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Human Capital and Economic Opportunity: A Global Working Group Working Paper Series Working Paper No. 2011-002

The Effects of Educational Choices on Labor Market, Health, and Social Outcomes James J. Heckman John Eric Humphries Sergio Urzua Gregory Veramendi

October, 2011

Human Capital and Economic Opportunity Working Group Economic Research Center University of Chicago 1126 E. 59th Street Chicago IL 60637 [email protected]

THE EFFECTS OF EDUCATIONAL CHOICES ON LABOR MARKET, HEALTH, AND SOCIAL OUTCOMES ∗ James J. Heckman University of Chicago, University College Dublin and the American Bar Foundation

Sergio Urz´ ua Northwestern University, NBER and IZA

John Eric Humphries University of Chicago

Gregory Veramendi Aarhus University

First version: October 10, 2009 This version: July 26, 2011



James Heckman: Department of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637; phone, 773-702-0634; fax, 773-702-8490; email, [email protected]. John Eric Humphries: Department of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637; phone, 773-980-6575; email, [email protected]. Sergio Urz´ ua, Department of Economics and Institute for Policy Research, Northwestern University, Handerson Hall, 2001 Sheridan Road, Evanston, IL 60208; phone, 847-491-8213; email, [email protected]. Gregory Veramendi, Aarhus University, School of Economics and Management, Bartholins Alle 10, Building 1322, DK-8000 Aarhus C, Denmark; phone, 45-8942-1546; email, [email protected]. We thank Chris Taber for comments on this draft. This research was supported by NIH R01-HD32058-3, NSF SES-024158, and NSF SES-05-51089, the J.B. and M.K. Pritzker Foundation, NIH R01 HD054702, NIH R01-HD065072-01, an INET grant to the Milton Friedman Institute, an ERC grant to the University College Dublin, and the American Bar Foundation. The Web Appendix for this paper is http://jenni.uchicago.edu/effects-school-labor.

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Abstract Using a sequential model of educational choices, we investigate the effect of educational choices on labor market, health, and social outcomes. Unobserved endowments drive the correlations in unobservables across choice and outcome equations. We proxy these endowments with numerous measurements and account for measurement error in the proxies. For each schooling level, we estimate outcomes for labor market, health, and social outcome. This allows us to generate counter-factual outcomes for dynamic choices and a variety of policy and treatment effects. In our framework, responses to treatment vary among observationally identical persons and agents may select into the treatment on the basis of their responses. We find important effects of early cognitive and socio-emotional abilities on schooling choices, labor market outcomes, adult health, and social outcomes. Education at most levels causally produces gains on labor market, health, and social outcomes. We estimate the distribution of responses to education and find substantial heterogeneity on which agents act. Keywords: education, early endowments, factor models, health, treatment effects. JEL codes: C32, C38, I12, I14, I21 James Heckman Department of Economics University of Chicago 1126 East 59th Street, Chicago, IL 60637 Phone: 773-702-0634 Email: [email protected]

John Eric Humphries Department of Economics University of Chicago 1126 East 59th Street, Chicago, IL 60637 Phone: 773-980-6575 Email: [email protected]

Sergio Urz´ ua Department of Economics and Institute for Policy Research Northwestern University Handerson Hall, 2001 Sheridan Road, Evanston, IL 60208 Phone: 847-491-8213 Email: [email protected]

Gregory Veramendi School of Economics and Management Aarhus University Bartholins Alle 10, Building 1322, DK-8000 Aarhus C, Denmark Phone: 45-8942-1546 Email: [email protected]

Contents 1 Introduction

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2 The 2.1 2.2 2.3

6 7 8

Model The Sequential Model of Educational Attainment . . . . . . . . . . . . Labor Market, Health, and Social Outcomes . . . . . . . . . . . . . . . . Measurement System for Unobserved Cognitive and Socio-emotional Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Estimation Strategy

9 10

4 Defining Treatment Effects 12 4.1 Gains from Changing Final Schooling Levels . . . . . . . . . . . . . . . . 13 4.2 Treatment Effect of Educational Decisions . . . . . . . . . . . . . . . . 13 5 Data and Estimation Strategy 5.1 Outcomes . . . . . . . . . . . . . . . . . . . . . 5.1.1 Schooling Levels . . . . . . . . . . . . . 5.1.2 Labor Market Outcomes . . . . . . . . . 5.1.3 Physical Health and Healthy Behaviors 5.1.4 Mental Health . . . . . . . . . . . . . . 5.2 Social Outcomes . . . . . . . . . . . . . . . . . 5.3 Early Adverse Behavior . . . . . . . . . . . . . 5.4 Measurement System . . . . . . . . . . . . . . . 5.5 Exogenous Observed Characteristics . . . . . .

