Michael Lechner. Swiss Institute for Empirical Economic Research (SEW), University of St. Gallen

Transcription

1 PARTIAL IDENTIFICATION OF WAGE EFFECT OF TRAINING PROGRAM Michael Lechner wiss Institute for Emirical Economic Research (EW), University of t. Gallen Blaise Melly Deartment of Economics, Brown University First version: July, 2007 This version: Aril, 2010 Date this version has been rinted: 12 Aril 2010 Abstract: In an evaluation of a job-training rogram, the influence of the rogram on the individual wages is imortant, because it reflects the rogram effect on human caital. Estimating these effects is comlicated because we observe wages only for emloyed individuals, and emloyment is itself an outcome of the rogram. Only usually imlausible assumtions allow identifying these treatment effects. Therefore, we suggest weaker and more credible assumtions that bound various average and quantile effects. For these bounds, consistent, nonarametric estimators are roosed. In a reevaluation of a German training rogram, we find that a considerable imrovement of the long-run otential wages of its articiants. Keywords: Bounds, treatment effects, causal effects, rogram evaluation JEL classification: C21, C31, J30, J68 Addresses for corresondence Michael Lechner, wiss Institute for Emirical Economic Research (EW), University of t. Gallen, Varnbüelstrasse 14, CH-9000 t. Gallen, witzerland; Michael.Lechner@unisg.ch; Blaise Melly, Deartment of Economics, Box B, Brown University, Providence, RI 02912, UA; Blaise_Melly@brown.edu, htt://

2 1 Introduction * For decades, many countries around the world have used active labor market olicies to imrove the labor market outcomes of the unemloyed. Training rograms are considered as most imortant comonents of this olicy. They should increase the emloyability of the unemloyed by adjusting their human caital to the demand in the labor market. The evaluation of these rather costly rograms has been the focus of a large literature in economics (e.g., see Friedlander, Greenberg, and Robins, 1997, Heckman, LaLonde, and mith, 1999, Kluve, 2006, and Martin and Grubb, 2001, for overviews). This literature has studied the effects of the various rograms on emloyment and realized earnings usually by setting earnings of the non-emloyed to zero. On the other hand, this literature was not able to measure the treatment effects on human caital because realized earnings are the roduct of the individual earnings otential times the robability of emloyment. Therefore, labor demand and labor suly influence them and we cannot distinguish how much of the effects is due to human caital changes. However, the aim of many countries when running these rograms is not to bring the rogram articiants into (otentially bad) jobs quickly but to increase long-term emloyability. Training rograms should increase the human caital of the articiants substantially and therefore hel unemloyed finding stable and qualified jobs. 1 For this reason, the effect of the * The first author has further affiliations with ZEW, Mannheim, CEPR, London, IZA, Bonn, PI, London, and IAB, Nuremberg. Financial suort from the Institut für Arbeitsmarkt- und Berufsforschung (IAB), Nuremberg, (roject 6-531a) is gratefully acknowledged. The data originated from a joint effort with tefan Bender, Annette Bergemann, Bernd Fitzenberger, Ruth Miquel, tefan eckesser, and Conny Wunsch to make the administrative data accessible for research. We thank Josh Angrist, Alan Manning, seminar articiants at Boston University and MIT and two anonymous referees for helful comments on a revious version of the aer. 1 In the case of Germany, 1 of the Work Promotion Act (Arbeitsförderungsgesetz) exlicitly states that the rograms should imrove the human caital stock of the individuals and counter low-quality emloyment. Lechner and Melly, revised

3 rograms on otential wages is a highly olicy relevant arameter and measures the success of these rograms. Furthermore, in the absence of long-term emloyment outcomes over 10 to 15 ost-rogram years, changes in human caital are robably the best redictor for long-term emloyability in these countries. Evaluating the effect of a training rogram on the otential wage is, however, a comlicated econometric roblem because of the selective observability of wages. Particiants in training rograms are tyically low skilled unemloyed with 'bad' emloyment histories and low reemloyment rates. Therefore, if we are interested in the wage effects of such rograms we have to deal with the fact that many articiants as well as comarable non-articiants will not receive any wage since they did not take-u emloyment in the first lace. To comlicate the issue further, we exect that those individuals who take-u jobs are not randomly selected. A convenient, but generally incorrect, aroach to estimate the otential wage effects is to comare the earnings for emloyed articiants and emloyed non-articiants. An alternative oular strategy is to use classical samle selection models (Heckman, 1979). Unfortunately, the identification of such models either requires a distributional assumtion or relies on a continuous instrument that determines the emloyment status but does not affect wages. Finding such a variable, however, is usually very difficult. It is imossible in our alication. Therefore, we follow another strategy: we bound average and quantile rogram effects on otential wages. After having derived the so-called worst-case bounds that are usually very wide, we consider how these bounds can be tightened by making further economically motivated, but rather weak behavioural assumtions that will be lausible in many alications. The suggested bounds do not deend on the way the selection roblem related to rogram articiation is controlled for: by a randomized exeriment, by matching, or by instrumental variables. In our articular alication, we use a matching strategy that is reasonable given the informative administrative database available. Lechner and Melly, revised

