Evaluating (weighted) dynamic treatment effects by double machine learning
EEvaluating (weighted) dynamic treatment effectsby double machine learning
Hugo Bodory*, Martin Huber**, and Luk´aˇs Laff´ers+ *University of St. Gallen, Dept. of Economics**University of Fribourg, Dept. of Economics+Matej Bel University, Dept. of Mathematics
Abstract:
We consider evaluating the causal effects of dynamic treatments, i.e. of multiple treatmentsequences in various periods, based on double machine learning to control for observed, time-varyingcovariates in a data-driven way under a selection-on-observables assumption. To this end, we make use ofso-called Neyman-orthogonal score functions, which imply the robustness of treatment effect estimationto moderate (local) misspecifications of the dynamic outcome and treatment models. This robustnessproperty permits approximating outcome and treatment models by double machine learning even underhigh dimensional covariates and is combined with data splitting to prevent overfitting. In addition toeffect estimation for the total population, we consider weighted estimation that permits assessing dynamictreatment effects in specific subgroups, e.g. among those treated in the first treatment period. Wedemonstrate that the estimators are asymptotically normal and √ n -consistent under specific regularityconditions and investigate their finite sample properties in a simulation study. Finally, we apply themethods to the Job Corps study in order to assess different sequences of training programs under a largeset of covariates. Keywords: dynamic treatment effects, double machine learning, efficient score.
JEL classification:
C21.
We have benefited from comments by Saraswata Chaudhuri, Yingying Dong, Arturas Juodis, Frank Kleibergen,Jonathan Roth, Vasilis Syrgkanis, and seminar participants at McGill University, the University of Amsterdam, the Uni-versity of Duisburg-Essen, the University of Bolzano/Bozen, and the University of California Irvine (all online). Ad-dresses for correspondence: Hugo Bodory, University of St. Gallen, Varnb¨uelstrasse 14, 9000 St. Gallen, Switzerland,[email protected]; Martin Huber, University of Fribourg, Bd. de P´erolles 90, 1700 Fribourg, Switzerland, [email protected];Luk´aˇs Laff´ers, Matej Bel University, Tajovskeho 40, 97411 Bansk´a Bystrica, Slovakia, lukas.laff[email protected]. Laff´ers ac-knowledges support provided by the Slovak Research and Development Agency under contract no. APVV-17-0329 andVEGA-1/0692/20. a r X i v : . [ ec on . E M ] F e b Introduction
In many empirical problems, policy makers and researchers are interested in the causal effectsof sequences of interventions or treatments, i.e. dynamic treatment effects. Examples includethe impact of sequences of training programs (for instance, a job application training followedby a language courses) on the employment probabilities of job seekers or the effect of sequentialmedical interventions (for instance,a surgery combined with rehabilitation training) on health.As treatment assignment is typically non-random, causal inference about distinct sequences oftreatments requires controlling for confounders jointly affecting the various treatments and theoutcome of interest. An assumption commonly imposed in the literature is sequential conditionalindependence, which implies that the treatment in each period is unconfounded conditional onpast treatment assignments, past outcomes, and the history of observed covariates up to therespective treatment assignment. Due to increasing data availability, the number of observedcovariates that may potentially serve as control variables to justify the sequential conditionalindependence assumption has been growing in many empirical contexts, which poses the questionof how to optimally control for such a wealth of information in the estimation process.This paper combines the semiparametrically efficient estimation of dynamic treatment ef-fects under sequential conditional independence with double machine learning (DML) frameworkoutlined in Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018) tocontrol for observed covariates in a data-driven way. More specifically, treatment effect estima-tion is based on the efficient score function, belonging to the class of doubly robust estimationas discussed in Robins, Rotnitzky, and Zhao (1994) and Robins and Rotnitzky (1995), and re-lies on plug-in estimates of the dynamic treatment propensity scores (the conditional treatmentprobabilities given histories of covariates and past treatments) and conditional mean outcomes(given histories of treatments, covariates, and past outcomes). We obtain these plug-in estimatesby machine learning, which permits algorithmically controlling for covariates with the highestpredictive power for the treatments and outcomes.To safeguard against overfitting bias due to correlations between the estimation steps, theplug-in models and the treatment effects are estimated in different parts of the data, whoserole is subsequently swapped to prevent not using parts of the data for effect estimation (andthereby increasing the variance). We show that our estimator satisfies the so-called Neyman11959) orthogonality discussed in Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey,and Robins (2018) and is thus asymptotically normal and √ n -consistent under specific regu-larity conditions despite the data-driven estimation of the plug-ins. One restriction is that theconvergence of the plug-in estimates to the true models as a function of the covariates is nottoo slow, which is satisfied if each of the estimators converges at a rate not slower than n − / .When using lasso as machine learner, this implies a form of approximate sparsity, meaning thatthe number of important covariates for obtaining a decent approximation of the plug-ins is smallrelative to the sample size. However, the set of these important confounders need not be knowna priori, which is particularly useful in high dimensional data with a vast number of covariatesthat could potentially serve as control variables.As a further contribution, we discuss the DML-based estimation of weighted dynamic treat-ment effects where the weight is defined as a function of the baseline covariates. This permits,for instance, assessing treatment sequences among those treated or not treated in the first periodand therefore provides a rather general framework for the definition of interesting subpopula-tions. Also for this estimator based on a weighted version of the efficient sore function, we showNeyman (1959) orthogonality and √ n -consistency under specific restrictions on the convergencerates of the plug-in estimators, which now also include the estimated weighting function.Furthermore, we investigate the methods’ finite sample behavior in a simulation study, andfind the point estimators to perform rather decently in the simulation designs considered. As anempirical contribution, we assess the effects of various treatment sequences in the U.S. Job Corpsstudy on an educational intervention for disadvantaged youth. We find that attending vocationaltraining in the two initial years of the program likely increases the employment probability fouryears after the start of Job Corps when compared to no instruction. In contrast, the relativeperformance of sequences of vocational vs. academic classroom training is less clear.The literature on dynamic treatment effects goes back to Robins (1986), who proposes adynamic causal framework along with an estimation approach known as g-computation forrecursively modeling outcomes at some point in time as functions of the (histories of) ob-served covariates and treatments under the sequential conditional independence assumption.G-computation was originally implemented by parametric maximum likelihood estimation ofnested structural models for the outcomes in all periods, requiring the (in general tedious) es-timation of the conditional densities of all time-varying covariates. Robins (1998) suggested2n alternative, less complex modeling approach based on so-called marginal structural modelsrepresenting outcomes in specific treatment states as functions of time-constant covariates only.In order to also control for time-varying confounding, such marginal models need (in the spiritof Horvitz and Thompson (1952)) to be combined with weighting by the inverse of the dynamictreatment propensity scores, see for instance Robins, Greenland, and Hu (1999) and Robins,Hernan, and Brumback (2000). The propensity scores in each period are typically estimatedby sequential logit regressions, but see Imai and Ratkovic (2015) for an alternative, empiricallylikelihood-based approach that aims at finding propensity score specifications that maximizecovariate balance. Lechner (2009) considers inverse probability weighting (IPW) by the dy-namic treatment propensity scores alone (i.e. without the use of marginal outcome models),while Lechner and Miquel (2010) apply propensity score matching and Blackwell and Strezhnev(2020) direct matching on the covariates.Doubly robust estimators of dynamic treatment effects comprise methods that are consistentif either the sequential treatment propensity scores or nested outcome models are correctlyspecified. This includes estimation based on the sample analog of the efficient influence function(underlying the semiparametric efficiency bounds) provided in Robins (2000), which is a functionof both the nested treatment and outcome models. In contrast, Bang and Robins (2005)propose a doubly robust estimator that is based on estimating potential outcomes by nestedmodels of conditional mean outcomes (given the covariate histories as well as past and currenttreatment assignments) in all periods, a form of g-computation that does not require tediouslikelihood estimations of conditional densities as initially proposed in Robins (1986). Here,doubly robustness comes from the fact that a weight based on the nested treatment propensityscores is included as additional covariate in conditional mean estimation.van der Laan and Gruber (2012) demonstrate that this approach fits the framework ofTargeted Maximum Likelihood Estimation (TMLE) of van der Laan and Rubin (2006), whichobtains doubly robustness through updating initial conditional outcome estimates by regressingthem on a function of the nested propensity scores in each period, and offer a refined estimator.Specifically, they suggest estimating nuisance parameters by the super learner of van der Laan,Polley, and Hubbard (2007), an ensemble method for machine learning. In contrast, the ap- Yu and van der Laan (2006) discuss an alternative doubly robust approach based on combining propensity scoreswith the estimation of marginal structural models. √ n -consistency is attained. Finally, Lewis and Syrgkanis (2020) propose an alternative DMLestimator of dynamic treatment effects. It is based on residualizing or debiasing the outcomeand the treatment by purging the effects of observed confounders using machine learning andregressing the debiased outcome on the debiased treatment in a specific period. This approachmay also be applied to continuous (rather than discrete) treatments, but in contrast to ourmethod assumes partial linearity in the outcome model.This paper proceeds as follows. Section 2 introduces the concepts of dynamic treatmenteffects in the potential outcome framework, presents the identifying assumptions and discussesidentification. Section 3 proposes an estimation procedure based on double machine learning andshows √ n -consistency and asymptotic normality under specific conditions. Section 4 extendsthe procedure to the evaluation of weighted dynamic treatment effects. Section 5 providesa simulation study. Section 6 presents an empirical application to data from Job Corps, aneducational program for disadvantaged youth. Section 7 concludes. We are interested in the causal effect of a sequence of discretely distributed treatments andwill for the sake of simplicity focus on the case of two sequential treatments in the subsequentdiscussion. To this end, denote by D t and Y t the treatment (e.g a training program) and theoutcome (e.g. employment) in period T = t . Therefore, D and D are the treatments in thefirst and second periods, respectively, and may take values d , d ∈ { , , ..., Q } , with 0 indicating4on-treatment and 1 , ..., Q the different treatment choices (where Q denotes the number of non-zero treatments). Let Y denote the outcome of interest measured in the second period afterthe realization of treatment sequence D and D . To define the dynamic treatment effects ofinterest, we make use of the potential outcome framework, see for instance Rubin (1974). Wedenote by d a specific treatment sequence ( d , d ) with d , d ∈ { , , ..., Q } , then D ≡ ( D , D )and let Y ( d ) denotes the potential outcome hypothetically realized when the treatments areset to that sequence d . We also define { , , ..., Q } = { , , ..., Q } × { , , ..., Q } .We aim at evaluating the average treatment effect (ATE) of two distinct treatment sequencesin the population, ∆( d , d ∗ ) = E [ Y ( d ) − Y ( d ∗ )] , (1)with d (cid:54) = d ∗ such that the sequences differ either in d or in both d . Examples are theevaluation of a sequence of two binary treatments vs. no treatment, e.g. d = (1 ,
1) and d ∗ =(0 , d = (1 , d )and d ∗ = (0 , d ), with d ∈ { , } . The latter parameter is known as the controlled direct effectin causal mediation analysis, see for instance Pearl (2001), assessing the net effect of the firsttreatment when setting the second treatment to be D = d for everyone. Throughout the paperwe assume that stable unit treatment value assumption (SUTVA, Rubin (1980)) holds such thatPr( D = d = ⇒ Y = Y ( d )) = 1 . This rules out interaction effects, general equilibrium effectsand implicitly assumes that treatments are uniquely defined.Identification relies on a sequential conditional independence assumption, requiring that thetreatment in each period is conditionally independent of the potential outcomes, conditional onprevious treatments and (histories of) observed covariates measured prior to treatment, whichmight include past outcomes, too. Let to this end X t denote the observed characteristics in In the case of d , d ∗ sharing the same d but differing in terms d , the identification problem collapses to thestandard case with one treatment period (namely T = 2) under the condition that D = d . The case of a singletreatment period also prevails when considering the effects on Y , i.e. the outcome in period T = 1, which onlypermits assessing the effect of D . In either case, the standard double machine learning (DML) framework forsingle treatment periods can be applied as e.g. outlined in Belloni, Chernozhukov, Fern´andez-Val, and Hansen(2017), such that we do not consider these scenarios in this paper. From the perspective of causal mediation analysis, our paper complements the study of Farbmacher, Huber,Laffers, Langen, and Spindler (2020), who apply DML to the estimation of so-called natural direct and indirecteffects. In the latter case, D is not prescribed to have the same value d for everyone, but corresponds to thepotential value D ( d ), i.e. the hypothetical treatment state of D that would be ‘naturally chosen’ (i.e. withoutprescription) as a consequence of D = d . T = t . X consists of pre-treatment characteristics measured prior to the first treatment D , while X (which may contain Y ) is measured prior to D , but may be influenced by D aswell as X . Covariates in a particular period may therefore be affected by previous covariatesand treatments, implying that confounding may be dynamic in the sense that identificationrelies on time varying observables rather than on baseline covariates alone. Figure 1 providesa graphical illustration using a directed acyclic graph, with arrows representing causal effects.Each of D , D , and Y might be causally affected by distinct and statistically independent setsof unobservables not displayed in Figure 1, but none of these unobservables may jointly affect D and Y given X or D and Y given D , X , and X .Formally, the first assumption invokes conditional independence of the treatment in the firstperiod D and the potential outcomes Y ( d ) given X as commonly invoked in the treatmentevaluation literature, see e.g. Imbens (2004). It rules out unobserved confounders jointly affecting D and Y ( d ) conditional on X . Assumption 1 (conditional independence of the first treatment): Y ( d ) ⊥ D | X , for d ∈ { , , ..., Q } .where ‘ ⊥ ’ denotes statistical independence.The second assumption invokes conditional independence of the second treatment D given thefirst treatment D and the (history of) covariates X and X , which we denote by X = ( X , X )to ease notation. It rules out unobserved confounders jointly affecting D and Y ( d ) conditionalon D and X . Assumption 2 (conditional independence of the second treatment): ( d ) ⊥ D | D , X , X , for d ∈ { , , ..., Q } .The third assumption imposes common support, meaning that the treatment in each period isnot a deterministic function of the respective observables in the conditioning set, which rules outconditional treatment probabilities (or propensity scores) of 0 or 1. This implies that conditionalon each value of the observables occuring in the population, subjects with distinct treatmentassignments { , , ..., Q } exist. Assumption 3 (common support):
Pr( D = d | X ) >
0, Pr( D = d | D , X ) > d , d ∈ { , ..., Q } .To ease notation, we henceforth denote the propensity scores by p d ( X ) = Pr( D = d | X ) and p d ( D , X ) = Pr( D = d | D , X ). Furthermore, we denote the conditional mean outcome inthe second period by µ Y ( D , X ) = E [ Y | D , X , X ] and the nested conditional mean outcomein the first period by ν Y ( D , X ) = (cid:90) E [ Y | D , X , X = x ] dF X = x | D ,X = E [ E [ Y | D , X , X ] | D , X ] , where F X = x | D ,X denotes the conditional distribution function of X given ( D , X ) at value x . As for instance discussed in Tran, Yiannoutsos, Wools-Kaloustian, Siika, van der Laan, andPetersen (2019), Assumptions 1-3 permit identifying the mean potential outcome E [ Y ( d )] basedon the following expression: E [ Y ( d )] = E [ ψ d ] , where ψ d = I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X )+ I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) + ν Y ( d , X ) . (2)This follows from the fact that ψ d − E [ Y ( d )], which corresponds to the efficient score functionof dynamic treatment effects as discussed in Robins (2000), has a zero mean property: E [ ψ d − E [ Y ( d )]] = 0. 7 Estimation of the counterfactual with K-fold Cross-Fitting
We subsequently propose an estimation strategy for the counterfactual E [ Y ( d )] with d ∈{ , , ..., Q } and show its √ n -consistency under specific regularity conditions. Define ψ d ( W, η, Ψ d ) = I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X )+ I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X )+ ν Y ( d , X ) − Ψ d , (3)where W = { W i | ≤ i ≤ N } with W i = ( Y i , D i , D i , X i , X i ) for all i denotes the set ofobservations and I {·} denotes the indicator function. The true nuisance parameters are denotedby η = ( p d ( X ) , p d ( D , X ) , µ Y ( D , X ) , ν Y ( D , X )), their estimates byˆ η = (ˆ p d ( X ) , ˆ p d ( D , X ) , ˆ µ Y ( D , X ) , ˆ ν Y ( D , X )). Let Ψ d = E [ Y ( d )] denotes the truecounterfactual.We suggest estimating the Ψ d using the following algorithm that combines orthogonal scoreestimation and sample splitting. Further below we will outline the conditions under which thisestimation strategy leads to √ n -consistent estimates for the counterfactual. Algorithm 1: Estimation of E [ Y ( d )]1. Split W in K subsamples. For each subsample k , let n k denote its size, W k the set ofobservations in the sample and W Ck the complement set of all observations not in k .2. For each k , use W Ck to estimate the model parameters of p d ( X ) and p d ( d , X ). Split W Ck into 2 nonoverlapping subsamples and estimate the model parameters of the conditionalmean µ Y ( d , X ) and the nested conditional mean ν Y ( d , X ) in the distinct subsamples.Predict the models among W k , where the predictions are denoted by ˆ p d k ( X ), ˆ p d k ( D , X ),ˆ µ Y k ( d , X ), ˆ ν Y k ( d , X ).3. For each k , obtain an estimate of the moment condition for each observation i in W k ,8enoted by ˆ ψ d i,k :ˆ ψ d i,k = I { D i = d } · I { D i = d } · [ Y i − ˆ µ Y k ( d , X i )]ˆ p d k ( X i ) · ˆ p d k ( d , X i )+ I { D i = d } · [ˆ µ Y k ( d , X i ) − ˆ ν Y k ( d , X i )]ˆ p d k ( X i ) + ˆ ν Y k ( d , X i ) .
