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PROPENSITY SCORE-MATCHING METHODS for NONEXPERIMENTAL CAUSAL STUDIES Rajeev H

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PROPENSITY SCORE-MATCHING METHODS for NONEXPERIMENTAL CAUSAL STUDIES Rajeev H PROPENSITY SCORE-MATCHING METHODS FOR NONEXPERIMENTAL CAUSAL STUDIES Rajeev H. Dehejia and Sadek Wahba* Abstract—This paper considers causal inference and sample selection bias treatment impact.1 The first generation of matching methods in nonexperimental settings in which (i) few units in the nonexperimental paired observations based on either a single variable or comparison group are comparable to the treatment units, and (ii) selecting a subset of comparison units similar to the treatment units is difficult weighting several variables. (See, inter alia, Bassi (1984), because units must be compared across a high-dimensional set of pre- Cave and Bos (1995), Czajka et al. (1992), Cochran and treatment characteristics. We discuss the use of propensity score-matching Rubin (1973), Raynor (1983), Rosenbaum (1995), Rubin methods, and implement them using data from the National Supported Work experiment. Following LaLonde (1986), we pair the experimental (1973, 1979), Westat (1981), and studies cited by Barnow treated units with nonexperimental comparison units from the CPS and (1987).) PSID, and compare the estimates of the treatment effect obtained using The motivation for focusing on propensity score- our methods to the benchmark results from the experiment. For both matching methods is that, in many applications of interest, comparison groups, we show that the methods succeed in focusing attention on the small subset of the comparison units comparable to the the dimensionality of the observable characteristics is high. treated units and, hence, in alleviating the bias due to systematic differ- With a small number of characteristics (for example, two ences between the treated and comparison units. binary variables), matching is straightforward (one would group units in four cells). However, when there are many variables, it is difficult to determine along which dimen- I. Introduction sions to match units or which weighting scheme to adopt. n important problem of causal inference is how to Propensity score-matching methods, as we demonstrate, are Aestimate treatment effects in observational studies, especially useful under such circumstances because they situations (like an experiment) in which a group of units is provide a natural weighting scheme that yields unbiased exposed to a well-defined treatment, but (unlike an experi- estimates of the treatment impact. ment) no systematic methods of experimental design are The key contribution of this paper is to discuss and apply used to maintain a control group. It is well recognized that propensity score-matching methods, which are new to the the estimate of a causal effect obtained by comparing a economics literature. (Previous papers include Dehejia and treatment group with a nonexperimental comparison group Wahba (1999), Heckman et al. (1996, 1998), Heckman, could be biased because of problems such as self-selection Ichimura, and Todd (1997, 1998). See Friedlander, Green- or some systematic judgment by the researcher in selecting berg, and Robins (1997) for a review.) This paper differs units to be assigned to the treatment. This paper discusses from Dehejia and Wahba (1999) by focusing on matching methods in detail, and it complements the Heckman et al. the use of propensity score-matching methods to correct for papers by discussing a different array of matching estima- sample selection bias due to observable differences between tors in the context of a different data set. the treatment and comparison groups. An important feature of our method is that, after units are Matching involves pairing treatment and comparison matched, the unmatched comparison units are discarded and units that are similar in terms of their observable character- are not directly used in estimating the treatment impact. Our istics. When the relevant differences between any two units approach has two motivations. First, in some settings of are captured in the observable (pretreatment) covariates, interest, data on the outcome variable for the comparison which occurs when outcomes are independent of assign- group are costly to obtain. For example, in economics, some ment to treatment conditional on pretreatment covariates, data sets provide outcome information for only one year; if matching methods can yield an unbiased estimate of the the outcome of interest takes place in a later period, possibly thousands of comparison units have to be linked across data Received for publication February 12, 1998. Revision accepted for sets or resurveyed. In such settings, the ability to obtain the publication January 24, 2001. needed data for a subset of relevant comparison units, * Columbia University and Morgan Stanley, respectively. discarding the irrelevant potential comparison units, is ex- Previous versions of this paper were circulated under the title “An Oversampling Algorithm for Nonexperimental Causal Studies with In- tremely valuable. Second, even if information on the