4 We also roose consistent, nonarametric estimators for all bounds and aly them to the evaluation of retraining rograms in West Germany. 2 uch rograms are an imortant (and exensive) tool of the German labor market olicy. Recent German administrative databases are very informative and allow us to credibly control for selective articiation into rograms, to cature imortant asects of the effect heterogeneity, and to follow the effects of training over a longer eriod. The methodological art of this aer builds on the existing literature on artial identification. Manski (1989, 1990, 1994, and 2003) and Robins (1989) contribute rominently to this aroach consisting in bounding the effects of interest using only weak assumtions. Horowitz and Manski (2000) bound treatment effects with missing covariate and outcome data. Blundell, Ichimura, Gosling, and Meghir (2007) introduce a restriction imosing ositive selection into work, while Lee (2009) uses an assumtion restricting the heterogeneity of the rogram effects on emloyment. We consider variants of these assumtions and show that they allow tightening the bounds on the treatment effects. Zhang and Rubin (2003) and Zhang, Rubin, and Mealli (2007) combine these two tyes of assumtions. Angrist, Bettinger, and Kremer (2006) use a similar combination of assumtions to bound the effects of school vouchers on test scores. 3 This aer contributes to the existing literature in four ways. First, we bound not only average but also quantile treatment effects. The effects of a treatment on the distribution of the outcome are of interest in many areas of emirical research. Policy-makers might be interested in the effects of the rogram on the disersion of the outcome, or its effect on the lower tail of the outcome distribution. Interestingly, the distribution is easier to bound than the 2 3 We concentrate on West Germany only, because East Germany faces unique transition roblems. They assume directly that the effects of the treatment on the otential test-taking status and on the otential score are ositive. Lechner and Melly, revised

5 mean. For instance, bounds on the suort of the outcome variable are required to bound the mean but not the quantiles of a random variable in resence of missing observations. The second contribution of this aer is to allow for a more general first ste selection rocess. The existing aers bounding the treatment effects have assumed that the treatment status was randomly determined. While this simlifies the derivation of the results, it does not corresond to the majority of the otential alications and therefore reduces the interest in these methods. Our theoretical results only require that the first ste selection roblem is solved for some suboulation. They do not deend on the secific method used to solve this roblem. In our alication, we assume unconfoundedness, i.e., we assume that the treatment status is indeendent of the outcome variables conditionally on a set of covariates. Third, in our alication we bound the treatment effects for the observable oulation consisting of the emloyed articiants. 4 Most of the existing literature bounds the effects for the unobserved oulation of individuals working irresectively of their rogram articiation status. However, results for an unobserved oulation are less intuitive and more difficult to communicate. For examle, simle descritive statistics cannot characterize such a oulation. Furthermore, from an economic oint of view, the oulation of working articiants is clearly the most interesting oulation because only they have realized otential wage effects that actually affect their consumtion ossibilities. Finally, we aly our results to a olicy relevant question. Using our referred combination of assumtions, we find substantial increases in the earnings caacity for the training rogram we consider. In fact, both average treatment effects and most quantile treatment effects significantly exclude a zero otential wage effect. This shows that our bounding strategy is 4 More generally, our theoretical results aly to the emloyed art of the oulation for which the first ste selection is solved. This oulation may (e.g. for selection on observables) or may not (e.g. for instrumental variables) be observed but its size is known and it can always be described by summary statistics. Lechner and Melly, revised

6 not only credible because it makes weak assumtions, but that it can be informative for olicy makers as well. The rest of the aer is organized as follows. The next section gives some institutional details about training rograms in Germany and discusses data issues. In ection 3, we define the notation and the treatment effects of interest. We also resent a unifying framework for analyzing average and quantile treatment effects. ection 4 contains the identification results. ection 5 rooses nonarametric estimators for the bounds derived in ection 4. ection 6 resents the emirical results and ection 7 concludes. An aendix contains the main roofs of the theorems. Further roofs are relegated to an aendix that can be downloaded from the web ages of the authors at 2 Training rograms in Germany 2.1 Active and assive labor market olicy Germany belongs to the OECD countries with the highest exenditure on labor market training measured as a ercentage of GDP after Denmark and the Netherlands, and it makes u the largest fraction of total exenditure on active labor market olicies. 5 Table 1 dislays the exenditures for active and assive labor market olicies and esecially for training rograms for the unemloyed in West Germany for the years As usual, training has the objective of udating and increasing the human caital. It is the most utilized instrument and reresents almost 50% of the total exenditure devoted to the active labor market olicy. In Germany, labor market training consists of heterogeneous instruments that differ in the form and in the intensity of the human caital investment, as well as in their duration. 6 In our 5 6 ee Wunsch (2005) for a detailed account of the German labor market olicy. For a recent (classical) evaluation study of the German training olicy, see Lechner, Miquel, and Wunsch (2005). This aer contains also more extensive descrition of the institutional environment and many details on the data. Lechner and Melly, revised

7 emirical alication, we concentrate on so-called Re-training courses. They are substantive investments suosed to enable unemloyed of working in a different rofession than the one currently held by awarding new vocational degrees. Their mean full-time duration is about 20 months. 2.2 Data and definition of the samle We use a database obtained by merging administrative data from three different sources: the IAB emloyment subsamle, the benefit ayment register, and the training articiant data. This is the most comrehensive database in Germany with resect to training conducted rior to We reconstruct the individual emloyment histories from 1975 to It also contains detailed ersonal, regional, emloyer, and earnings information. Thus, it allows controlling for many, if not all, imortant factors that determine selection into rograms and labor market outcomes. Moreover, recise measurements of the interesting outcome variables are available u to We consider rogram articiation between 1993 and A erson is included in our oulation of interest if he starts an unemloyment sell between 1993 and The grou of articiants consists of all ersons entering a re-training rogram between the beginning of this unemloyment sell and the end of We require that all individuals were emloyed at least once and that they received unemloyment benefits or assistance before the start of the rogram. Finally, we imose an age restriction (25-55 years) and exclude trainees, home workers, arentices and art-time workers. The resulting samle comrises about 9000 nonarticiants and 4000 articiants in re-training courses. 7 7 We use the same data as Lechner, Miquel, and Wunsch (2005). We also follow their definitions of oulations, rograms, articiation, non-articiation and their otential start dates, outcomes, and selection variables. ee this aer for much more detailed information on all these toics. Lechner and Melly, revised

