Bounds on Population Average Treatment E ects with an Instrumental Variable

Transcription

1 Bounds on Population Average Treatment E ects with an Instrumental Variable Preliminary (Rough) Draft Xuan Chen y Carlos A. Flores z Alfonso Flores-Lagunes x October 2013 Abstract We derive nonparametric sharp bounds on population average treatment e ects with an instrumental variable. The main assumptions employed are monotonicity in the treatment of the average outcomes of speci ed subpopulations, and mean dominance assumptions across the potential outcomes of these subpopulations. Importantly, the direction of the mean dominance assumptions can be informed from data and some of our bounds do not require an outcome with a bounded support. We illustrate the informativeness of our bounds by assessing the e ect of the Job Corps program. Our empirical results suggest that enrolling into the program increases weekly earnings and employment by at least $24:61 and 4:3 percentage points, respectively, and decreases yearly dependence on public welfare bene ts by at least $84:29. Key words and phrases: Sharp Bounds; Average treatment e ects; Instrumental variables JEL classi cation: C13, C21, J30 We are grateful for comments from seminar participants at The Seventh New York Econometrics Camp at Syracuse University, and Midwest Econometrics Goup Meetings at University of Kentucky. y xchen11@ruc.edu.cn. School of Labor and Human Resources, Renmin University of China. z c ore32@calpoly.edu. Department of Economics, California Polytechnic State University at San Luis Obispo. x a ores@binghamton.edu. Department of Economics, State University of New York at Binghamton, and IZA.

2 1 Introduction Instrumental variable (IV) approaches have been widely used in the literature of program evaluation due to its high internal validity. An in uential framework for studying causality within the IV framework was developed by Imbens and Angrist (1994) and Angrist, Imbens and Rubin (1996) (hereafter IA and AIR, respectively). They show that allowing heterogeneous e ects, IV estimators point identify the local average treatment e ect (LAT E) for compliers, those whose treatment status is a ected by the instrument. A common criticism of their approach is the focus on the e ect for a subpopulation (e.g., Heckman, 1996; Robins and Greenland, 1996; Deaton, 2010; Heckman and Urzua, 2010), and the instrument-speci c interpretation of the LAT E stimulates recently growing literature on IV approaches in pursuit of external validity. Point identi cation of the population treatment e ects usually requires an instrument to be strong enough to drive the probability of being treated from zero to one (e.g., Heckman, 2010), which is hard to satisfy in practice. An alternative way relies on stable IV estimates revealed empirically, which inspire the use of multiply instruments for the same causal relationship (e.g., Angrist and Fernandez-Val, 2010). In contrast with point identi cation, Manski (1990) pioneered partial identi cation of the population average treatment e ect (AT E) under the mean independence assumption of the instrument. Our paper derives nonparametric sharp bounds for the population AT E by extending the work of IA and AIR. There has been a growing literature on partial identi cation of the AT E with IV methods since Manski (1990). One strand of this literature endeavors to improve Manski s bounds by assuming di erent versions of monotonicity of the outcome. Manski and Pepper (2000) introduce the monotonicity of the treatment response (MTR) assumption and the monotonicity of the treatment selection (MTS) assumption. Combining with the mean independence assumption, Chiburis (2010a) derives the bounds under both MTR and MTS assumptions without specifying the direction of the monotonicity a priori. Another strand of the literature imposes structural models on the treatment or the outcome. Under the statistical independence assumption of the instrument, Heckman and Vytlacil (2000) impose a threshold crossing model with a separable error on the treatment. Focusing on a binary outcome, Shaikh and Vytlacil (2011) impose threshold crossing models on both the treatment and the outcome, while Chiburis (2010b) considers a threshold crossing model on the outcome. Instead of assuming the threshold crossing model with separable errors, Chesher (2010) imposes a non-separable structural model on the outcome and assumes the structural function is weakly increasing in the non-separable error. Comparison of identi cation power among these assumptions are also discussed in the existing literature on partial identi cation with IV methods. The assumptions employed by IA and AIR are also cited despite their purpose of point identi cation. First, the monotonicity 1

3 assumption on the treatment (e.g., IA; AIR; Balke and Pearl, 1997; Huber and Mellace, 2010) and the structural model assumptions on the treatment (e.g., Heckman and Vytlacil, 2000) do not improve Manski s bounds derived under the mean independence assumption. Second, monotonicity assumptions on the outcome (e.g., Manski and Pepper, 2000) and the structural model assumptions on the outcome do improve Manski s bounds (e.g., Bhattacharya, Shaikh and Vytlacil, 2008, hereafter BSV; Chiburis, 2010a; Chiburis, 2010b; Chesher 2010; Shaikh and Vytlacil, 2011). Third, partial identi cation with IV methods usually requires bounded support of the outcome. This might also be the reason why quite a few papers focus on binary outcomes (e.g., Balke and Pearl, 1997; BSV; Hahn, 2010; Chiburis, 2010b; Shaikh and Vytlacil, 2011). It s worth noting that for a binary dependent variable, the monotonicity assumptions and the structural model assumptions are equivalent. Vytlacil (2002) shows the equivalence between the monotonicity assumption and the threshold crossing model on the treatment. Machado et al. (2011) notice the equivalence between the MTR assumption and the threshold crossing model on the outcome. In the absence of covariates, Chiburis (2010b) observe the equivalence between the threshold crossing model with a separable error and the non-separable structural function being weakly increasing in the non-separable error. BSV show that in the absence of covariates, the bounds for a binary outcome under MTR and the mean independence assumptions are equal to the ones derived from the threshold crossing models on both the treatment and the outcome. Afterwards, Chiburis (2010b) notice that his bounds under the threshold crossing model only on the outcome are equal to the ones under MTR and the mean independence assumptions. This paper improves Manski s nonparametric bounds by extending the work of IA and AIR. We consider the setup consisting of a binary instrument and a binary treatment, which is quite common in the existing literature on partial identi cation of the AT E with IV methods. We add to the literature by considering two di erent sets of assumptions that can be useful in practice. The rst is monotonicity in the treatment of the average outcomes of subpopulations (strata) de ned by the joint potential values of the treatment status under each value of the instrument. Just as BSV and Shaikh and Vytlacil (2011), we do not require prior knowledge about the direction of the monotonicity. However, in contrast to the existing literature (e.g., Manski and Pepper, 2000; BSV; Shaikh and Vytlacil, 2011), we impose monotonicity on the average outcomes of the strata rather than on the individuals outcomes. This makes the assumption more plausible in practice by allowing some individuals to experience a treatment e ect that has the opposite sign to the AT E. The second set of assumptions involves mean dominance assumptions across the potential outcomes of di erent strata, which have been shown to have signi cant identifying power in other settings (e.g., Zhang, Rubin and Mealli, 2008; Flores and Flores-Lagunes, 2010). The direction of the mean dominance assumptions can be informed by comparing average baseline characteristics across strata that are likely to be highly correlated with the outcome. Importantly, some of our bounds do not require a bounded outcome. 2

