) and thus independent of treatment effect:

This expression resembles the Rosenbaum and Rubin (1983) result on the sufficiency of the propensity score except that we now condition on $U$ in addition to $P(Z)$. Thus, to characterize selection bias, it would be sufficient to model treatment effect as a bivariate function of the propensity score (rather than the entire vector of covariates) and the latent variable $U$. We redefine MTE as the expected treatment effect given $P(Z)$ and $U$:

Compared with the original MTE, $\widetilde{\mathrm{MTE}}(p,u)$ has two immediate advantages. First, because it conditions on the propensity score $P(Z)$ rather than the whole vector of $X$, it captures all the treatment-effect heterogeneity that is relevant for selection bias in a more parsimonious way. Second, by discarding treatment effect variation that is orthogonal to the two-dimensional space spanned by $P(Z)$ and $U$, $\widetilde{\mathrm{MTE}}(p,u)$ is a bivariate function and thus easier to visualize than $\mathrm{MTE}(x,u)$.

As with $\mathrm{MTE}(x,u)$, $\widetilde{\mathrm{MTE}}(p,u)$ also can be used as a building block for constructing standard causal estimands such as ATE and TT. However, compared with the weights on $\mathrm{MTE}(x,u)$, the weights on $\widetilde{\mathrm{MTE}}(p,u)$ are simpler, more intuitive, and easier to compute. The weights for ATE, TT, and TUT are shown in the first three rows of Table 2. To construct $\mathrm{ATE}(p)$, we simply integrate $\widetilde{\mathrm{MTE}}(p,u)$ against the marginal distribution of $U$—a standard uniform distribution. To construct $\mathrm{TT}(p)$, we integrate $\widetilde{\mathrm{MTE}}(p,u)$ against the truncated distribution of $U$ given $U<p$. Likewise, to construct $\mathrm{TUT}(p)$, we integrate $\widetilde{\mathrm{MTE}}(p,u)$ against the truncated distribution of $U$ given $U\ge p$. To obtain population-level ATE, TT, and TUT, we further integrate $\mathrm{ATE}(p)$, $\mathrm{TT}(p)$, and $\mathrm{TUT}(p)$ against appropriate marginal distributions of $P(Z)$. For example, to construct TT, we integrate $\mathrm{TT}(p)$ against the marginal distribution of the propensity score among treated units.

Table 2. Weights for Constructing ATE, TT, TUT, PRTE, and MPRTE from $\widetilde{\mathrm{MTE}}(p,u)$

Table 2. Weights for Constructing ATE, TT, TUT, PRTE, and MPRTE from $\widetilde{\mathrm{MTE}}(p,u)$

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In practice, $\widetilde{\mathrm{MTE}}(p,u)$ can be estimated as a byproduct of $\mathrm{MTE}(x,u)$. Under Assumptions 1, 2, and 3,⁹$\widetilde{\mathrm{MTE}}(p,u)$ can be written as

A proof of Equation 14 is given in Appendix A. Comparing Equation 14 with Equation 9, we see that the only difference between the original MTE and $\widetilde{\mathrm{MTE}}(p,u)$ is that the first component of $\widetilde{\mathrm{MTE}}(p,u)$ is now the conditional expectation of ${\mathrm{\mu }}_1(X)-{\mathrm{\mu }}_0(X)$ given the propensity score rather than ${\mathrm{\mu }}_1(X)-{\mathrm{\mu }}_0(X)$. Therefore, to estimate $\widetilde{\mathrm{MTE}}(p,u)$, we need only one more step after implementing the procedures described in Section 2.3: fit a nonparametric curve of $({\hat{\mathrm{\beta }}}_1-{\hat{\mathrm{\beta }}}_1{)}^{T}X$ with respect to $\hat{P}$ (e.g., using a local linear regression) and combine it with existing estimates of $K\prime (u)$.