8 Our outcome variables are annual emloyment and earnings during the seventh year after rogram start. It is a weakness of this database that there is no information on hours available. Therefore, we cannot construct wages but have to stick to annual (or monthly) earnings. Looking at the effects seven years after rogram start allows us to concentrate on the long-run effects, which are more interesting olicy arameters than the short-term effects, because the former are closer to the ermanent effects of the rogram. Particularly for this rather long rogram, the short-run effects are much influenced by the so-called lock-in effects (Van Ours, 2004), meaning that unemloyed reduce their job search activities while being in the rogram. 2.3 Descritive statistics Table 2 shows descritive statistics for selected socio-economic variables in the sub-samles defined by training articiation and emloyment (emloyed / non-emloyed) status. This illustrates the 'double selection roblem' for the estimation of rogram effects on wages. Concerning selection into the rograms (comare the two columns total), the results can be summarized as follows: Particiants in re-training are younger comared to non-articiants, which is line with the idea that human caital investments are more beneficial if the roductive eriod of the new human caital is longer. Interestingly the share of foreigners in retraining is only about half the share of foreigners in the grou of non-articiants. Particiants in re-training are less educated and less skilled. Nevertheless, ast earnings are somewhat higher for articiants in re-training than for non-articiants. As exected, we observe a ositive selection into emloyment: Emloyed individuals are better educated, younger, and received higher salaries during their last occuation than nonemloyed individuals. Interestingly, they reside less frequently in a big city (reflecting the higher unemloyment rates in German cities). Thus, there is a clear non-random selection into rograms as well as into emloyment. Understanding and correcting for these two selection rocesses is the key to recover the 'ure' human caital effects of these training rograms. Lechner and Melly, revised

9 3 Notation, definitions, and effects 3.1 The standard model of otential outcomes We observe N random draws, indexed by i, from a large oulation. The binary variable D i indicates articiation of individual i in the training rogram. Individual characteristics are catured by X i which is defined over the suort χ. We follow the standard aroach in the microeconometric literature to use otential outcomes to define causal effects of interest. Rubin (1974), among others, oularized this aroach. Let Y i () 1 be the earnings individual i would earn after rogram articiation and let Y ( 0) i be the earnings individual i would obtain () otherwise. imilarly, we define the otential emloyment statuses 1 and 0, where i i =1 for emloyed individuals and 0 otherwise. We assume that the four otential outcomes are well-defined even if we do not observe all of them in ractice. Assuming the validity of the stable-unit-treatment-value assumtion (see Rubin, 1980) allows us to relate the different otential outcomes to the observable outcomes: = D(1) + 1 D (0), Y = DY(1) + 1 D Y(0) if = 1 and Y is missing if = 0. While the definition of is standard in the treatment effect literature, the double selection roblem aears in the definition of Y. For instance, we observe Y 1 only if D = 1 and () 1 = 1. This exlains why most of the literature has considered only effects on gross or i i () i total earnings, ( d) Y d 8, that are always observed. 8 Here earnings are simly set to zero for non-working individuals. Alternatively, they could also contain some non-wage income like unemloyment or retirement benefits. In the former case, the causal effect would measure some roductivity gain due to the rogram, whereas in the latter case we would estimate the imact Lechner and Melly, revised

10 Following the literature, we base our analysis on causal arameters that can be deduced from the differences of the marginal distributions of otential outcomes. 9 First, consider average and quantile treatment effects on Y caused by D. To define the quantile effects, let F ( v; w) be the distribution function of the random variable V conditional on W evaluated at VW v and w. W may be a vector of random variables. The corresonding θ th (0 θ 1) quantile of VW VW 1 F ( v; w) is denoted by F ( θ; w). Using this definition, we obtain the following earnings effects of articiating in a rogram: () () ( 1 1 ) ( ( 0) ( 0) ) Y ATE T t E Y T t E Y T t = = = = ; 1 1 () 1 () 1 ( 0) ( 0) Y QTEθ T = t = F θ; t F ( θ; t). Y T Y T The random variable T defines the target oulation for which we define the effect. For instance, if T is the treatment variable D and t = 1 we obtain the so-called treatment effect on the treated, whereas for t = 0 we obtain the treatment effect on the non-treated. If we identify the treatment effects using an instrumental variable strategy, T can be an indicator for being a comlier as defined in Imbens and Angrist (1994) and we obtain the local average or local quantile treatment effect arameter. In the framework of Heckman and Vytlacil (2005, 2007), T can be the error term in the treatment selection equation and we obtain a family of marginal treatment effects. These arameters, which are defined for various outcome variables, are the usual objects of investigation in emirical evaluation studies. They are interesting in their own right and are 9 of the rogram on a measure of disosable income. In some alications, setting the unobserved outcome to any value may not make sense, for instance when the outcome is a test score or a measure of health. We do not investigate issues related to the joint distribution of otential outcomes, e.g. F ( y ), since the Y(1) Y(0) latter is very hard to in down with reasonable assumtions. For a thorough discussion of these issues, see Heckman, mith, and Clemens (1997). Of course, this distinction does not matter for linear oerators like the exectation, for examle, since the exectation of the difference equals the difference of the exectations. Lechner and Melly, revised