4. Average the estimated scores ˆ ψ d i,k over all observations across all K subsamples to obtainan estimate of Ψ d in the total sample, denoted by ˆΨ d = 1 /n (cid:80) Kk =1 (cid:80) n k i =1 ˆ ψ d i,k .In order to achieve √ n -consistency for counterfactual estimation, we make the followingassumption on the prediction quality of the machine learners when estimating the nuisance pa-rameters. Closely following Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, andRobins (2018), we introduce some further notation. Let ( δ n ) ∞ n =1 and (∆ n ) ∞ n =1 denote sequencesof positive constants with lim n →∞ δ n = 0 and lim n →∞ ∆ n = 0 . Furthermore, let c, (cid:15), C and q be positive constants such that q > , and let K ≥ Z = ( Z , ..., Z l ), let (cid:107) Z (cid:107) q = max ≤ j ≤ l (cid:107) Z l (cid:107) q , where (cid:107) Z l (cid:107) q = ( E [ | Z l | q ]) q . In order to easenotation, we assume that n/K is an integer. For the sake of brevity we omit the dependence ofprobability Pr P , expectation E P ( · ) , and norm (cid:107)·(cid:107) P,q on the probability measure P. Assumption 4 (regularity conditions and quality of plug-in parameter estimates):
For all probability laws P ∈ P the following conditions hold for the random vector ( Y , D , D , X , X )for all d , d ∈ { , , ..., Q } :(a) (cid:107) Y (cid:107) q ≤ C, (cid:13)(cid:13) E [ Y | D = d , D = d , X ] (cid:13)(cid:13) ∞ ≤ C ,(b) Pr( (cid:15) ≤ p d ( X ) ≤ − (cid:15) ) = 1 , Pr( (cid:15) ≤ p d ( d , X ) ≤ − (cid:15) ) = 1 , (c) (cid:13)(cid:13)(cid:13) Y − µ Y ( d , X ) (cid:13)(cid:13)(cid:13) = E (cid:104) (cid:16) Y − µ Y ( d , X ) (cid:17) (cid:105) ≥ c (d) Given a random subset I of [ n ] of size n k = n/K, the nuisance parameter estimatorˆ η = ˆ η (( W i ) i ∈ I C ) satisfies the following conditions. With P -probability no less than9 − ∆ n : (cid:107) ˆ η − η (cid:107) q ≤ C, (cid:107) ˆ η − η (cid:107) ≤ δ n , (cid:13)(cid:13)(cid:13) ˆ p d ( X ) − / (cid:13)(cid:13)(cid:13) ∞ ≤ / − (cid:15), (cid:13)(cid:13)(cid:13) ˆ p d ( D , X ) − / (cid:13)(cid:13)(cid:13) ∞ ≤ / − (cid:15), (cid:13)(cid:13)(cid:13) ˆ µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) ˆ p d ( X ) − p d ( X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) ˆ µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) ˆ p d ( D , X ) − p d ( D , X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) ˆ ν Y ( D , X ) − ν Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) ˆ p d ( X ) − p d ( X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / . The only non-primitive condition is the condition (d). It puts restrictions on the quality ofthe nuisance parameter estimators. Condition (a) states that the distribution of the outcomedoes not have unbounded moments. (b) refines the common support condition such that thepropensity scores are bounded away from 0and 1. Finally, (c) states that the covariates X donot perfectly predict the conditional mean outcome.For demonstrating the √ n -consistency of our estimator of the mean potential outcome, weshow that it satisfies the requirements of the DML framework in Chernozhukov, Chetverikov,Demirer, Duflo, Hansen, Newey, and Robins (2018) by first verifying linearity and Neyman or-thogonality of the score (see Appendix A.1). Then, as ψ d ( W, η, Ψ d ) is smooth in ( η, Ψ d ), it issufficient that the plug-in estimators converge with rate n − / for achieving n − / -convergencefor the estimation of ˆΨ d as postulated in Theorem 1. This convergence rate of n − / has beenshown to be achieved by many commonly used machine learners under specific conditions, suchas lasso, random forests, boosting and neural nets, see for instance Belloni, Chernozhukov, andHansen (2014), Luo and Spindler (2016), Wager and Athey (2018), and Farrell, Liang, and Misra(2018). Theorem 1
Under Assumptions 1-4, it holds for estimating E [ Y ( d )] based on Algorithm 1: √ n (cid:16) ˆΨ d − Ψ d (cid:17) → N (0 , σ ψ d ), where σ ψ d = E [( ψ d − Ψ d ) ].The proof of Theorem 1 is provided in Appendix A.1.10 Evaluation of weighted dynamic treatment effects
Lechner and Miquel (2010) show that under our assumptions, one may identify treatment effectsfor specific subgroups that are defined as a function of the distribution of the baseline covariates X . To this end, let S denote a binary indicator for belonging to the subgroup of interest and g ( X ) = Pr( S = 1 | X ) the probability of being in that group conditional on X . Interestingexamples for such subgroups are the treated or non-treated populations in the first period, ob-tained by defining S = I { D = d } with d ∈ { , , ..., Q } . Mean potential outcomes conditionalon S = 1 are identified based on reweighting by g ( X ), see e.g. Hirano, Imbens, and Ridder(2003) who use this approach for weighted ATE evaluation based on IPW. That is, E [ Y ( d ) | S = 1] = (cid:34) S · Y ( d )Pr( S = 1) (cid:35) = E (cid:34) S Pr( S = 1) · E [ Y ( d ) | X ] (cid:35) (4)= E (cid:34) g ( X )Pr( S = 1) · E [ Y ( d ) | X ] (cid:35) = E (cid:34) g ( X )Pr( S = 1) · Y ( d ) (cid:35) , where the first equality follows from basic probability theory and the remaining ones from thelaw of iterated expectations. This suggests the following identification approach: E [ Y ( d ) | S = 1] = E [ ψ d ,S =1 ] , where ψ d ,S =1 = g ( X )Pr( S = 1) · I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (5)+ g ( X )Pr( S = 1) · I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) + S Pr( S = 1) · ν Y ( d , X ) . Note that the term S Pr( S =1) · ν Y ( d , X ) in (5) corresponds to S Pr( S =1) · E [ Y ( d ) | X ] in (4). Ap-pendix A.2 shows that the moment condition E [ ψ d ,S =1 − E [ Y ( d ) | S = 1]] = 0 holds, such that E [ ψ d ,S =1 ] identifies the weighted mean potential outcome, and proves Neyman orthogonality. Itdemonstrates that DML is √ n -consistent and asymptotically normal under Assumption 5 below.The latter formalizes the rate restrictions on the plug-in estimates, which now also contain anestimate of g ( X ) denoted by ˆ g ( X ). To this end, Algorithm 1 outlined in Section 3 is appliedto estimate E [ Y ( d ) | S = 1] by using modified moment conditions in steps 3 and 4.11ore specifically, the previously used ˆ ψ d i,k computed in some subsample k is replaced byˆ ψ d ,S =1 i,k = ˆ g k ( X i ) · I { D i = d } · I { D i = d } · [ Y i − ˆ µ Y k ( d , X i )]ˆ p d k ( X i ) · ˆ p d k ( d , X i )+ ˆ g k ( X i ) · I { D i = d } · [ˆ µ Y k ( d , X i ) − ˆ ν Y k ( d , X i )]ˆ p d k ( X i ) + S i · ˆ ν Y k ( d , X i ) . (6)In step 4, the estimated scores ˆ ψ d ,S =1 i,k are averaged over all observations across all K subsamplesand divided by an estimate of Pr( S = 1) to obtain an estimate of Ψ d ,S =10 = E [ Y ( d ) | S = 1]based on ˆΨ d ,S =1 = (cid:104) (cid:80) Kk =1 (cid:80) n k i =1 ˆ ψ d i,k (cid:105)(cid:46)(cid:104) (cid:80) Kk =1 (cid:80) n k i =1 S i (cid:105) .The following assumption refines the conditions of Assumption 4 such that asymptotic nor-mality holds for the DML estimator based on (5). Assumption 5 (regularity conditions and quality of plug-in parameter estimates):
For all probability laws P ∈ P the following conditions hold for the random vector ( Y , D , D , X , X , S )for all d , d ∈ { , , ..., Q } :(a) (cid:107) Y (cid:107) q ≤ C, (cid:13)(cid:13) E [ Y | D = d , D = d , X ] (cid:13)(cid:13) ∞ ≤ C ,(b) Pr( (cid:15) ≤ p d ( X ) ≤ − (cid:15) ) = 1Pr( (cid:15) ≤ p d ( d , X ) ≤ − (cid:15) ) = 1(c) (cid:13)(cid:13)(cid:13) Y − µ Y ( d , X ) (cid:13)(cid:13)(cid:13) = E (cid:104) (cid:16) Y − µ Y ( d , X ) (cid:17) (cid:105) ≥ c (d) Given a random subset I of [ n ] of size n k = N/K, the nuisance parameter estimatorˆ χ = ˆ χ (( W i ) i ∈ I C ) satisfies the following conditions. With P -probability no less than12 − ∆ n : (cid:107) ˆ χ − χ (cid:107) q ≤ C, (cid:107) ˆ χ − χ (cid:107) ≤ δ n , (cid:107) ˆ g ( X ) − / (cid:107) ∞ ≤ / − (cid:15), (cid:13)(cid:13)(cid:13) ˆ p d ( X ) − / (cid:13)(cid:13)(cid:13) ∞ ≤ / − (cid:15), (cid:13)(cid:13)(cid:13) ˆ p d ( D , X ) − / (cid:13)(cid:13)(cid:13) ∞ ≤ / − (cid:15), (cid:13)(cid:13)(cid:13) ˆ χ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) ˆ p d ( X ) − p d ( X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) ˆ µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) ˆ p d ( D , X ) − p d ( D , X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) ˆ ν Y ( D , X ) − ν Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) ˆ p d ( X ) − p d ( X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / . (cid:13)(cid:13)(cid:13) ˆ µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:107) ˆ g ( X ) − g ( X ) (cid:107) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) ˆ ν Y ( D , X ) − ν Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:107) ˆ g ( X ) − g ( X ) (cid:107) ≤ δ n n − / . Assumption 5 can be satisfied if the plug-in estimator ˆ g ( X ) converges to its’ true value g ( X ) with rate n − / just like the estimators of the other nuisance terms. Then, the averagetreatment effect in the subgroup, denoted by∆( d , d ∗ , S = 1) = E [ Y ( d ) − Y ( d ∗ ) | S = 1] , (7)is √ n -consistently estimated, as postulated in Theorem 2. Theorem 2
Under Assumptions 1-3 and 5, it holds for estimating E [ Y ( d ) | S = 1] based on Algorithm 1: √ n (cid:16) ˆΨ d ,S =1 − Ψ d ,S =10 (cid:17) → N (0 , σ ψ d ,S =1 ), where σ ψ d ,S =1 = E [( ψ d ,S =1 − Ψ d ,S =10 ) ].The proof of Theorem 2 is provided in Appendix A.2. This section provides a simulation study to investigate the finite sample behavior of our doublemachine learning method for dynamic treatment effects based on the following data generating13rocess: Y = D + D + X (cid:48) β X + X (cid:48) β X + U,D = I { X (cid:48) β X + V > } ,D = I { . D + X (cid:48) β X + X (cid:48) β X + W > } ,X ∼ N (0 , Σ ) , X ∼ N (0 , Σ ) ,U, V, W ∼ N (0 , , independently of each other . Outcome Y is a function of the observed variables D , D , X , X , and the unobserved scalar U . The treatment effects of both D and D are equal to 1. D is a function of X and theunobserved scalar V . D is a function of both pre- and post-treatment covariates X and X ,the first treatment D , and the unobservable scalar W . Both X and X are vectors of covariatesof dimension p , drawn from a multivariate normal distribution with zero mean and covariancematrices Σ and Σ , respectively. U, V, W are random and standard normally distributed. Weconsider two sample sizes of n = 2500 and 10000, running 1000 simulations for the smaller and250 simulations for the larger sample sizes.In our simulations, we set p , the number of covariates in X and X , respectively, to 50 or100. Σ and Σ are defined based on setting the covariance of the i th and j th covariate in X or X to 0 . | i − j | . The coefficients β X and β X gauge the impacts of the covariates on Y , D ,and D , respectively, and thus, the magnitude of confounding. The i th element in the coefficientvectors β X and β X is set to 0 . /i for i = 1 , ..., p , implying a quadratic decay of covariateimportance in terms of confounding. As reported in Table 1, this specification implies that the R statistic based on linearly predicting Y by X ranges from 36 to 41%, depending on thenumber of covariates and the sample size. Furthermore, the Nagelkerke (1991) pseudo- R whenpredicting D by X and D by D , X based on probit models ranges from 13 to 17% and26 to 33%, respectively. These figures point to a substantial level of confounding as it may bereasonably encountered in empirical applications.We investigate the performance of ATE estimation when comparing the sequences of obtain-ing both treatments ( d = ( d = 1 , d = 1)) vs. no treatment ( d ∗ = ( d = 0 , d = 0)) in the totalpopulation based on Theorem 1 and in the treated in the first period based on Theorem 2. The14able 1: Confounding based on β X = β X = 0 . /i number of sample pseudo- R (%) pseudo- R (%) R (%)covariates size ˆ p d ( X ) ˆ p d ( d , X ) ˆ µ Y ( X )50 2500 15 29 3850 10000 17 33 41100 2500 13 26 36100 10000 14 27 37nuisance parameters, i.e. the linear and probit specifications of the outcome and treatment equa-tions, are estimated by lasso regressions using the default options of the SuperLearner packageprovided by van der Laan, Polley, and Hubbard (2007) for the statistical software R . 3-fold cross-fitting is used for the estimation of the treatment effects. We drop observations whose productsof estimated treatment propensity scores in the first and second period, ˆ p d ( X ) · ˆ p d ( D , X ),are close to zero, namely smaller than a trimming threshold of 0 .
01 (or 1%). This avoids anexplosion of the propensity score-based weights and thus of the variance when estimating themean potential outcomes by the sample analogue of identification result (6), where the productof the propensity scores enters the denominator for reweighing the outcome. Our estimationprocedure is available in the causalweight package for R by Bodory and Huber (2018).Table 2: Simulation results based on β X = β X = 0 . /i covar- sample true absolute standard average RMSE coverageiates size effect bias deviation SE in %ATE: ˆ∆( d , d ∗ ) (all)50 2500 2 0.027 0.07 0.049 0.075 8050 10000 2 0.007 0.035 0.024 0.036 82100 2500 2 0.04 0.072 0.048 0.083 74100 10000 2 0.011 0.035 0.024 0.037 78ATE on selected: ˆ∆( d , d ∗ , S = 1)50 2500 2 0.027 0.076 0.06 0.081 8550 10000 2 0.006 0.037 0.029 0.038 90100 2500 2 0.042 0.079 0.06 0.089 81100 10000 2 0.01 0.039 0.029 0.04 88 Notes: SE and RMSE denote the standard error and the root mean squared error, respectively. Coverage is based on 95%confidence intervals.
