A Simple Matching Method for Estimating Sample Selection Models Using Experimental Data

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1 ANNALS OF ECONOMICS AND FINANCE 6, (2005) A Simple Matcing Metod for Estimating Sample Selection Models Using Experimental Data Songnian Cen Te Hong Kong University of Science and Tecnology and Yaong Zou Te Sangai University of Economics and Finance In tis paper estimation of sample selection models using experimental data is considered wit some weak restriction imposed on te error distribution. Under a normality setting, te most popular approac is te two-step metod proposed by Heckman (1979). But Heckman s approac relies on te nonlinearity of te probit function (i.e. te nonlinearity of te selection correction function ) unless some exclusion restriction is imposed. Furtermore, Heckman s metod is sensitive to te underlying distributional assumption. Following tis two-step metod, several semiparametric estimators ave been proposed for sample selection models by explicitly imposing te exclusion restriction. Using experimental data, tis paper proposes a simple semiparametric matcing metod. Tere are certain advantages of our estimator over Heckman s estimator and te existing semiparametric estimators under eiter te parametric setting and semiparametric setting. We do not rely on te nonlinearity of te selection correction function or te exclusion restriction. In addition, unlike oter semiparametric metods, we can also estimate te intercept term in te equation of interest. Te estimator is sown to be consistent and asymptotically normal under some regularity conditions. A small monte carlo study illustrates te usefulness of te new estimator. c 2005 Peking University Press Key Words: Matcing metod; Experimental data. JEL Classification Numbers: C30, C INTRODUCTION In tis paper estimation of a regression equation (te outcome equation) subject to a sample selection rule and random assignment is considered /2005 Copyrigt c 2005 by Peking University Press All rigts of reproduction in any form reserved.

2 156 SONGNIAN CHEN AND YAHONG ZHOU based on experimental data. Te experimental data in question ere is generated as follows. In te first stage te selection rule identifies te group of nonparticipants for wom te regression equation is observable. In te second stage, some randomization sceme is applied to te remaining individuals, among wom te regression equation is only observable for te randomized-out group. In te context of job training program evaluations (see e.g., Heckman et. al 1998), we are interested in estimating te earnings equation for nontrainees using experimental data (encefort te baseline earnings equation). Tis is an important step in determining various aspects of te program benefits and caracterizing te selection bias. In te first stage, a selection rule classifies te wole sample into nontrainees (or nonparticipants) and prospective trainees. In te second stage only a fraction of te prospective trainees receive training according to certain random assignment, wile te remaining portion of te latter group are randomized out for te training program. Tus we observe te baseline earnings equation for te nonparticipants and te randomized-out control group. Estimation of sample selection models in te context of evaluating various training programs as mainly been based on tecniques developed for nonexperimental data sets. In recognizing deficiencies of te conventional metods and limitations of te nonexperimental data, many researcers ave turned to te available experimental data to recover various aspects of training programs (see, e.g., Lelonde (1986), Fraker and Maynard(1987), Heckman et. al (1998)). Tere ave been numerous social experiments, especially for te purpose of evaluating te impact of federal job training on earnings and employment. In tis paper we propose a new approac to estimating sample selection models by taking advantage of unique features of experimental data. For sample selection models, te least squares metod would produce inconsistent estimates due to te presence of te selection correction term in te outcome equation. Te usual approac is to specify te distribution of te underlying errors parametrically, normality in particular, and independent of te explanatory variables. Ten te parameters of te model can be consistently estimated by maximum likeliood or oter likeliood based metods. Te two-step metod proposed by Heckman (1974,1976) is by far te most popular approac by including a consistent estimate of te selection correction term as part of te regressors in te outcome equation. One important limitation of Heckman s approac is its reliance on te nonlinearity of te probit function (i.e. te nonlinearity of te selection correction function ) unless some exclusion restriction is imposed. More significantly, misspecification of te error distribution in sample selection models will in general render likeliood-based estimators inconsistent. Since a parametric form of error distributing can not generally be