11 frequently estimated in emirical studies (e.g. Lechner, Miquel, and Wunsch, 2005). However, they fail to answer the imortant question whether the rogram lead to roductivity increases. The failure of answering this imortant olicy question comes from the fact that these earnings outcomes mix emloyment and ure earnings effects. Therefore, to answer questions about the otential earnings effects, we comare otential outcomes for different articiation states in a (otential) world in which all individuals had found a job, which is not observable for non-working individuals. In articular, we investigate the otential earnings effects for those individuals who would find a job under the treatment: ( 1 ), ( 1) 1 ( 0 ), ( 1) = = = = = =1 ; Y ATE T t E Y T t E Y T t 1 () ()( ;,1) 1 ( ;,1) Y QTEθ T t F θ t F θ t Y 1 T, 1 Y 0 T, 1 = =. Note that the roblem is symmetric with resect to the definition of the treatment. The technical arguments would be almost identical if we were interested in the same effects for those individuals who would find job when non-treated. Note also that we consider treatment effects for suboulations defined by the otential emloyment status () 1, not by the observed. The causal interretation of effects conditional on is unclear, because art of the effect of D on Y is already 'taken away' by the conditioning variable (see Lechner, 2008). Therefore, we will not consider the effect of D on those articiants and nonarticiants who actually found a job. In section 4, we derive the bounds for a generic oulation defined by T t and 1 = 1. In = () our alication, we are more secific and consider the effects for the doubly treated oulation with D = 1 and () 1 = 1. We could also consider the treatment effects for the whole oulation (irresectively of whether individuals have found a job or not). However, such Lechner and Melly, revised

12 effects may be of less olicy interest than the effects for the effectively treated oulation, articularly in the context of a narrowly targeted rogram. The effects for other oulations have been considered in the literature as well. Card, Michalooulos, and Robins (2001) considered earnings effects for those workers who were induced to work by rogram articiation. imilarly, Zhang and Rubin (2003) and Lee (2009) consider wage effects for individuals who would work irresective whether they articiate in a rogram or not. Of course, both such oulations are unobserved and, thus, difficult to describe. They cannot be characterized, for examle, by simle descritive statistics. Furthermore, Card, Michalooulos, and Robins (2001) and Lee (2009) severely restrict the heterogeneity of the treatment effect. While Card, Michalooulos, and Robins (2001) assume that the treatment effect on emloyment is ositive for all observations, Lee (2009) assumes that this treatment effect is either ositive for everybody or negative for everybody. However, heterogeneous effects are a tyical finding in rogram evaluation studies, as confirmed by our alication. 3.2 Unified notation for average and quantile effects In this aer, we consider exlicitly the identification and estimation of average and quantile treatment effects. To do so, we introduce a notation that encomasses both tyes of effects to avoid redundancies in our formal arguments. Let g() be a function maing Y into the real line. We will show below that we only need to consider (artial) identification of ( ) () E g Y d X = x, T = t, 1 = 1 for d { 0,1} and x χ T = t to examine the identification of the average and quantile treatment effects. Letting = Y g Y, we obtain the ATEs defined above. Letting g Y = 1 Y y, we identify the Lechner and Melly, revised

13 distribution function of Y evaluated at y. 10 The distribution function can then be inverted to get the quantiles of interest and to obtain all QTEs defined above. Define b inf g g y as lower bound of and y g () b su g g y y as its uer bound. These bounds may or may not be finite deending on g () and the suort of Y. If we estimate the distribution function, g () is an indicator function, which is naturally bounded between 0 and 1. If we estimate the exected value of Y, g is the identity function and () b Y and b Y are the bounds of the suort of Y. If we estimate the variance of Y, ( = g Y Y E Y ) 2. In this case, () and in the absence of further information on EY, the lower bound on g is 0 and the uer bound is ( b b ) Y Y Lemma 1 shows that we obtain shar unconditional bounds by integrating shar conditional bounds. 11 The roofs of all lemmas can be found in the Internet Aendix. Lemma 1 (bounds on the unconditional exected value of g () ) Let bg ( x ) and bg ( x ) be shar lower and uer bounds on E g Y X = x. Then E b g ( X) and E b g X are shar lower and uer bounds on E g ( Y ). This result holds in the oulation and all suboulations defined by values of T and. Naturally, if b ( x) b ( x) g g ( ) = for x χ, then E g Y is identified. For instance, in the Assumtion 1 defined below, we assume that () () E Y 1 X = x, T = t, 1 = 1 is oint identified. = : Y Therefore, we obtain the following bounds on ATE T t 10 The indicator function 1() equals one if its argument is true. 11 We define shar (or tight) bounds as bounds that cannot be imroved uon without further information. Lechner and Melly, revised

14 XT = t, 1 = 1 ( () 1 =, =, ( 1) = 1) Y 0 Y ( () 1 ( 0),, Y ) E E Y b X X x T t () ATE T = t E E Y b X X = x T = t 1 = 1. XT = t, () 1 = 1 imilarly, by letting = 1( ) g Y Y y and using the same rinciles, we obtain bounds on the unconditional distribution function. Lemma 2 shows how the bounds on the unconditional distribution function can be inverted to get bounds on the unconditional quantile function. Lemma 2 (bounds on the quantile function) Let b1 ( Y y ) and b1 ( Y y ) be shar lower and uer bounds on the distribution function of Y evaluated at y. Let 0 θ 1 and define < < b ( θ ) and b QY QY θ as follows: b { θ} ( θ) inf b1 ( ) QY y Y y if lim b1 ( Y y ) y > θ, b Y otherwise; b ( θ) su { b1 θ } QY Y y y lim b Y y if 1( ) y < θ, b Y otherwise. The shar lower and uer bounds for the θ th quantile of Y are b θ and QY b θ. QY The imlication of Lemmas 1 and 2 is that we only need to determine tight bounds of the conditional exected value of g( Y ( 0) ) for the oulation of interest to bound sharly the ATEs and QTEs. This is what we do in the next section. 4 Identification 4.1 First ste assumtions To concentrate on the secial issues related to the 'double selection roblem' into rograms and emloyment, we assume that the treatment effects would be oint identified if there was Lechner and Melly, revised

