Treatment Effects: A Bayesian Perspective

Abstract

This paper contributes to the emerging Bayesian literature on treatment effects. It derives treatment parameters in the framework of a potential outcomes model with a treatment choice equation, where the correlation between the unobservable components of the model is driven by a low-dimensional vector of latent factors. The analyst is assumed to have access to a set of measurements generated by the latent factors. This approach has attractive features from both theoretical and practical points of view. Not only does it address the fundamental identification problem arising from the inability to observe the same person in both the treated and untreated states, but it also turns out to be straightforward to implement. Formulae are provided to compute mean treatment effects as well as their distributional versions. A Monte Carlo simulation study is carried out to illustrate how the methodology can easily be applied.

Keywords:

• Bayesian
• Counterfactual distributions
• Potential outcomes
• Treatment effects

JEL Classification:

• C11
• C15
• C31

ACKNOWLEDGMENTS

This research was supported in part by the American Bar Foundation, the JB & MK Pritzker Family Foundation, Susan Thompson Buffett Foundation, NICHD R37HD065072, R01HD54702. We acknowledge the support of a European Research Council grant hosted by University College Dublin, DEVHEALTH 269874, a grant to the Becker Friedman Institute for Research and Economics from the Institute for New Economic Thinking (INET), and an anonymous funder. We thank the guest editor, Ehsan Soofi, and two anonymous referees for helpful comments. The views expressed in this paper are those of the authors and not necessarily those of the funders or commentators mentioned here.

Notes

See, e.g., Carneiro et al. (2003), Hansen et al. (2004), and Heckman et al. (2006, 2011)

Applying the proposed methodology to other types of treatment parameters would be straightforward to achieve. Heckman and Vytlacil (1999, 2000) and Carneiro et al. (2010, 2011) show how a variety of treatment parameters can be derived from the MTE.

See Heckman (1990) for one discussion of the more general Roy model and its identification. Heckman and Vytlacil (2007b) provide a general discussion. This model has been used in econometrics since Heckman (1974).

Heckman (1990) and Heckman and Vytlacil (2007b) present conditions for nonparametric identification of the model.

This approach at best would also only be able to estimate marginal distributions of Y ₀ and Y ₁.

Papers dealing with Bayesian inference of treatment effects include Poirier and Tobias (2003), Li et al. (2004), Tobias (2006), and Li and Tobias (2008), but none of them assume a latent structure with factors.

In the examples in Heckman et al. (1997), the nonparametric bounds are wide.

For a general introduction to MCMC methods see, for example, Gamerman and Lopes (2006).

In some applications, this restriction is relaxed, and other centerings are used. See Heckman et al. (2013).

E.g., if the factors are assumed to follow a mixture of normals then ψ_θ is the set of mixture means, variances and weights.

E.g., treatment effects on the untreated (see Heckman et al., 1998), local average treatment effects (Imbens and Angrist, 1994; Heckman and Vytlacil, 1999), and policy-relevant treatment effects (Heckman and Vytlacil, 2001).

The conditioning on the covariates Z and X, as well as and on the model parameters Γ, is kept implicit and therefore omitted.

In our simulation study, we sample 1, 000 draws of η from a normal distribution with mean and variance truncated to the interval [0, + ∞). g(η) is therefore the probability density function of the corresponding truncated normal distribution.

The same J draws are used across the M iterations to damp numerical instabilities generated by the integration of the latent factor (Lee, 1992). Hence the notation θ^(j) where the superscript m has been dropped.

Notes: 1,000 Monte Carlo replications. SE = MCMC standard error, RMSE=MCMC root mean squared error. MC2 = factors integrated out numerically, along with model parameters; MC3 = numerical integration of the factors embedded in the integration of model parameters.

Notes: 1,000 Monte Carlo replications. SE = MCMC standard error, RMSE=MCMC root mean squared error. MC2 = factors integrated out numerically, along with model parameters; MC3 = numerical integration of the factors embedded in the integration of model parameters.

The results of MC2 and MC3 with J = 10, 000 are virtually identical, to the 4-digit level of precision used to report the results in the table.

As noted in Abbring and Heckman (2007, the “auxiliary measurements” can include vectors of the realized potential outcomes and need not be qualitatively different from the outcomes being studied. In principle, one can also dispense with the choice equation, although the resulting model is difficult to interpret.