The relationship between treatment parameters within a latent variable framework

Transcription

1 Economics Letters 66 (2000) locate/ econbase The relationship between treatment parameters within a latent variable framework James J. Heckman *,1, Edward J. Vytlacil 2 Department of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637, USA Received 18 January 1999; accepted 8 July 1999 Abstract If responses to a treatment vary among people, a variety of parameters can be defined [Heckman, J., Robb, R Alternative methods for evaluating the impact of interventions. In: Heckman J., Singer B. (Eds.), Longitudinal Analysis of Labor Market Data. Cambridge University Press, New York, pp ; Heckman, J., 1997, first draft 1995, Instrumental variables: a study of implicit behavioral assumptions used in making program evaluations. Journal of Human Resources 32, ]. We show a simple relationship between various treatment parameters when the treatment parameters are defined within a common, latent variable framework Elsevier Science S.A. All rights reserved. Keywords: Local instrumental variables; Latent variable models; Social program evaluation JEL classification: C50; H43 1. Introduction If responses to a treatment vary among people, a variety of parameters can be defined (Heckman and Robb, 1985, Heckman, 1997). We show a simple relationship between various treatment parameters when the treatment parameters are defined within the latent variable framework used in 3 sample selection models. *Corresponding author. Tel.: ; fax: address: jjh@uchicago.edu (J.J. Heckman) 1 James J. Heckman is a Henry Schultz Distinguished Service Professor of Economics at the University of Chicago and a Senior Fellow at the American Bar Foundation. 2 Edward Vytlacil is a Sloan Fellow at the University of Chicago. 3 Heckman (1976) introduced sample selection models based on a latent variable. Semiparametric and nonparametric developments of the model include Powell (1987), Ahn and Powell (1993), and Das and Newey (1998) / 00/ $ see front matter 2000 Elsevier Science S.A. All rights reserved. PII: S (99)

2 34 J.J. Heckman, E.J. Vytlacil / Economics Letters 66 (2000) Latent variable framework Let D denote the receipt of treatment and assume that D is binary valued. Let Y be the outcome variable where Y 5 DY11 (1 2 D)Y 0. D could be an indicator variable for receipt of job training, and Y could be a labor market outcome, such as employment, length of time until employed, or level of earnings. Values for Y and Y are defined for everyone and independence of the variables (Y, Y, D) across persons is assumed. Let D denote the person-specific treatment effect: D 5 Y 2 Y. 1 0 We specify a latent-variable, discrete-choice framework and define different treatment parameters within this framework. The following decision rule for program participation is cast in terms of latent variable D*: D* 5 m(z) 2 U D 5 1ifD* $ 0, 5 0 otherwise, where Z are observed random variables and U is an unobserved random variable. We make the following assumptions: (1) (i) m(z) is a nondegenerate random variable. (ii) U is absolutely continuous with respect to Lebesgue measure on R. (iii) Z is independent of (U, Y 1, Y 0). (iv) Y1 and Y0 have finite first moments. Assumptions (i) and (iii) require an exclusion restriction that there exists a variable that determines the treatment decision but is known not to directly affect the outcome. For any random variable A, let FA be the distribution of A. Let denote Pr(D 5 1uZ 5 z). We do not explicitly consider observed covariates X determining the potential outcomes, (Y 0, Y 1), in order to reduce notation. Everything in this paper is conditional on X. 4 Without loss of generality, we assume that U Unif[0,1], in which case m(z) 5. To see that there is no loss of generality, note that if the underlying index is D* 5 n(z) 2V, with assumptions (i) and (iii) satisfied for V, then taking m(z) 5 F V(n(Z)) and U 5 F V(V(Z)) equates the two models. This transformation is innocuous, since any CDF is left-continuous and non-decreasing and thus m(z) $ U n(z) $V. In addition, since U is distributed Unif[0,1] and independent of Z, we have m(z) 5. Note that the latent variable assumption is not testable. If we take Z to include all observed covariates, and define m(z) 5 Pr(D 5 1uZ 5 z) and U Unif[0, 1], the model imposes no restrictions on the data. From the analysis of Vytlacil (1999), we have that the latent variable assumption is equivalent to the independence and monotonicity assumptions of Imbens and Angrist (1994), including in the case of a continuous instrument. 3. Definition of parameters We examine four different mean parameters within this framework: the average treatment effect 4 This representation is used in Das and Newey (1998).