The redefined MTE can be used not only to construct traditional causal estimands but also, in the context of program evaluation, to draw implications for how the program should be revised in the future. To predict the impact of an expansion (or contraction) in program participation, one needs to examine treatment effects for individuals who would be affected by such an expansion (or contraction). To formalize this idea, Heckman and Vytlacil (2001b, 2005) define the policy-relevant treatment effect (PRTE) as the mean effect of moving from a baseline policy to an alternative policy per net person shifted into treatment, that is,

They further show that under the generalized Roy model, the PRTE depends on a policy change only through its effects on the distribution of the propensity score $P(Z)$. Specifically, conditional on $X=x$, the PRTE can be written as a weighted average of $\mathrm{MTE}(x,u)$, where the weights depend only on the distribution of $P(Z)$ before and after the policy change. Within this framework, Carneiro and colleagues (2010) further define the marginal policy-relevant treatment effect (MPRTE) as a directional limit of the PRTE as the alternative policy converges to the baseline policy. Denoting by $F(·)$, the cumulative distribution function of $P(Z)$, they consider a set of alternative policies indexed by a scalar $\mathrm{\alpha }$, $\{F_{\mathrm{\alpha }}$: $\mathrm{\alpha }\in \mathbb{R}$}, such that $F_0$ corresponds to the baseline policy. The MPRTE is defined as

We follow their approach to analyzing policy effects but without conditioning on $X$. Whereas Carneiro and colleagues (2010) assume that the effects of all policy changes are through shifts in the conditional distribution of $P(Z)$ given $X$, we focus on policy changes that shift the marginal distribution of $P(Z)$ directly. In other words, we consider policy interventions that incorporate individual-level treatment-effect heterogeneity by values of $P(Z)$, whether their differences in $P(Z)$ are induced by their baseline characteristics $X$ or the instrumental variables $Z\backslash X$. In Section 3.5, we compare these two approaches in more detail and discuss some major advantages of our new approach.

Specifically, let us consider a class of policy changes under which the $i$th individual’s propensity of treatment is boosted by $\mathrm{\lambda }(p_i)$ (in a way that does not change his or her treatment effect), where $p_i$ denotes the individual’s propensity score $P(z_i)$ and $\mathrm{\lambda }(·)$ is a positive, real-valued function such that $p+\mathrm{\lambda }(p)\le 1$ for all $p\in [0,1)$. The policy change thus nudges everyone in the same direction, and two persons with the same baseline probability of treatment share an inducement of the same size. For such a policy change, the PRTE given $P(Z)=p<1$ and $\mathrm{\lambda }(p)$ becomes

As with standard causal estimands, $\mathrm{PRTE}(p,\mathrm{\lambda }(p))$ can be expressed as a weighted average of $\widetilde{\mathrm{MTE}}(p,u)$:

Here, the weight on $u$ is constant (i.e., $1/\mathrm{\lambda }(p)$) within the interval of $[p,p+\mathrm{\lambda }(p))$ and zero elsewhere.

To examine the effects of marginal policy changes, let us consider a sequence of policy changes indexed by a real-valued scalar $\mathrm{\alpha }$. Given $P(Z)=p$, we define the MPRTE as the limit of $\mathrm{PRTE}(p,\mathrm{\alpha }\mathrm{\lambda }(p))$ as $\mathrm{\alpha }$ approaches zero:

Hence, we have established a direct link between $\mathrm{MPRTE}(p)$ and $\widetilde{\mathrm{MTE}}(p,u)$: At each level of the propensity score, the MPRTE is simply the $\widetilde{\mathrm{MTE}}$ at the margin where $u=p$. As shown in the last row of Table 2, $\mathrm{MPRTE}(p)$ can also be expressed as a weighted average of $\widetilde{\mathrm{MTE}}(p,u)$ using the Dirac delta function.