Table 2 presents the main findings when estimating the ATE in the total population, ˆ∆( d , d ∗ ),and among the subgroup of treated in the first period, ˆ∆( d , d ∗ , S = 1). Irrespective of the num-ber of covariates, the absolute biases go to zero as the sample size increases. Furthermore, thestandard deviations and root mean squared errors (RMSE) of the ATE estimators are roughly15ut by half when quadrupling the sample size, as implied by √ n -consistency. The levels of thestandard deviations and RMSEs are somewhat higher for ˆ∆( d , d ∗ , S = 1) than for ˆ∆( d , d ∗ ),which comes from the additional weighting step due to targeting the treated subpopulation with S = 1. We also observe that the average standard errors (average SE) based on the asymptoticvariance approximations appear to converge at √ n -rate, however, they underestimate the truestandard deviations. This results in under-coverage of the true effects when constructing 95%confidence intervals based on those standard errors, an issue that decreases in the sample size. We apply our double machine learning approach to evaluate the effects of training sequencesprovided by the Job Corps program on employment. Job Corps is the largest U.S. program offer-ing vocational training and academic classroom instruction for disadvantaged individuals aged16 to 24. It is financed by the U.S. Department of Labor and currently has about 50,000 partic-ipants every year. Besides vocational credentials, students may obtain a high school diploma orequivalent qualifications. Individuals meeting specific low-income requirements can participatein Jobs Corps without any costs.A range of studies analyze the impact of Job Corps based on an experimental study withrandomized access to the program between November 1994 and February 1996, see e.g. Schochet,Burghardt, and Glazerman (2001). Schochet, Burghardt, and Glazerman (2001) and Schochet,Burghardt, and McConnell (2008) discuss in detail the study design and report the average effectsof random program assignment on a broad range of outcomes. Their findings suggest that JobCorps increases educational attainment, reduces criminal activity, and increases employmentand earnings, at least for some years after the program. Flores, Flores-Lagunes, Gonzalez,and Neumann (2012) assess the impact of a continuously defined treatment, namely the lengthof exposure to academic and vocational instruction on earnings and find positive effects. Asthe length of the treatment is (in contrast to program assignment) not random, they imposea conditional independence assumption and control for baseline characteristics at Job Corpsassignment. Colangelo and Lee (2020) suggest double machine learning-based estimation ofcontinuous treatment effects and apply it to assess the employment effects of Job Corps. Incontrast to these contributions on continuous treatment doses of Job Corps, we consider discrete16equences of multiple treatments and also control for post-treatment confounders rather thanbaseline covariates only.Several contributions assess specific causal mechanisms of the program. Flores and Flores-Lagunes (2009) find a positive direct effect of program assignment on earnings when controllingfor work experience which they assume to be conditionally independent given observed covari-ates. Also Huber (2014) imposes a conditional independence assumption and estimates a positivedirect health effect when controlling for the mediator employment. Using a partial identificationapproach permitting mediator endogeneity, Flores and Flores-Lagunes (2010) compute boundson the direct and indirect effects of Job Corps assignment on employment and earnings medi-ated by obtaining a GED, high school degree, or vocational degree. Under their strongest setof assumptions, the results point to a positive direct effect net of obtaining a degree. Fr¨olichand Huber (2017) use an instrumental variable strategy based on two instruments to disentanglethe earnings effect of being enrolled in Job Corps into an indirect effect via hours worked and adirect effect, likely related to a change in human capital. Their results point to the existence ofan indirect rather than a direct mechanism. Even though our framework of analyzing sequencesof treatments is in terms of statistical issues somewhat related to the evaluation of causal mecha-nisms, it relies on distinct identifying assumptions than the previously mentioned studies, whiche.g. do not consider controlling for post-treatment confounders.Our sample consists of 11313 individuals with completed follow-up interviews four years afterrandomization, out of which 6828 and 4485 were randomized in and out of Job Corps, respec-tively. We exploit the sequential structure of academic education and vocational training in theprogram to define dynamic treatment states. Since most of the education and training activitieswere taken in the first two years, we focus on the latter when generating a sequence of binarytreatments for each observation. The treatment states in our application can take four differentvalues: d , d , d ∗ , d ∗ ∈ { , , , } . State 0 refers to no instruction offered due to being randomizedout of Job Corps (control group), 1 to no instruction despite being randomized in (never takersin the denomination of Angrist, Imbens, and Rubin (1996)), 2 to academic education amongprogram participants, and 3 to vocational training among program participants. If individualsparticipate in both academic education and vocational training in a specific year, we assign thecode of the treatment that was attended to a larger extent in terms of completed hours.Table 3 reports various sequences of treatments in the data along with the corresponding17able 3: Sequences of binary treatmentsdynamic treatments in Job Corps observationscode year 1 year 200 no educ/train no educ/train no 448511 no educ/train no educ/train yes 32012 no educ/train acad educ yes 4313 no educ/train voc train yes 4221 acad educ no educ/train yes 132822 acad educ acad educ yes 34123 acad educ voc train yes 18331 voc train no educ/train yes 127932 voc train acad educ yes 10933 voc train voc train yes 573missings 2610 Notes: no educ/train means not participating in any Job Corps program related to education or training measures. acadeduc and voc train stand for academic education and vocational training, respectively, offered by Job Corps. number of observations. For instance, the treatment sequence 00 refers to those 4485 controlgroup members that were randomized out and did not participate in any education activitiesoffered by Job Corps. Furthermore, 320 individuals assigned to Job Corps do not participatein any form of education either, as indicated by the sequence 11. We also note that for 2610out of the 11313 individuals, information on the treatment sequences is missing. The literatureexplains the missing values by a random skip logic error, due to which asking questions abouttreatment participation was randomly omitted for a subset of survey participants, see page J.5 inSchochet, Bellotti, Cao, Glazerman, Grady, Gritz, McConnell, Johnson, and Burghardt (2003).In our analysis, we drop the control group with treatment sequence 00, but make use of it in aplacebo test outlined further below. Furthermore, for several potential comparisons of treatmentsequences, small sample issues and/or problems of a lack of common support in propensity scores(and thus, covariates) arise. For this reason, we confine our evaluation to comparing treatmentsequence 33 (vocational training in both years) to either 22 (academic education in both years),21 (academic education in the first year), or 11 (no participation in either year).Our outcome variable is a binary employment indicator measured four years after random-ization. Table 4 reports the mean outcome across various treatment sequences, which rangesfrom 77 to 89 percent. It also provides the sequence-specific numbers of cases with missingoutcomes that are dropped from the analysis, which appear quite low. We aim at estimatingthe ATE of treatment sequences d vs. d ∗ among individuals whose treatment in the first year18able 4: Mean outcome conditional on treatment sequencetreatment code employmentmean missings00 0.78 4611 0.78 721 0.78 1722 0.82 133 0.89 3missings 0.77 37 Notes: The first column 1 provides the codes of the treatment sequences, see Table 3. The second column gives the averageemployment per sequence, the third one the number of missing observations. corresponds to the first-year-treatment of either d or d ∗ . An alternative would be to assess theATE in the total sample randomized into Job Corps (which would thus also include individualswith different first-year treatments than the ones evaluated), but this proved to be problematicdue to lacking common support in terms of treatment propensity scores.We make use of a large set of potential control variables that also include covariates whichhave been identified as important confounders in several articles assessing the sensitivity ofprogram evaluations to the inclusion and omission of such confounders in observational labormarket studies. Biewen, Fitzenberger, Osikominu, and Paul (2014), for instance, conclude thatimposing conditional independence assumptions requires the availability of rich data on em-ployment and benefit histories, and socio-economic characteristics. Lechner and Wunsch (2013)point to the importance of factors like health, caseworker assessments, regional information,timing of unemployment and program start, pre-treatment outcomes, job search behavior, andlabor market histories. In line with these findings, our covariates comprise information aboutsocio-economic characteristics, pre-treatment labor market histories, education and training, jobsearch activities, welfare receipt, health, crime, and how one learnt about the existence of JobCorps. Table B.1 in the Appendix B reports more details on these features, including variabledescriptions and distributions across treatment sequences.We condition on observed characteristics X t in periods t ∈ { , } . X denotes control vari-ables measured at baseline prior to the first treatment D , whereas X is observed one year afterrandomization but prior to the second treatment D . Table 5 provides the number and types ofvariables assigned to X and X . Our raw data include 1188 characteristics. After some datamanipulations based on generating dummies for values of categorical variables and missing itemsin dummy or categorical variables, we end up with all in all 2336 regressors. Missing observations19n numerical variables were replaced by the mean values of the non-missing items. Furthermore,we standardized numerical covariates to have a zero mean and a standard deviation of 0.5.Table 5: Number of covariatesraw variables X X dummy 295 575categorical 53 13numeric 26 226total 374 814processed variables X X dummy 884 1200numeric 26 226total 910 1426 Note: X and X denote regressors measured prior to the first and second periods, respectively. We estimate ∆( d , d ∗ , S = 1), with S = 1 if the first treatment corresponds to either thefirst treatment in d or d ∗ , based on 3-fold cross-fitting and the random forest (see Breiman(2001)) as machine learner of the nuisance parameters. To this end, we use the SuperLearner package with default options provided by van der Laan, Polley, and Hubbard (2007) for thestatistical software R . Our motivation for choosing the random forest is that it is (in the spirit ofkernel regression) a nonparametric estimator that does not impose functional form assumptions(like linearity) on the conditional outcome or treatment models. As in our simulation study, wedrop observations whose products of propensity scores in the first and second period are smallerthan 0 .
01 to impose common support in our sample and avoid an explosion in the propensityscore-based weights. For a visual assessment of the common support, Appendix C providesplots with the propensity score distributions across all treatment sequences considered in thisapplication. In general, common support is rather decent for the first period propensity scoresˆ p d ( X ), while the overlap is weaker for the scores in the second period ˆ p d ( D , X ), especiallyat the boundaries of the distributions.Table 6: Effect estimates with a trimming threshold of 0.01 d d ∗ ˆ E [ Y ( d ∗ ) | S = 1] ˆ∆( d , d ∗ , S = 1) SE p-value observations trimmed33 22 0.77 0.08 0.07 0.23 3783 51233 21 0.82 0.04 0.03 0.15 3783 4333 11 0.81 0.08 0.03 0.02 2346 27 Notes: d and d ∗ indicate the treatment sequences under treatment and non-treatment, respectively. ˆ E [ Y ( d ∗ ) | S = 1]denotes the mean potential outcome under non-treatment conditional on S = 1, where S is an indicator for the first treatmentcorresponding to either the first treatment in d or d ∗ . ˆ∆( d , d ∗ , S = 1) provides the ATE estimate, SE the standard error.The last column gives the number of observations dropped according to the trimming rule p d ( X ) · p d ( D , X ) < . d , d ∗ , S = 1) suggests anincrease of 8 and 4 percentage points in the employment probability (starting from a counterfac-tual probability of 77% and 82%, respectively), the p-values are beyond any conventional levelof statistical significance. For the comparison of vocational training to no training in either yearpresented in the third row, however, the effect of 8 percentage points is statistically significant atthe 5% level. We therefore conclude that vocational training appears to increase the employmentprobability 4 years after randomization into Job Corps, while it is less clear whether it performsrelatively better than academic classroom training. The results are qualitatively similar whenincreasing the trimming threshold for the products of the propensity scores to 0 .