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6 Empirical Estimates 6.1 Estimates from the Measurement System . . . . . . . . . . . . . . . . . 6.2 The Effect of Cognitive and Socio-emotional Endowments on Schooling Decision, Labor Market, and Health Outcomes . . . . . . . . . . . . . . 6.3 Sorting into Schooling Level . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Treatment Effects: Comparison of Outcomes for Different Final Schooling Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Treatment Effects: Pair-wise Comparison by Decision Node . . . . . . . 6.7 Treatment Effects: Continuation Values in the Choice to Graduate from High School or Enroll in College . . . . . . . . . . . . . . . . . . . . . .

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7 Conclusions

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23 25 25 25 27 29

1

Introduction

This paper investigates the causal effect of education on labor market, health, and social outcomes. A positive association between education and labor market outcomes has long been noted (Mincer, 1958; Becker, 1964; Mincer, 1974). For example, a positive correlation between schooling and health is a well-established finding in the social sciences (Grossman, 1972, 2000, 2006). More recently, it has been noted that there is a positive association between education and social outcomes, such as welfare use and civic participation. To what extent these positive associations reflect causal effects of education is still subject to debate. Our analysis contributes to the literature on the causal effects of education on labor market outcomes (Card, 2001; Willis and Rosen, 1979; Carneiro, Heckman, and Vytlacil, 2010), health (Adams, 2002; Arendt, 2005; Lleras-Muney, 2005; Silles, 2009; Spasojevic, 2003; Arkes, 2003; Auld and Sidhu, 2005; Grossman, 2008; Grossman and Kaestner, 1997; Cutler and Lleras-Muney, 2010; Conti, Heckman, and Urzua, 2010), and participation in society (Coelli, Green, and Warburton, 2007; Milligan, Moretti, and Oreopoulos, 2004). We estimate a model of sequential schooling decisions in which individuals make their educational decisions based on expected returns and costs, which are determined by observed and unobserved characteristics (see Keane and Wolpin, 1997; Cameron and Heckman, 1998, 2001). Individuals are endowed with cognitive and socio-emotional abilities (Heckman, Stixrud, and Urzua, 2006; Urzua, 2008) and these endowments determine, in part, schooling attainment. We adjoin to our dynamic model of schooling choice data on labor market, health, and social outcomes, observed after the final schooling level is reached. We assume these outcomes are determined, in part, by unobserved characteristics, which can be correlated with the unobserved variables in the schooling choice model. Ours is a model of heterogenous dynamic treatment effects (Heckman and Vytlacil, 2007; Heckman, Urzua, and Vytlacil, 2006). Therefore, under our model, two observationally equivalent individuals might experience different treatment effects of education. We estimate a

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variety of different treatment effects and estimate differences in treatment effects across individuals with different levels of unobserved abilities. One contribution of this paper is that we estimate educational continuation values. Each educational choice opens up additional educational options. We estimate returns to schooling, both as the direct causal benefit between two final schooling levels, that is the traditional focus in the human capital literature (see, e.g., Becker, 1964, and the discussion in Heckman, Lochner, and Todd, 2006), as well as returns through continuation values, created by the options opened up by schooling. Our analysis contributes to the growing literature documenting the impact of cognition on health (Grossman, 1975; Shakotko, Edwards, and Grossman, 1982; Hartog and Oosterbeek, 1998; Elias, 2005; Auld and Sidhu, 2005; Kenkel, Lillard, and Mathios, 2006; Cutler and Lleras-Muney, 2010; Kaestner, 2008; Whalley and Deary, 2001; Gottfredson and Deary, 2004) and labor market outcomes (Cawley, Conneely, Heckman, and Vytlacil, 1997; Herrnstein and Murray, 1994; Neal and Johnson, 1996; Carneiro and Heckman, 2002; Glewwe, 2002). Furthermore, our analysis relates to the literature documenting the impact of socio-emotional development on health and labor market outcomes (Hampson and Friedman, 2008; Kaestner, 2008; Heckman, Stixrud, and Urzua, 2006; Cutler and Lleras-Muney, 2010). Our main empirical findings are: • We find substantial upward biases in effects of education that do not control for unobserved cognitive and noncognitive traits. • For most outcomes, the causal gain from education is increasing in school levels. • For a variety of outcomes measures, we find different effects of education for high and low-ability people. • Decomposing the return to education into its direct effect (the payment to a given level of education) and its effect on creating options for further education, we see that much of the difference in returns to education by ability levels arises from option values.