3 A SIMPLE MATCHING METHOD 157 justified by economic teory, following Heckman s two-step approac, several semiparametric estimation metods ave been proposed recently for sample selection models (see, e.g., Andrews, Newey (1988), Powell (1989), Heckman et. al (1998), among oters), wic only assume weak restriction on te error distribution to guard against possible misspecification. Tese semiparametric estimators, owever, require tat te exclusion be satisfied. In addition, te intercept in te outcome equation is absorbed into te selection correction term in tese semiparametric approaces, tus can not be estimated along te slope parameters. In te paper we propose a new semiparametric estimator by taking advantage of unique features of experiment data. Te idea beind our estimator is based on te following observation. In te experimental data wit mild conditions tere exist pairs of individuals wit offsetting selection biases. Simple matcing of suc pairs would eliminate te selection bias in a straigtforward way. Our estimator does not rely on te nonlinearity of te selection correction function or exclusion restriction. Furtermore, unlike oter semiparametric metods, we can also consistently estimate te intercept term in te outcome equation. Tis paper is organized as follows. Te next section describes te model and motivates te proposed estimator. Section 3 gives regularity conditions and investigates te large sample properties of te estimator. Tey are sown to be consistent and asymptotically normal. Section 4 reports a small monte Carlo study. Te final section concludes. 2. THE MODEL AND ESTIMATORS We consider estimation of te sample selection model wit experimental data defined by y = xβ 0 + u (1) d = 1{wδ 0 v > 0} (2) were we wis to estimate β 0 R K2 and δ 0 R K1 based on observations of (d, (1 d)y, d(1 R)y, x, w)), and R is an random variable independent of te oter variables in te model tat can take on values 0 and 1. Here y is te potential outcome equation. d is a discrete coice variable, (x, w) are vectors of exogenous variables wic may ave components in common and R is an randomization indicator. Let z be a vector consist of te distinct components in (x, w). In te first stage, te selection equation (2) determines te subsample of nonparticipants wit d = 0 for wom te equation (1) is observable. For te remaining individuals, te potential outcome equation is observable according as te randomization indicator R is equal to 0 or not. Consequently te potential outcome equation is

4 158 SONGNIAN CHEN AND YAHONG ZHOU observable if d 1 = 1 or d 2 = 1 were d 1 = (1 d) and d 2 = d(1 R). In te context of training programs evaluations wit experiment data (see, e.g., Heckman et. al. 1998), ere we are interested in estimating te baseline earnings equation for nontrainees. d = 1 indicates tat a person applies and is provisionally accepted into te program before te act of randomization. R = 1 if a person for wom d = 1 is randomly assigned into te program, and R = 0 if te person is denied access to te program. Terefore we observe te baseline earnings equation for te individuals wit d 1 = 1 or d 2 = 1. Wen te error terms are normally distributed, te model can be estimated by maximum likeliood. But te usual approac is te computationally simpler two-step metod first proposed by Heckman (1974,1979). Under normality we ave and E(u d 1 = 1, z) = E(u d = 0, z) = σ 12 σ 1 1 λ 1(wδ 0 /σ 1 ) E(u d 2 = 1, z) = E(u d = 1, R = 0, z) = E(u d = 1, z) = σ 12 σ 1 1 λ 2(wδ 0 /σ 1 ) were σ 12 =cov(u, v), σ 1 =var(v), λ 1 (t) = φ(t)/φ(t), and λ 2 (t) = φ(t)/(1 Φ(t)) wit φ(t) and Φ(t) denoting te density and distribution functions for te standard normal random variable. Define d = d 1 + d 2, so we ave E[u d 1 λ 1 (wδ 0 ) d 2 λ 2 (wδ 0 ) d = 1, z] = 0 Heckman s two step estimator is based on te following moment equation for te subsample d = 1, y = xβ 0 σ 12 σ 1 1 (d 1λ 1 (wδ 0 ) + d 2 λ 2 (wδ 0 )) + ɛ 1 (3) suc tat E(ɛ 1 d 1, d 2, z) = 0. Given a first step estimator ˆδ for δ 0, β 0 can be consistently estimated by regressing y on (x, d 1 λ 1 (wˆδ) + d 2 λ 2 (wˆδ)) for te subsample d =1. Note tat wen x = w, tis approac will depend on te nonlinearity of te λ 1 (wδ 0 ) and λ 2 (wδ 0 ). However, as pointed out by Leung and Yu (1996), Nawata (1994), and Vella (1995), among oters, tese functions can be close to be linear in certain ranges, wic migt lead to unreliable estimates for β 0. Anoter potentially more serious drawback to tis and oter likelioodbased metods is teir sensitivity to te assumed parametric distribution of te unobservable error terms in te model. Recently several semiparametric estimators (e.g., Andrews (1991), Newey (1988), Powell (1989), among oters) ave been proposed for sample selection models wic do not impose