03, see Table 7.However, the p-value of the effect of vocational vs. no training is now somewhat higher (6%),while the effect of vocational training (in both periods) vs. classroom training in the first periodonly is borderline significant at the 10% level.Table 7: Effect estimates with a trimming threshold of 0.03 d d ∗ ˆ E [ Y ( d ∗ ) | S = 1] ˆ∆( d , d ∗ , S = 1) SE p-value observations trimmed33 22 0.79 0.05 0.06 0.4 3783 193233 21 0.82 0.05 0.03 0.1 3783 57833 11 0.81 0.07 0.04 0.06 2346 157 Notes: d and d ∗ indicate the treatment sequences under treatment and non-treatment, respectively. ˆ E [ Y ( d ∗ ) | S = 1]denotes the mean potential outcome under non-treatment conditional on S = 1, where S is an indicator for the first treatmentcorresponding to either the first treatment in d or d ∗ . ˆ∆( d , d ∗ , S = 1) provides the ATE estimate, SE the standard error.The last column gives the number of observations dropped according to the trimming rule p d ( X ) · p d ( D , X ) < . To partially assess the validity of the conditional independence assumptions imposed in thisapplication, we conduct a placebo test based on comparing the outcomes of two control groups asfor instance discussed in Athey and Imbens (2017). The first control group are the never takers,i.e. those randomized into Job Corps who never attended any form of instruction with treatmentsequence 11. The second control group are those randomized out and thus without access to JobCorps instruction with treatment sequence 00. We estimate the pseudo-treatment effect of JobCorps on the employment outcome using the double machine learning approach for assessingstatic (rather than dynamic) treatments as for instance discussed in Chernozhukov, Chetverikov,Demirer, Duflo, Hansen, Newey, and Robins (2018). To this end, we consider sequence 11 as21seudo-treatment and sequence 00 as non-treatment and control for the baseline covariates X based on the random forest as machine learner of the nuisance parameters. As neither groupattended any training, the true ATE is equal to zero. As shown in Table 8, the estimated ATE isindeed approximately zero with a p-value of 93%. This provides some statistical support for thesatisfaction of the conditional independence assumption, at least w.r.t. the baseline covariates X . Table 8: Placebo test with a trimming threshold of 0.01ATE estimate SE p-value observations trimmed-0.00 0.02 0.93 4752 196 Notes: The ATE estimate provides the pseudo-treatment effect when comparing the employment outcomes of never takers(treatment sequence 11) and those randomized out (treatment sequence 00) conditional on baseline covariates X . The lastcolumn states the number of observations dropped according to the trimming rule: p ( X ) < . In this paper, we combined dynamic treatment evaluation with double machine learning undersequential selection-on-observables assumptions which avoids adhoc pre-selection of control vari-ables. This approach appears particularly fruitful in high-dimensional data with many potentialcontrol variables. We suggested estimators for the (weighted) average effects of sequences oftreatments (with the so-called controlled direct effect being a special case) based on Neyman or-thogonal score functions, sample splitting, and machine learning-based plug-in estimates of con-ditional mean outcomes and treatment propensity scores. We demonstrated the √ n -consistencyand asymptotic normality of the treatment effect estimators under specific regularity conditionsand analyzed their finite sample behavior in a Monte Carlo simulation. Finally, we appliedour method to the Job Corps data to analyze the effects of distinct sequences of educationalprograms and found positive employment effects for vocational training when compared to noprogram participation. References
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A Proofs
A.1 Proof of Theorem 1
For the proof of Theorem 1 it is sufficient to check the conditions of Assumptions 3.1 and 3.2 fromTheorem 3.1 and 3.2 and Corollary 3.2 from Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey,and Robins (2018). All bounds hold uniformly over P ∈ P , where P is the set of all possible probabilitylaws, and we omit P for brevity.Define the nuisance parameters to be the vector of functions η = ( p d ( X ) , p d ( D , X ) , µ Y ( D , X )), ν Y ( D , X ), with p d ( X ) = Pr( D = d | X ), p d ( D , X ) = Pr( D = d | D , X , X ), µ Y ( D , X ) = E [ Y | D , X , X ], and ν Y ( D , X ) = (cid:82) E [ Y | d , X , X = x ] dF X = x | D ,X , where F X = x | D ,X denotesthe conditional distribution function of X at value x . The score function for the counterfactual Ψ d = E [ Y ( d )] is given by: ψ d ( W, η, Ψ d ) = I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X )+ I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X )+ ν Y ( d , X ) − Ψ d . Let T n be the set fo all η = ( p d , p d , µ Y , ν Y ) consisting of P -square integrable functions p d , p d , µ Y and ν Y such that (cid:107) η − η (cid:107) q ≤ C, (A.1) (cid:107) η − η (cid:107) ≤ δ n , (cid:13)(cid:13) p d ( X ) − / (cid:13)(cid:13) ∞ ≤ / − (cid:15), (cid:13)(cid:13) p d ( D , X ) − / (cid:13)(cid:13) ∞ ≤ / − (cid:15), (cid:13)(cid:13)(cid:13) µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d ( X ) − p d ( X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d ( D , X ) − p d ( D , X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) ν Y ( D , X ) − ν Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d ( X ) − p d ( X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / . We furthermore replace the sequence ( δ n ) n ≥ by ( δ (cid:48) n ) n ≥ , where δ (cid:48) n = C (cid:15) max( δ n , n − / ) , where C (cid:15) issufficiently large constant that only depends on C and (cid:15). ssumption 3.1: Linear scores and Neyman orthogonalityAssumption 3.1(a)Moment Condition: The moment condition E (cid:104) ψ d ( W, η , Ψ d ) (cid:105) = 0 is satisfied, which follows fromthe law of iterated expectations: E (cid:104) ψ d ( W, η , Ψ d ) (cid:105) = E (cid:34) = E [ E [ Y − µ Y ( d ,X ) | D = d ,X ] | D = d ,X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) E (cid:34) I { D = d } · I { D = d } p d ( X ) · p d ( d , X ) · [ Y − µ Y ( d , X )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (cid:35) (cid:35) + E (cid:34) = (cid:82) E (cid:2) µ Y ( d ,x ) − ν Y ( d ,x ) (cid:12)(cid:12) D = d ,X = x (cid:3) dF X x | D d ,X =0 (cid:122) (cid:125)(cid:124) (cid:123) E (cid:34) I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (cid:35) (cid:35) + E (cid:2) ν Y ( d , X ) (cid:3) − Ψ d = Ψ d − Ψ d = 0To better see this result, note that E (cid:34) I { D = d } · I { D = d } p d ( X ) · p d ( d , X ) · [ Y − µ Y ( d , X )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (cid:35) = E (cid:34) I { D = d } p d ( d , X ) · [ Y − µ Y ( d , X )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D = d , X (cid:35) = E (cid:34) E (cid:34) I { D = d } p d ( d , X ) · [ Y − µ Y ( d , X )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D = d , X ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D = d , X (cid:35) = E [ E [ Y − µ Y ( d , X ) | D = d , X ] | D = d , X ]= E [ µ Y ( d , X ) − µ Y ( d , X ) | D = d , X ] = 0 , where the first and third equalities follow from basic probability theory and the second from the law ofiterated expectations. Furthermore, E (cid:34) I { D = d } · [ µ Y ( d , X ) − ν Y ( d , x )] p d ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X = x (cid:35) = E (cid:2) µ Y ( d , X ) − ν Y ( d , x ) (cid:12)(cid:12) D = d , X = x (cid:3) = (cid:90) E (cid:2) µ Y ( d , x ) − ν Y ( d , x ) (cid:12)(cid:12) D = d , X = x (cid:3) dF X = x | D = d ,X = x = (cid:90) E (cid:2) µ Y ( d , x ) (cid:12)(cid:12) D = d , X = x (cid:3) dF X = x | D = d ,X = x − ν Y ( d , x )= ν Y ( d , x ) − ν Y ( d , x ) = 0 . where the first equality follows from basic probability theory, the second from conditioning on and inte-grating over X , and the third from the fact that ν Y ( d , X ) is not a function of X . ssumption 3.1(b)Linearity: The score ψ d ( W, η , Ψ d ) is linear in Ψ d : ψ d ( W, η , Ψ d ) = ψ d a ( W, η ) · Ψ d + ψ d b ( W, η )with ψ d a ( W, η ) = − ψ d b ( W, η ) = I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X )+ I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) + ν Y ( d , X ) . Assumption 3.1(c)Continuity:
The expression for the second Gateaux derivative of a map η (cid:55)→ E [ ψ d ( W, η, Ψ d )] iscontinuous. ssumption 3.1(d)Neyman Orthogonality : For any η ∈ T N , the Gateaux derivative in the direction η − η = ( p d ( X ) − p d ( X ) , p d ( D , X ) − p d ( D , X ) , µ Y ( d , X ) − µ Y ( d , X ) , ν Y ( d , X ) − ν Y ( d , X )) is given by: ∂E (cid:2) ψ d ( W, η, Ψ d ) (cid:3)(cid:2) η − η (cid:3) = − E (cid:34) I { D = d } · I { D = d } · [ µ Y ( d , X ) − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (cid:35) ( ∗ )+ E (cid:34) I { D = d } · [ µ Y ( d , X ) − µ Y ( d , X )] p d ( X ) (cid:35) ( ∗∗ ) − E (cid:34) E [ ·| X ]= E [ E [ Y − µ Y ( d ,X ) | D = d ,X ] | D = d ,X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) · [ p d ( X ) − p d ( X )] p d ( X ) (cid:35) − E (cid:34) E [ ·| X ]= (cid:82) E (cid:2) µ Y ( d ,x ) − ν Y ( d ,x ) (cid:12)(cid:12) D = d ,X = x (cid:3) dF X x | D d ,X =0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) · [ p d ( X ) − p d ( X )] p d ( X ) (cid:35) − E (cid:34) E [ ·| X ]= E [ E [ Y − µ Y ( d ,X ) | D = d ,X ] | D = d ,X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) · [ p d ( d , X ) − p d ( d , X )] p d ( d , X ) (cid:35) − E (cid:34) I { D = d } · [ ν Y ( d , X ) − ν Y ( d , X )] p d ( X ) (cid:124) (cid:123)(cid:122) (cid:125) E [ ·| X ]= pd
10 ( X pd
10 ( X · [ ν Y ( d ,X ) − ν Y ( d ,X )] (cid:35) + E [ ν Y ( d , X ) − ν Y ( d , X )] (cid:124) (cid:123)(cid:122) (cid:125) =0 = 0The Gateaux derivative is zero because expressions ( ∗ ) and ( ∗∗ ) cancel out. To see this, note that E (cid:34) I { D = d } · [ µ Y ( d , X ) − µ Y ( d , X )] p d ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X = x (cid:35) = E (cid:2) µ Y ( d , X ) − µ Y ( d , X ) (cid:12)(cid:12) D = d , X = x (cid:3) = (cid:90) E (cid:2) µ Y ( d , x ) − µ Y ( d , x ) (cid:12)(cid:12) D = d , X = x (cid:3) dF X = x | D = d ,X = x , where the first equality follows from basic probability theory and the second from conditioning on and ntegrating over X . Furthermore, E (cid:34) I { D = d } · I { D = d } · [ µ Y ( d , X ) − µ Y ( d , X )] p d ( x ) · p d ( d , X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X = x (cid:35) = E (cid:34) I { D = d } · [ µ Y ( d , X ) − µ Y ( d , X )] p d ( d , X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D = d , X = x (cid:35) = (cid:90) E (cid:34) I { D = d } · [ µ Y ( d , x ) − µ Y ( d , X )] p d ( d , x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D = d , x = x (cid:35) dF X = x | D = d ,X = x = (cid:90) E (cid:2) µ Y ( d , x ) − µ Y ( d , x ) (cid:12)(cid:12) D = d , D = d , X = x (cid:3) dF X = x | D = d ,X = x = (cid:90) E (cid:2) µ Y ( d , x ) − µ Y ( d , x ) (cid:12)(cid:12) D = d , X = x (cid:3) dF X = x | D = d ,X = x , where the first equality follows from basic probability theory, the second from conditioning on andintegrating over X , the third from basic probability theory, and the fourth from simplification as µ Y ( d , x ) = E [ Y | D = d , D = d , X = x ] is already conditional on D = d . ∂E (cid:2) ψ d ( W, η, Ψ d ) (cid:3)(cid:2) η − η (cid:3) = 0proving that the score function is orthogonal. Assumption 3.1(e)Singular values of E [ ψ d a ( W ; η )] are bounded: Holds trivially, because ψ d a ( W ; η ) = − . Assumption 3.2: Score regularity and quality of nuisance parameter estimatorsAssumption 3.2(a)
This assumption directly follows from the construction of the set T n and the regularity conditions(Assumption 4). Assumption 3.2(b)Bound for m n : (cid:13)(cid:13)(cid:13) µ Y ( D , X ) (cid:13)(cid:13)(cid:13) q = (cid:16) E (cid:104)(cid:12)(cid:12)(cid:12) µ Y ( D , X ) (cid:12)(cid:12)(cid:12) q (cid:105)(cid:17) q = (cid:88) d ∈{ , ,...,Q } E (cid:104)(cid:12)(cid:12)(cid:12) µ Y ( d , X ) (cid:12)(cid:12)(cid:12) q Pr P ( D = d | X ) (cid:105) q ≥ (cid:15) /q (cid:88) d ∈{ , ,...,Q } E (cid:104)(cid:12)(cid:12)(cid:12) µ Y ( d , X ) (cid:12)(cid:12)(cid:12) q (cid:105) q ≥ (cid:15) /q (cid:18) max d ∈{ , ,...,Q } E (cid:104)(cid:12)(cid:12)(cid:12) µ Y ( d , X ) (cid:12)(cid:12)(cid:12) q (cid:105)(cid:19) q = (cid:15) /q (cid:18) max d ∈{ , ,...,Q } (cid:13)(cid:13)(cid:13) µ Y ( d , X ) (cid:13)(cid:13)(cid:13) q (cid:19) , here the first equality follows from definition, the second from the law of total probability, and the thirdline from the fact that Pr( D = d | X ) = p d ( X ) · p d ( d , X ) ≥ (cid:15) . Similarly, we obtain (cid:13)(cid:13)(cid:13) ν Y ( D , X ) (cid:13)(cid:13)(cid:13) q ≥ (cid:15) /q (cid:18) max d ∈{ , ,...,Q } (cid:13)(cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13)(cid:13) q (cid:19) . Notice that by Jensen’s inequality (cid:13)(cid:13)(cid:13) µ Y ( D , X ) (cid:13)(cid:13)(cid:13) q ≤ (cid:107) Y (cid:107) q and (cid:13)(cid:13)(cid:13) ν Y ( D , X ) (cid:13)(cid:13)(cid:13) q ≤ (cid:107) Y (cid:107) q andhence (cid:13)(cid:13)(cid:13) µ Y ( d , X ) (cid:13)(cid:13)(cid:13) q ≤ C/(cid:15) /q and (cid:13)(cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13)(cid:13) q ≤ C/(cid:15) /q , by conditions (A.1). Similarly, for any η ∈ T N : (cid:13)(cid:13)(cid:13) µ Y ( d , X ) − µ Y ( d , X ) (cid:13)(cid:13)(cid:13) q ≤ C/(cid:15) /q and (cid:13)(cid:13)(cid:13) ν Y ( d , X ) − ν Y ( d , X ) (cid:13)(cid:13)(cid:13) q ≤ C/(cid:15) /q , because (cid:13)(cid:13)(cid:13) µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) q ≤ C and (cid:13)(cid:13)(cid:13) ν Y ( D , X ) − ν Y ( D , X ) (cid:13)(cid:13)(cid:13) q ≤ C. Consider E (cid:104) ψ d ( W, η, Ψ d ) (cid:105) = E (cid:34) I { D = d } · I { D = d } p d ( X ) · p d ( d , X ) · Y (cid:124) (cid:123)(cid:122) (cid:125) = I + I { D = d } p d ( X ) · (cid:18) − I { D = d } p d ( d , X ) (cid:19) · µ Y ( d , X ) (cid:124) (cid:123)(cid:122) (cid:125) = I + (cid:18) − I { D = d } p d ( X ) (cid:19) ν Y ( d , X ) (cid:124) (cid:123)(cid:122) (cid:125) = I − Ψ d (cid:35) and thus (cid:13)(cid:13)(cid:13) ψ d ( W, η, Ψ d ) (cid:13)(cid:13)(cid:13) q ≤ (cid:107) I (cid:107) q + (cid:107) I (cid:107) q + (cid:107) I (cid:107) q + (cid:13)(cid:13)(cid:13) Ψ d (cid:13)(cid:13)(cid:13) q ≤ (cid:15) (cid:107) Y (cid:107) q + 1 − (cid:15)(cid:15) (cid:13)(cid:13) µ Y ( d , X ) (cid:13)(cid:13) q ++ 1 − (cid:15)(cid:15) (cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13) q + | Ψ d |≤ C (cid:18) (cid:15) + 2(1 − (cid:15) ) (cid:15) /q (cid:18) (cid:15) + 1 (cid:15) (cid:19) + 1 (cid:15) (cid:19) , because of triangular inequality and because the following set of inequalities hold: (cid:13)(cid:13) µ Y ( d , X ) (cid:13)(cid:13) q ≤ (cid:13)(cid:13)(cid:13) µ Y ( d , X ) − µ Y ( d , X ) (cid:13)(cid:13)(cid:13) q + (cid:13)(cid:13)(cid:13) µ Y ( d , X ) (cid:13)(cid:13)(cid:13) q ≤ C/(cid:15) /q , (A.2) (cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13) q ≤ (cid:13)(cid:13)(cid:13) ν Y ( d , X ) − ν Y ( d , X ) (cid:13)(cid:13)(cid:13) q + (cid:13)(cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13)(cid:13) q ≤ C/(cid:15) /q , | Ψ d | = | E [ ν Y ( d , X )] | ≤ E (cid:104) (cid:12)(cid:12)(cid:12) ν Y ( d , X ) (cid:12)(cid:12)(cid:12) (cid:105) = (cid:13)(cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13)(cid:13) ≤ (cid:107) Y (cid:107) /(cid:15) / q> (cid:122)(cid:125)(cid:124)(cid:123) ≤ (cid:107) Y (cid:107) q /(cid:15) ≤ C/(cid:15). which gives the upper bound on m n in Assumption 3.2(b) of Chernozhukov, Chetverikov, Demirer, Duflo, ansen, Newey, and Robins (2018). Bound for m (cid:48) n : Notice that (cid:16) E [ | ψ d a ( W, η ) | q ] (cid:17) /q = 1and this gives the upper bound on m (cid:48) n in Assumption 3.2(b). Assumption 3.2(c)