15 no samle selection. We state Assumtion 1 in general terms because the bounding strategy that we roose does not deend on the way this first selection roblem is solved. Below, we discuss two examles that satisfy this assumtion: selection on observables and instrumental variables. Assumtion 1 (Identification of conditional emloyment and wages given emloyment) xt, F y ; x, t,1 and F are identified for d XT Y d X, T, d XT ( x; t ) d { 0,1}, for x χ T = t (,, ) 12 and y,. Lemma 3 illustrates the emirical content of Assumtion 1. It states that these conditions are sufficient to identify the causal effects of D on earnings and emloyment outcomes. Lemma 3 (Assumtion 1 identifies effects of D on emloyment and earnings) If Assumtion 1 holds, then () 1 ( 0) Y Y E T = t, ATE ( T = t), and QTEθ ( T = t) for θ (0,1) are identified. Examle 1: effects on the treated, selection on observables This examle corresonds to our alication in section 6. Here, we assume indeendence of treatment, D, and otential Y( d ) outcomes,, d, conditional on confounders, X, as in the standard matching literature (see ection 6 for a brief justification in the secific context of that alication). T is equal to the treated (we could as well consider the effect on the non-treated, the whole oulation, etc.). Furthermore, to be able to recover the necessary information from the data, common suort assumtions are added in art b) of Assumtion 1. As we are interested in the effects on the double treated, combinations of characteristics that lead to a zero treatment robability 12 When the elements of V are binary, the robability of all elements of V jointly being equal to one conditional on W is denoted by w w. = VW Lechner and Melly, revised

16 do not matter. Lemma 4 verifies that this traditional matching assumtion satisfies the conditions of Assumtion 1. Assumtion 1 (conditional indeendence assumtion for first stage) { } a) Conditional indeendences: ( 0) ( 0 ), ( 0) b) Common suort: P( D X x) Y D X = x for x χ D = 1 ; 13 = 1 = < 1 for x χ. Lemma 4: Assumtion 1 imlies Assumtion 1 for the treated oulation. Imbens (2004) rovides an excellent survey of estimators for ATEs consistent under Assumtion 1. Efficient estimation of such average treatment effects is discussed for examle in Hahn (1998), Heckman, Ichimura, and Todd (1998), Hirano, Imbens, and Ridder (2003), and Imbens, Newey, and Ridder (2005). Firo (2007) and Melly (2006) discuss efficient estimation of quantile treatment effects. Note that by restricting X to be a constant we obtain the secial case of a random exeriment, which is analyzed by Lee (2009). Examle 2: binary instrumental variable, effects for the comliers In this examle, we show that the assumtions of Imbens and Angrist (1994) made for both outcomes (emloyment and earnings) satisfy the conditions stated in Assumtion 1 for the oulation of comliers. Let Z be a binary variable and define the otential treatment variables indexed D () against Z, 0 and 1. We follow the original article and call comliers the individuals i D i ( () ) who react to a change in the value of Z: C = 1 D 1 > D 0. Assumtion 1 states standard conditions as found in Abadie, Angrist, and Imbens (2002), for instance. Lemma 5 shows that these assumtions satisfy the requirements of Assumtion 1 for the oulation of comliers (T is C, and ) t = 1 Y 13 This notation means that the joint distribution of 0 0 and 0 is indeendent of D conditional on X. Lechner and Melly, revised

17 Assumtion 1 (conditional instrumental variable assumtion for first stage) { } a) Conditional indeendences: ( 0) ( 0, ) ( 1) ( 1, ) ( 0, ) ( 1, ) ( 0, ) ( 1) Y Y D D Z X = x for x χ ; ( i() i) = E D() 1 > E D( 0) b) Monotonicity and first-stage: Pr D 1 D 0 1 and c) Common suort: 0 < PZ ( = 1 X= x) <1 for x χ ; ; Lemma 5: Assumtion 1 imlies Assumtion 1 for the comliers. Therefore, the bounds derived in this aer are also useful in cases where the earnings effects are identified using an instrumental variable strategy. A recent examle of such a case occurs in Engberg, Ele, Imbrogno, ieg, and Zimmer (2009). They are interested in the effects of magnet schools on education outcomes, where a lottery determines magnet school assignment. The roblem is that some households move to another school district if they lose the lottery and the outcomes are no longer observed in this case. We could bound the causal effect of magnet schools on the schooling outcomes using the strategy suggested in this aer. These two examles do not exhaust all cases where Assumtion 1 is satisfied. For instance, in the resence of a continuous instrument, Heckman and Vytlacil (2005, 2007) discuss the identification and estimation of the marginal treatment effects. Our results can be alied to bound these effects when the outcome is not observed only for a selected oulation. 4.2 Worst case bounds Assumtion 1 is not sufficient to identify the effects of D on the otential earnings, Y( d) course, we oint identify these effects if we assume that selection into emloyment is indeendent from otential earnings. Another alternative to identify the treatment effects on otential earnings is the resence of a continuous instrument for the articiation decision. The nonarametric identification of the resulting samle selection models is discussed in Das,. Of Lechner and Melly, revised

18 Newey, and Vella (2003). However, since assuming indeendent samle selection is not lausible in our alication (and robably the majority of alications), and no continuous instrument is available for the second stage selection rocess, we give u on trying to achieve oint identification. Instead, we bound the treatment effects using weaker assumtions that aear to be more reasonable in our emirical study (and many other alications). Theorem 1 shows that knowing the effects of D on emloyment and realized earnings reduces the uncertainty. The bounds given in Theorem 1 as well as all other bounds derived in this aer are shar (see Internet Aendix for roofs). To state this theorem concisely, we denote the exected value of Y in the fraction of the oulation with the smallest value of Y by E ( Y ). If Y is a continuous random variable, min indicator variable, min otherwise. imilarly, E min E Y = 0 if Pr Y 1 1 and max 1 ( y) FY fy = E Y y dy. If Y is an ( = ) < E ( Y) ( ( Y ) ) min = Pr = / Y denotes the exected value of Y in the fraction of the oulation with the largest value of Y. 14 Theorem 1 (worst-case bounds) Assumtion 1 holds. If ( 0 ) ( x, t) +, () ( x, t) > 1, then the lower and uer bounds on X T 1 X, T ( ( 0 )),, () 1 E g Y X = x T = t = 1 are given resectively by X T x, t + X T x, t 1 min ( 0 ), () 1, ( 0 ) X, T( x, t) ( ( ( 0 )) =, =, ( 0) = 1) E g Y X x T t ( x, t) + ( x, t) 1 () 0 X, T 1 X, T () 1 X, T (,) x t 1 ( 0 ) ( x, t) X, T, + bg and ( x, t) () 1 X, T 14 We use the notation introduced by Zhang and Rubin (2003), who bounded the effects for the unobserved oulation of individuals working irresectively of their rogram articiation status. Lechner and Melly, revised