4 A recent paper by Huber and Mellace (2010) also derives bounds on the AT E within the IV framework. The main di erence between our work and theirs is that we consider the monotonicity assumption on the average outcomes of the strata, which results in narrower bounds and can be justi ed by economic theory in many applications. Also, we avoid imposing the direction of the monotonicity a priori, while its direction can be inferred from data. In addition, the mean dominance assumptions we consider not only di er from theirs, but the direction of our mean dominance can be informed by comparing the average baseline characteristics across strata, which are estimated by solving an overidenti ed nonparametric GMM problem. To illustrate the identi cation power of our bounds, we analyze the e ect of enrolling into the Job Corps program, which is the largest federally-funded job training program for disadvantaged youth in the United States. Using randomization into the program as an instrument, the narrowest bounds on the AT Es derived by our assumptions are [24:61, 201:04] for weekly earnings, [:042, :163] for employment, and [ 142:76, 84:29] for public bene ts. These bounds are signi cantly narrower than the IV bounds proposed by Manski (1990), Heckman and Vytlacil (2000) and Kitagawa (2009), when applied to our setting, and the ones by Huber and Mellace (2010). The width of our bounds is also smaller than that under the IV and MTR assumptions of Manski and Pepper(2000), especially for public bene ts. Our bounds on employment are also narrower than the ones proposed by Balke and Pearl (1997), BSV, Chesher (2010), Chiburis (2010b) and Shaikh and Vytlacil (2011) for the case of a binary outcome. Our lower bounds for weekly earnings and employment are 10 percent higher than their respective intention-to-treat (IT T ) e ects (22:19 and :038), while the upper bound for public bene ts is equal to its IT T e ect. 1 Meanwhile, the LAT Es for compliers on the three outcomes fall within our narrowest bounds. In sum, our empirical results suggest that enrolling into the Job Corps program increases weekly earnings by at least $24:61 and employment by at least 4:3 percentage points, and decreases yearly dependence on public welfare bene ts by at least $84:29. 2 Framework 2.1 Setup and Benchmark Bounds Consider a random sample of size n from a population. Let D i 2 f0; 1g indicate whether unit i is treated (D i = 1) or not (D i = 0), and let Z i 2 f0; 1g be an instrument for treatment. Let D i (1) and D i (0) denote the treatment individual i would receive if Z i = 1 or Z i = 0, respectively. Our outcome of interest is Y. Denote by Y i (1) and Y i (0) the potential outcomes as a function of D, i.e., the outcomes individual i would experience if she received the treatment or not, respectively. Finally, let Y i (z; d) be the potential outcome as a function of the instrument and the treatment. 1 This is because our bounds derive di erent numerical results dependent on the sign of the LAT E for compliers. 3

5 Our parameter of interest is the population average treatment e ect, AT E = E[Y i (1) Y i (0)]. For each unit, we observe fz i ; D i (Z i ); Y i (Z i ; D i (Z i ))g. We omit the subscript i unless necessary for clarity. This setting has received considerable attention in the literature (e.g., AIR, BSV). AIR partition the population into four strata based on the values of fd i (0) ; D i (1)g: f1; 1g, f0; 0g, f0; 1g and f1; 0g. AIR and the subsequent literature refer to these strata as always-takers (at), never-takers (nt), compliers (c), and de ers (d), respectively. AIR impose the following assumptions which we adopt: Assumption 1 (Randomized Instrument). fy (0; 0); Y (0; 1); Y (1; 0); Y (1; 1); D(0); D(1)g is independent of Z. Assumption 2 (Exclusion Restriction). Y i (0; d) = Y i (1; d) = Y i (d); d 2 f0; 1g for all i. Assumption 3 (Nonzero First Stage). E[D(1) D(0)] 6= 0. Assumption 4 (Individual-Level Monotonicity of D in Z). Either D i (1) D i (0) for all i or D i (1) D i (0) for all i. Assumptions 1 through 3 are standard assumptions in the IV literature (e.g., IA, AIR, Huber and Mellace, 2010; Blanco, Flores, and Flores-Lagunes, 2012). Assumption 4 rules out the existence of de ers (compliers) when the monotonicity is non-decreasing (non-increasing). The direction of the monotonicity can be inferred from the data given the independence of Z. Following BSV, we order Z so that E[DjZ = 1] E[DjZ = 0] to simplify notation. To partially identify the AT E, we write it as a function of the average e ects for the existing strata. Let LAT E k = E[Y (1) Y (0)jk] and k denote, respectively, the local average treatment e ect and the stratum proportion in the population, for stratum k, with k = at, nt, c. Let Y zd = E[Y jz = z; D = d] and p djz = Pr(D = djz = z). Under Assumptions 1 through 4, the following quantities are point identi ed: at = p 1j0, nt = p 0j1, c = p 1j1 p 1j0, E[Y (1)jat] = Y 01, E[Y (0)jnt] = Y 10 and LAT E c = (E[Y jz = 1] E[Y jz = 0])=(p 1j1 p 1j0 ). As shown in IA and AIR, LAT E c is point identi ed for compliers whose treatment status is a ected by the instrument, and equals the conventional IV estimand in the absence of covariates. Decomposing the population AT E as a weighted average of the LAT Es for always-takers, nevertakers, and compliers, we have: AT E = at LAT E at + nt LAT E nt + c LAT E c (1) = p 1j1 Y 11 p 0j0 Y 00 + p 0j1 E[Y (1)jnt] p 1j0 E[Y (0)jat]; (2) where E[Y jz = z] = E[E[Y jz = z; D = d]jz = z] is used in the second equality. By equation (2), since Y (1) for never-takers and Y (0) for always-takers are never observed in the data, 4