In the following, we omit arguments for the sake of brevity and use p d = p d ( X ) , p d = p d ( d , X ) , ν Y = ν Y ( d , X ) , µ Y = µ Y ( d , X ) and similarly for p d , p d , ν Y , µ Y . Bound for r n : For any η = ( p d , p d , µ Y , ν Y ) we have (cid:12)(cid:12)(cid:12) E (cid:16) ψ d a ( W, η ) − ψ d a ( W, η ) (cid:17)(cid:12)(cid:12)(cid:12) = | − | = 0 ≤ δ (cid:48) N , and thus we have the bound on r n from Assumption 3.2(c). Bound for r (cid:48) n : (cid:13)(cid:13)(cid:13) ψ d ( W, η, Ψ d ) − ψ d ( W, η , Ψ d ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I { D = d } · I { D = d } · Y · (cid:32) p d p d − p d p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (A.3)+ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I { D = d } · I { D = d } (cid:32) µ Y p d p d − µ Y p d p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I { D = d } (cid:32) µ Y p d − µ Y p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I { D = d } (cid:32) ν Y p d − ν Y p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ν Y − ν Y (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y · (cid:32) p d p d − p d p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ Y p d p d − µ Y p d p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ Y p d − µ Y p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ν Y p d − ν Y p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ν Y − ν Y (cid:13)(cid:13)(cid:13) ≤ C(cid:15) δ n (cid:18) (cid:15) (cid:19) + δ n (cid:18) (cid:15) + C + C(cid:15) (cid:19) + δ n (cid:18) (cid:15) + C(cid:15) (cid:19) + δ n (cid:18) (cid:15) + C(cid:15) (cid:19) + δ n (cid:15) ≤ δ (cid:48) n as long as C (cid:15) in the definition of δ (cid:48) n is sufficiently large. This gives the bound on r (cid:48) n from Assump-tion 3.2(c). Here we made use of the fact that (cid:13)(cid:13)(cid:13) µ Y − µ Y (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) µ Y ( d , X ) − µ Y ( d , X ) (cid:13)(cid:13)(cid:13) ≤ δ n /(cid:15), (cid:13)(cid:13)(cid:13) ν Y − ν Y (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ν Y ( d , X ) − ν Y ( d , X ) (cid:13)(cid:13)(cid:13) ≤ δ n /(cid:15) and (cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) p d ( d , X ) − p d ( d , X ) (cid:13)(cid:13)(cid:13) ≤ δ n /(cid:15) using similar steps as in Assumption 3.1(b).The last inequality in (A.3) is satisfied because we can bound the first term by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y · (cid:32) p d p d − p d p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p d p d − p d p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C(cid:15) (cid:13)(cid:13)(cid:13) p d p d − p d p d (cid:13)(cid:13)(cid:13) = C(cid:15) (cid:13)(cid:13)(cid:13) p d p d − p d p d + p d p d − p d p d (cid:13)(cid:13)(cid:13) ≤ C(cid:15) (cid:16)(cid:13)(cid:13)(cid:13) p d ( p d − p d ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) p d ( p d − p d ) (cid:13)(cid:13)(cid:13) (cid:17) ≤ C(cid:15) (cid:16)(cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) (cid:17) ≤ C(cid:15) δ n (cid:18) (cid:15) (cid:19) , here the first inequality follows from the second inequality in Assumption 4(a). The second term in(A.3) is bounded by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ Y p d p d − µ Y p d p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:13)(cid:13)(cid:13) p d p d µ Y − p d p d µ Y (cid:13)(cid:13)(cid:13) = 1 (cid:15) (cid:13)(cid:13)(cid:13) p d p d µ Y − p d p d µ Y + p d p d µ Y − p d p d µ Y (cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:16)(cid:13)(cid:13)(cid:13) p d p d ( µ Y − µ Y ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) µ Y ( p d p d − p d p d ) (cid:13)(cid:13)(cid:13) (cid:17) ≤ (cid:15) (cid:16)(cid:13)(cid:13)(cid:13) µ Y − µ Y (cid:13)(cid:13)(cid:13) + C (cid:13)(cid:13)(cid:13) p d p d − p d p d (cid:13)(cid:13)(cid:13) (cid:17) ≤ (cid:15) (cid:18) δ n (cid:15) + C (cid:13)(cid:13)(cid:13) p d p d − p d p d (cid:13)(cid:13)(cid:13) (cid:19) ≤ δ n (cid:18) (cid:15) + C + C(cid:15) (cid:19) , where the third inequality follows from E [ Y | D = d , D = d , X ] ≥ ( E [ Y | D = d , D = d , X ]) = µ ( d , X ) by the conditional Jensen’s inequality and therefore (cid:13)(cid:13)(cid:13) µ Y ( d , X ) (cid:13)(cid:13)(cid:13) ∞ ≤ C . For the third term we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ Y p d − µ Y p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 1 (cid:15) (cid:13)(cid:13)(cid:13) p d µ Y − p d µ Y (cid:13)(cid:13)(cid:13) = 1 (cid:15) (cid:13)(cid:13)(cid:13) p d µ Y − p d µ Y + p d µ Y − p d µ Y (cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:16)(cid:13)(cid:13)(cid:13) p d ( µ Y − µ Y ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) µ Y ( p d − p d ) (cid:13)(cid:13)(cid:13) (cid:17) ≤ (cid:15) (cid:16)(cid:13)(cid:13)(cid:13) µ Y − µ Y (cid:13)(cid:13)(cid:13) + C (cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) (cid:17) ≤ δ n (cid:18) (cid:15) + C(cid:15) (cid:19) , and similarly, for the fourth term we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ν Y p d − ν Y p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ δ n (cid:18) (cid:15) + C(cid:15) (cid:19) , where we used Jensen’s inequality twice to get (cid:13)(cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13)(cid:13) ∞ ≤ C . Bound for λ (cid:48) n : Now consider f ( r ) := E [ ψ ( W ; Ψ d , η + r ( η − η )] . or any r ∈ (0 ,
1) : ∂ f ( r ) ∂r = E (cid:34) I { D = d } · I { D = d } ( −
2) ( µ Y − µ Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) (A.4)+ E (cid:34) I { D = d } · I { D = d } ( −
2) ( µ Y − µ Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } · I { D = d } Y − µ Y − r ( µ Y − µ Y ))( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } · I { D = d } Y − µ Y − r ( µ Y − µ Y ))( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } · I { D = d } Y − µ Y − r ( µ Y − µ Y ))( p d − p d )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } ( −
2) ( µ Y − µ Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } ν Y − ν Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } r ( µ Y − µ Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } r ( ν Y − ν Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } µ Y − ν Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) Note that because E [ Y − µ Y ( d , X ) | D = d , D = d , X ] = 0 , | p d − p d | ≤ , | p d − p d | ≤ (cid:13)(cid:13)(cid:13) µ Y (cid:13)(cid:13)(cid:13) q ≤ (cid:107) Y (cid:107) q /(cid:15) /q ≤ C/(cid:15) /q (cid:13)(cid:13)(cid:13) ν Y (cid:13)(cid:13)(cid:13) q ≤ (cid:107) Y (cid:107) q /(cid:15) /q ≤ C/(cid:15) /q (cid:13)(cid:13)(cid:13) µ Y − µ Y (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) ≤ δ n n − / /(cid:15), (cid:13)(cid:13)(cid:13) µ Y − µ Y (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) ≤ δ n n − / /(cid:15) , (cid:13)(cid:13)(cid:13) ν Y − ν Y (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) ≤ δ n n − / /(cid:15). we get that for some constant C (cid:48)(cid:48) (cid:15) that only depends on C and (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) ∂ f ( r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:48)(cid:48) (cid:15) δ n n − / ≤ δ (cid:48) n n − / and this gives the upper bound on λ (cid:48) n in Assumption 3.2(c) of Chernozhukov, Chetverikov, Demirer, uflo, Hansen, Newey, and Robins (2018) as long as C (cid:15) ≥ C (cid:48)(cid:48) (cid:15) . We used the following inequalities (cid:13)(cid:13)(cid:13) µ Y − µ Y (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) µ Y ( d , X ) − µ Y ( d , X ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) /(cid:15) (cid:13)(cid:13)(cid:13) ν Y − ν Y (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ν Y ( d , X ) − ν Y ( d , X ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ν Y ( D , X ) − ν Y ( D , X ) (cid:13)(cid:13)(cid:13) /(cid:15) (cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) p d ( d , X ) − p d ( d , X ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) p d ( D , X ) − p d ( D , X ) (cid:13)(cid:13)(cid:13) /(cid:15), and these can be shown using similar steps as in Assumption 3.1(b).To verify that (cid:12)(cid:12)(cid:12) ∂ f ( r ) ∂r (cid:12)(cid:12)(cid:12) ≤ C (cid:48)(cid:48) (cid:15) δ n n − / holds, note that by the triangular inequality it is sufficient tobound the absolute value of each of the ten terms in (A.4) separately. We illustrate it for the first, third,and last terms. For the first term: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) I { D = d } · I { D = d } ( −
2) ( µ Y − µ Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) ( µ Y − µ Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) ( p d + r ( p d − p d ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) ( µ Y − µ Y )( p d − p d ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) δ N (cid:15) n − / . For the second inequality we used the fact that for i ∈ { , } : 1 ≥ p d i + r ( p d i − p d i ) = (1 − r ) p d i + rp d i ≥ (1 − r ) (cid:15) + r(cid:15) = (cid:15) and in the third Holder’s inequality. For the third term, we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) I { D = d } · I { D = d } Y − µ Y − r ( µ Y − µ Y ))( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) I { D = d } · I { D = d } ( Y − µ Y − r ( µ Y − µ Y ))( p d − p d ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) I { D = d } · I { D = d } ( Y − µ Y ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) r ( µ Y − µ Y )( p d − p d ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ · (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) · ( µ Y − µ Y ))( p d − p d ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) δ N (cid:15) n − / , where in addition we made use of conditions (A.1). or the last term, we have E (cid:34) I { D = d } µ Y − ν Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) = E (cid:34) (cid:82) E (cid:2) µ Y ( d ,x ) − ν Y ( d ,x ) (cid:12)(cid:12) D = d ,X = x (cid:3) dF X x | D d ,X x =0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } ( µ Y − ν Y ) p d · p d ( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) = 0 . The remaining terms in (A.4) are bounded similarly.
Assumption 3.2(d) E (cid:104) ( ψ d ( W, η , Ψ d )) (cid:105) = E (cid:34)(cid:32) I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (cid:124) (cid:123)(cid:122) (cid:125) = I + I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) (cid:124) (cid:123)(cid:122) (cid:125) = I + ν Y ( d , X ) − Ψ d (cid:124) (cid:123)(cid:122) (cid:125) = I (cid:33) (cid:35) = E [ I + I + I ] ≥ E [ I ]= E (cid:34) I { D = d } · I { D = d } · (cid:32) [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (cid:33) (cid:35) ≥ (cid:15) E (cid:34)(cid:32) [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (cid:33) (cid:35) ≥ (cid:15) c (1 − (cid:15) ) > . because Pr( D = d | X ) = p d ( X ) · p d ( d , X ) ≥ (cid:15) , p d ( X ) ≤ − (cid:15) and p d ( d , X ) ≤ − (cid:15). where the second equality follows from E (cid:104) I · I (cid:105) = E (cid:34) E [ ·| X ]= E [ E [ Y − µ Y ( d ,X ) | D = d ,X ] | D = d ,X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } ( p d ( X )) · p d ( d , X ) · [ Y − µ Y ( d , X )] · [ µ Y ( d , X ) − ν Y ( d , X )] (cid:35) ,E (cid:104) I · I (cid:105) = E (cid:34) E [ ·| X ]= (cid:82) E (cid:2) µ Y ( d ,x ) − ν Y ( d ,x ) (cid:12)(cid:12) D = d ,X = x (cid:3) dF X x | D d ,X x =0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } p d ( X ) · [ µ Y ( d , X ) − ν Y ( d , X )] · [ ν Y ( d , X ) − Ψ d ] (cid:35) ,E (cid:104) I · I (cid:105) = E (cid:34) E [ ·| X ]= E [ E [ Y − µ Y ( d ,X ) | D = d ,X ] | D = d ,X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } p d ( X ) · p d ( d , X ) · [ Y − µ Y ( d , X )] · [ ν Y ( d , X ) − Ψ d ] (cid:35) . .2 Proof of Theorem 2 The proof follows in a similar manner than the one of Theorem 1 (Section A.1). All bounds hold uniformlyover all probability laws P ∈ P where P is the set of all possible probability laws, and we omit P forbrevity.Denote by S a binary indicator for being selected into the target population and by g ( X ) = Pr( S =1 | X ) the selection probability as a function of X . Define the nuisance parameter to be χ = ( g, η ) =( g ( X ) , p d ( X ) , p d ( D , X ) , µ Y ( D , X ) , ν Y ( D , X )). The score function for the weighted counterfac-tual Ψ d ,S =10 = E [ Y ( d ) | S = 1] = E [ g ( X ) · Y ( d ) / Pr( S = 1)] is given by: ψ d ,S =1 ( W, χ, Ψ d ,S =10 ) = g ( X )Pr( S = 1) · I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X )+ g ( X )Pr( S = 1) · I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X )+ S Pr( S = 1) · ν Y ( d , X ) − Ψ d ,S =10 . Let T ∗ n be the set fo all χ = ( g, η ) = ( g, p d , p d , µ Y , ν Y ) consisting of P -square integrable functions g, p d , p d , µ Y and ν Y such that (cid:107) χ − χ (cid:107) q ≤ C, (A.5) (cid:107) χ − χ (cid:107) ≤ δ n , (cid:107) g ( X ) − / (cid:107) ∞ ≤ / − (cid:15), (cid:13)(cid:13) p d ( X ) − / (cid:13)(cid:13) ∞ ≤ / − (cid:15), (cid:13)(cid:13) p d ( D , X ) − / (cid:13)(cid:13) ∞ ≤ / − (cid:15), (cid:13)(cid:13)(cid:13) µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d ( X ) − p d ( X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d ( D , X ) − p d ( D , X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) ν Y ( D , X ) − ν Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) p d ( X ) − p d ( X ) (cid:13)(cid:13)(cid:13) ≤ δ n n − / . (cid:13)(cid:13)(cid:13) µ Y ( D , X ) − µ Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:107) g ( X ) − g ( X ) (cid:107) ≤ δ n n − / , (cid:13)(cid:13)(cid:13) ν Y ( D , X ) − ν Y ( D , X ) (cid:13)(cid:13)(cid:13) × (cid:107) g ( X ) − g ( X ) (cid:107) ≤ δ n n − / . We furthermore replace the sequence ( δ n ) n ≥ by ( δ (cid:48) n ) n ≥ , where δ (cid:48) n = C (cid:15) max( δ n , n − / ) , where C (cid:15) issufficiently large constant that only depends on C and (cid:15). ssumption 3.1: Linear scores and Neyman orthogonalityAssumption 3.1(a) Moment Condition: The moment condition E (cid:104) ψ d ,S =1 ( W, χ , Ψ d ,S =10 ) (cid:105) = 0 holdsby the law of iterated expectations: E (cid:104) ψ d ,S =1 ( W, χ , Ψ d ,S =10 ) (cid:105) = E (cid:34) g ( X )Pr( S = 1) · = E [ E [ Y − µ Y ( d ,X ) | D = d ,X ] | D = d ,X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) E (cid:34) I { D = d } · I { D = d } p d ( X ) · p d ( d , X ) · [ Y − µ Y ( d , X )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (cid:35) (cid:35) + E (cid:34) g ( X )Pr( S = 1) · = (cid:82) E (cid:104) µ Y ( d ,x ) − ν Y ( d ,x ) (cid:12)(cid:12) D = d ,X = x (cid:3) dF X x | D d ,X x =0 (cid:122) (cid:125)(cid:124) (cid:123) E (cid:34) I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (cid:35) (cid:35) + E (cid:34) S Pr( S = 1) · ν Y ( d , X ) (cid:35) − Ψ d ,S =10 = Ψ d ,S =10 − Ψ d ,S =10 = 0 . Assumption 3.1(b)Linearity:
The score ψ d ,S =1 ( W, χ , Ψ d ,S =10 ) is linear in Ψ d ,S =10 : ψ d ,S =1 ( W, χ , Ψ d ,S =10 ) = ψ d ,S =1 a ( W, χ ) · Ψ d ,S =10 + ψ d ,S =1 b ( W, χ ) with ψ d ,S =1 a ( W, χ ) = − ψ d ,S =1 b ( W, χ ) = g ( X )Pr( S = 1) · I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X )+ g ( X )Pr( S = 1) · I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X )+ S Pr( S = 1) · ν Y ( d , X ) . Assumption 3.1(c) Continuity:
We may observe that the expression for the second Gateaux derivativeof a map χ (cid:55)→ E [ ψ d ,S =1 ( W, χ, Ψ d ,S =1 )], given in (A.6), is continuous. Assumption 3.1(d)Neyman Orthogonality : For any η ∈ T ∗ N , the Gateaux derivative in the direction χ − χ = ( g ( X ) − g ( X ) , p d ( X ) − p d ( X ) , p d ( D , X ) − p d ( D , X ) , µ Y ( d , X ) − µ Y ( d , X ) , ν Y ( d , X ) − ν Y ( d , X )) s given by ∂E (cid:2) ψ d ,S =1 ( W, χ, Ψ d ,S =10 ) (cid:3)(cid:2) χ − χ (cid:3) = − E (cid:34) g ( X )Pr( S = 1) · I { D = d } · I { D = d } · [ µ Y ( d , X ) − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (cid:35) ( ∗ )+ E (cid:34) g ( X )Pr( S = 1) · I { D = d } · [ µ Y ( d , X ) − µ Y ( d , X )] p d ( X ) (cid:35) ( ∗∗ ) − E (cid:34) g ( X )Pr( S = 1) · E [ ·| X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) · [ p d ( X ) − p d ( X )] p d ( X ) (cid:35) − E (cid:34) g ( X )Pr( S = 1) · E [ ·| X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) · [ p d ( X ) − p d ( X )] p d ( X ) (cid:35) − E (cid:34) g ( X )Pr( S = 1) · E [ ·| X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) · [ p d ( d , X ) − p d ( d , X )] p d ( d , X ) (cid:35) − E (cid:34) g ( X )Pr( S = 1) · I { D = d } · [ ν Y ( d , X ) − ν Y ( d , X )] p d ( X ) (cid:124) (cid:123)(cid:122) (cid:125) E [ ·| X ]= pd
10 ( X pd
10 ( X · [ ν Y ( d ,X ) − ν Y ( d ,X )] (cid:35) ( ∗ ∗ ∗ )+ E (cid:104) E [ ·| X ]= g X S =1) (cid:122) (cid:125)(cid:124) (cid:123) S Pr( S = 1) · [ ν Y ( d , X ) − ν Y ( d , X )] (cid:3) ( ∗ ∗ ∗∗ )+ E (cid:34) E [ ·| X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) · [ g ( X ) − g ( X )]Pr( S = 1) (cid:35) + E (cid:34) I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) (cid:124) (cid:123)(cid:122) (cid:125) E [ ·| X ]=0 · [ g ( X ) − g ( X )]Pr( S = 1) (cid:35) = 0 , where terms ( ∗ ) and ( ∗∗ ) as well as ( ∗ ∗ ∗ ) and ( ∗ ∗ ∗∗ ) cancel out. Assumption 3.1(e)Singular values of E [ ψ d ,S =1 a ( W ; χ )] are bounded: Holds trivially, because ψ d ,S =1 a ( W ; χ ) = − . ssumption 3.2: Score regularity and quality of nuisance parameter estimatorsAssumption 3.2(a) This assumption directly follows from the defition of T ∗ n and the regularity conditions (Assumption5). Assumption 3.2(b)Bounds for m n : We start by rearranging the terms in the Neyman score function (A.5) E (cid:104) ψ d ,S =1 ( W, χ, Ψ d ,S =10 ) (cid:105) = E (cid:34) g ( X )Pr( S = 1) · I { D = d } · I { D = d } p d ( X ) · p d ( d , X ) · Y (cid:124) (cid:123)(cid:122) (cid:125) = I + g ( X )Pr( S = 1) · I { D = d } p d ( X ) · (cid:18) − I { D = d } p d ( d , X ) (cid:19) · µ Y ( d , X ) (cid:124) (cid:123)(cid:122) (cid:125) = I + (cid:18) S Pr( S = 1) − g ( X )Pr( S = 1) · I { D = d } p d ( X ) (cid:19) ν Y ( d , X ) (cid:124) (cid:123)(cid:122) (cid:125) = I − Ψ d ,S =10 (cid:35) and then, Following the same steps as in (A.1), we get (cid:13)(cid:13)(cid:13) ψ d ,S =1 ( W, χ, Ψ d ,S =10 ) (cid:13)(cid:13)(cid:13) q ≤ (cid:107) I (cid:107) q + (cid:107) I (cid:107) q + (cid:107) I (cid:107) q + (cid:13)(cid:13)(cid:13) Ψ d ,S =10 (cid:13)(cid:13)(cid:13) q ≤ (cid:15) (cid:107) Y (cid:107) q + 1 − (cid:15)(cid:15) (cid:13)(cid:13) µ Y ( d , X ) (cid:13)(cid:13) q ++ 1 − (cid:15)(cid:15) (cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13) q + | Ψ d ,S =10 |≤ C (cid:18) (cid:15) + 2(1 − (cid:15) ) (cid:15) /q (cid:18) (cid:15) + 1 (cid:15) (cid:19) + 1 (cid:15) (cid:19) because of triangular inequality and because the following set of inequalities hold (similarly to (A.2)): (cid:13)(cid:13) µ Y ( d , X ) (cid:13)(cid:13) q ≤ C/(cid:15) /q , (cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13) q ≤ C/(cid:15) /q , | Ψ d ,S =10 | = (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) S Pr( S = 1) ν Y ( d , X ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:104) (cid:12)(cid:12)(cid:12) ν Y ( d , X ) (cid:12)(cid:12)(cid:12) (cid:105) /(cid:15) = (cid:13)(cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13)(cid:13) /(cid:15) ≤ (cid:13)(cid:13)(cid:13) ν Y ( d , X ) (cid:13)(cid:13)(cid:13) /(cid:15) ≤ (cid:107) Y (cid:107) /(cid:15) / q> (cid:122)(cid:125)(cid:124)(cid:123) ≤ (cid:107) Y (cid:107) q /(cid:15) ≤ C/(cid:15) . which gives the upper bound on m n in Assumption 3.2(b) of Chernozhukov, Chetverikov, Demirer, Duflo,Hansen, Newey, and Robins (2018). Bounds for m (cid:48) n : Notice that (cid:16) E [ | ψ d ,S =1 a ( W, χ ) | q ] (cid:17) /q = 1 nd this gives the upper bound on m (cid:48) n in Assumption 3.2(b) of Chernozhukov, Chetverikov, Demirer,Duflo, Hansen, Newey, and Robins (2018). Assumption 3.2(c)Bound for r n : For any χ = ( g, p d , p d , µ Y , ν Y ) we have (cid:12)(cid:12)(cid:12) E (cid:16) ψ d ,S =1 a ( W, χ ) − ψ d ,S =1 a ( W, χ ) (cid:17)(cid:12)(cid:12)(cid:12) = | − | = 0 ≤ δ (cid:48) N , and thus we have the bound on r n from Assumption 3.2(c) of Chernozhukov, Chetverikov, Demirer,Duflo, Hansen, Newey, and Robins (2018). Bound for r (cid:48) n : (cid:13)(cid:13)(cid:13) ψ d ,S =1 ( W, χ, Ψ d ,S =10 ) − ψ d ,S =1 ( W, χ , Ψ d ,S =10 ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I { D = d } · I { D = d } Pr( S = 1) · Y · (cid:32) gp d p d − g p d p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I { D = d } · I { D = d } Pr( S = 1) (cid:32) g · µ Y p d p d − g · µ Y p d p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I { D = d } Pr( S = 1) (cid:32) g · µ Y p d − g · µ Y p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I { D = d } Pr( S = 1) (cid:32) g · ν Y p d − g · ν Y p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) S Pr( S = 1) ( ν Y − ν Y ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y · (cid:32) gp d p d − g p d p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 1 (cid:15) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g · µ Y p d p d − g · µ Y p d p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 1 (cid:15) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g · µ Y p d − g · µ Y p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 1 (cid:15) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g · ν Y p d − g · ν Y p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 1 (cid:15) (cid:13)(cid:13)(cid:13) ν Y − ν Y (cid:13)(cid:13)(cid:13) ≤ C(cid:15) δ n (cid:18) (cid:15) (cid:19) + δ n (cid:15) (cid:18) (cid:15) + 2 C + C(cid:15) (cid:19) + δ n (cid:15) (cid:18) (cid:15) + 2 C (cid:19) + δ n (cid:15) (cid:18) (cid:15) + 2 C (cid:19) + δ n (cid:15) ≤ δ (cid:48) n as long as C (cid:15) in the definition of δ (cid:48) n is sufficiently large. This gives the bound on r (cid:48) n from Assumption3.2(c) of Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018).The last inequality holds because we can bound the first term by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y · (cid:32) gp d p d − g p d p d (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) gp d p d − g p d p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C(cid:15) (cid:13)(cid:13)(cid:13) p d p d g − p d p d g (cid:13)(cid:13)(cid:13) = C(cid:15) (cid:13)(cid:13)(cid:13) p d p d g − p d p d g + p d p d g − p d p d g (cid:13)(cid:13)(cid:13) ≤ C(cid:15) (cid:16) (cid:107) g − g (cid:107) + 1 · (cid:13)(cid:13)(cid:13) p d p d − p d p d (cid:13)(cid:13)(cid:13) (cid:17) ≤ C(cid:15) (cid:18) δ n + 1 · δ n (cid:18) (cid:15) (cid:19)(cid:19) ≤ C(cid:15) δ n (cid:18) (cid:15) (cid:19) , he second term is bounded by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g · µ Y p d p d − g · µ Y p d p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:13)(cid:13)(cid:13) p d p d g · µ Y − p d p d g · µ Y (cid:13)(cid:13)(cid:13) = 1 (cid:15) (cid:13)(cid:13)(cid:13) p d p d g · µ Y − p d p d g · µ Y + p d p d g · µ Y − p d p d g · µ Y (cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:16)(cid:13)(cid:13)(cid:13) p d p d ( g · µ Y − g · µ Y ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) g · µ Y ( p d p d − p d p d ) (cid:13)(cid:13)(cid:13) (cid:17) ≤ (cid:15) (cid:16)(cid:13)(cid:13)(cid:13) g · µ Y − g · µ Y (cid:13)(cid:13)(cid:13) + 1 · C (cid:13)(cid:13)(cid:13) p d p d − p d p d (cid:13)(cid:13)(cid:13) (cid:17) ≤ (cid:15) (cid:18) δ n (cid:18) C + 1 (cid:15) (cid:19) + Cδ n (cid:18) (cid:15) (cid:19)(cid:19) = δ n (cid:15) (cid:18) (cid:15) + 2 C + C(cid:15) (cid:19) where (cid:13)(cid:13)(cid:13) g · µ Y − g · µ Y (cid:13)(cid:13)(cid:13) is bounded similarly as (cid:13)(cid:13)(cid:13) p d µ Y − p d µ Y (cid:13)(cid:13)(cid:13) and we also used bounds for (cid:13)(cid:13)(cid:13) p d p d − p d p d (cid:13)(cid:13)(cid:13) derived in section (A.1).while for the third term we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g · µ Y p d − g · µ Y p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 1 (cid:15) (cid:13)(cid:13)(cid:13) p d g · µ Y − p d g · µ Y (cid:13)(cid:13)(cid:13) = 1 (cid:15) (cid:13)(cid:13)(cid:13) p d g · µ Y − p d g · µ Y + p d g · µ Y − p d g · µ Y (cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:16)(cid:13)(cid:13)(cid:13) p d ( g · µ Y − g · µ Y ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) g · µ Y ( p d − p d ) (cid:13)(cid:13)(cid:13) (cid:17) ≤ (cid:15) (cid:18) δ n (cid:18) C + 1 (cid:15) (cid:19) + C (cid:13)(cid:13)(cid:13) p d − p d (cid:13)(cid:13)(cid:13) (cid:19) ≤ (cid:15) (cid:18) δ n (cid:18) C + 1 (cid:15) (cid:19) + Cδ n (cid:19) = δ n (cid:15) (cid:18) (cid:15) + 2 C (cid:19) . and similarly, for the fourth term we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ν Y p d − ν Y p d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ δ n (cid:15) (cid:18) (cid:15) + 2 C (cid:19) . Bound for λ (cid:48) n : Now consider f ( r ) := E [ ψ d ,S =1 ( W ; Ψ d ,S =10 , χ + r ( χ − χ )] or any r ∈ (0 ,
1) : ∂ f ( r ) ∂r = E (cid:34) I { D = d } Pr( S = 1) · g − g ) (cid:16) ( µ Y − µ Y ) − ( ν Y − ν Y ) (cid:17) p d + r ( p d − p d ) (cid:35) (A.6)+ E (cid:34) I { D = d } Pr( S = 1) · ( − (cid:16) g + r ( g − g ) (cid:17)(cid:16) ( µ Y − µ Y ) − ( ν Y − ν Y ) (cid:17) ( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } Pr( S = 1) · g − g ) (cid:16) ( µ Y + r ( µ Y − µ Y )) − ( ν Y + r ( ν Y − ν Y )) (cid:17)(cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } Pr( S = 1) · (cid:16) g + r ( g − g ) (cid:17)(cid:16) ( µ Y + r ( µ Y − µ Y )) − ( ν Y + r ( ν Y − ν Y )) (cid:17) ( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } I { D = d } Pr( S = 1) · ( − g − g ) (cid:16) Y − ( µ Y + r ( µ Y − µ Y )) (cid:17) ( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } I { D = d } Pr( S = 1) · ( − g − g ) (cid:16) Y − ( µ Y + r ( µ Y − µ Y )) (cid:17) ( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } I { D = d } Pr( S = 1) · ( − g − g )( µ Y − µ Y ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } I { D = d } Pr( S = 1) · (cid:16) g + r ( g − g ) (cid:17)(cid:16) Y − ( µ Y + r ( µ Y − µ Y )) (cid:17) ( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } I { D = d } Pr( S = 1) · (cid:16) g + r ( g − g ) (cid:17)(cid:16) Y − ( µ Y + r ( µ Y − µ Y )) (cid:17) ( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } I { D = d } Pr( S = 1) · (cid:16) g + r ( g − g ) (cid:17)(cid:16) Y − ( µ Y + r ( µ Y − µ Y )) (cid:17) ( p d − p d )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } I { D = d } Pr( S = 1) · (cid:16) g + r ( g − g ) (cid:17) ( µ Y − µ Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) + E (cid:34) I { D = d } I { D = d } Pr( S = 1) · (cid:16) g + r ( g − g ) (cid:17) ( µ Y − µ Y )( p d − p d ) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:16) p d + r ( p d − p d ) (cid:17) (cid:35) here we follow the same procedure as in bounding (A.4), with the only exception that now we haveto make use of the last two inequalities from the regularity conditions (A.5).As an example, consider the first term: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) I { D = d } Pr( S = 1) · g − g ) (cid:16) ( µ Y − µ Y ) − ( ν Y − ν Y ) (cid:17) p d + r ( p d − p d ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) I { D = d } Pr( S = 1) · g − g )( µ Y − µ Y ) p d + r ( p d − p d ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34) I { D = d } Pr( S = 1) · g − g )( ν Y − ν Y ) p d + r ( p d − p d ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) δ N (cid:15) n − / + 2 (cid:15) δ N (cid:15) n − / = 4 δ N (cid:15) n − / . We may bound all the remaining terms similarly and get that for some constant C (cid:48)(cid:48) (cid:15) that only dependson C and (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) ∂ f ( r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:48)(cid:48) (cid:15) δ n n − / ≤ δ (cid:48) n n − / and this gives the upper bound on λ (cid:48) n in Assumption 3.2(c) of Chernozhukov, Chetverikov, Demirer,Duflo, Hansen, Newey, and Robins (2018) as long as C (cid:15) ≥ C (cid:48)(cid:48) (cid:15) . Assumption 3.2(d) E (cid:104) ( ψ d ,S =1 ( W, χ , Ψ d ,S =10 )) (cid:105) = E (cid:34)(cid:32) g ( X )Pr( S = 1) · I { D = d } · I { D = d } · [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (cid:124) (cid:123)(cid:122) (cid:125) = I + g ( X )Pr( S = 1) · I { D = d } · [ µ Y ( d , X ) − ν Y ( d , X )] p d ( X ) (cid:124) (cid:123)(cid:122) (cid:125) = I + S Pr( S = 1) · ν Y ( d , X ) − Ψ d (cid:124) (cid:123)(cid:122) (cid:125) = I (cid:33) (cid:35) = E [ I + I + I ] ≥ E [ I ]= E (cid:34) g ( X )Pr( S = 1) · I { D = d } · I { D = d } · (cid:32) [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (cid:33) (cid:35) ≥ (cid:15) − (cid:15) E (cid:34)(cid:32) [ Y − µ Y ( d , X )] p d ( X ) · p d ( d , X ) (cid:33) (cid:35) ≥ (cid:15) c (1 − (cid:15) ) > . because Pr( D = d | X ) = p d ( X ) · p d ( d , X ) ≥ (cid:15) , p d ( d , X ) ≤ − (cid:15) and g ( X ) ≥ (cid:15). here the second equality follows from E (cid:104) I · I (cid:105) = E (cid:34) (cid:18) g ( X )Pr( S = 1) (cid:19) · E [ ·| X ]= E [ E [ Y − µ Y ( d ,X ) | D = d ,X ] | D = d ,X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } ( p d ( X )) · p d ( d , X ) · [ Y − µ Y ( d , X )] · [ µ Y ( d , X ) − ν Y ( d , X )] (cid:35) ,E (cid:104) I · I (cid:105) = E (cid:34) g ( X ) · S (Pr( S = 1)) · E [ ·| X ]= (cid:82) E (cid:2) µ Y ( d ,x ) − ν Y ( d ,x ) (cid:12)(cid:12) D = d ,X = x (cid:3) dF X x | D d ,X x =0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } p d ( X ) · [ µ Y ( d , X ) − ν Y ( d , X )] · [ ν Y ( d , X ) − Ψ d ] (cid:35) ,E (cid:104) I · I (cid:105) = E (cid:34) g ( X ) · S (Pr( S = 1)) · E [ ·| X ]= E [ E [ Y − µ Y ( d ,X ) | D = d ,X ] | D = d ,X ]=0 (cid:122) (cid:125)(cid:124) (cid:123) I { D = d } · I { D = d } p d ( X ) · p d ( d , X ) · [ Y − µ Y ( d , X )] · [ ν Y ( d , X ) − Ψ d ] (cid:35) . Covariates
Table B.1 provides information on the covariates in our empirical application across treatmentsequences. The first two columns contain the name and a description of each variable. Thevariable type in the third column can take the value 1, 2, or 3, which stands for dummy,categorical, or continuous variable, respectively. The fourth column presents an indicator whichis 0 for covariates observed prior to the first treatment ( X ) and 1 for covariates observed afterthe first treatment ( X ). The number of missing values is given in the fifth column, and columns6-15 display the mean values of the covariates for the treatment sequences in Table 3.48 a b l e B . : D e s c r i p t i o n o f r e g r e ss o r s a nd m e a n s a c r o ss t r e a t m e n t s e q u e n ce s V a r i a b l e d e s c r i p t i o nS t a t i s t i c s M e a n v a l u e s a c r o ss tr e a t m e n t s e q u e n ce s n a m e d e s c r i p t i o n t y p e x m i ss - j c m s a M S A C A T E G O R Y . . . . . . . . . . .