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• We find significant gains in labor market outcomes from graduating high school and going to college. These are larger for high-ability people. The GED has no significant benefit in the labor market or on other outcomes. • High school and college attainment causally reduce the probability of being a daily smoker. They improve physical health. High school and college enrollment reduce the probability of being a heavy drinker. Graduating from high school and from a four-year college improve reported physical health. College attainment improves mental health with the effect being much larger for low-ability individuals. • We find evidence of the impact of education on social behavior. Graduating from high school, enrolling in college, and graduating from college increase the probability of voting and decrease the probability of being divorced and the probability of being on welfare. The paper is organized as follows: Section 2 presents our model for measuring the returns to schooling. Section 3 describes our estimation strategy. Section 4 presents a detailed analysis of our data. Section 5 discusses the main empirical findings. Section 6 concludes.

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The Model

We estimate a model of sequential schooling decisions in which individuals make decisions about future schooling levels given their current state. After agents complete their educational decisions, we observe adult outcomes. If unobserved components driving schooling decisions are correlated with unobserved variables determining individual outcomes, it is necessary to control for such selection effects to identify the causal effects of education. We address the selection problem by analyzing a model of potential outcomes with unobserved heterogeneity.1 We present the model in the following way: 1

See the survey of dynamic discrete choice by Abbring and Heckman (2007) and the analysis of Heckman and Navarro (2007).

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• We first describe our sequential decision model for educational attainment. • We identify the schooling model using a version of matching on mismeasured covariates with proxies for the true covarites. This is a conditional independence assumption, previously used in Aakvik, Heckman, and Vytlacil (2005) and Carneiro, Hansen, and Heckman (2003). • Adult outcomes are defined separately by schooling level.

2.1

The Sequential Model of Educational Attainment

Following Cameron and Heckman (2001), each agent makes schooling decisions using a sequential choice model. The choices available to the agent are limited by their previous schooling decisions. Let an individual’s current schooling attainment be represented by j ∈ J , where J is the set of all possible schooling states. An individual with schooling attainment j makes his next educational decision out of choice set Cj . Let Dj,c = 1 if the individual with education state j chooses c ∈ Cj . We assume that individuals make optimal decisions at each educational state. The optimal choice, cˆ, is

cˆ = arg max{Ij,c }, c∈Cj

where Ij,c is the value of choice c for a person with educational attainment j. Thus, an individual’s next educational state j 0 is determined by his optimal educational decision, j 0 = cˆ. Finally, let D represent the set of educational decisions taken by an individual over his life cycle. We assume a binary decision model at each decision node. In particular, we assume that at a particular node, defined by schooling level j, the agent considers Cj = {j 0 , j 00 }. Thus, Dj,j 00 = 1 − Dj,j 0 , and we can fully analyze the individual decision by simply considering a discrete choice model of the form

Dj,j 00 =

   1 if Ij,j 00 ≥ 0   0 otherwise

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.