6 additional assumptions are needed to bound AT E. The most basic assumption considered in the previous literature is the bounded support of the outcome. Assumption 5 (Bounded Outcome). Y (0); Y (1) 2 [y l ; y u ]. This assumption states that the potential outcomes under the two treatment arms have a bounded support. Replacing E[Y (1)jnt] and E[Y (0)jat] in equation (2) with y l and y h, we obtain sharp bounds on the AT E under Assumptions 1 through 5. Proposition 1 Under Assumptions 1 through 5 the bounds LB AT E UB are sharp, where LB = Y 11 p 1j1 Y 00 p 0j0 + y l p 0j1 y u p 1j0 UB = Y 11 p 1j1 Y 00 p 0j0 + y u p 0j1 y l p 1j0 : The bounds in Proposition 1, which we present for reference, coincide with the IV bounds in Manski (1990), Heckman and Vytlacil (2000) and Kitagawa (2009), when applied to our setting, and with those in Huber and Mellace (2010). When the outcome is binary, these bounds also coincide with those in Balke and Pearl (1997). 2.2 Bounds under Monotonicity Now let us introduce the monotonicity assumption we employ to improve the identi cation power of the bounds in Proposition 1. Assumption 6 (Monotonicity in D of Average Outcomes of Strata). (i) Either E[Y (1)jk] E[Y (0)jk] for all k = at; nt; c; or E[Y (1)jk] E[Y (0)jk] for all k = at; nt; c. (ii) E[Y (1) Y (0)jc] 6= 0. Assumption 6 requires that the LAT Es of the three existing strata are all either non-negative or non-positive. This assumption is similar to that in BSV, with the important distinction that we impose it on the LAT Es rather than on the individual e ects, which makes it more plausible in practice by allowing some individuals to have a treatment e ect of the sign di erent from that of the AT E. Since we ordered Z so that E[DjZ = 1] E[DjZ = 0], the direction of the monotonicity is identi ed from the sign of the IV estimand (LAT E c ) under the current assumptions. The following proposition presents sharp bounds on the AT E under the additional Assumption 6. 5

7 Proposition 2 Under Assumptions 1 through 6 the bounds LB AT E UB are sharp, where, if E[Y jz = 1] E[Y jz = 0] > 0, LB = E[Y jz = 1] E[Y jz = 0] UB = Y 11 p 1j1 Y 00 p 0j0 + y u p 0j1 y l p 1j0 ; and if E[Y jz = 1] E[Y jz = 0] < 0, LB = Y 11 p 1j1 Y 00 p 0j0 + y l p 0j1 y u p 1j0 UB = E[Y jz = 1] E[Y jz = 0]: Depending on the sign of LAT E c, either the lower or the upper bound in Proposition 2 improves upon the corresponding bound in Proposition 1. When LAT E c > 0, the lower bounds on LAT E at and LAT E nt become zero; otherwise, their upper bounds become zero. Consequently, either the lower or upper bound on the AT E equals the IT T e ect dependent on the sign of LAT E c. When the outcome is binary, the bounds in Proposition 2 coincide with those in BSV and Chiburis (2010b), which both equal the bounds in Shaikh and Vytlacil (2011) and Chesher (2010) when there are no exogenous covariates other than the binary instrument. Moreover, if LAT E c is positive (negative) and Assumptions 1 through 6 hold, then the bounds in Proposition 2 equal the bounds obtained by imposing the mean independence assumption of the instrument and increasing (decreasing) MTR assumptions in Manski and Pepper (2000). MTR imposes monotonicity of the outcome in the treatment at the individual level, and it requires one to know the direction of the e ect a priori. Depending on the sign of the individual e ect, BSV shows the equivalence of their bounds to those under the IV and MTR assumptions for the case of a binary outcome. Thus, in our setting along with the relaxed version of the monotonicity assumption, our results can be seen as an extension of those in BSV to the case of a non-binary outcome Bounds under Mean Dominance In practice, some strata are likely to have more favorable characteristics and thus better mean potential outcomes than others. The three alternative assumptions below formalize the notion that under the same treatment status, never-takers tend to have the best average potential outcome among the three strata, while always-takers tend to have the worst one. Assumption 7a. E[Y (d)jat] E[Y (d)jnt] for d = 0; 1. Assumption 7b. E[Y (0)jat] E[Y jz = 0; D = 0] and E[Y (1)jnt] E[Y jz = 1; D = 1]. Assumption 7c. E[Y (0)jat] E[Y (0)jc] and E[Y (1)jnt] E[Y (1)jc]. 2 For a discussion of the trade-o between the MTR assumption and assuming monotonicity of the treatment in the instrument, see BSV. 6