out- complete Matching and Missing Outcome Variables” (1995) and as come is available for all comparison units (as it is in our National Bureau of Economic Research working paper no. 6829. We thank data), the process of searching for the best subset from the Robert Moffitt and two referees for detailed comments and suggestions that have improved the paper. We are grateful to Gary Chamberlain, Guido comparison group reveals the extent of overlap between the Imbens, and Donald Rubin for their support and encouragement, and treatment and comparison groups in terms of pretreatment greatly appreciate comments from Joshua Angrist, George Cave, and Jeff characteristics. Because methods that use the full set of Smith. Special thanks are due to Robert LaLonde for providing, and helping to reconstruct, the data from his 1986 study. Valuable comments were received from seminar participants at Harvard, MIT, and the Man- 1 More precisely, to estimate the treatment impact on the treated, the power Demonstration Research Corporation. Any remaining errors are the outcome in the untreated state must be independent of the treatment authors’ responsibility. assignment. The Review of Economics and Statistics, February 2002, 84(1): 151–161 © 2002 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/003465302317331982 by guest on 02 October 2021 152 THE REVIEW OF ECONOMICS AND STATISTICS comparison units extrapolate or smooth across the treatment where Ti ϭ 1(ϭ 0) if the ith unit was assigned to treatment and comparison groups, it is extremely useful to know how (control).3 The problem of unobservability is summarized many of the comparison units are in fact comparable and by the fact that we can estimate E(Yi1͉Ti ϭ 1), but not hence how much smoothing one’s estimator is expected to E(Yi0͉Ti ϭ 1). e perform. The difference, ␶ ϭ E(Yi1͉Ti ϭ 1) Ϫ E(Yi0͉Ti ϭ 0), The data we use, obtained from LaLonde (1986), are can be estimated, but it is potentially a biased estimator of from the National Supported Work (NSW) Demonstration, a ␶. Intuitively, if Yi0 for the treated and comparison units labor market experiment in which participants were ran- systematically differs, then in observing only Yi0 for the domized between treatment (on-the-job training lasting be- comparison group we do not correctly estimate Yi0 for the tween nine months and a year) and control groups. Follow- treated group. Such bias is of paramount concern in nonex- ing LaLonde, we use the experimental controls to obtain a perimental studies. The role of randomization is to prevent benchmark estimate for the treatment impact and then set this: them aside, wedding the treated units from the experiment to comparison units from the Population Survey of Income Yi1, Yi 0 ʈTi f E͑Yi0͉Ti ϭ 0͒ ϭ E͑Yi0͉Ti ϭ 1͒ ϭ E͑Yi͉Ti ϭ 0͒, Dynamics (PSID) and the Current Population Survey 2 (CPS). We compare estimates obtained using our nonex- where Yi ϭ TiYi1 ϩ (1 Ϫ Ti)Yi0 (the observed value of the perimental methods to the experimental benchmark. We outcome) and ʈ is the symbol for independence. The treated show that most of the nonexperimental comparison units are and control groups do not systematically differ from each not good matches for the treated group. We succeed in other, making the conditioning on Ti in the expectation selecting the comparison units that are most comparable to unnecessary (ignorable treatment assignment, in the termi- e 4 the treated units and in replicating the benchmark treatment nology of Rubin (1977)), and yielding ␶͉Tϭ1 ϭ␶. impact. The paper is organized as follows. In section II, we B. Exact Matching on Covariates discuss the theory behind our estimation strategy. In section III, we discuss propensity score-matching methods. In sec- To substitute for the absence of experimental control tion IV, we describe the NSW data, which we then use in units, we assume that data can be obtained for a set of section V to implement our matching procedures. Section potential comparison units, which are not necessarily drawn VI tests the matching assumption and examines the sensi- from the same population as the treated units but for whom tivity of our estimates to the specification of the propensity we observe the same set of pretreatment covariates, Xi. The score. Section VII concludes the paper. following proposition extends the framework of the previ- ous section to nonexperimental settings: II. Matching Methods Proposition 1 (Rubin, 1977). If for each unit we observe ʈ ͉ @ A. The Role of Randomization a vector of covariates Xi and Yi0 Ti Xi, i, then the popu- lation treatment effect for the treated, ␶͉Tϭ1, is identified: it A cause is viewed as a manipulation or treatment that is equal to the treatment effect conditional on covariates and brings about a change in the variable of interest, compared on assignment to treatment, ␶͉Tϭ1,X, averaged over the 5 to some baseline, called the control (Cox, 1992; Holland, distribution X͉Ti ϭ 1 . 1986). The basic problem in identifying a causal effect is that the variable of interest is observed under either the 3 In a nonexperimental setting, the treatment and comparison samples treatment or control regimes, but never both.
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