(1)

In the empirical implementation of our model, we assume a linear-in-the-parameters form for Ij,j 00 that approximates the underlying decision structure, as in Cameron and Heckman (2001): S S Ij,j 00 = Xj,j 00 βj,j 00 + αj,j 00 θ − νj,j 00 ,

(2)

where Xj,j 00 is a vector of observed variables relevant to the schooling decision of the agent with schooling level j, and θ is a vector of unobserved endowments. These endowments are unobserved to the econometrician but are known to the agent. θ links the unobservables in schooling choices and outcomes, discussed below. νj,j 00 represents an idiosyncratic error term and satisfies νj,j 00 ⊥⊥ (Xj,j 00 , θ), where “⊥⊥” denotes statistical independence. Therefore, νj,j 00 is assumed to be independent across agents and states. From the sequential decision model one can define a set of final schooling levels. Let s denote a final schooling level in the set of final schooling levels S = {s0 , s1 , ..., s¯}. Define a binary indicator, Hs , such that Hs = 1 if the individual attains the final schooling level s, and 0 otherwise. Thus,

Hs =

   1 if D1,j = Dj,j 0 = ... = Dj 00 ,s = 1, Ds,j 000 = 0

(3)

  0 otherwise.

2.2

Labor Market, Health, and Social Outcomes

We seek to estimate the causal effects of education on a variety of adult outcomes. We distinguish between continuous and discrete (binary) outcomes. • Continuous outcomes are approximated by a linear-in-the-parameters model. Let Ysk denote the outcome k(= {1, .., K}) associated with final schooling level s ∈ S. Thus, Ysk = Xks βsk + αsk θ + νsk ,

(4)

where Xks is the vector of observed controls relevant for outcome k, and θ is the vector of unobserved endowments. νsk represents an idiosyncratic error term such

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that νsk ⊥ ⊥ (Xks , θ). The νsk are mutually independent across s. Equations (3) and (4) can be used to define observed outcome Y k , using the conventional switching regression framework: Yk =

X

Hs Ysk .

(5)

s∈S

• We model binary outcomes using a latent index structure. Let Vsk denote the latent utility and outcome k associated with final schooling level s. The latent utility is given by a linear-in-the-parameters specification:

Vsk = Xks β˜sk + α ˜sk θ + ν˜sk ,

(6)

where Xks , θ, and ν˜sk have analogous definitions to the continuous outcome case. We can define a binary outcome variable, Bsk :

Bsk =

   1 if V k ≥ 0 s

.

(7)

  0 otherwise The observed outcome can be expressed as in the continuous case:

Bk =

X

Hs Bsk .

(8)

s∈S

2.3

Measurement System for Unobserved Cognitive and

Socio-emotional Endowments Given θ and condition on X, all outcomes and choices are statistically independent. If we could measure θ, we could condition on it (along with X) and do matching. (See Carneiro, Hansen, and Heckman, 2003, and Abbring and Heckman, 2007.) We do not directly measure θ, but we can proxy it and estimate and correct for the effects of any measurement error in the proxy. We follow Carneiro, Hansen, and Heckman (2003) and Heckman, Stixrud, and Urzua (2006) and identify the schooling choice model and the models for outcomes

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using information from a measurement system. Using this system allows us to interpret unobserved endowments as cognitive and socio-emotional abilities. Before introducing the measurement system, let θC and θSE denote the levels of cognitive and social-emotional abilities, respectively, so that θ = (θC , θSE ). We allow θC and θSE to be correlated. Let TsC be a vector of cognitive test scores, TsSE a set of variables that measure by socio-emotional abilities, and TsC,SE a set of variables influenced by cognitive and socioemotional abilities, all measured at schooling level s. We posit a linear measurement system for these variables. More precisely,

TsC

=

C C C C XC s βs + αs θ + es

TsSE

=

SE SE SE XSE + eSE s βs + αs θ s

(10)

XC,SE βsC,SE + α ˜ sC θC + α ˜ sSE θSE + eC,SE . s s

(11)

TsC,SE =

(9)

The structure assumed in (9), (10), and (11), when allowing for correlated factors, is identified if the model has one measure which depends only on cognitive ability (TsC ), one measure which depends only on socio-emotional ability (TsSE ), and several equations loading both on cognitive ability and socio-emotional ability (Tsc,SE ). A proof of nonparametric identification of the distribution of θ for our model is provided in the Web Appendix.2

3

Estimation Strategy

We estimate this model in two stages. The distribution of latent endowments and the schooling choice equations are estimated in the first stage, and equations governing adult outcomes are estimated in the second stage using estimates from the first stage. In this fashion, the measurement system is estimated separately from the outcome sys-

2

See Section A in the Web Appendix.