8 The direction of these assumptions can be inverted depending on the application in question. The always-takers and never-takers are likely to be the most extreme" groups in many applications, so Assumption 7a may be viewed as the weakest of the three. Assumption 7b compares the mean Y (0) and Y (1) of the always-takers and never-takers, respectively, to those of a weighted average of the other two strata, while Assumption 7c compares them to those of the compliers. Although none of these assumptions is directly testable, it is possible to obtain indirect evidence about their plausibility by comparing relevant average pre-treatment characteristics of the di erent strata (e.g., Flores and Flores-Lagunes, 2010; Frumento et al., 2012). For Assumption 7c, the direction may also be inferred by comparing point identi ed quantities, E[Y (1)jat] to E[Y (1)jc] and E[Y (0)jnt] to E[Y (0)jc], if these inequalities also hold under the alternative treatment status. We present bounds under Assumptions 1 through 5 and each of the three versions of Assumption 7. In each case, the lower bound is higher than that in Proposition 1. Proposition 3 Let UB = Y 11 p 1j1 Y 00 p 0j0 + y u p 0j1 y l p 1j0. (a) Under Assumptions 1 through 5 and 7a the bounds LB AT E UB are sharp, where LB = Y 11 p 1j1 Y 00 p 0j0 + Y 01 p 0j1 Y 10 p 1j0 : (b) Under Assumptions 1 through 5 and 7b the bounds LB AT E UB are sharp, where LB = Y 11 Y 00 : (c) Under Assumptions 1 through 5 and 7c the bounds LB AT E UB are sharp, where LB = Y 11 p 1j1 Y 00 p 0j0 + Y 11 p 1j1 Y p 1j0 Y p0j0 Y 10 p 0j1 p p 1j1 p 0j1 p 1j0 p 1j1 p 1j0 : 1j0 Assumptions 7a through 7c have testable implications when combined with Assumption 6, if LAT E c < 0. The following inequalities are expected to hold: Y 01 Y 10 (7a); Y 01 Y 00 and Y 11 Y 10 (7b); and, Y 01 E[Y (0)jc] and E[Y (1)jc] Y 10 (7c). If some (or all) of these inequalities are not rejected in applications, then their corresponding assumptions are expected to hold. The following three propositions provide bounds when Assumptions 6 and 7 are combined. Proposition 4 Under Assumptions 1 through 6 and 7a the bounds LB AT E UB are sharp, where, if E[Y jz = 1] E[Y jz = 0] > 0, LB = Y 11 p 1j1 Y 00 p 0j0 + maxfy 10 ; Y 01 gp 0j1 minfy 10 ; Y 01 gp 1j0 UB = Y 11 p 1j1 Y 00 p 0j0 + y u p 0j1 y l p 1j0 ; 7

9 and if E[Y jz = 1] E[Y jz = 0] < 0, LB = Y 11 p 1j1 Y 00 p 0j0 + Y 01 p 0j1 Y 10 p 1j0 UB = E[Y jz = 1] E[Y jz = 0]: Proposition 5 Under Assumptions 1 through 6 and 7b the bounds LB AT E UB are sharp, where, if E[Y jz = 1] E[Y jz = 0] > 0, LB = Y 11 p 1j1 Y 00 p 0j0 + maxfy 10 ; Y 11 gp 0j1 minfy 01 ; Y 00 gp 1j0 UB = Y 11 p 1j1 Y 00 p 0j0 + y u p 0j1 y l p 1j0 ; and if E[Y jz = 1] E[Y jz = 0] < 0, LB = Y 11 Y 00 UB = E[Y jz = 1] E[Y jz = 0]: Proposition 6 Under Assumptions 1 through 6 and 7c the bounds LB AT E UB are sharp, where, if E[Y jz = 1] E[Y jz = 0] > 0, LB = Y 11 p 1j1 Y 00 p 0j0 + maxfy 10 ; Y 11 p 1j1 Y 01 p 1j0 p 1j1 p 1j0 gp 0j1 minfy 01 ; Y 00 p 0j0 Y 10 p 0j1 gp p 1j1 p 1j0 1j0 UB = Y 11 p 1j1 Y 00 p 0j0 + y u p 0j1 y l p 1j0 ; and if E[Y jz = 1] E[Y jz = 0] < 0, LB = Y 11 p 1j1 Y 00 p 0j0 + Y 11 p 1j1 Y p 1j0 Y p0j0 Y 10 p 0j1 p p 1j1 p 0j1 p 1j0 p 1j1 p 1j0 1j0 UB = E[Y jz = 1] E[Y jz = 0]: Note that, if LAT E c < 0, the bounds in Propositions 4 through 6 do not require boundedness of the outcome, because Assumption 6 improves upon the upper bound in Proposition 1, while Assumption 7 improves upon the lower bound. In contrast, if LAT E c > 0, Assumptions 6 and 7 each improve only upon the lower bound in Proposition 1. The bounds in Propositions 4 through 6 are narrower compared with the bounds in Proposition 2 and the corresponding bounds in Proposition 3. This is because under the combined assumptions, the monotonicity assumption improves upon further either the lower or upper bound in Proposition 3, depending on the sign of LAT E c, while the mean dominance assumptions further improve upon the lower bound in Proposition 2. 8

10 Proposition 5 overlaps with the bounds recently derived by Chiburis (2010a) under the MTR assumption without specifying a priori direction and the decreasing MTS assumption, as well as the mean independence assumption of the instrument. This is because Assumption 7b coincides with the decreasing MTS assumptions imposed on the counterfactual average outcomes for always-takers and never-takers (i.e., E[Y (0)jat] and E[Y (1)jnt]). The form of Chiburis bounds, however, cannot simplify to Proposition 6, in that his monotonicity assumptions also involve the counterfactual average outcome for the mixture of never-takers and compliers and that for the mixture of always-takers and compliers (i.e., E[Y (1)jZ = 0; D = 0] and E[Y (0)jZ = 1; D = 1]), which are not involved in our setting. It is important to note that the bounds in Proposition 6 are also the sharp bounds for AT E if we replace Assumption 7c with the assumption, E[Y (d)jat] E[Y (d)jc] E[Y (d)jnt] for d = 0; 1. However, since E[Y (d)jc] may su er from the potential issue of a weak IV (i.e., p 1j1 p 1j0 is close to zero), and thus be more di cult to estimate than E[Y jz = d; D = d]. Consequently, the estimated bounds in Proposition 5 may produce narrower con dence intervals than those in Proposition 6 if p 1j1 p 1j0 is close to zero. 2.4 Estimation and Inference Some of our bounds involve minimum (min) or maximum (max) operators, which create complications for estimation and inference. First, because of the concavity (convexity) of the min (max) function, sample analog estimators of the bounds can be severely biased in small samples. Second, closed-form characterization of the asymptotic distribution of estimators for parameters involving min or max functions are very di cult to derive and, thus, usually unavailable. Furthermore, Hirano and Porter (2012) show that there exist no locally asymptotically unbiased estimators and no regular estimators for parameters that are nonsmooth functionals of the underlying data distribution, such as those involving min or max operators. These issues have generated a growing literature on inference methods for partially identi ed models of this type (see Tamer, 2010, and the references therein). We employ the methodology proposed by Chernozhukov, Lee and Rosen (2011) (hereafter CLR) to obtain con dence regions for the true parameter value, as well as half-median unbiased estimators for the lower and upper bounds. The half-median-unbiasedness property means that the upper (lower) bound estimator exceeds (falls below) the true value of the upper (lower) bound with probability at least one half asymptotically. This is an important property because achieving local asymptotic unbiasedness is not possible, implying that "bias correction procedures cannot completely eliminate local bias, and reducing bias too much will eventually cause the variance of the procedure to diverge" (Hirano and Porter, 2012). For details on the procedure of applying CLR s method to our bounds see the Appendix. For the bounds without min or max operators, we use sample analog estimators and construct the con dence regions 9