5 A SIMPLE MATCHING METHOD 159 parametric forms on error distributions. In te context of experimental data, tese semiparametric estimators are based on te following observation. Under te condition te error term (u, v) is independent of te regressors (or te sligtly weaker assumption of te index restriction), we ave te following partial linear setup for te subsample d = 1 y = xβ 0 + d 1 K 1 (wδ 0 ) + d 2 K 2 (wδ 0 ) + ɛ 2 (4) were K 1 (wδ 0 ) = E(u d 1 = 1, z), K 2 (wδ 0 ) = E(u d 2 = 1, z) are unknown selection correction terms, and E(ɛ 2 d 1, d 2, z) = 0. Te objective of tese approaces is to eliminate te contaminating selection correction terms in (4). Notice, owever, tat under te setup of equation (4), it is necessary for w to ave component not included in x for identification. In addition, an explicit intercept term is not allowed in β 0 since it would be absorbed in te selection correction terms. Instead of taking te equation (4) as te departure point, our estimation approac will rely on te zero mean restriction E(u) = 0. Te idea beind our estimator is based on te following observation. If β 0 were known, ten we can observe u given v > wδ 0 for individuals wit d 1 = 1 and u given v < wδ 0 for individuals wit d 2 = 1. Under random sampling te combination of te error terms u i 1{v i > t} + u j 1{v j < t} will ave te same moments as u i, for any constant t. If tere exist pairs of observations i and j wit w i δ 0 = w j δ 0, ten te zero mean restriction and a matcing of tese two observations lead to te following zero mean condition i.e. E{(1 R j )[d 1i u i + d 2j u j ] w i δ 0 = w j δ 0 } = E(1 R j )E[d 1i u i + d 2j u j ] w i δ 0 = w j δ 0 ] = 0 E{(1 R j )[(1 d i )(y i x i β 0 ) + d j (y j x j β 0 )] w i δ 0 = w j δ 0 } = 0 (5) Terefore estimation of β 0 can be based on te moment equation (5). An instrumental variables approac is adopted ere. Tis approac could be directly implemented if δ 0 were known and tere exist pairs of observations in a random sample wit exactly identical indices wit positive probability. Neverteless, given a consistent estimator ˆδ for δ 0, if te nuisance function E[d 1 u wδ 0 =t] is sufficiently smoot, te preceding matcing arguments would old approximately for pairs of observations wit approximately similar pairs of w iˆδ and wj ˆδ, i.e.wiˆδ wj ˆδ. As several metods exist in te econometric literature for semiparametric estimation of te binary coice model, (see, for example, Cosslett (1983),

6 160 SONGNIAN CHEN AND YAHONG ZHOU Han (1987), Icimura (1987), Klein and Spady (1993), and Powell, Stock and Stoker (1989) ), in tis article we assume a consistent estimator of δ 0 exists, and terefore, we will concentrate on te estimation of β 0. Consequently, te estimator β of β 0 is defined as a weigted instrumental variables estimator were Ŝ xx = Ŝ xy = 2 n(n 1) 2 n(n 1) i<j i<j β = Ŝ 1 xx Ŝxy (6) (1 R j )(x i + x j ) ((1 d i )x i + d j x j ) 1 K(w iˆδ w j ˆδ (1 R j )(x i + x j ) ((1 d i )y i + d j y j ) 1 K(w iˆδ w j ˆδ were te kernel weigt 1 K( wi ˆδ w j ˆδ ) gives declining weigt to pairs wit large values of w iˆδ wj ˆδ. ) ) 3. LARGE SAMPLE PROPERTIES OF THE ESTIMATOR In tis subsection we derive te asymptotic properties of te proposed estimator. We begin by making te following assumptions. Assumption 1. Te vectors (y i, x i, d i, w i ) generated from (1) and (2 ) are independent and identically distributed across i, wit finite sixt order moments for eac component. Te randomization indicator R is independent of te oter variables. Te error term (u, v) is independent of z, wit E(u) = 0. Assumption 2. Te preliminary estimator δ of δ 0 is n-consistent, and as te following asymptotic linear representation δ = δ ψi + o p (n 1/2) n for some ψ i = ψ(d i, w i ), suc tat Eψ(d i, w i ) = 0 and E ψ(d i, w i ) 2 <. Assumption 3. Define g x (u) = E(x i w i β 0 = u), wit g w (u) similarly defined, ten eac component of g x (u) and g w (u) are four times continuously differentiable.