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tem, so that we do not force predictive power of the latent factors on adult outcomes in our estimation procedure. We assume νj,j 00 , νsk , ν˜sk , and es are mutually independent, mean-zero, unit variance, normal variates. Additionally, we assume that these errors are independent conditional on the observables and the unobserved factors. The factor structure is assumed to explain all of the correlations in unobservables across outcomes, conditional on Xi . Identification of the factors comes from the schooling and measurement system. This approach follows that from the analysis of Carneiro, Hansen, and Heckman (2003). Conditional on θ and X, all potential outcomes are independent of each other. As previously noted, our procedure is a version of matching where we do not measure a subset of the conditioning variables but instead match on proxies for θ and account for the effects of measurement error in the proxies in generating our estimates. The likelihood, assuming independence across observations, is

L =

Y

f (Yi , Bi , Di , Ti |Xi )

i

=

YZ

f (Yi , Bi |Di Xi θ)f (Di , Ti |Xi θ)f (θ)dθ,

i

where the last two steps are justified from the assumptions that θ ⊥⊥ Xi and that the outcomes are independent once we condition on θ and Xi . For the first stage, the sample likelihood is

1

L =

YZ i

f (Di , Ti |Xi , θ = z)dFθ (z),

(12)

θ∈Θ

where we integrate over the distributions of the latent factors. The goal of the first stage is to secure estimators, fˆ(Di , Ti |Xi , θ) and fˆ(θ), for f (Di , Ti | Xi , θ) and f (θ), respectively. In the second stage, we use first stage estimates to express the likelihood as 2

L =

YZ i

f (Yi , Bi |Di , Xi , θ = z)fˆ(Di , Ti |Xi , θ = z)dFˆθ (z).

(13)

θ∈Θ

Since Yi , Bi are independent from the first stage conditional on Xi , θ, Di under stan-

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dard conditions, we can obtain a consistent estimate of the parameters for the adult outcome models. Each stage is estimated using maximum-likelihood. Standard errors and confidence intervals are calculated by estimating two hundred bootstrap samples.

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Defining Treatment Effects

The estimated model generates the causal effect of education and ability on labor market, health, and social outcomes. Since the model can be used to produce counterfactual outcomes, we can create a variety of average and distributional treatment effects. They can be used to predict how causally manipulating education affects people at different ability levels and allows us to understand the effectiveness of policy for different segments of the population. The traditional literature on the returns to schooling defines its parameters in terms of the returns generated by going from one final schooling level to another (Becker, 1964). This approach ignores the sequential nature of schooling and the options created by going to an additional level of schooling. For example, consider the gains in going from being a GED to becoming a four-year college graduate. The GED may enter community college. The GED may complete community college. From community college, the GED may go on to a four year college and so forth. Each decision opens up further possibilities. There are many choices at multiple nodes of education. We analyze sequential decisions made by the individuals. We identify treatment effects at each binary decision node. For example, we estimate the treatment effect for deciding to graduate from high school or drop out (D0,1 ). But once agents graduate from high school, agents have the option of going to college and even graduating from college. Similarly, once agents drop out, they have the option of getting a GED. All of these schooling decisions are options that emerge from a dynamic model of schooling. We estimate the traditional gains from choosing between final schooling levels. Such gains are calculated relative to the return from being a high school dropout. In this way we can compare our results with other methods used in the literature. In addition,

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we estimate treatment effects for each sequential decision node. This method takes into account future options opened up by educational choices.

4.1

Gains from Changing Final Schooling Levels

Let Y0 be defined as the outcome for the final schooling level of a high school dropout and Ys is the final schooling level being studied. The average treatment effect in this case is measured in the full population:

E ∆AT s

ZZ ≡

Eν (Ys − Y0 |X = x, θ = z)dFX,θ (x, z),

(14)

where Eν is the expectation over idiosyncratic shocks to outcome Yj , j ∈ {0, s}. The average effect of the treatment on the treated is measured only for those who attain the final schooling being studied (s):

∆Ts T

ZZ ≡

Eν (Ys − Y0 |X = x, θ = z)dFX,θ|Hs =1 (x, z),

(15)

and the average effect of the treatment on the untreated is measured only for those who are high school dropouts (s = 0):

∆Ts U T

4.2

ZZ ≡

Eν (Ys − Y0 |X = x, θ = z)dFX,θ|H0 =1 (x, z).