Figure 1 illustrates the relationships between ATE, TT, TUT, and MPRTE. Panel a shows a shaded gray plot of $\widetilde{\mathrm{MTE}}(p,u)$ for heterogeneous treatment effects in a hypothetical setup. In this plot, both the propensity score $p$ and the latent resistance $u$ (both ranging from 0 to 1) are divided into 10 equally spaced strata, yielding 100 grids, and a darker grid indicates a higher treatment effect. The advantage of such a shaded gray plot is that we can use subsets of the 100 grids to represent meaningful subpopulations. For example, we present the grids for treated units in Panel b, untreated units in Panel c, and marginal units in Panel d. Thus, evaluating ATE, TT, TUT, and MPRTE simply means taking weighted averages of $\widetilde{\mathrm{MTE}}(p,u)$ over the corresponding subsets of grids.

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Figure 1. Demonstration of treatment-effect heterogeneity by propensity score $P(Z)$ and latent variable $U$.

Note: A darker color means a higher treatment effect.

For policymakers, a key question of interest would be how $\mathrm{MPRTE}(p)$ varies with the propensity score $p$. From Equations 14 and 15, we see that $\mathrm{MPRTE}(p)$ consists of two components:

The first component reflects how treatment effect varies by the propensity score, and the second component reflects how treatment effect varies by the latent resistance $U$. Among marginal entrants, $P(Z)$ is equal to $U$ so that these two components fall on the same dimension.

To see how the two components combine to shape $\mathrm{MPRTE}(p)$, let us revisit the classic example on economic returns to college. In the labor economics literature, researchers often have found a negative association between $\mathrm{\eta }$ and $U$, suggesting a pattern of positive selection, that is, individuals who benefit more from college are more motivated than their peers to attend college in the first place (e.g., Blundell, Dearden, and Sianesi 2005; Carneiro et al. 2011; Heckman, Humphries, and Veramendi 2018; Moffitt 2008; Willis and Rosen 1979). In this case, the second component of Equation 16 would be a decreasing function of $p$. On the other hand, the literature has not paid much attention to the first component, that is, whether individuals who by observed characteristics are more likely to attend college also benefit more from college. A number of observational studies suggest that nontraditional students, such as racial and ethnic minorities or students from less educated families, experience higher returns to college than do traditional students, although interpretation of such findings remains controversial due to potential unobserved selection biases (e.g., Attewell and Lavin 2007; Bowen and Bok 1998; Dale and Krueger 2011; Maurin and McNally 2008; for a review, see Hout 2012).¹⁰ However, if the downward slope in the second component were sufficiently strong, $\mathrm{MPRTE}(p)$ would also decline with $p$. In this case, we would, paradoxically, observe a pattern of negative selection (Brand and Xie 2010): Among students who are at the margin of attending college, those who by observed characteristics are less likely to attend college would actually benefit more from college.

To better understand the paradoxical implication of self-selection, let us revisit Figure 1. From Panel a, we see that in the hypothetical data, treatment effect declines with $u$ at each level of the propensity score, suggesting unobserved self-selection. In other words, individuals may have self-selected into treatment on the basis of their anticipated gains. On the other hand, at each level of the latent variable $u$, treatment effect increases with the propensity score, indicating that individuals who by observed characteristics are more likely to be treated also benefit more from the treatment. This relationship, however, is reversed among the marginal entrants. As shown in Panel d, among the marginal entrants, individuals who appear less likely to be treated (bottom left grids) have higher treatment effects. This pattern of negative selection at the margin, interestingly, is exactly due to an unobserved positive selection into treatment.