19 x, t + x, t 1 max ( 0 ) X, T () 1 X, T ( 0 ) X, T( x, t) ( ( ( 0 )) =, =, ( 0) = 1) E g Y X x T t (,) x t + (,) x t 1 () 0 X, T 1 X, T () 1 X, T (,) x t 1 ( 0 ) ( x, t) X, T + bg. ( x, t) () 1 X, T ( x, t) + ( x, t) 1, then the bounds are b g and b g. If () 0 X, T 1 X, T Note that the bounds are identified by Assumtion 1. Theorem 1 shows that we can learn art of the nonarticiation-emloyment outcome of emloyed articiants from the emloyment outcomes of non-articiants. However, there remains uncertainty because the emloyed articiants could be unemloyed when non-treated. In addition, Assumtion 1 is silent about which wage the always-emloyed articiants would receive when non-treated. The imortance of these two sources of uncertainty decreases as the emloyment robabilities increase. Obviously, these bounds will be very wide if the emloyment robabilities are low. In our alication, the emloyment robabilities are small because we consider a samle of ersons who are unemloyed when treatment starts. The emloyment robability at the end of our samle eriod is below 60%. Therefore, we cannot obtain informative bounds without further restricting the selection rocess into emloyment. uch restrictions will be imosed below Positive selection into emloyment In the standard labor suly model individuals accet a job offer if the offered wage is higher R R R than the reservation wage, denoted by, i.e. 0 = 1 Y 0 Y ). If Y is indeendent Y ( from Y ( 0) this model trivially imlies that emloyed individuals are ositively selected from R the oulation. This is more generally true as long as the difference between Y 0 and Y is 15 The resence of a discrete instrument for emloyment is an examle of such a restriction. We do not resent the imlied bounds because we do not have any lausible exclusion restriction in our alication. Moreover, the bounds are straightforward to derive as the intersections of the bounds conditionally on the instrument. Lechner and Melly, revised

20 ( X x Y y) 16 ositively associated with Y 0. This relation motivates the assumtion, suggested by Blundell, Gosling, Ichimura, and Meghir (2007), that Pr ( 0) 1, ( 0) ( = X = x Y > y ) Pr 0 1, 0 = =. uch a condition is equivalent to assuming that the wage distribution of the emloyed first-order stochastically dominates the wage distribution of nonemloyed: 17 Assumtion 2 (ositive selection into emloyment) F F ( y ; x, t,0 ) t 0,, (0) Y 0 X, T, (0)( y ; x,,1 Y X T ) for x χ T = tand y. Blundell, Gosling, Ichimura, and Meghir (2007) are interested in the conditional distribution of wages for the whole oulation. We are more secifically interested in the (unconditional) treatment effects for the emloyed treated oulation. In our alication, their bounds are uninformative while we obtain tighter bounds because we consider a oulation with a higher emloyment robability and we use the treatment effect structure. We obtain esecially informative bounds by combining ositive selection with an assumtion about the treatment choice defined below that cannot be used to bound outcomes for the whole oulation. Assumtion 2 tightens one of the bounds derived in Theorem 1: Theorem 2 (ositive selection into emloyment) a) If Assumtions 1 and 2 hold, and g() is a monotone increasing function, then: ( ) t, () 1 = max, () 1 X, T( x t) E g Y 0 X = x, T = 1 E g Y 0 X = x, T = t, 0 = 1. b) If Assumtions 1 and 2 hold and g() is a monotone decreasing function, then: ( ) t, () 1 min, () 1 X, T( x t) E g Y 0 X = x, T = = 1 E g Y 0 X = x, T = t, 0 = 1. Y is of course allowed to be ositively correlated with Y R 17 ee Blundell, Gosling, Ichimura, and Meghir (2007) for a roof. Lechner and Melly, revised

21 Thus, assuming ositive selection allows tightening the uer bound on the exected value and the lower bound on the distribution function of the otential outcome. There is no contradiction because the distribution function must be inverted to obtain the quantile function, which means that adding ositive selection has a similar effect on the quantiles and the mean. 4.4 Conditional monotonicity of the treatment effect on emloyment Lee (2009) restricts the individual treatment effect on the emloyment robability to have the same sign for all of the oulation. 18 Lee's assumtion is similar to the monotonicity assumtion of Imbens and Angrist (1994), but they restrict the effect of the instrument on the treatment status, while Lee restricts the effect of the treatment on samle selection. 19 Lee's (2009) assumtion aears to be overly restrictive for the tye of alication we consider. For instance, it excludes the ossibility that a training rogram has ositive effects on long-term unemloyed but negative effects on short-term unemloyed. However, this tye of heterogeneity is tyically found in the literature. Thus, we imose the weaker assumtion that the direction of the effect on emloyment is the same for all individuals with the same characteristics X. This assumtion is satisfied if the vector of characteristics is rich enough to cature the rogram effect heterogeneity on emloyment. A second difference with Lee (2009) is that we consider a broader range of identifying assumtions for the first ste of the selection rocess, thus making the aroach alicable outside the setting of random exeriments. A third difference with Lee (2009) is that we bound the effect for an observable oulation as long as assumtion 1 is satisfied for an observable oulation. In our alication, we bound 18 The same assumtion is also made by Zhang and Rubin (2003). 19 These assumtions are fundamentally different from the monotone-treatment-resonse assumtion of Manski (1997) and from the monotone instrumental variables assumtion of Manski and Peer (2000), because those authors assume certain functions to be monotone. Lechner and Melly, revised