7 A SIMPLE MATCHING METHOD 161 Define S xx = E(x i + E(x i w i δ 0 ) (x i + E(x i w i δ 0 )p(w i δ 0 )E(1 R i ) Assumption 4. Te matrix S xx is nonsingular. Assumption 5. Te kernel function K ( ) as a bounded support. It is four times continuously differentiable and satisfies K (u) du = 1 and u l K (u) du = 0 for l = 1, 2, 3. Assumption 6. Te bandwidt sequence satisfies n 6 / ln(n), and n 8 0 as n. Assumption 1 describes te model and te data. Te independence assumption between (u, v) and x can be relaxed to allow x to include endogenous variables, and te distribution of (u, v) can be allowed to depend on w troug te linear index wδ 0. Several estimators for δ 0 mentioned in te previous section (Han (1987), Icimura (1993), Klein and Spady(1993), and Powell, Stock and Stoker(1989)) satisfy assumption 2. Assumption 4 is an identification condition. Notice tat S xx = E(x ix i + 3E(x i w i δ 0 )E(x i w i δ 0 )p(w i δ 0 )E(1 R i ). Assume tat te support of w i δ 0 is te wole line, ten te nonsingularity of S xx is implied by te nonsingularity of Ex i x i. Assumption 5 is a iger order bias reduction kernel condition, wic, togeter wit te rate of convergence condition on te bandwidt sequence in Assumption 6, ensures tat te estimator proposed is asymptotically unbiased. Assumption 3 is a boundedness and smootness conditions, wic can be justified by some primitive conditions on te distributions of te variables in te model (see Lee (1994) and Serman (1994) for some discussions on similar conditions). Rewriting (6) as n( β β0 ) = Ŝ 1 xx Ŝxu were 2 Ŝ xu = (1 R j )(x i + x j )((1 d i )u i + d j u j ) 1 n(n 1) K(w iˆδ w j ˆδ ) i<j

8 162 SONGNIAN CHEN AND YAHONG ZHOU we will establis te asymptotic results for β in two steps; first, we sow tat Ŝxx converges in probability to a nonsingular matrix; second, we establis an asymptotic linear representation for Ŝxu. Lemma 1. Under Assumptions 1 troug 6 above, as n, Ŝxx S xx were S xx is defined as in assumption 3. p Proof. mimic te proof of Lemma 5.1 of Powell (1989). By assumption 3, Lemma 1 implies tat te matrix inverse in te definition of β is well defined in large samples. Next we consider Ŝxu. Te following lemma establises an asymptotic linear representation for Ŝxu, Similar to te proof of Teorem 5.1 of Powell (1989) we can establis te follow lemma. Lemma 2. Under conditions 1 troug 6, we ave n Ŝ xu = 1 n n i=1 [ζ i Ωψ i ] + o p (1) were ζ i = [x i + E(x i w i δ 0 )]u i wit u i = (E(1 R i))((1 d i )u i λ(w i δ 0 ))+(1 R i )(d i u i +λ(w i δ 0 ))p(w i δ 0 ), and Ω = 2E(1 R i )E[λ (w i δ 0 )E(x w i δ 0 ) (w i E(w w i δ 0 )p(w i δ 0 )]. Combining lemmas 1 and 2, we obtain te main teorem by te central limit teorem. Teorem 1. Under conditions 1-6, te estimator β is consistent for β 0, and asymptotically normal, n( β β0 ) d N(0, Σ) were Σ = Sqq 1 [C ζζ + ΩC ψζ + C ζψ Ω + ΩC ψψ Ω ]Sqq 1 for C ζζ = E[ζ i ζ i ], C ψψ = E[ψ i ψ i ] and C ψζ = E[ψ i ζ i ] = C ζψ.