In Section 3.2, we used $\mathrm{\lambda }(p)$ to denote the increment in treatment probability at each level of the propensity score $p$. Because $\mathrm{MPRTE}(p)$ is defined as the pointwise limit of $\mathrm{PRTE}(p,\mathrm{\alpha }\mathrm{\lambda }(p))$ as $\mathrm{\alpha }$ approaches zero, the mathematical form of $\mathrm{\lambda }(p)$ does not affect $\mathrm{MPRTE}(p)$. However, in deriving the population-level (i.e., unconditional) MPRTE, we need to use $\mathrm{\lambda }(p)$ as the appropriate weight, that is,

Here $F_P(·)$ is the marginal distribution function of the propensity score, and $C=1/{\int }_0^1\mathrm{\lambda }(p)d{F}_P(p)$ is a normalizing constant (see Appendix B for a derivation). Thus, given the estimates of $\mathrm{MPRTE}(p)$, a policymaker could use the previous equation to design a formula for $\mathrm{\lambda }(·)$ to boost the population-level MPRTE. This is especially useful if $\mathrm{MPRTE}(p)$ varies systematically with the propensity score $p$. For example, if one found that the marginal return to college declines with the propensity score $p$, a college expansion program targeted at students with relatively low values of $p$ (e.g., a means-tested financial aid program) would yield higher average marginal returns than would a universal expansion of college enrollment regardless of student characteristics.¹¹

In practice, for a given policy $\mathrm{\lambda }(p)$, we can evaluate the aforementioned integral directly from sample data using

where ${\hat{p}}_i$ is the estimated propensity score for unit $i$ in the sample. When the sample is not representative of the population by itself, sampling weights need to be incorporated into these summations.

In the previous discussion, PRTE and MPRTE are defined for a class of policy changes in which the intensity of policy intervention depends on the individual’s propensity score $P(Z)$. In other words, inducements are differentiated between individuals with different values of $P(Z)$, whether their differences in $P(Z)$ are determined by the baseline covariates $X$ or the instrumental variables $Z\backslash X$. This approach to defining MPRTE contrasts sharply with the approach taken by Carneiro and colleagues (2010, 2011), who stipulate that all policy changes have to be “conditioned on $X$.” In their approach, inducements are allowed to vary across individuals with different values of $Z\backslash X$ but not across individuals with different values of $X$. For convenience, we call Carneiro et al’s approach the conditional approach and our approach the unconditional approach. Compared with the conditional approach, the unconditional approach to studying policy effects has several major advantages.

First, as noted earlier, preferential policies under the conditional approach distinguish individuals with different instrumental variables ($Z\backslash X$) but not individuals with different baseline characteristics ($X$). To see the limitation of such policies, let us revisit the college education example and consider a simplistic model where the only baseline covariate $X$ is family income and the only instrumental variable $Z\backslash X$ is the presence of four-year colleges in the county of residence. In this case, an “affirmative” policy—a policy that favors students with lower values of $P(Z)$—would be a policy that induces students who happen to live in a county with no four-year colleges, regardless of family income. Given that $P(Z)$ equals $U$ at the margin, this policy benefits students with relatively low $U$s at all levels of family income. To the extent that there is self-selection into college (i.e., $\mathrm{Cor}(\mathrm{\eta },U)<0$), this policy would yield a larger MPRTE than would a neutral policy. However, if $P(Z)$ is largely determined by family income rather than the local presence of four-year colleges (a plausible scenario), the variation of $P(Z)$ conditional on $X$ would be very limited, as would the gain in MPRTE from a preferential policy. In contrast, the unconditional approach distinguishes individuals with different values of $P(Z)$, most of which may be driven by $X$ rather than $Z\backslash X$. Because $P(Z)$ equals $U$ at the margin, this approach can sort out marginal entrants with different levels of $U$ effectively. Therefore, preferential policies under the unconditional approach are more effective in exploiting unobserved heterogeneity in treatment effects.

Second, because treatment effect in general depends on the observed covariates $X$ as well as the latent resistance $U$, an ideal policy intervention should exploit the variation of treatment effect along both dimensions. The conditional approach, however, differentiates individuals with different $U$s but not individuals with different observed characteristics (at least in practice). In contrast, by focusing on the propensity score $P(Z)$, the unconditional approach accounts for treatment-effect heterogeneity in both observed and unobserved dimensions. Because $P(Z)$ equals $U$ at the margin, the bivariate function $\widetilde{\mathrm{MTE}}(p,u)$ degenerates into a univariate function of $p$ among marginal entrants (see Equation 16). Thus, by weighting individuals with different values of $P(Z)$, the unconditional approach captures the “collision” of observed and unobserved heterogeneity at the margin. To see why the latter is more effective, consider an extreme scenario where there is no unobserved sorting (i.e., $E(\mathrm{\eta }|U)$ is constant) but treatment effect varies considerably by $X$. In this case, the unconditional approach can partly exploit the variation of treatment effect by $X$ (through the first component of Equation 16), whereas the conditional approach cannot (because it focuses exclusively on the second component of Equation 16).