22 the effects for the emloyed treated while Lee bounds the effects for the oulation who would work with or without the rogram. Therefore, if the rogram has a ositive effect on emloyment, then he bounds the effects for the non-treated oulation, while if the rogram has a negative effect on emloyment, he bounds the effects on the treated. When the emloyment effect is heterogeneous with resect to X, he estimates the effects for a oulation that is a mixture of treated and non-treated, which is difficult to interret. The formal definition of monotonicity is given in Assumtion 3: Assumtion 3 (conditional monotonicity of the treatment effect on emloyment) For each x χ T = t, either a) P (1) (0) X = x, T = t = 1, or b) P (1) (0) X = x, T = t = 1. Theorem 3 shows that Assumtion 3 allows tightening the bounds considerably: Theorem 3 (conditional monotonicity of the treatment effect on emloyment) a) Assumtions 1 and 3-a) hold. The bounds are given by the following exressions: xt, xt, xt E g Y X = x T = t = + b x t x t ( ( 0 )),, ( 0) 1 () E g( Y( 0 )) X x, T t, () 1 1 ( ( 0 )) =, =, ( 0) = 1 + () () (, ) () 0 X, T 1 X, T 0 X, T g,, 1 X, T 1 X, T = = = xt, xt, xt E g Y X x T t b x t x t () (, ) () 0 X, T 1 X, T 0 X, T g,, 1 X, T 1 X, T b) Assumtions 1 and 3-b) hold. The bounds are given by the following exressions: min max () 1 X, T( x, t) ( 0 ) X, T( x, t) () 1 X, T( x, t) ( 0 ) X, T( x, t) ( ) ( ) () E g Y 0 X = x, T = t, 0 = 1 E g Y 0 X = x, T = t, 1 = 1 ( ) E g Y 0 X = x, T = t, 0 = 1.. Lechner and Melly, revised

23 Interestingly, we obtain oint identification if () ( xt, ) = 1, ( 0 ) ( xt, ). The reason is that X T X, T under the monotonicity assumtion, both treatment and control grous are comrised of individuals whose samle selection was unaffected by the assignment to treatment, and therefore the two grous are comarable. amle selection correction rocedures are similar in this resect because they condition on the articiation robability, as discussed by Angrist (1997). However, they require continuous exclusion restrictions to achieve nonarametric identification. In the absence of such exclusion restrictions, there is only identification if the emloyment robabilities are, by chance, the same. Theorem 4-b) comrises the result of Proosition 1 in Lee (2009) as a secial case. This result has the aealing feature that the bounds do not deend on the suort of g(). Thus, the bounds are finite even when the suort of Y is infinite. Obviously, this is irrelevant for the distribution function or if the suort of Y is naturally bounded. 4.5 Combination of assumtions Adding ositive selection as defined in Assumtion 2 to conditional monotonicity as defined in Assumtion 3 tightens one of the bounds on ( ( 0 )),, () 1 E g Y X = x T = t = 1 : Theorem 4 (ositive selection into emloyment and conditional monotonicity) a) Assumtions 1, 2, and 3-a) hold. If g() is an monotone increasing function, then, the uer bound given in Theorem 3-a) tightens to: E g( Y( 0 )) X = x, T = t, ( 0) = 1 () 1, (, ) ( 0 ), (, ) 1 ( 0 ) X, T( x, t) X T x t X T x t max ( x, t 0 X, T ) () 1, ( x, t X T ) ( ) + E g Y 0 X = x, T = t, 0 = 1 () ( xt, ) 1, ( 0 ) ( xt, ) X T X, T. ( x, t) () 1 X, T b) Assumtions 1, 2, and 3-a) hold. If g() is a monotone decreasing function, then the lower bound given in Theorem 3-a) tightens to: Lechner and Melly, revised

24 ( ( ( 0 )) =, =, ( 0) = 1) E g Y X x T t () 1, (, ) ( 0 ), (, ) 1 ( 0 ) X, T( x, t) X T x t X T x t min ( x, t 0 X, T ) () 1, ( x, t X T ) ( ( ( 0 )),, ( 0) 1) + E g Y X = x T = t =. () ( xt, ) 1, ( 0 ) ( xt, ) X T X, T. ( x, t) () 1 X, T The bounds obtained in Theorem 4 are only slightly more informative than those of Theorem 3. The reason is that the ositive selection assumtion comares suressing the deendence on T = t - F ( y ; x,0) and F ( y ; x,1), but not Y 0 X, (0) Y 0 X, (0) F ( y ; x,0,1) Y 0 X, (0), (1) and F ( y ; x,1,1). Therefore, it is ossible that ( ;,0,1) Y y x dominates 0 X, F (0), (1) Y 0 X, (0), (1) F F ( y ; x,1,1), although at the same time F ( y ; x,1) dominates, (0), (1) Y 0 X, (0) Y 0 X Y 0 X ( y ; x, 0). We rule out this imlausible scenario in Assumtion 4:, (0) Assumtion 4 (ositive selection into emloyment conditionally on 1 = 1) F ( y ; x, t,0,1 ) F ( y ; x, t,1,1 Y 0 X, T, 0, 1 Y 0 X, T, 0, 1 ). () Combining Assumtion 4 with the conditional monotonicity assumtion leads to simle and intuitive bounds that are given in Theorem 5: Theorem 5 (ositive selection into emloyment conditionally on (1) = 1 and monotonicity) a) Assumtions 1, 3-a), and 4 hold. If g() is a monotone increasing function, then: ( ) ( ) () E g Y 0 X = x, T = t, 0 = 1 E g Y 0 X = x, T = t, 1 = 1. b) Assumtions 1, 3-a), and 4 hold. If g() is a monotone decreasing function, then: ( ) ( ) () E g Y 0 X = x, T = t, 0 = 1 E g Y 0 X = x, T = t, 1 = 1. The intuition for this result is simle. uose that a rogram has a ositive effect on emloyment. This means that the otential wage of the emloyed articiants is lower than that Lechner and Melly, revised