3 J.J. Heckman, E.J. Vytlacil / Economics Letters 66 (2000) (), the effect of treatment on the treated (), the local average treatment effect (), and the 5 local instrumental variable () parameter. The average treatment effect is given by: D ; E(D) (2) The expected effect of treatment on the treated is the most commonly estimated parameter (see 6 Heckman and Robb, 1985), and we define it as: D (D 5 1) ; E(DuD 5 1) (3) 7 It will be useful to define so that D (, D 5 1) ; E(DuP(Z) 5, D 5 1) (4) D (D 5 1) 5E D (, D 5 1) df. (5) P(Z )ud51 We define a version of the parameter of Imbens and Angrist (1994) where we use P(Z) as the 8 instrument. Assume. P(z9). Then, E(YuP(Z) 5 ) 2 E(YuP(Z) 5 P(z9)) D (, P(z9) ;]]]]]]]]]]]]. (6) The final parameter that we will consider is the local IV parameter introduced in Heckman (1997), and defined as: E(YuP(Z) 5 ) D () ;]]]]]]. (7) 9 Local IV is the limit form of the parameter, 10 lim D (, P(z9)) 5 D (). P(z9) We have defined each of the parameters in terms of the index P(Z) instead of Z. For each of these parameters, defining the parameters in terms of Z or P(Z) results in equivalent expressions. Z enters the model only through the m(z) index, so that for any measurable set A, 5 From assumption (iv), we have that E(D) exists and is finite. 6 From assumption (iv), we have that D (D 5 1) exists and is finite. 7 From assumption (iv), we have that D (, D 5 1) exists and is finite a.e. F P(Z )ud51. 8 From assumption (iv), we have that D (, P(z9)) is well defined and is finite a.e. FP(Z ) 3 F P(Z ). 9 As we show in the next section, D () exists and is finite a.e. FP(Z ) under our previous assumptions. 10 The limit form of the parameter was introduced independent work by Angrist et al. (1995, unpublished); and in parallel work by Heckman (1997, published; first draft, 1995) and Heckman and Smith (1998, published; first draft, 1995). These papers do not develop the relationships among the parameters that are the focus of this paper.

4 36 J.J. Heckman, E.J. Vytlacil / Economics Letters 66 (2000) Pr(Y j[ AuZ 5 z, D 5 1) 5 Pr(Y j[ AuU # m(z)) Pr(Y j[ AuZ 5 z, D 5 0) 5 Pr(Y j[ AuU. m(z)) (8) and m(z) 5 from U Unif[0,1]. 4. Relationship between parameters The interpretation of the and local IV parameters is not immediately clear, nor is the relationship between these parameters and the more conventional and parameters. However, a simple relationship can be shown among the four parameters, and the relationship helps to interpret the parameters. Each parameter can be shown to be the average value of D conditional on U lying within different intervals. We have immediately, D (, D 5 1) 5 E(DuU # ). (9) Now consider D (, P(z9)). Note that so that E(YuP(Z) 5 ) 5 E(Y1 up(z) 5, D 5 1) 1 (1 2 )E(Y0 up(z) 5, D 5 0) 1 (10) 5 E E(Y uu 5 u)du 1E E(Y uu 5 u)du, E(YuP(Z) 5 ) 2 E(YuP(Z9) 5 P(z9)) 5 E E(Y uu 5 u)du 2 E E(Y uu 5 u)du 1 0 P(z9) P(z9) (11) D (, P(z9)) 5 E(DuP(z9) # U # ). (12) We now examine D. Consider Eq. (10). E(Y1uU ) and E(Y0uU ) are integrable against df U. We thus have the standard result that E(Y1uP(Z) 5 ) and E(Y0uP(Z) 5 ) are differentiable a.e. with respect to, and thus that E(YuP(Z) 5 ) is differentiable a.e. with respect to with derivative 11 given by E(YuP(Z) 5 ) ]]]]]] 5 E(DuU 5 ) (13) 11 From assumption (iv), we have that (13) is finite a.e. F. U

5 J.J. Heckman, E.J. Vytlacil / Economics Letters 66 (2000) is the average treatment effect for an individual chosen at random. is the average treatment effect for someone who selected into participation. is the average treatment effect for someone who would not participate if P(Z) # P(z9) and would participate if P(Z) $. Local IV is the average effect for persons who are just indifferent between participation or not for a given value of the instrument (are indifferent at P(Z) 5 ). We can now rewrite these relationships as D 1 5 E E(DuU 5 u)du 0 D (, D 5 1) 5 E E(DuU 5 u)du 0 ()D (, P(z9)) 5 E E(DuU 5 u)du P(z9) D () 5 E(DuU 5 ). (14) The relationship between the parameters is thus very simple. First, to see the relationship between and, note that lim D (, D 5 1) 5 D. 1 Thus, and coincide when the effect of treatment on the treated is evaluated at a selection index such that the agent selects into the program with probability 1. For such an index value, there is no selection, all individuals with that value of the selection index will select into the program, and the two parameters coincide. This type of argument is the basis for the identification at the limit method for showing identification of selection models used in Heckman (1990) and Heckman and Honore 12 (1990). We can also relate to and. This relationship is more general because can exist even when does not. To see the relationship between and, note that P(z9) D (, P(z9)) 5]]]] D (, D 5 1) 2]]]] D (P(z9), D 5 1). Thus, can be seen as a weighted difference between two parameters. For appropriate choices of (, P(z9)), is equivalent to and. We have 12 Chamberlain (1986) also uses an identification at the limit method for identification. However, he analyzes a fundamentally different identification problem he exploits linearity in the outcome equation to show that the slope parameters of the outcome equation are identified by an identification at infinity argument even in the absence of any exclusion restrictions. He does not examine identification of the intercept term, and identification of the intercept term is crucial for identification of average treatment effects.