Finally, the unconditional approach is computationally simpler. $\mathrm{MPRTE}(p)=\widetilde{\mathrm{MTE}}(p,p)$, so no further step is needed to estimate $\mathrm{MPRTE}(p)$ once we have estimates of $\widetilde{\mathrm{MTE}}(p,u)$. The conditional approach, by contrast, needs to build $\mathrm{MPRTE}(x)$ on $\mathrm{MTE}(x,u)$ using policy-specific weights. As shown in Table 3, these policy-specific weights generally involve the conditional density of $P(Z)$ given $X$. Because $X$ is usually a high-dimensional vector, estimation of these weights is difficult and often tackled with ad hoc methods (see Section 2.3).

Table 3. Weights for Constructing $\mathrm{MPRTE}(x)$ from $\mathrm{MTE}(x,u_D)$

Table 3. Weights for Constructing $\mathrm{MPRTE}(x)$ from $\mathrm{MTE}(x,u_D)$

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We reanalyze a sample of white males (N = 1,747) who were 16 to 22 years old in 1979, drawn from the NLSY 1979. Treatment ($D$) is college attendance defined by having attained any postsecondary education by 1991. Under this definition, the treated group consists of 865 individuals, and the comparison group consists of 882 individuals. The outcome $Y$ is the natural logarithm of hourly wage in 1991.¹² Following the original study, we include in pretreatment variables (in both $X$ and $Z$) linear and quadratic terms of mother’s years of schooling, number of siblings, the Armed Forces Qualification Test (AFQT) score adjusted by years of schooling, permanent local log earnings at age 17 (county log earnings averaged between 1973 and 2000), and permanent local unemployment rate at age 17 (state unemployment rate averaged between 1973 and 2000) as well as a dummy variable indicating urban residence at age 14 and cohort dummies. Also following Carneiro and colleagues (2011), we use the following instrumental variables ($Z\backslash X$): (1) the presence of a four-year college in the county of residence at age 14, (2) local wage in the county of residence at age 17, (3) local unemployment rate in the state of residence at age 17, and (4) average tuition in public four-year colleges in the county of residence at age 17 as well as their interactions with mother’s years of schooling, number of siblings, and the adjusted AFQT score. In addition, four variables are included in $X$ but not in $Z$: years of experience in 1991, years of experience in 1991 squared, local log earnings in 1991, and local unemployment rate in 1991. More details about the data can be found in Carneiro and colleagues’ (2011) online appendix.

To estimate the bivariate function $\widetilde{\mathrm{MTE}}(p,u)$, we first need estimates of $\mathrm{MTE}(x,u)$. In Section 2, we discussed a parametric and a semiparametric method for estimating $\mathrm{MTE}(x,u)$. Here, we examine treatment-effect heterogeneity with the semiparametric estimates of $\mathrm{MTE}(x,u)$ (Equation 12) and thus $\widetilde{\mathrm{MTE}}(p,u)$.¹³ Figure 2 presents our key results for the estimated $\widetilde{\mathrm{MTE}}(p,u)$, with a shaded gray plot in which a darker grid indicates a higher treatment effect. The effect heterogeneity by the two dimensions—the propensity score and the latent resistance to treatment—exhibits an easy-to-interpret but surprising pattern. On the one hand, at each level of the propensity score, a higher level of the latent variable $u$ is associated with a lower treatment effect, indicating the presence of self-selection based on idiosyncratic returns to college. This pattern of “sorting on gain” echoes the classic findings reported in Willis and Rosen (1979) and Carneiro and colleagues (2011). On the other hand, the color gradient along the propensity score suggests that given the latent resistance to attending college, students who by observed characteristics are more likely to go to college also tend to benefit more from attending college.