25 of the emloyed non-articiants by the ositive selection assumtion. If, desite this ositive effect on emloyment, the rogram has also a ositive effect on the observed wages, this imlies that the rogram has a ositive effect on otential wages. Formally, () 1 ( 0 ) =, =, () 1 = 1 () 1,, () 1 1 ( 0 ),, ( 0) E Y Y X x T t E Y X = x T = t = E Y X = x T = t = 1. 5 Estimation This aer focuses on the identification issues as well as on the emirical study that motivated the methodological innovation. Naturally, we bridge the ga between the identification results and the emirical study by roosing some estimators as well but kee this art of the aer brief. We consider only selection on observables as defined by Assumtion 1 and the effects on the treated because we use this strategy in our alication. We start by roosing consistent, nonarametric estimators. However, the combination of the dimension of the control variables and the samle sizes in this alication are such that a fully nonarametric estimation strategy would lead to very imrecise estimators. Therefore, in ection 5.2 we suggest to use a (arametric) roensity score to reduce the dimension of the estimation roblem and so to gain recision. 5.1 Nonarametric estimators Here, we rovide consistent, nonarametric estimators for all elements aearing in the different bounds of Theorems 1 to 5. ince we are interested in average as well as quantile effects, we consider two secial cases of the g-function, namely = 1( ) g Y Y y. g( Y) = Y and nonarametrically using Nadaraya-Watson or local linear regression. However, a local nonli- The conditional emloyment robabilities ( xd, ) for could be estimated X, D d {0,1} Lechner and Melly, revised

26 near estimator (Fan, Heckman, and Wand, 1995), like a local robit for instance, should be more suited for binary deendent variables. 20 E 1 Y y X = x, D = 0, = 1 could be estimated by a local robit as well. However, for the QTEs we need to estimate the conditional distribution function evaluated at a large number of y, which is comutationally very intensive. Moreover, since we need to estimate the comlete conditional distribution anyway, it is natural and faster to estimate the whole distribution by using locally weighted quantile regressions (Chaudhuri, 1991). By exloiting the linear rogramming reresentation of the quantile regression roblem, it is ossible to estimate all quantile regression coefficients efficiently (see Koenker, 2005, ection 6.3). The estimated conditional quantiles, though not necessarily monotonous in finite samles, may be inverted using the strategy roosed by Chernozhukov, Fernández-Val and Galichon (2009). The conditional exectations of earnings, E( Y X = x, D = 0, = 1), is estimated by local linear least squares regression. The majority of the bounds for the mean contain conditional, asymmetrically trimmed means like E ( Y X x, D 0, 1) max = = = and E ( Y X = x, D= 0, = 1). 21 Lee (2009) rooses an min estimator for the case with discrete X. We roose a new estimator allowing for discrete and continuous X. Koenker and Portnoy (1987) suggest an estimator based on linear quantile regression that allows estimating conditional trimmed means. They consider estimators of the form 1 ˆ J ( θ ) β ( θ ) dθ, where ˆ( β θ ) is the θ th quantile regression coefficient vector. We aly a 0 nonarametric version of their estimator with a articular weight function, J ( θ ), that selects 20 Moreover, in Frölich (2006) the local arametric estimator aears to have better small samle roerties. 21 For the distribution function, E ( ( Y y) ) max 1 = 1 if E 1 ( Y y) < ( ) and E ( 1( Y y) ) = E( 1( Y y) ) max otherwise, such that we do not need to estimate a trimmed mean. A similar result holds for the lower bound. Lechner and Melly, revised

27 quantiles only above or below. We estimate E ( Y X = x, D = 0, =1) by max 1 1 x ˆ( β x, θ ) dθ and E ( Y X = x, D = 0, =1) by min x ˆ( β x, θ ) dθ where ˆ( β x, θ ) is the θ th local linear quantile 0 regression evaluated at x in the suboulation D=0, =1. Lemma 1 shows that the bounds of the unconditional exected values equal the exected values of the conditional bounds. Thus, we estimate the bounds on the ATE by the mean of the conditional bounds evaluated at the treated observations. imilarly, for the QTE, we estimate the unconditional distribution by integrating the bounds on the conditional distribution. These bounds, which are monotone, are inverted to obtain the bounds of the QTE as shown in Lemma Using the roensity score to reduce the dimensionality of the roblem In our alication, the number of control variables X necessary to make Assumtion 1 lausible is too high to attemt a fully nonarametric estimation strategy, even with large samles. Rosenbaum and Rubin (1983) show that the roensity score reresents a useful dimension reduction device, because conditional indeendence of assignment and treatment (Assumtion 1 a) holds conditional on the (one-dimensional) roensity score as well: { } ( 0) ( 0) { } Y 0 0, 0 D X = x Y, 0 D ( x). DX imilarly, if Assumtion 2 and 4 (ositive selection into emloyment) are valid conditionally on X, they are also valid conditionally on the roensity score. In fact, these two assumtions are less restrictive conditional on the roensity score, as the score is less fine than X. In contrast, conditioning only on the roensity score instead of X would considerably strengthen Assumtion 3. The monotonicity assumtion states that the sign of the rogram effect on emloyment is the same for all observations with the same value of X. Therefore, the Lechner and Melly, revised