90 ag e A G E I NY E A R S A T B A S E L I N E . . . . . . . . . . . R A C EE T H R A C E O R E T HN I C I T Y . . . . . . . . . . . N T V L AN G NA T I V E L AN G UA G E . . . . . . . . . . . HH H E A D O F H O U S E W H E N S A M P L E M E M B E R W A S . . . . . . . . . . . W E L F K I D F A M I L Y O N W E L F A R E W H E N G R O W I N G U P . . . . . . . . . . . H G C M O T HH I G H E S T G R A D E M O T H E R C O M P L E T E D . . . . . . . . . . . M W O R K M O T H E R W O R K E D W H E N S A M P L E M E M B E R W A S . . . . . . . . . . . O CC M O T H O CC U P A T I O N O F M O T H E R W H E N S A M P L E W A S . . . . . . . . . . . H G C F A T HH I G H E S T G R A D E F A T H E R C O M P L E T E D . . . . . . . . . . . F W O R K F A T H E R W O R K E D W H E N S A M P L E M E M B E R W A S . . . . . . . . . . . O CC F A T H O CC U P A T I O N O FF A T H E R W H E N S A M P L E W A S . . . . . . . . . . . m a rr i ag e m a r i t a l s t a t u s . . . . . . . . . . . h a s c h l dh a d c h il d r e n a tr a nd o m a ss i g n m e n t . . . . . . . . . . . p r o p li v e P R O P O R T I O N O F C H I L D R E N W H O L I V E W I T H . . . . . . . . . . . P R E G N R A C U RR E N T L Y P R E G NAN T . . . . . . . . . . .
00 o l d A G E O F O L D E S TC H I L D . . . . . . . . . . . y n g A G E O F Y O UN G E S TC H I L D . . . . . . . . . . .
17 o t h w i t h P L A C E W H E R E A B S E N TC H I L D R E N L I V E . . . . . . . . . . . n c h l dnu m b e r o f c h il d r e n . . . . . . . . . . .
48 ag e p a r n t A G E W H E N S A M P L E M E M B E R B E C A M E A P A R E N T . . . . . . . . . . . NU M B HHNU M B E R I NH O U S E H O L D . . . . . . . . . . . R H E A D S A M P L E M E M B E R I S H E A D O F H O U S E H O L D . . . . . . . . . . . hh m e m b H O U S E H O L D M E M B E R S H I P . . . . . . . . . . . H O U S A RR C U RR E N T H O U S I N G A RR AN G E M E N T . . . . . . . . . . . P AY R E N T S A M P L E M E M B E R C O N T R I B U T E S T O R E N T . . . . . . . . . . . h g c h i g h e s t g r a d ec o m p l e t e d . . . . . . . . . . . H S D HA D H S D I P L O M AA T R AN D O M A SS I G N M E N T . . . . . . . . . . . G E DD HA D G E D A T R AN D O M A SS I G N M E N T . . . . . . . . . . . V O C D HA D V O C D E G R EE A T R AN D O M A SS I G N M E N T . . . . . . . . . . . O T H D E G HA D O T H E R D E G R EE A T R AN D O M A SS I G N M E N T . . . . . . . . . . . i n s c h oo l I N S C H OO L M O N T H P R I O R T O J C A PP L I C A T I O N . . . . . . . . . . . ANY E D A TT E N D E D ANY E D O R T R G I N P A S T Y E A R . . . . . . . . . . . N E D C A T NU M B E R O F E D P R O G R A M S I N P A S T Y E A R . . . . . . . . . . . m o n i n e d M O T H E R S I N E D P R O G R A M S I N P A S T Y E A R . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - r e a s l e f t M A I N R E A S O N L E F T S C H OO L . . . . . . . . . . . R E C E D M O S T R E C E N T E D U C A T I O N O R T R A I N I N G P R O G R A M . . . . . . . . . . . T Y PEE D R T Y PE O F M O S T R E C E N T E D P R O G R A M . . . . . . . . . . . NH R S E D R U S UA L H O U R S / W K I N M O S T R E C E N T E D P R O G R A M . . . . . . . . . . . R E A S E D R M A I N R E A S O N L E F T M O S T R E C E N T P R O G R A M . . . . . . . . . . . nu m b j o b s NU M B E R O F J O B S I N P A S T Y E A R . . . . . . . . . . . e v w o r k b E V E R HA D F U LL O R P A R TT I M E J O B . . . . . . . . . . . Y R W O R K HA D J O B I N P A S T Y E A R . . . . . . . . . . . E A R NY R E A R N I N G S I N P A S T Y E A R . . . . . . . . . . . m o s i n j o b m o n t h s e m p l o y e d i np a s t y e a r . . . . . . . . . . . R E C J O B M O S T R E C E N T J O B . . . . . . . . . . . O CC R O CC U P A T I O NA T M O S T R E C E N T J O B . . . . . . . . . . . H R S W K J R U S UA L W EE K L YH O U R S A T M O S T R E C E N T J O B . . . . . . . . . . . h r w ag e r h o u r l y w ag e a t m o s tr ece n t j o b . . . . . . . . . . . C OO P R M O S T R E C E N T J O B P A R T O F C O - O PP R O G R A M . . . . . . . . . . . G O V P R G R M O S T R E C E N T J O B P A R T O F G O V T P R O G R A M . . . . . . . . . . . l e f t j o b r L E F T M O S T R E C E N T J O B P R I O R T O R A . . . . . . . . . . . r s l f t j r M A I N R E A S O N L E F T M O S T R E C E N T J O B . . . . . . . . . . . M O S A F D C M O S T S R E C E I V E D A F D C I N P A S T Y E A R . . . . . . . . . . . M O S O T H W M O N T H S R E C E I V E D O T H E R W E L F A R E I N P Y . . . . . . . . . . . M O S F S M O N T H S R E C E I V E D F OO D S T A M P S I N P A S T Y E A R . . . . . . . . . . . G O T ANY W R E C E I V E D ANY P U B L I C A SS I S T AN C E I N P Y . . . . . . . . . . . M O S ANY W M O N T H S R E C E I V E D ANY P U B L I C A SS I S T I N P Y . . . . . . . . . . . G O T A F D C R E C E I V E D A F D C I N P A S T Y E A R . . . . . . . . . . . G O T O T H W R E C E I V E D O T H E R W E L F A R E I N P A S T Y E A R . . . . . . . . . . . G O T F S R E C E I V E D F OO D S T A M P S I N P A S T Y E A R . . . . . . . . . . . HH I N CT O T A L H O U S E H O L D I N C O M E . . . . . . . . . . . PE R S I N CT O T A L PE R S O NA L I N C O M E . . . . . . . . . . . h e a l t h H E A L T H S T A T U S . . . . . . . . . . . s i c k HA D H E A L T H P R O B L E M T HA T L I M I T E D W O R K . . . . . . . . . . . t y p e h l t h T Y PE H E A L T H P R O B L E M T HA T L I M I T E D W O R K . . . . . . . . . . . t i m e s i c k T I M E HA D H E A L T H P R O B L E M T HA T L I M I T E D W O R K . . . . . . . . . . . E V C I G E V E R S M O K E D C I G A R E TT E S . . . . . . . . . . . P Y C I G S M O K E D C I G A R E TT E S I N P A S T Y E A R . . . . . . . . . . . E VA L C H L E V E R D R AN K A L C O H O L . . . . . . . . . . . P YA L C H L D R AN K A L C O H O L I N P A S T Y E A R . . . . . . . . . . . E V P O T E V E R S M O K E D M A R I J UANA . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - P Y P O T S M O K E D M A R I J UANA I N P A S T Y E A R . . . . . . . . . . . E V C O K EE V E R S N O R T E D C O C A I N E . . . . . . . . . . . P Y C O K E S N O R T E D C O C A I N E I N P A S T Y E A R . . . . . . . . . . . E V C R A C K E V E R S M O K E D C R A C K . . . . . . . . . . . P Y C R A C K S M O K E D C R A C K I N P A S T Y E A R . . . . . . . . . . . E VH R O I N E V E R U S E D H E R O I N . . . . . . . . . . . P YH R O I NU S E D H E R O I N I N P A S T Y E A R . . . . . . . . . . . E V S PEE D E V E R U S E D S PEE D . . . . . . . . . . . P Y S PEE D U S E D S PEE D I N P A S T Y E A R . . . . . . . . . . . E V L S D E V E R U S E D L S D . . . . . . . . . . . P Y L S D U S E D L S D I N P A S T Y E A R . . . . . . . . . . . E V O T H D R E V E R U S E D O T H E R I LL E G A L D R U G S . . . . . . . . . . . P Y O T H D R U S E D O T H E R I LL E G A L D R U G S I N P A S T Y E A R . . . . . . . . . . . E V I N J CT E V E R I N J E CT E DD R U G S W I T HN EE D L E . . . . . . . . . . . D R U G T R T E V E R I N D R U G O R A L C O H O L T R E A T M E N T . . . . . . . . . . . M O U TT R T M O N T H S P R I O R T O R A I N D R U G T R E A T M E N T . . . . . . . . . . . M O S T R T R M O N T H S I N M O S T R E C E N T D R U G T R E A T M E N T . . . . . . . . . . . F R Q C I G H O W O F T E N S M O K E D C I G A R E TT E S I N P A S T Y E A R . . . . . . . . . . . F R Q A L C H O W O F T E N D R AN K A L C O H O L I N P A S T Y E A R . . . . . . . . . . . F R Q P O T H O W O F T E N S M O K E D M A R I J UANA I N P A S T Y E A R . . . . . . . . . . . F R Q C O K E H O W O F T E N S N O R T E D C O C A I N E I N P A S T Y E A R . . . . . . . . . N a N . F R Q C R A C H O W O F T E N S M O K E D C R A C K I N P A S T Y E A R . . . . . . . . . N a N . F R Q H E R NH O W O F T E NU S E D H E R O I N I N P A S T Y E A R . . . . . . . N a N . N a N . F R Q S PE D H O W O F T E NU S E D S PEE D I N P A S T Y E A R . . . . . . . . . . . F R Q L S D H O W O F T E NU S E D L S D I N P A S T Y E A R . . . . . . . . . . . F R Q I N J H O W O F T E N I N J E CT E DD R U G S I N P A S T Y E A R . . . N a NN a N . . N a N . N a N . F R QO T HH O W O F T E NU S E D O T H E R I LL E G A L D R U G S I N P Y . . . N a NN a N . N a N . . N a N . n a rr c a t NU M B E R O F A RR E S T S . . . . . . . . . . . E VA RR S T E V E R A RR E S T E D . . . . . . . . . . . R C A RR S T M O S T R E C E N T A RR E S T . . . . . . . . . . . M A RR C A T M O N T H SS I N C E M O S T R E C E N T A RR E S T . . . . . . . . . . .
65 ag e a r c a t A G E A T F I R S T A RR E S T . . . . . . . . . . . bu r g l a r y E V E R A RR E S T E D F O R B U R G L A R Y . . . . . . . . . . . r o bb e r y E V E R A RR E S T E D F O RR O BB E R Y . . . . . . . . . . .
00 a ss a u l t E V E R A RR E S T E D F O R M U R D E R O R A SS AU L T . . . . . . . . . . . l a r ce n y E V E R A RR E S T E D F O R L A R C E NY . . . . . . . . . . . d r u g v i o l E V E R A RR E S T E D F O R D R U G V I O L A T I O N S . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - t hp e r s E V E R A RR E S T E D F O R O T H E R PE R S O NA L C R I M E S . . . . . . . . . . .
02 o t h m i s c E V E R A RR E S T E D F O R O T H E R M I S CC R I M E S . . . . . . . . . . . S E R C R S M U R D E R O R A SS AU L T W A S M O S T S E R I O U S C R I M E . . . . . . . . . . . S E R C R S R O BB E R Y W A S M O S T S E R I O U S C R I M E . . . . . . . . . . . S E R C R S B U R G L A R Y W A S M O S T S E R I O U S C R I M E . . . . . . . . . . . S E R C R S L A R C E NY W A S M O S T S E R I O U S C R I M E . . . . . . . . . . . S E R C R S D R U G V I O L A T I O N S W A S M O S T S E R I O U S C R I M E . . . . . . . . . . . S E R C R S O T H E R PE R S C R I M E S W E R E M O S T S E R I O U S C R I M E . . . . . . . . . . . S E R C R S O T H E R M I S CC R I M E S W E R E M O S T S E R I O U S C R I M E . . . . . . . . . . . N G U I L T YNU M B E R O F T I M E S C O NV I CT E D . . . . . . . . . . . G U I L T Y E V E R C O NV I CT E D O R P L E D G U I L T Y . . . . . . . . . . . w k s j a il T O T A L W EE K SS PE N T I N J A I L . . . . . . . . . . . PE N D I N G HA S A RR E S TC HA R G E S PE N D I N G . . . . . . . . . . . C O PP L E A E V E R M A D E A D E A L O R C O PPE D A P L E A . . . . . . . . . . . S E R C R C M O S T S E R I O U S C O NV I CT I O N - M U R D E R O R A SS AU L T . . . . . . . . . . . S E R C R C M O S T S E R I O U S C O NV I CT I O N - R O BB E R Y . . . . . . . . . . . S E R C R C M O S T S E R I O U S C O NV I CT I O N - B U R G L A R Y . . . . . . . . . . . S E R C R C M O S T S E R I O U S C O NV I CT I O N - L A R C E NY . . . . . . . . . . . S E R C R C M O S T S E R I O U S C O NV I CT I O N - D R U G V I O L A T I O N S . . . . . . . . . . . S E R C R C M O S T S E R I O U S C O NV I CT I O N - O T H E R PE R S C R I M E S . . . . . . . . . . . S E R C R C M O S T S E R I O U S C O NV I CT I O N - O T H E R M I S CC R I M E S . . . . . . . . . . . A SS L TC E V E R C O NV I CT E D O F M U R D E R O R A SS AU L T . . . . . . . . . . . R O B C E V E R C O NV I CT E D O F R O BB E R Y . . . . . . . . . . . B U R G L C E V E R C O NV I CT E D O F B U R G L A R Y . . . . . . . . . . . L A R C NY C E V E R C O NV I CT E D O F L A R C E NY . . . . . . . . . . . D R V I O L C E V E R C O NV I CT E D O F D R U G V I O L A T I O N S . . . . . . . . . . . O T H PE R C E V E R C O NV I CT E D O F O T H E R PE R S C R I M E S . . . . . . . . . . . O T H M S CC E V E R C O NV I CT E D O F O T H E R M I S CC R I M E S . . . . . . . . . . . E V J A I L E V E R S E R V E D T I M E I N J A I L . . . . . . . . . . . P A R O L E E V E R P U T O N P R O B A T I O N O R P A R O L E . . . . . . . . . . . H E A R J C H O W F I R S T H E A R D A B O T U J O B C O R P S . . . . . . . . . . . F R O M O A F I R S T H E A R D A B O U T J C F R O M O A C O UN S E L O R . . . . . . . . . . . K N E W J C K N E W S O M E O N E W H O A TT E N D E D J O B C O R P S . . . . . . . . . . . I N F O J C H O W G O T M O S T I N F O A B O U T W HA T J C I S L I K E . . . . . . . . . . . R H O M E J O I N E D J CT O G E T A W AY F R O M H O M E . . . . . . . . . . . R C O MM J O I N E D J CT O G E T A W AY F R O M C O MM UN I T Y . . . . . . . . . . . R T R A I N J O I N E D J CT O G E T J O B T R A I N I N G . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - R C R G O A L J O I N E D J CT O A C H I E V E C A R EE R G O A L . . . . . . . . . . . R G E T G E D J O I N E D J CT O G E T A G E D . . . . . . . . . . . R N O W O R K J O I N E D J CT O B E A B L E T O F I N D W O R K . . . . . . . . . . . R O T H E R J O I N E D J C F O R O T H E R I M P O R T AN T R E A S O N S . . . . . . . . . . . m o s t i m p r M O S T I M P O R T AN T R E A S O N F O R J O I N I N G J C . . . . . . . . . . .