11 for the true parameter value proposed by Imbens and Manski (2004). 3 Bounds on Population Average Treatment E ects of Job Corps 3.1 Data Job Corps (JC) is the largest and most comprehensive education and job training program in the United States. It o ers to economically disadvantaged youth aged 16 to 24 years old academic education, vocational training, residential living, health care and health education, counseling and job placement assistance. According to the U.S. Department of Labor (2005), a typical JC student lives at a local JC center for eight months and receives about 1100 hours of academic and vocational instruction, which is equivalent to approximately one year in high school. We employ data from the National Job Corps Study (NJCS), a randomized experiment undertaken in the mid-to-late nineties. The study examined the impacts of JC on labor market outcomes, welfare dependence and several other outcomes to help assess whether the program achieved its goals of helping students become more responsible and productive citizens. Eligible applicants were randomly assigned to treatment and control groups. Individuals in the control group were embargoed from the program for a period of three years. The research sample was interviewed at random assignment and at 12, 30, and 48 months after random assignment. Taking advantage of randomization, most of previous works on JC study IT T e ects or LAT Es for compliers (e.g., Burghardt et al., 2001; Schochet, Burghardt, and Glazerman, 2001; Schochet, Burghardt, and McConnell, 2008; Lee, 2009; Blanco, Flores, and Flores-Lagunes, 2012). The noncompliance behavior, however, tends to dilute the impacts of JC. In our sample, 73% of individuals of the treatment group actually enrolled in JC, while 4% of individuals of the control group also enrolled. Even adjusting to noncompliance by examining LAT E c, that e ect is representative for a subpopulation accounting for 69% in the population of our interest. We make inference about the population AT E, which is of great public interest, using data on individuals who responded to the 48-month interview. The outcome variables we consider are weekly earnings and employment at Week 208 and public assistance bene ts received during the fourth year after randomization 3. Given the objectives and services provided by JC (e.g., academic and vocational training, job search assistance), it tends to have positive e ects on participants labor market outcomes, though the direction of these e ects may be inverted in the short run. On one hand, individuals who enroll in JC could be less likely to be employed while undergoing training, which is usually referred to as the lock-in e ect (van Ours, 2004). On the other hand, some participants may raise their reservation wages after training and 3 Bene ts include Aid to Families with Dependent Children (AFDC) or Temporary Assistance for Needy Families (TANF), food stamps, Supplemental Security Income (SSI) or Social Security Retirement, Disability, or Survivor (SSA), and General Assistance. 10

12 choose to be unemployed by rejecting some job o ers. Both potential threats make participants possibly experience lower earnings than those who do not enroll in JC in the short run. Given a long enough period of time, however, trained individuals are no longer locked-in away from employment and those raising reservation wages nd jobs, and both of them are more likely to have higher earnings after training. Consistent with this view, Schochet, Burghardt, and Glazerman (2001) and Lee (2009) nd negative e ects of JC on weekly earnings and employment in the short run, but positive e ects in the long run. Participation in JC may also a ect welfare dependence di erently in the short and long runs. In the short run, participants may experience a reduction in welfare receipt while they enroll in JC, because the program provides shelter (except to nonresidential students), food, and a stipend. In the long run, after they leave JC, participants may receive less public income support because of higher earnings. In contrast with this expectation, Schochet, Burghardt, and Glazerman (2001) report that the reductions in bene t receipt persisted throughout 4 years after randomization. Therefore, we focus our analysis on the latest measures available of labor market outcomes and welfare dependence in the NJCS. The treatment variable indicates whether or not the individual ever enrolled in JC during the 208 weeks (i.e., four years) after random assignment. The random assignment indicator serves as an instrument for JC enrollment. Two samples are obtained by dropping individuals with missing relevant variables from the survey. 4 The sample for weekly earnings and employment involves 10,520 individuals (4,187 and 6,333 in the control and treatment groups, respectively), while for public bene ts 10,976 individuals (4,387 and 6,589 in the control and treatment groups, respectively). Finally, due to both design and programmatic reasons, some subpopulations were randomized in the NJCS with di erent (but known) probabilities (Schochet, Burghardt, and Glazerman, 2001). Hence, we employ design weights throughout our analysis. 5 Table 1 reports the average baseline characteristics of both samples by treatment assignment status along with the percentage of missing values for each of those variables. The pretreatment variables include demographic characteristics, education and background variables, employment, earnings and public bene ts dependency at baseline, as well as labor market outcomes in the year prior to randomization. As one would expect, the average pre-treatment characteristics of the treatment and control groups are similar in both our samples due to randomization, with the di erence in means being statistically di erent from zero at the ve percent level for only one variable (personal income: 3,000-6,000). Thus, both our samples maintain the balance of baseline variables between the control and treatment groups. 4 We derive two samples because individuals with missing labor market outcomes and with missing public bene ts are di erent. 5 Speci cally, the weight we employ addresses sample design, 48-month interview design, and 48-month interview non-response. 11

