Estimating Average and Marginal Treatment Effects in Heterogeneous Populations

This paper proposes a nonparametric method of estimating average and marginal treatment effects in heterogeneous populations. Building upon an insight of Heckman and Vytlacil, the conventional treatment effects model with heterogeneous effects is shown to imply that outcomes are a nonlinear function of participation probabilities. The degree of this nonlinearity, and hence the shape of the marginal response curve, can be estimated with series estimation methods (e.g., power series or splines). An illustration is provided for the returns to higher education in the U.K, indicating that the average and marginal returns to higher education fall as the proportion of the population with higher education rises, thus providing evidence of heterogeneity in returns. The possible existence of individual heterogeneity in the effect of a treatment on outcomes in a population has been a focus of much work in the causal effects literature. In economics, heterogeneity in the effect of a binary endogenous regressor was introduced in the literature on switching regression models by Quandt (1972), Heckman (1978), and Lee (1979), while in the statistics literature the causal model of potential outcomes of Rubin (1974) also allowed full heterogeneity in treatment effects. This heterogeneity was reformulated as a random coefficient by Heckman and Robb (1985) and by Bjorklund and Moffitt (1987), who also introduced the concept of the marginal treatment effect (termed the ‘marginal gain’ by Bjorklund and Moffitt) in the context of a multivariate-normal switching regression model and showed that it was observationally equivalent to the Lee switching regression model. Imbens and Angrist (1994) showed that the treatment effect in a heterogeneous population across two points in the distribution, termed the Local Average Treatment Effect (LATE), could be nonparametrically estimated with instrumental variables (IV) and Angrist et al. (1996) elaborated and clarified this method. Heckman and Vytlacil (1999, 2005) have clarified the distinctions between the marginal treatment effect (MTE), the LATE, and other treatment effects of interest. In this paper, we build upon a remark by Heckman and Vytlacil (2005, p.691) that the treatment effects model with heterogeneous effects of a binary treatment implies that outcomes are simply a nonlinear function of participation probabilities. A model is set up in this paper which demonstrates that point in a slightly reformulated random coefficients model which makes minimal identifying assumptions for the identification of the nonlinearity. A simple series 2 estimation method is proposed to nonparametrically estimate the shape of the outcomeparticipation-probability relationship, and hence marginal returns to treatment, which can be implemented with widely-available software packages. An empirical illustration is provided for the effect of a binary higher education indicator on earnings in the UK using the data from a study by Blundell et al. (2005). The literature on the effect of education on earnings has seen the largest number of discussions of heterogeneity in the return, a concept discussed in the Woytinsky Lecture of Becker (1975) and in Mincer (1974). Surveys of the empirical literature by Card (1999, 2001) have emphasized the impact of possible heterogeneity in the return on the interpretation of the estimates in that literature (see also Lang (1993)). The large majority of these estimates use only a binary instrument and hence only one piece of the marginal return function can be nonparametrically identified, whereas in this paper a wider portion of the return function is estimated because multiple, multi-valued instruments are used. Carneiro et al. (2003a, 2003b) have also used a wider range of instruments, together with other identifying assumptions, and have been able to estimate the full range of returns to education. Oreopoulos (2006) has examined heterogeneity in returns to education by comparing LATE estimates based on compulsory schooling laws between two countries which have different fractions of the population affected by the laws, which implicitly uses a three-valued instrument rather than a binary one. The next section lays out the model and estimation method, and the subsequent section provides the illustration. A summary appears at the end.