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Figure 2. Treatment-effect heterogeneity based on semiparametric estimates of $\widetilde{\mathrm{MTE}}(p,u)$.

Note: MTE = marginal treatment effect.

If we read along the diagonal of Figure 2, however, we find that among students who are at the margin of indifference for attending college, those who appear less likely to attend college would benefit more from a college education, that is, $\mathrm{MPRTE}(p)$ declines with the propensity score $p$. Figure 3 shows smoothed estimates of $\mathrm{MPRTE}(p)$ as well as its two components (see Equation 16). The negative association between $\mathrm{\eta }$ and the latent resistance $U$ more than offsets the positive association between $({\mathrm{\beta }}_1-{\mathrm{\beta }}_0{)}^{T}X$ and the propensity score $P(Z)$, resulting in the downward slope of $\mathrm{MPRTE}(p)$. Echoing our discussion in Section 3.3, it is unobserved “sorting on gain” that leads to the negative association between the propensity score and returns to college among students at the margin.

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Figure 3. Decomposition of $\mathrm{MPRTE}(p)$ based on semiparametric estimates of $\widetilde{\mathrm{MTE}}(p,u)$.

Note: MPRTE = marginal policy-relevant treatment effect; MTE = marginal treatment effect.

We use weights given in Table 2 to estimate ATE, TT, and TUT at each level of the propensity score. Figure 4 shows smoothed estimates of $\mathrm{ATE}(p)$, $\mathrm{TT}(p)$, $\mathrm{TUT}(p)$, and $\mathrm{MPRTE}(p)$. Several patterns are worth noting. First, there is a sharp contrast between $\mathrm{ATE}(p)$ and $\mathrm{MPRTE}(p)$: A higher propensity of attending college is associated with a higher return to college on average (solid line) but a lower return to college among marginal entrants (dot-dash line). Second, $\mathrm{TT}(p)$ (dashed line) is always larger than $\mathrm{TUT}(p)$ (dotted line), suggesting that at each level of the propensity score, individuals are positively self-selected into college based on their idiosyncratic returns to college. Finally, $\mathrm{TT}(p)$ and $\mathrm{TUT}(p)$ converge to $\mathrm{ATE}(p)$ and $\mathrm{MPRTE}(p)$ at the extremes of the propensity score. When $p$ approaches 0, $\mathrm{TT}(p)$ converges to $\mathrm{MPRTE}(p)$ and $\mathrm{TUT}(p)$ converges to $\mathrm{ATE}(p)$. At the other extreme, when $p$ approaches 1, $\mathrm{TT}(p)$ converges to $\mathrm{ATE}(p)$ and $\mathrm{TUT}(p)$ converges to $\mathrm{MPRTE}(p)$. Looking back at Figure 1, we see that these relationships simply reflect compositional shifts in the treated and untreated groups as the propensity score changes from 0 to 1.

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Figure 4. Heterogeneity in ATE, TT, TUT, and $\mathrm{MPRTE}(p)$ by propensity score based on semiparametric estimates of $\widetilde{\mathrm{MTE}}(p,u)$.

Note: ATE = average treatment effect; TT = treatment effect of the treated; TUT = treatment effect of the untreated; MPRTE = marginal policy-relevant treatment effect; MTE = marginal treatment effect.