13 3.2 Assessment of Assumptions Table 2 shows some relevant point identi ed averages for both samples. The noncompliance behavior is similar between the two samples. As we have already mentioned, 73% of individuals of the treatment group actually enrolled in JC, while 4% of individuals of the control group also enrolled during the 208 weeks after randomization. The IT T e ects on weekly earnings, employment and public bene ts are 22:19, :038 and 84:29, respectively. These e ects are all statistically signi cant, with their signs as expected. The LAT E c estimates for compliers on earnings, employment and public bene ts are 32:29, :055 and 122:28, respectively, 45 percent higher than their corresponding IT T estimates. By Assumption 6, the sign of LAT E c identi es the sign of the LAT Es for the other two strata. Thus, our estimates of LAT E c indicate positive population average treatment e ects on weekly earnings and employment and a negative population e ect on public bene ts. The middle part of Table 2 shows the proportion of each stratum. In both samples, the proportion of compliers is the largest, :69, followed by never-takers, :27, and always-takers, :04. And by Assumption 6, there are no de ers in our samples. The end part of Table 2 reports the point identi ed averages cited in Assumptions 6 and 7. 6 These estimates are all statistically signi cant and follow a certain pattern in both samples: under the treated status, the average outcome for always-takers is the smallest, followed by the average for the mixture of always-takers and compliers, and the average for compliers, while under the untreated status, the average outcome for compliers is the smallest, followed by the average for the mixture of never-takers and compliers, and the average for never-takers. As mentioned previously, di erences across these point identi ed averages may provide a preliminary hint for Assumption 7. To begin, we may infer the direction of Assumption 7c by comparing the identi ed averages of always-takers and never-takers to those of compliers under the same treatment status. The hypotheses that E[Y (1)jat] E[Y (1)jc] and E[Y (0)jc] E[Y (0)jnt] are not rejected for all of the three outcomes. Thus, if the same relationship with compliers also hold under the alternative treatment status, we may expect Assumption 7c to hold. Furthermore, since the IT T e ect on public bene ts is negative, testable implications are available when Assumption 7 is combined with Assumption 6, as discussed in Section 2.3. These testable implications are not rejected in our application. More importantly, indirect evidence of Assumption 7 are obtained by estimating the average baseline characteristics across strata from an overidenti ed nonparametric GMM problem. For details on the GMM procedure see Appendix. Tables 3 and 4 show these estimates and their di erences across strata for our samples. The average characteristics across strata are similar 6 As in Lee (2009), we use a transformed measure to estimate the sample averages of weekly earnings and public bene ts to minimize the e ect of outliers. Speci cally, the entire observed outcome distribution (for either weekly earnings or public bene ts) is split into 20 percentile groups (5 th ; 10 th ; : : : ; 95 th ), and then the mean outcome within each of the 20 groups is assigned to each individuals. 12

14 between the two samples. Among the three strata, never-takers are more likely to be female, older, married, to have children, higher level of education, personal income above $9,000 (less likely to have personal income below $3,000), higher weekly earnings at baseline, and to have better labor market outcomes the year before randomization. By contrast, always-takers tend to be male, younger, to have lower level of education at baseline, and to have lower earnings in the previous year. Furthermore, the statistical signi cance of the di erence between alwaystakers and never-takers indirectly supports Assumption 7a, while the statistical signi cance of the di erences compared to compliers (i.e., columns 4 and 5) tends to support Assumption 7c. Meanwhile, when the di erences across the three strata are all statistically signi cant, Assumption 7b are more likely to hold. Note that these di erences become more convincing evidence of Assumption 7 on the labor market outcomes under the untreated arm than those under the treated arm. However, the di erences across the strata in the public bene t dependency at the baseline are not statistically signi cant. 7 We conclude from these results that the data do not provide indirect evidence against Assumption 7, and that the point estimates of the di erences suggest that the assumption is plausible. 3.3 Empirical Results Table 5 shows our bounds on the AT Es on the labor market outcomes and the public dependency under Proposition 1 through Proposition 6. Under each pair of the estimated bounds, we report a 95% level con dence interval for the true parameter. Since the bounds for weekly earnings and employment in Propositions 4 through 6 involve max or min operators, we report the half-median unbiased estimators and the corresponding con dence intervals proposed by CLR. The bounds without max or min operators are estimated with sample analogs and their con dence intervals are obtained by the method of Imbens and Manski (2004). We begin with the AT E on weekly earnings in the rst two columns. Proposition 1 provides the bounds in the AIR setting under the bounded-outcome assumption (A.5). The estimated bounds are rather wide and fail to identify the sign of the AT E. Note that these bounds are also the IV bounds proposed by Manski (1990), Heckman and Vytlacil (2000) and Kitagawa (2009), when applied to our setting, and the ones by Huber and Mellace (2010). The AT E in Proposition 2 under the monotonicity assumption (A.6) is bounded between [22:19, 201:02], obtained by identifying positive LAT Es for always-takers and never-takers. Note that they are also the ones under the IV and MTR assumptions proposed by Manski and Pepper (2000). The mean dominance assumptions (A.7) improve upon the lower bound in Proposition 1, and negative e ects are ruled out in Propositions 3b and 3c in the absence of assuming the sign of LAT Es. When we impose Assumptions 1 through 7 together, all of our bounds and their corresponding con dence intervals lie in the positive region. In particular, the bounds on the 7 Unfortunately, information about the amount of public bene ts in dollars is unavailable at the baseline. 13