9 A SIMPLE MATCHING METHOD 163 In order for large-sample inference on β 0 to be carried out using te estimator β, a consistent estimator of Σ needs to be constructed. In te following one suc estimator is proposed following Powell (1989). Lemma 1 sows Ŝxx is a consistent estimator for S xx. An analogous estimator for Ω is Ω = 2 n(n 1) i<j (1 R j )((1 d i )û i +d j û j ) 1 2 K ( w iˆδ w j ˆδ )(x i +x j ) (w i w j ) were û i = y i x i ˆβ. To estimate C ψψ and C ψζ, it is useful to assume tat a sequence of { ψ i } exists wic satisfies 1 n n ψ i ψ i 2 = o p (1). i=1 Tis sequence, of course, depends upon te particular first-step estimator of δ 0 ; an example of an appropriate sequence { ψ i } for a particular preliminary estimator δ is given by Powell, Stock and Stoker (1989). Similar sequences can be constructed for te estimators proposed by Icimura (1993) and Klein and Spady (1993). As for te sequence {ζ i }, let ζ i = 1 (n 1) j i (1 R j )(x i + x j ) ((1 d i )û i + d j û j ) 1 K(w i δ w j δ Lemma 3. : Under Assumptions 1-7 above, as n, 1 n n ζ i ζ i 2 = o p (1). i=1 ). Proof. mimic Lemma 6.2 of Powell (1989). 4. A MONTE CARLO STUDY In te section we present a small Monte Carlo study to illustrate te usefulness of te proposed estimator. Te data is generated according to te following model y = x 1 + x 2 + u

10 164 SONGNIAN CHEN AND YAHONG ZHOU d = 1{w 1 + w 2 + v > 0} and R = 1{r > 0}, were v as te standard normal distribution. Te regressors x 1 and x 2 are draw from a normal N(0, 1) distribution and a uniform U( 2, 2) distribution, respectively. Different designs are constructed by varying te distributions of te error terms and te structure of (w 1, w 2 ). In all cases, r is draw from U( 0.5, 0.5) independent of te rest of te variables in te model. u= 0.5 v v wit v draw from N(1, 0) independent of te oter variables in te model. We consider two different designs for te regressors wit (w 1, w 2 )=(x 1, x 2 ) and w 1 = x 1 and w 2 = x 2 /2+x 3 /2 respectively, wit x 3 is drawn from a uniform U( 2, 2) distribution independent of (x 1, x 2 ). Data on v are also generated from tree different distributions, namely, normality, nonnormality and eteroscedasticity. Consequently we ave six designs from tese variations of te regressors and error terms. Here we consider te finite sample performance of our estimator, along wit te Heckman s two step estimator; te estimators proposed by Newey (1988) and Powell (1989) are also considered wen te exclusion restriction applies. Te first-step estimator is cosen to be te probit maximum likeliood estimator. Te results from 300 replications from eac design are presented wit sample size of 100. For eac estimator under consideration, we report te mean value (Mean), te standard Deviation (SD), and te root mean square error (RMSE). For Powell s estimator, we use te standard normal density as te kernel function. For te Newey s estimator, te approximating series is. Te bandwidt and te number of series are cosen by generalized cross-validation (G. Waba 1979). We also te standard normal density as te kernel function for our estimator wic te bandwidt is cosen by minimizing MM were MM = i<j (1 R j )(x i +x j )((1 d i )(y i x i β)+d j (y j x j β)) 1 K(w iˆδ w j ˆδ ). since our estimated is based on a related moment condition. Table 1 reports te simulation results for our estimator and Heckman s two-step estimator for te first design were v is a standard normal N(0, 1). In te case were (w 1, w 2 )=(x 1, x 2 ), even toug normality is correctly specified, our estimator is superior due to weak nonlinearity of te probit function in te relevant range. Wen tere is an exclusion restriction, i.e., w 1 = x 1 and w 2 = x 2 /2 + x 3 /2, Heckman s estimator is sligtly better. In Table 2 we consider te same design te error term departing from joint normality; v=2v 3 + v 2 1 were v is drawn from a standard normal N(0, 1) independent of te oter variables. In tis case, joint normality is misspecified. Heckman s metod produces inconsistent estimates for bot designs of te regressors. Our approac still provides reasonably good