6 38 J.J. Heckman, E.J. Vytlacil / Economics Letters 66 (2000) and lim D (, P(z9)) 5 D (, D 5 1), P(z9) 0 lim D (, P(z9)) 5 D. 1,P(z9) 0 While the above expressions are written in terms of D (, P(z9)), they can equivalently be written as a weighted sum of parameters. Note that for any z1 such that P(z9), P(z 1),, we have P(z 1) 2 P(z9) 2 P(z 1) D (, P(z9)) 5]]]] D (P(z 1), P(z9)) 1]]]] D (, P(z 1)) and thus the above expressions for and can be rewritten in terms of a weighted sum of parameters. Following the same logic, the left hand side of the above expressions can also be written as a weighted sum of parameters for any sequence of z values, (z,...,z ), s.t. P(z9), P(z ), 1 k 1???,P(z k),. Recall from (13) that D ( p) is the average effect for persons who are just indifferent between participation or not at the given value of the instrument (are indifferent at P(Z) 5 p). D ( p) for values of p close to zero is the average effect for individuals with unobservable characteristics that make them the most inclined to participate, and D ( p) for values of p close to one is the average effect for individuals with unobservable characteristics that make them the least inclined to participate We can view D, D and D as integrated versions of D ( p). In particular: (i) D integrates D ( p) over the entire support of U (from p50 top51). (ii) D (,D 5 1) integrates D ( p) uptop5. As a result, it is primarily determined by the average effect for individuals whose unobservable characteristics make them the most inclined to participate in the program. (iii) D (,P(z9)) integrates D ( p) from p 5 P(z9) top 5. If individuals participate in the program based in part on their idiosyncratic gain from the program, it would be reasonable to assume that D ( p) is monotonically decreasing in p (i.e., that the individuals who would receive the largest treatment effect are the ones most inclined to participate in the program). In that case, D (, D 5 1) $ D (, P(z9)) and D (, D 5 1) $ D. Heckman and Vytlacil (1999) show that the relationship between the parameters established in this paper can be used to show the conditions under which and are identified and to bound these parameters when they are not identified. Aakvik et al. (1999) empirically estimate and bound these parameters for the Norwegian vocational rehabilitation program and find wide differences between the parameter values for that social program. References Aakvik, A., Heckman, J., Vytlacil, E., Training effects on employment when the training effects are heterogeneous: an application to Norwegian vocational rehabilitation programs, working paper, University of Chicago.

7 J.J. Heckman, E.J. Vytlacil / Economics Letters 66 (2000) Ahn, H., Powell, J., Semiparametric estimation of censored selection models with a nonparametric selection mechanism. Journal of Econometrics 58, Angrist, J., Graddy, K., Imbens, G., Non-parametric demand analysis with an application to the demand for fish, working paper, Massachusetts Institute of Technology. Chamberlain, G., Asymptotic efficiency in semi-parametric models with censoring. Journal of Econometrics 32, Das, M., Newey, W., Non-parametric estimation of the sample selection model, working paper, Massachusetts Institute of Technology. Heckman, J., Varieties of selection bias. American Economic Review 80, Heckman, J., Instrumental variables: a study of implicit behavioral assumptions used in making program evaluations. Journal of Human Resources 32, , First draft Heckman, J., Honore, B., The empirical content of the Roy Model. Econometrica 58, Heckman, J., Robb, R., Alternative methods for evaluating the impact of interventions. In: Heckman, J., Singer, B. (Eds.), Longitudinal Analysis of Labor Market Data, Cambridge University Press, New York, pp Heckman, J., Smith, J., In: Strom, S. (Ed.), Evaluating the welfare state, presented in the Ragner Frisch Centenery, Oslo, March 1995, Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial, Econometric Society Monograph Series, Cambridge University Press, New York, First draft Heckman, J., Vytlacil, E., Local instrumental variables and latent variable models for estimating and bounding treatment effects. Proceedings of the National Academy of Sciences of the USA 96, Imbens, G., Angrist, J., Identification and estimation of local average treatment effects. Econometrica 62, Powell, J., Semiparametric estimation of bivariate latent variable models, unpublished manuscript, University of Wisconsin. Vytlacil, E., Independence, monotonicity, and latent index models: an equivalence result, working paper, University of Chicago.