(16)

Treatment Effect of Educational Decisions

The treatment effect of an educational decision is calculated by looking at the difference in expected outcomes when changing a single educational decision in the sequential schooling model. Since a given educational decision can open up further educational choices to be made in the future, in order to calculate the full effect of a given educational decision, the treatment effect needs to include the probability weighted benefit of further educational choices. Let the expected value of an educational decision

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(Dj,j 00 = 1) to an individual with X = x and θ = z be   X Pr s|X = x, θ = z, Dj,j 00 = 1 ×E (Ys |X = x, θ = z) , E Y |X = x, θ = z, Dj,j 00 = 1 ≡ s

where the expectation (E) is over future educational choices and idiosyncratic shocks,  Pr s|X = x, θ = z, Dj,j 00 = 1 is the probability that the individual stops at education level s, and Ys is the value of the outcome if the individual stops at education level s.3 Of course, Pr(s|Dj,j 00 = 1) = 0 if s is not accessible given Dj,j 00 = 1. Let the person-specific treatment effect for an individual changing his decision at decision node j be defined as the difference between the expected value of the decisions:

∆j,j 00 [Y |X = x, θ = z] ≡ E(Y |X = x, θ = z, Dj,j 00 = 1) − E(Y |X = x, θ = z, Dj,j 00 = 0).

This person-specific treatment effect takes into account not only the direct effect of the decision, but also includes the value of possible additional schooling.4 3

For example, the choice to graduate from high school opens up the possibility of enrolling in college and possibly graduating from college. Let s indicate the level of final schooling, where 0 corresponds to dropping out of high school, 1 to graduating high school, 2 to attaining a GED, 3 to attaining some college, and 4 for graduating college. Then let D0,1 represent the decision to graduate from high school and D0,2 represent the decision to get the GED once an individual has chosen to drop out (D0,1 = 0). The expected wage (Y ) for an individual, who chooses to graduate from high school (D0,1 = 1)is then E(Y |D0,1 = 1) =

Pr(s = 1|D0,1 = 1) × Y1 + Pr(s = 3|D0,1 = 1) × Y3 + Pr(s = 4|D0,1 = 1) × Y4 ,

where Pr() is the probability that an individual has a given final educational level and the wage Y depends on the final schooling level. Of course, Pr(s = 1|D0,1 = 1) + Pr(s = 3|D0,1 = 1) + Pr(s = 4|D0,1 = 1) = 1. Likewise, the expected value for someone who decides to drop out of high school (D0,1 = 0) is then E(Y |D0,1 = 0) = Pr(s = 0|D0,1 = 0) × Y0 + Pr(s = 2|D0,1 = 0) × Y2 , where Pr(s = 0|D0,1 = 0) + Pr(s = 2|D0,1 = 0) = 1. 4 The treatment effect can be broken up into the direct effect and the continuation value. The continuation value of graduating from high school is the probability that they enroll in college times the wage benefit of having some college plus the probability of then completing college times the wage benefit of completing college. For the high school graduation decision, the continuation value is: CV (Y |D0,1 = 1) = [(Y4 − Y3 ) × Pr(D3,4 = 1|D1,3 = 1) + (Y3 − Y1 )] × Pr(D1,3 = 1|D0,1 = 1) where in this case Pr represents the probability of making an educational decision as opposed to terminating in a final educational state as before, D1,3 represents the decision to enroll in college and D3,4 represents the decision to graduate from college. The direct treatment effect of graduating from high school is: DT E(Y |D0,1 = 1) = Y1 − [Y0 + (Y2 − Y0 ) × Pr(D0,2 = 1|D0,1 = 0)]

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Thus, the average treatment effect is

E ∆AT j,j 00

ZZ ≡

∆j,j 00 [Y |X = x, θ = z]dFX,θ (x, z),

(17)

the average effect of the treatment on the treated is ZZ

∆Tj,jT00 ≡

∆j,j 00 [Y |X = x, θ = z]dFX,θ|Ij,j 00 ≥0 (x, z),

(18)

and the average effect of the treatment on the untreated is

∆Tj,jU00T

ZZ ≡

∆j,j 00 [Y |X = x, θ = z]dFX,θ|Ij,j 00 12)

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