Given the estimates of $\mathrm{ATE}(p)$, $\mathrm{TT}(p)$, and $\mathrm{TUT}(p)$, we construct their population averages using appropriate weights across the propensity score. For example, to estimate TT, we simply integrate $\mathrm{TT}(p)$ against the marginal distribution of the propensity score among individuals who attended college. The estimates of $\mathrm{MPRTE}(p)$ allow us to construct different versions of MPRTE, depending on how the policy change weights students with different propensities of attending college (see Equation 18). Table 4 reports our estimates of ATE, TT, TUT, and MPRTE under different policy changes from the parametric and semiparametric estimates of $\mathrm{MTE}(x,u)$. To compare our new approach with Carneiro and colleagues’ (2011) original approach, we show estimates built on $\widetilde{\mathrm{MTE}}(p,u)$ and those built on $\mathrm{MTE}(x,u)$. Following Carneiro and colleagues (2011), we annualize the returns to college by dividing all parameter estimates by four, which is the average difference in years of schooling between the treated and untreated groups.

Table 4. Estimated Returns to One Year of College

Table 4. Estimated Returns to One Year of College

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The first three rows of Table 4 indicate that TT > ATE > TUT ≈ 0. That is, returns to college are higher among individuals who actually attended college than among those who attended only high school, for whom the average returns to college are virtually zero. Using either the parametric or semiparametric estimates of $\mathrm{MTE}(x,u)$, our new approach and the original approach yield nearly identical point estimates and bootstrapped standard errors. This consistency affirms our argument that $\widetilde{\mathrm{MTE}}(p,u)$ preserves all of the treatment-effect heterogeneity that is consequential for selection bias. Although the redefined MTE seems to contain less information than the original MTE (as it projects $({\mathrm{\beta }}_1-{\mathrm{\beta }}_0{)}^{T}X$ onto the dimension of $P(Z)$), the discarded information does not contribute to identification of average causal effects.

The last four rows of Table 4 present our estimates of MPRTE under four stylized policy changes: (1) $\mathrm{\lambda }(p)=\mathrm{\alpha }$, (2) $\mathrm{\lambda }(p)=\mathrm{\alpha }p$, (3) $\mathrm{\lambda }(p)=\mathrm{\alpha }(1-p)$, and (4) $\mathrm{\lambda }(p)=\mathrm{\alpha }\mathbb{I}(p<0.3)$. Put in words, the first policy change increases everyone’s probability of attending college by the same amount, the second policy change favors students who appear more likely to go to college, the third policy change favors students who appear less likely to go to college, and the last policy change only targets students whose observed likelihood of attending college is less than 30 percent. For each policy change, the MPRTE is defined as the limit of the corresponding PRTE as $\mathrm{\alpha }$ goes to zero. The first policy change is also the first policy change considered by Carneiro and colleagues (2011:2760), that is, $P_{\mathrm{\alpha }}=P+\mathrm{\alpha }$ (see also the first row of Table 3). For this case, we estimated the MPRTE using both the original approach and our new approach. As expected, the two approaches yield the same results. However, the other three policy changes considered here cannot be readily accommodated within the original framework (see Section 3.5). Thus, we evaluate their effects using only our new approach, that is, via Equation 18.

Although the estimates of TUT are close to zero, all four policy changes imply substantial marginal returns to college. For example, under the first policy change, the semiparametric estimate of MPRTE is .083, suggesting that one year of college would translate into an 8.3 percent increase in hourly wages among marginal entrants. However, the exact magnitude of MPRTE depends heavily on the form of the policy change, especially under the semiparametric model. Whereas the marginal return to a year of college is about 5 percent if we expand everyone’s probability of attending college proportionally (policy change two), it can be as high as 15.5 percent if we only increase enrollment among students whose observed likelihood of attending college is less than 30 percent (policy change four). Figure 5 shows how different policy changes produce different compositions of marginal college entrants. Because students who benefit the most from college are located at the low end of the propensity score, a college expansion program targeted at those students will yield the highest marginal returns to college. Fortuitously, earlier research that did not account for the presence of unobserved selection reached similar policy implications (Brand and Xie 2010).

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Figure 5. Semiparametric estimates of marginal policy-relevant treatment effect under four policy changes.

Note: MTE = marginal treatment effect.