Inverse Probability Tilting Estimation of Average Treatment E ¤ ects in Stata

Graham, Pinto and Egel (forthcoming) introduce a new inverse probability weighting type estimator for general moment condition models with data missing at random. Estimation of the Average Treatment E¤ect (ATE) under exogenous treatment assignment is included in their framework. In this paper we describe a Stata implementation of their method for ATE estimation: the iptATE command. Keywords: iptATE, Inverse Probability Tilting, Program Evaluation, Average Treatment E¤ect When comparing subpopulations of treatedand controlunits it is generally desirable to adjust for di¤erences in observed baseline characteristics. Under strong conditions Rubin (1977) showed that such comparisons have a causal interpretation, however they may be of interest even in settings where this interpretation is untenable (cf., Imbens, 2004). In recent years a large literature has emerged proposing various methods of covariate adjustment. Imbens and Wooldridge (2011) provide a recent and comprehensive review. When the sample size is small, and/or the number of observed baseline characteristics large, it is common for di¤erent methods of covariate adjustment to result in di¤erent average treatment e¤ect (ATE) estimates. Furthermore, researcher chosen tuning parameters, such as the number of matches to use per unit in matching procedures, may nontrivially a¤ect the location and estimated precision of point estimates (Abadie et al., 2004). Angrist and Pischke (2009), in reaction to the perceived sensitivity of ATE point estimates to researcher chosen implementation details, advocate for simple linear regression-based methods of covariate adjustment (pp. 80 91). A di¤erent response to this perception is embodied in the work of Kalyanaraman (2008), who develops automated tuning parameter choice procedures for a particular ATE estimator. Graham, Pinto and Egel (forthcoming) propose a semiparametric method of covariate adjustment called Inverse Probability Tilting (IPT). Implementing their method requires the researcher to formulate two models. The rst is for the probability of assignment to treatment conditional on characteristics or the propensity score. The second is actually a pair of models: for the conditional expectation functions (CEFs) of the potential outcomes under active and control treatment given characteristics. While the IPT point estimate of the ATE will be sensitive to them, these modelling decisions are arguably less abstract than that of, say, tuning parameter choice in the context of a c 2001 StataCorp LP iptATE 2 Inverse Probability Tilting fully nonparametric method of covariate adjustment. In addition to its conceptual simplicity the IPT estimator has a number of desirable theoretical properties. First, to use the language of Bang and Robins (2005), it is locally e¢ cient and doubly robust. Second, relative to other estimators with similar rst order asymptotic properties, such as the class of augmented inverse probability weighting (AIPW) estimators introduced by Robins, Rotnitzky and Zhao (1994), IPT has low higher order asymptotic bias.1 Third, it provides a operational extension of methods of direct adjustmentor post-strati cation from discretelyto continuouslyvalued covariates. To understand this last feature consider a setting with a single baseline covariate, gender. If men and women di¤erentially select into treatment, the gender composition of both the treated and control subsamples will depart from that of the full sample. In such a situation it is straightforward to reweight the two subsamples such that the distribution of gender in them mirrors that in the full sample (i.e., to restore balance; cf., Rosenbaum, 1987). When covariates are continuously-valued reweighting to restore balance is considerably more complex. IPT uses a reweighting of the data that balances a nite number of sample moments across treatment and control units. If, for example, units di¤erentially select into treatment on the basis of their baseline earnings, then IPT can reweight the data such that the mean and variance of baseline earnings is identical across treatment and control units. IPT implements the idea that reweighting should make the treatment and control subsamples more comparable in terms of the distribution of baseline characteristics in a very concrete and aesthetically attractive way. The underlying theory of IPT for general moment condition problems is developed in Graham, Pinto and Egel (forthcoming). The appendix to that paper also provides computational details. Other variants of inverse probability weighting (IPW) are discussed by Rosenbaum (1987), Wooldridge (2007) and Hirano, Imbens and Ridder (2003). This article presumes familiarity with the notation and language of the econometric program evaluation literature. Imbens and Wooldridge (2011) is a convenient reference for this material. 1 Review of inverse probability tilting We seek to estimate the average e¤ect of a binary treatment, D, on the scalar outcome Y . We let Y1 denote a randomly sampled units potential outcome given assignment to the active treatment (D = 1) and Y0 the corresponding potential outcome under control (D = 0). The average treatment e¤ect (ATE) is ATE 0 = E [Y1 Y0] : (1) 1. For an implementation of AIPW in Stata see Emsley, Lunt, Pickles and Dunn (2008). Graham, Pinto and Egel 3 If both Y1 and Y0 were observed for all sampled units, then an analog estimate of (1) would be straightforward to construct. In practice we only observe Y = (1 D)Y0 +DY1; or Y1 for treated units and Y0 for control units. Let X denote a vector of baseline characteristics and p0 (x) = Pr (D = 1jX = x) the propensity score or probability of assignment to the active treatment given characteristics. We assume the availability of the random sample f(Di; Xi; Yi)gNi=1 from the population of interest and use N1 and N0 to respectively denote the number of treated and control units. If (i) D is independent of (Y0; Y1) conditional on X = x for all x 2 X R and (ii) < p0 (x) < 1 for some > 0, then it is straightforward to show that 0 is identi ed (e.g., Rosenbaum and Rubin, 1983) and estimable at parametric rates (e.g., Robins, Rotnitzky and Zhao, 1994). The following approach to identi cation is useful for our purposes. Rewriting (1) in integral form we have ATE 0 = Z Z y1fY1;X (y1; x) dy1dx Z Z y0fY0;Xf (y0; x) dy0dx: (2) From condition (i) above we have the equalities fY1jX (y1jx) = fY jX;D (yjx; d = 1) ; fY0jX (y0jx) = fY jX;D (yjx; d = 0) : (3) BayesLaw gives the additional pair of equalities fX (x) = fXjD (xj d = 1) Q0 p0 (x) = fXjD (xj d = 0) 1 Q0 1 p0 (x) ; (4) where Q0 = E [D] is the marginal frequency of treatment. Substituting (3) and (4) into (2) and consolidating terms we get the following representation of the ATE ATE 0 = Z Z yfY;XjD (y; xj d = 1) Q0 p0 (x) dydx Z Z yfY;XjD (y; xj d = 0) 1 Q0 1 p0 (x) dydx: Each component on the right-hand-side of the above expression is identi ed by the joint distribution of the observed data (D;X; Y ). To construct an analog estimator based on the above representation we replace fY;XjD (y; xj d = 1) and fY;XjD (y; xj d = 0) with the empirical measures of the treated and control subsamples. These measures, which may be viewed as nonparametric maximum likelihood estimates (NPMLEs), place weight N 1 1 and N 1 0 on each treated and 4 Inverse Probability Tilting control unit (Owen, 2001). Replacing Q0 and p0 (x) with the estimates b Q and b p (x) we get b ( b Q;b p) = N1 X i=N0+1 b Q N1 1 b p (Xi) N0 X i=1 1 b Q N0 1 1 b p (Xi) (5)