15 AT E on the weekly earnings in Proposition 6 are the narrowest, [24:61, 201:04], with the lower bound 10 percent higher than the IT T e ect (22:19), while the LAT E c for compliers (32:29) falls within the bounds. It turns out that informing the unobserved terms (i.e., E[Y (0)jat] and E[Y (1)jnt]) from the point identi ed outcomes of compliers provide a sharper lower bound on the AT E on the weekly earnings than that obtained by identifying the sign of LAT E at and LAT E nt under the monotonicity assumption (A.6). The next two columns show the bounds on the AT E on the employment, whose value is between 0 and 1. A similar pattern to the bounds for the weekly earnings is also found in the bounds for the employment. Note that the bounds in Proposition 1, [ :015, :163], unable to identify the sign of the AT E, also coincide with those in Balke and Pearl (1997) for the case of a binary outcome. The bounds in Proposition 2 derive a positive AT E, which varies between [:038, :163], and also equal the ones proposed by BSV, Chesher (2010), Chiburis (2010b) and Shaikh and Vytlacil (2011), all of which analyze a binary outcome. Again, Proposition 6 provides the narrowest bounds on the AT E on the employment under Assumptions 5, 6 and 7c, [:042, :163], with the lower bound 10 percent higher than the IT T e ect (:038), while the LAT E c for compliers (:055) falls within the bounds in Proposition 6. The nal two columns report the bounds on the AT E on the public bene ts. Di erent from the labor market outcomes, since the IT T e ect on the public bene ts is negative, imposing only the monotonicity assumption improves upon its upper bound in Proposition 1, while imposing only the mean dominance assumptions improves upon its lower bound. The bounds in Proposition 1 are extremely wide and uninformative. The monotonicity assumption (A.6) has a strong identi cation power compared with the mean dominance assumptions (A.7) in the case of the public bene ts, in that the former identi es the negative sign of the AT E on the public bene ts, though the latter greatly sharpen the lower bound in Proposition 1 by at least 70 percent. However, once we consider the two types of assumptions together, the bounded outcome assumption is no longer necessary and the width of the bounds shrink dramatically. Under the combined assumptions, the estimated bounds and their corresponding con dence intervals lie in the negative region. Proposition 6 provides the narrowest bounds on the AT E on public bene ts, [ 142:76, 84:29], with the upper bound equal to the IT T e ect, while the LAT E c on compliers ( 122:28) falls within the bounds. 4 Conclusion This paper derives sharp nonparametric bounds on the population average treatment e ects by extending the work of Imbens and Angrist (1994) and Angrist, Imbens and Rubin (1996). The favorable bounds are derived by combining two sets of assumptions. The rst is monotonicity in the treatment of the average outcomes of strata without specifying a priori direction. The second 14

16 is mean dominance assumptions on average potential outcomes across strata. Importantly, some of our bounds do not require a bounded support. And empirically, the direction of the mean dominance assumption can be inferred from data by estimating an overidenti ed nonparametric GMM problem. The application to the Job Corps program illustrates the informativeness of our bounds. Our empirical results suggest that enrolling into the program increases weekly earnings by at least $24:61 and employment by at least 4:3 percentage points at Week 208 after randomization, and decreases the dependence on public welfare bene ts by at least $84:29 during the fourth year after randomization. We close by pointing out that a similar analytical strategy to the one followed here can be used to bound the AT E when the instrument does not satisfy the exclusion restriction, in which case the local average treatment e ect for compliers can be bounded as in Flores and Flores-Lagunes (2012). In addition to solving the endogeneity issue, the strategy can also be applied to the identi cation of direct and indirect e ects (e.g., Rubin, 2004; Sjölander, 2009; VanderWeele, 2011) 15

17 Table 1: Summary Statistics of Baseline Variables Sample for Labor Market Outcomes Sample for Public Assistance Bene ts Missing Z = 1 Z = 0 Di.(Std.Err.) Missing Z = 1 Z = 0 Di.(Std.Err.) Prop. Prop. Female (.010) (.010) Age at Baseline (.042) (.041) White, Non-hispanic (.009) (.009) Black, Non-Hispanic (.010) (.010) Hispanic (.008) (.007) Other Race/Ethnicity Never Married Married (.003) (.003) Living Together (.004) (.004) Separated (.003) (.003) Has Child (.008) (.008) Number of Children (.012) (.012) Personal Education (.031) (.030) Mother s Education (.058) (.057) Father s Education (.073) * (.072) Ever Arrested (.009) (.009) Household Inc.: < (.011) (.011) (.010) (.010) (.008) (.008) (.011) (.011) > (.010) (.010) Personal Inc.: < (.008) (.008) (.007) (.007) ** ** (.004) > (.004) (.004) Have Job (.008) (.008) Weekly Hours Worked (.417) (.407) Weekly Earnings (4.482) (4.328) Had Job, Prev. Yr (.010) (.009) Months Employed,Prev.Yr (.085) (.083) Earnings, Prev.Yr (109.10) (107.01) Received Public Bene ts (.010) (.010) Months Received Bene ts (.125) (.122) Numbers of Observations Note: Z denotes whether the individual was randomly assigned to participate (Z = 1) or not (Z = 0) in the program. Bene ts include AFDC/TANF, food stamps, SSI/SSA, and General Assistance. Numbers in parentheses are standard errors. ** and * denote that di erence is statistically di erent from 0 at 5% and 10% level, respectively. Computations use the weights that account for sample and interview design and interview non-response. 16

18 Table 2: Point Identi ed Average Outcomes after Random Assignment Sample for Labor Market Outcomes at Week 208 Sample for Public Bene ts in Year 4 Variables: Enrollment Earnings Employment Enrollment Public bene ts Averages for Z = 1.730** Averages for Z = 0.043** (.003) IT T Effects.687** ** (3.004) ** (3.552) 22.19** (4.652) LAT E c 32.29** (7.007).608**.570** (.008).038** (.010).055** (.015).732**.043** (.003).689** ** (23.40) ** (30.28) ** (38.27) ** (56.78) Proportions of Strata under Assumptions 1 to 4 nt.270** c.687** (.007) at.043** (.003).268**.689**.043** (.003) Other Point Identi ed Average Outcomes under Assumptions 1 to 4 E[Y (1)jat] ** (14.94).393** (.037) E[Y (0)jnt] ** (5.967) E[Y (1)jc] ** (4.022) E[Y (0)jc] ** (5.655) E[Y jz = 1; D = 1] ** (3.614) E[Y jz = 0; D = 0] ** (3.709).600** (.012).624** (.008).569** (.012).611** (.007).578** (.008) ** (110.12) ** (47.98) ** (28.26) ** (49.69) ** (25.87) ** (33.18) Note: Z denotes whether the individual was randomly assigned to participate (Z = 1) or not (Z = 0) in the program. D denotes whether the individual was ever enrolled in the program (D = 1) or not (D = 0) during the 4 years (208 weeks) after randomization. Bene ts include AFDC/TANF, food stamps, SSI/SSA, and General Assistance. Numbers in parentheses are standard errors. ** denotes estimate is statistically di erent from 0 at 5% level. Computations use the weights that account for sample and interview design and interview non-response. The standard errors of LAT Es, proportions of strata and other identi ed average outcomes are calculated by 5000-repetition bootstrap. 17