11 A SIMPLE MATCHING METHOD 165 TABLE 1. Results of simulation wit normal errors x w x = w Estimator Coeff. Mean SD RMSE Mean SD RMSE Heckman α β β Matcing α β β TABLE 2. Results of simulation wit nonnormal errors x w x = w Estimator Coeff. Mean SD RMSE Mean SD RMSE Heckman α β β Matcing α β β estimates. Existence of an exclusion restriction improves te performance of bot estimators. Te results wit a eteroscedastic error are reported in Table 3. Te error term v=exp(wδ 0 )v, were v is a standard normal N(0, 1) independent of te oter variables. It is obvious Heckman s estimate is biased, wile te oter tree estimators are still consistent. As expected, all te estimators perform better wen tere is an exclusion restriction. 5. CONCLUSION In tis paper we consider semiparametric estimation of sample selection models using experimental data. We propose a new estimator by matcing pairs of observations wit offsetting selection bias. wic does not rely on te nonlinearity of te selection correction function or some exclusion restrictions. We improve upon Heckman s two-step under parametric setting in tat our estimator does not rely on te nonlinearity of te selection correction function wen tere is no exclusion restriction. We also improve upon te existing semiparametric estimators in tat exclusion restriction is not needed for our procedure. Te estimator is sown to be consistent and

12 166 SONGNIAN CHEN AND YAHONG ZHOU asymptotically normal under some regularity conditions. A small monte carlo study illustrates te usefulness of te new estimator. Our estimator is based on access to random samples. Frequently, available data are in te form of coice based samples. It migt be possible to modify te current approac to still consistently estimate te outcome equation using coice based samples. It is a topic for future researc. REFERENCES Andrews, D.W.K. 1991, Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrica 59, Cosslett, S.R., 1983, Distribution-free maximum likeliood estimator of te binary coice model. Econometrica 51, Cosslett,S.R.,1991, Seimparametric estimation of a regression model wit sample selectivity. In: W.A. Barnett, J.L. Powell, and G.Taucen, eds, Nonparametric and Semiparametric Metods in Econometrics and Statistics. (Cambridge University Press, Cambridge). Donald, S.G. 1995, Two-step estimation of eteroskedastic sample selection models. Journal of Econometrics 65, Fraker, T. and R. Maynard, 1984, An Assessment of Alternative Comparison Group Metodologies for Evaluating Employment and Training Programs. Princeton, NJ: MPR, Inc. Han, A.K., 1987, Nonparametric analysis of a generalized regression model: Te maximum rank correlation estimator. Journal of Econometrics 35, Heckman, J.J., 1974, Sadow prices, market wages, and labor supply. Econometrica 42, Heckman, J.J., 1976, Te common structure of statistical models of truncation, sample selection, and limited dependent variables and a simple estimator for suc models. Annals of Economic and Social Measurement 5, Heckman, J.J. 1990, Varieties of selection bias. American Economic Review 80, 2, Heckman, J.J., H., Icimura, J., Smit, and P. Todd, 1998, Caracterizing selection bias using experimental data. Econometrica 66, Horowitz, J.L., 1992, A smoot maximum score estimator for te binary response model. Econometrica 60, Icimura, H., 1993, Semiparametric least squares (SLS) and weigted SLS estimation of single-index models 58, Klein, R.W. and R.S. Spady, 1993, An efficient semiparametric estimator of te binary response model. Econometrica 61, Lee, L.F. 1994, Semiparametric instrumental variable estimation of simultaneous equation sample selection models. Journal of Econometrics 63, Leung, S.F. and S.Yu, 1996, On te coice between sample selection and two-part models. Journal of Econometrics 72, LaLonde, R., 1986, Evaluating te econometric evaluations of training programs wit experimental data. American Economic Review 76,

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