62 o t h i m p r M A I N O T H E RR E A S O N S F O R J O I N I N G J C . . . . . . . . . . . E M A T H E X PE CT J CT O I M P R O V E M A T H S K I LL S . . . . . . . . . . . E R E A D E X PE CT J CT O I M P R O V E R E A D I N G S K I LL S . . . . . . . . . . . E A L O N G E X PE CT J CT O H E L P G E T A L O N G W I T H O T H E R S . . . . . . . . . . . E C O N T R L E X PE CT J CT O I M P R O V E S E L F C O N T R O L . . . . . . . . . . . EE S T EE M E X PE CT J CT O I M P R O V E S E L F E S T EE M . . . . . . . . . . . E S P C J O B E X PE CT J CT O G I V E T R G F O R S PE C I F I C J O B . . . . . . . . . . . E F R I E N D E X PE CT J CT O L E A D T O N E W F R I E N D S H I P S . . . . . . . . . . . k n e w c n tr K N E W J CC E N T E R T HA T W AN T E D T O A TT E N D . . . . . . . . . . . i m p r c n tr M A I N R E A S O N W AN T S T O A TT E N D C E N T E R . . . . . . . . . . . k n e w j o b K N E WW HA T J O B W AN T E D T O T R A I N F O R . . . . . . . . . . . t y p e j o bb T Y PE O F J O B W AN T E D T O T R A I N F O R . . . . . . . . . . . E A R N C M PE X PE CT E D E A R N I N G S PE R H R A F T E R J C . . . . . . . . . . . h a d w o rr y W O RR I E D A B O U T A TT E N D I N G J C . . . . . . . . . . . t y p e w o rr M A I N T Y PE O F W O RR YA B O U T A TT E N D I N G J C . . . . . . . . . . . T A L K P A R T A L K E D T O P A R E N T S A B O U T A TT E N D I N G J C . . . . . . . . . . . I M PP A R P A R E N T A D V I C E W A S I M P O R T AN T . . . . . . . . . . . E N C R P A R P A R E N T E N C O U R A G E D A TT E N D I N G J C . . . . . . . . . . . T A L K R E L T A L K E D T O R E L A T I V E A B O U T A TT E N D I N G J C . . . . . . . . . . . I M P R E L R E L A T I V E A D V I C E W A S I M P O R T AN T . . . . . . . . . . . E N C RR E L R E L A T I V EE N C O U R A G E D A TT E N D I N G J C . . . . . . . . . . . T A L K F R D T A L K E D T O F R I E N D S A B O U T A TT E N D I N G J C . . . . . . . . . . . I M P F R D F R I E N D S A D V I C E W A S I M P O R T AN T . . . . . . . . . . . E N C R F R D F R I E N D S E N C O U R A G E D A TT E N D I N G J C . . . . . . . . . . . T A L K TC H T A L K E D T O T E A C H E R A B O U T A TT E N D I N G J C . . . . . . . . . . . I M P TC H T E A C H E R A D V I C E W A S I M P O R T AN T . . . . . . . . . . . E N C R TC H T E A C H E R E N C O U R A G E D A TT E N D I N G J C . . . . . . . . . . . T A L K C W T A L K E D T O C A S E W O R K E R A B O U T A TT E N D I N G J C . . . . . . . . . . . I M P C W C A S E W O R K E R A D V I C E W A S I M P O R T AN T . . . . . . . . . . . E N C R C W C A S E W O R K E R E N C O U R A G E D A TT E N D I N G J C . . . . . . . . . . . T A L K P R O T A L K E D T O P R O B A T I O N O FF I C E R A B O U T J C . . . . . . . . . . . I M PP R O P R O B A T I O N O FF I C E R A D V I C E W A S I M P O R T AN T . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - E N C R P R O P R O B A T I O N O FF I C E R E N C O U R A G E D A TT E N D I N G J C . . . . . . . . . . . T A L K C H L T A L K E D T O C HU R C H L E A D E R A B O U T J C . . . . . . . . . . . I M P C H L C HU R C H L E A D E R A D V I C E W A S I M P O R T AN T . . . . . . . . . . . E N C R C H L C HU R C H L E A D E R E N C O U R A G E D A TT E N D I N G J C . . . . . . . . . . . T A L K A D L T A L K E D T OO T H E R A D U L T A B O U T A TT E N D I N G J C . . . . . . . . . . . I M P A D L O T H E R A D U L T A D V I C E W A S I M P O R T AN T . . . N a NN a N . . . . . . E N C R A D L O T H E R A D U L T E N C O U R A G E D A TT E N D I N G J C . . N a NN a NN a N . . . . . . E N C R J C R G O T E N C O U R A G E M E N T F R O M O A C O UN S E L O R . . . . . . . . . . . h o w s p o k e H O W F I R S T S P O K E T OO A C O UN S E L O R . . . . . . . . . . . t e l e m o d e M O D E O F T E L EP H O N E C O N T A CT W I T H O A C O UN S . . . . . . . . . . . p l a ce i p c P L A C E O F I N - PE R S O N C O N T A CT W I T H O A C O UN S . . . . . . . . . . . t a l k s t a y T O L D H O W L O N G E X PE CT E D T O S T AYA T J C . . . . . . . . . . . W AY S T AY W AY T HA T D I S C U SS E D L E N G T H O F S T AYA T J C . . . . . . . . . . . r s t a y c a t M O S TC O MM O N R AN G E S O F E X PE CT E D S T AYA T J C . . . . . . . . . . . t s t a y c a t P R E C I S E T I M EE X PE CT E D T O S T AYA T J C . . . . . . . . . . . t a l kv s t f T O L D W H E N C O U L D F I R S T V I S I T F A M I L Y . . . . . . . . . . . t a l k t o l d T O L D H O W L O N G UN T I L C E N T E R A SS I G N M E N T . . . . . . . . . . . tr a d w a n t T O L D C HAN C E S O F G E TT I N G D E S I R E D T R A D E . . . . . . . . . . . c hn ce tr d C HAN C E S O F G E TT I N G D E S I R E D T R A D E . . . . . . . . . . . t o t a l h r s T O T A L H O U R SS PE N T W I T H O A C O UN S E L O R . . . . . . . . . . . V S T F C A T M O N T H S UN T I L C O U L D F I R S T V I S I T F A M I L Y . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . S C H L S C H OO L P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . T R N G T R A I N I N G P R O G R A M S Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . W O R K W O R K E X PE R I E N C E Y E A R B E F O R E R A - W EE K . . . . . . . . . . . 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E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D T NX I NAN E D U C / T R N I N G / J C P R G R M I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D AXH I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . D G T R H I NA D R U G T R E A T M E N T P R G R M I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . W O R K H E M P L O Y M E N T S T A T U S I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E M P U S E D C H I L D C A R E W H I L EE M P L O Y E D I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . CC E D T U S E D C H I L D C A R E W H I L E I N E D / T R N P R G R M I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . CC J C U S E D C H I L D C A R E W H I L E I N J O B C O R P S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . U I H G O T U I B E N E F I T S I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . H W H H O U R S W O R K E D I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E A R NH E A R N I N G S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . 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E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - E D T A H R S I NA C A D E M I C S I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . E D T V H R S I NV O CT R N G I NN O N - J C P R G R M S I N W EE K . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . A F D C H G O T A F D C I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . F S H G O T F OO D S T A M P S I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . SS I H G O T SS I / SS A I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . G AH G O T G E N E R A L A SS I S T . I N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . ANY P H G O T A F D C / F S / G A / SS II N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . W I C H G O T W I C I N M O N T H . . . . . . . . . . . ff m a r i t a l s t a t u s . . . . . . . . . . . f g78 r e s p & k i d s c o v e r e db y pub l h l t h i n s u r . . . . . . . . . . . f g80 i n t c h k : r s pnd t li v e s i n j c / i n s t i t u t i o n . . . . . . . . . . . f g811/0 r e s p o nd e n t li v e s i npub li c h o u s i n g1115640 . . . . . . . . . . . f g821/2 r e s p o nd e n t o w n s o rr e n t s h o m e . . . . . . . . . . . e v a rr q e v e r a rr e s t e d i n q tr . . . . . . . . . . . e v a rr q e v e r a rr e s t e d i n q tr . . . . . . . . . . . e v a rr q e v e r a rr e s t e d i n q tr . . . . . . . . . . . e v a rr q e v e r a rr e s t e d i n q tr . . . . . . . . . . . n a rr y nu m b e r o f a rr e s t s i n y e a r . . . . . . . . . . . n a r c a rr e s t s y r : c h r g e d w / m u r d e r . . . . . . . . . . . n a r c a rr e s t s y r : c h r g e d w /agg . a ss a u l t . . . . . . . . . . . n a r c a rr e s t s y r : c h r g e d w / r o bb e r y . . . . . . . . . . . n a r c a rr e s t s y r : c h r g e d w / bu r g l a r y . . . . . . . . . . . n a r c a rr e s t s y r : c h r g e d w / t h e f t . . . . . . . . . . . n a r c a rr e s t s y r : c h r g e d w / d r u g v i o l. . . . . . . . . . . . n a r c a rr e s t s y r : c h r g e d w /o t hp e r s o n a l . . . . . . . . . . . n a r c a rr e s t s y r : c h r g e d w /o t h m i s cc r i m e s . . . . . . . . . . .
02 a n y c r a12a n y c r i m e s aga i n s tr e s p : m i . . . . . . . . . . . m o n e y a12a m t m o n e y r e s p t l o s t f r o m c r i m e s i np y . . . . . . . . . . . nu m v i c t i m e s v i c t i m i ze d : m i n t . . . . . . . . . . . nu m c r a12 t o t a l c r i m e s ag n s tr e s p i np y : m . . . . . . . . . . . c a r s t c a r w a ss t o l e n : m i n t . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - bu r g d h o m e w a s bu r g l a r i ze d : m i n t . . . . . . . . . . .
02 a ss l d r a ss a u l t e d ( agg r a v a t e d ) . . . . . . . . . . . r o bbd rr o bb e d : m i n t . . . . . . . . . . . r i ppd v i c t i m o f t h e f t / p i c k p o k e t / e x t o rt i o n . . . . . . . . . . . n a ss l d t i m e s r a ss a u l t e d ( agg r a v a t e d ) : m . . . . . . . . . . . nbu r g d t i m e s h o m e w a s bu r g l a r i ze d : m i n t . . . . . . . . . . . n r o bbd t i m e s rr o bb e d : m i n t . . . . . . . . . . . n r i ppd t i m e s v i c t i m o f t h e f t / p i c k p / e x t o rt : m . . . . . . . . . . . n c a r s t t i m e s c a r w a ss t o l e n : m i n t . . . . . . . . . . . c i g12 s m o k e d c i ga r e tt e s i n m n t hb e f m i n t . . . . . . . . . . . d r i n k d r a n k a l c o h o li n m n t hb e f m n t h i n t . . . . . . . . . . . p o t u s e dp o t i n m o n t hb e f m i n t . . . . . . . . . . . c o k e u s e d c o k e i n m n t hb e f m n t h i n t . . . . . . . . . . . c r a c k u s e d c r a c k i n m n t hb e f m n t h i n t . . . . . . . . . . . h e r o i n u s e dh e r o i n i n m n t hb e f m i n t . . . . . . . . . . . s p ee d u s e d s p ee d i n m n t hb e f m i n t . . . . . . . . . . . l s d u s e d l s d i n m n t hb e f m i n t . . . . . . . . . . . i n j ec t i n j ec t e dd r u g s i n m n t hb e f m i n t . . . . . . . . . . .
00 o t hd r g12 u s e d o t hd r u g s i n m n t hb e f m i n t . . . . . . . . . . . f c i g12 f r e q s m o k e d c i g s i n m n t hb e f m i n t . . . . . . . . . . . f p o t f r e q u s e dp o t i n m n t hb e f m i n t . . . . . . . . . . . f c o k e f r e q u s e d c o k e i n m n t hb e f m i n t . . . . . . . . . . . f c r a c k f r e q u s e d c r a c k i n m n t hb e f m i n t . . . . . . . . . . . f h e r n f r e q u s e dh e r o i n i n m n t hb e f m i n t . . . . . . . . . . . f s p ee d f r e q u s e d s p ee d i n m n t hb e f m i n t . . . . . . . . . . . fl s d f r e q u s e d l s d i n m n t hb e f m i n t . . . . . . . . . . . fi n j c t f r e q i n j ec t e dd r u g s i n m n t hb e f m i n t . . . . . . . . . . . f o t hd g12 f r e q u s e d o t hd r u g s i n m n t hb e f m i n t . . . . . . . . . . . h a r d u s e dh a r dd r u g s i n m n t hb e f m i n t . . . . . . . . . . .
02 a n y d r u s e d a n y d r u g s i n m n t hb e f m i n t . . . . . . . . . . . f d r i n k f r e q d r a n k a l c o h o li n m n t hb e f m i n t . . . . . . . . . . . h e a l t h = e x c h e a l t h = goo d = f a i r = p oo r . . . . . . . . . . . p e p r b = ph y s / e m o t p r o b s a t m t h s = n o p r o b . . . . . . . . . . . w h o u r s y cc h o u r s / w k , a ll t y p e s i n y r . . . . . . . . . . . e p a r e n t cc b y p a r e n t s i n y r . . . . . . . . . . . e g r a nd cc b y g r a ndp a r e n t s i n y r . . . . . . . . . . . e o tr e l cc b y o t h r e li n y r . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e a b l e B . c o n t i nu e d f r o m p r e v i o u s p a g e n a m e d e s c r i p t i o n t y p e x m i ss - e pn r e l p a i d cc b y n o n r e l a t i v e i n y r . . . . . . . . . . . e un r e l unp a i d cc b y n o n r e l a t i v e i n y r . . . . . . . . . . . e d a y c a r cc b y d a y c a r e / p r e s c h oo li n y e a r . . . . . . . . . . . e s c h oo l cc b yk i nd e r g/ e l e m e n t a r y i n y r . . . . . . . . . . . e n r e l n y cc b y n o n r e l a t i v e i n y r . . . . . . . . . . . e v r e l e v e r u s e d r e l a t i v ec h il d c a r e i n y e a r . . . . . . . . . . . C o n t i nu e d o nn e x t p ag e Propensity score plots
The following figures display the overlap of the sequential treatment propensity scores acrosstreatment states in the empirical application by means of kernel density plots. Each figure isdivided into four windows. The upper windows show the first and second period propensityscores ˆ p d ( X ) and ˆ p d ( d , X1