19 Table 3: Average Baseline Characteristics in the Sample for Labor Market Outcomes Variable nt c at nt c c at nt at Female.467** (.011).397** (.007).324** (.035).070** (.015).073** (.037).143** (.036) Age at Baseline 18.74** (.052) White, Non-hispanic.284** (.011) Black, Non-Hispanic.472** (.012) Married.035** (.004) Has Child.237** (.010) Personal Education 10.27** (.035) Household Inc.:< ** (.008) > ** (.007) Personal Inc.: < ** (.010) > ** Have Job at Baseline.224** (.010) Weekly Hours Worked 22.07** (.488) Weekly Earnings ** (2.989) Had Job, Prev. Yr..667** (.010) Months Employed, Prev.Yr ** (.102) Earnings, Prev.Yr ** (101.80) Received Public Bene ts.607** (.011) Months Received Bene ts 6.744** (.122) 18.32** (.029).263**.484** (.007).016** (.002).162** 10.05** (.020).255**.181** (.004).799**.030** (.002).208** 21.29** (.272) ** (2.041).640** 3.527** (.057) ** (63.58).588** 6.503** (.073) 17.64** (.133).296** (.034).488** (.037) ** (.028) 9.637** (.095).187** (.021).233** (.027).843** (.026).015* (.008).216** (.031) 20.44** (1.652) 92.63** (7.986).651** (.035) 3.120** (.310) ** (251.57).596** (.037) 6.518** (.414).428** (.063).021* (.013) (.015).019**.075** (.012).224** (.044).012 (.010).000 (.009) -.049** (.012).012*.015 (.012).775 (.585) 11.03** (3.989).027** (.013).157 (.125) ** (127.99).020 (.013).240 (.153).674** (.137) (.036) (.039).011**.015 (.030).408** (.101).068** (.022) -.052* (.028) (.027).015* (.009) (.033).853 (1.734) (8.328) (.036).407 (.324) ** (263.42) (.038) (.437) 1.102** (.143) (.036) (.039).030**.089** (.029).632** (.100).080** (.022) -.052* (.027) -.093** (.027).027** (.009).008 (.032) (1.700) 21.15** (8.562).016 (.035).563* (.325) ** (273.94).011 (.037).226 (.424) Note: Bene ts include AFDC/TANF, food stamps, SSI/SSA, and General Assistance. Numbers in parentheses are standard errors. ** and * denote that estimate is statistically di erent from 0 at 5% and 10% level, respectively. Computations use the weights that account for sample and interview design and interview non-response. Missing values for each of the baseline variables were imputed with the mean of the variable. 18

20 Table 4: Average Baseline Characteristics in the Sample for Public Assistance Bene ts Variable nt c at nt c c at nt at Female.464** (.011).396**.330** (.035).069** (.014).066* (.037).134** (.036) Age at Baseline 18.75** (.049) White, Non-hispanic.289** (.011) Black, Non-Hispanic.461** (.012) Married.036** (.004) Has Child.234** (.009) Personal Education 10.27** (.034) Household Inc.:< ** (.008) > ** (.007) Personal Inc.: < ** (.010) > ** Have Job at Baseline.227** (.010) Weekly Hours Worked 21.80** (.460) Weekly Earnings ** (2.890) Had Job, Prev. Yr..667** (.011) Months Employed, Prev.Yr ** (.103) Earnings, Prev.Yr ** (99.19) Received Public Bene ts.601** (.010) Months Received Bene ts 6.684** (.122) 18.31** (.027).265**.480** (.007).016** (.002).163** 10.05** (.020).253** (.004).184** (.004).797**.030** (.002).211** 21.41** (.291) ** (2.180).642** 3.553** (.057) ** (65.20).581** 6.433** (.076) 17.68** (.126).289** (.035).503** (.037) ** (.029) 9.663** (.091).198** (.020).233** (.028).840** (.024).015** (.007).213** (.028) 20.63** (1.426) 94.21** (7.394).668** (.031) 3.060** (.282) ** (233.19).583** (.033) 6.395** (.378).435** (.061).024* (.014) (.015).020**.072** (.012).225** (.043).009 (.010).000 (.009) -.051** (.012).012**.016 (.012).392 (.594) 9.025** (4.094).025* (.013).091 (.130) ** (130.21).020 (.013).251 (.158).635** (.135) (.037) (.039).010** (.031).382** (.096).055** (.022) -.050* (.029) -.043* (.026).015* (.008) (.030).774 (1.548) (7.954) (.033).492 (.302) * (250.73) (.035).038 (.408) 1.070** (.133) (.036) (.039).030**.071** (.030).607** (.094).064** (.021) -.049* (.028) -.094** (.025).027** (.009).014 (.029) (1.494) 18.37** (7.804) (.032).584* (.299) ** (249.72).019 (.034).289 (.385) Note: Bene ts include AFDC/TANF, food stamps, SSI/SSA, and General Assistance. Numbers in parentheses are standard errors. ** and * denote that estimate is statistically di erent from 0 at 5% and 10% level, respectively. Computations use the weights that account for sample and interview design and interview non-response. Missing values for each of the baseline variables were imputed with the mean of the variable. 19