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Abstract:
We are interested in estimating the average effect of a binary treatment on a scalar outcome. If assignment to the treatment is unconfounded, that is, independent of the potential outcomes given covariates, biases associated with simple treatment-control average comparisons can be removed by adjusting for differences in the covariates. Rosenbaum and Rubin (1983a) show that adjusting solely for differences between treated and control units in a scalar function of the covariates, the propensity score, also removes all biases associated with differences in covariates. Although adjusting for the propensity score removes all the bias, this can come at the expense of efficiency, as shown by Hahn (1998), Heckman, Ichimura, Todd (1998), and Rotnitzky and Robins (1995). We show that weighting by the inverse of a nonparametric estimate of the propensity score, rather than the true propensity score, leads to efficient estimates of the average treatment effect. We provide intuition for this result by showing that this estimator can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score.

Abstract:
We are interested in estimating the average effect of a binary treatment on a scalar outcome. If assignment to the treatment is independent of the potential outcomes given pretreatment variables, biases associated with simple treatment-control average comparisons can be removed by adjusting for differences in the pre-treatment variables. Rosenbaum and Rubin (1983, 1984) show that adjusting solely for differences between treated and control units in a scalar function of the pre-treatment, the propensity score, also removes the entire bias associated with differences in pre-treatment variables. Thus it is possible to obtain unbiased estimates of the treatment effect without conditioning on a possibly high-dimensional vector of pre-treatment variables. Although adjusting for the propensity score removes all the bias, this can come at the expense of efficiency. We show that weighting with the inverse of a nonparametric estimate of the propensity score, rather than the true propensity score, leads to efficient estimates of the various average treatment effects. This result holds whether the pre-treatment variables have discrete or continuous distributions. We provide intuition for this result in a number of ways. First we show that with discrete covariates, exact adjustment for the estimated propensity score is identical to adjustment for the pre-treatment variables. Second, we show that weighting by the inverse of the estimated propensity score can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score. Finally, we make a connection to results to other results on efficient estimation through weighting in the context of variable probability sampling.

Abstract:
Many social experiments are run in multiple waves, or replicate earlier social experiments. In principle, the sampling design can be modified in later stages or replications to allow for more efficient estimation of causal effects. We consider the design of a two-stage experiment for estimating an average treatment effect, when covariate information is available for experimental subjects. We use data from the first stage to choose a conditional treatment assignment rule for units in the second stage of the experiment. This amounts to choosing the propensity score, the conditional probability of treatment given covariates. We propose to select the propensity score to minimize the asymptotic variance bound for estimating the average treatment effect. Our procedure can be implemented simply using standard statistical software and has attractive large-sample properties.

experimental design, propensity score, efficiency bound

Abstract:
In certain auction, search, and related models, the boundary of the support of the observed data depends on some of the parameters of interest. For such nonregular models, standard asymptotic distribution theory does not apply. Previous work has focused on characterizing the nonstandard limiting distributions of particular estimators in these models. In contrast, we study the problem of constructing efficient point estimators. We show that the maximum likelihood estimator is generally inefficient, but that the Bayes estimator is efficient according to the local asymptotic minmax criterion for conventional loss functions. We provide intuition for this result using Le Cam's limits of experiments framework.

Abstract:
In many fields researchers wish to consider statistical models that allow for more complex relationships than can be inferred using only cross-sectional data. Panel or longitudinal data where the same units are observed repeatedly at different points in time can often provide the richer data needed for such models. Although such data allows researchers to identify more complex models than cross-sectional data, missing data problems can be more severe in panels. In particular, even units who respond in initial waves of the panel may drop out in subsequent waves, so that the subsample with complete data for all waves of the panel can be less representative of the population than the original sample. Sometimes, in the hope of mitigating the effects of attrition without losing the advantages of panel data over cross-sections, panel data sets are augmented by replacing units who have dropped out with new units randomly sampled from the original population. Following Ridder (1992), who used these replacement units to test some models for attrition, we call such additional samples refreshment samples. We explore the benefits of these samples for estimating models of attrition. We describe the manner in which the presence of refreshment samples allows the researcher to test various models for attrition in panel data, including models based on the assumption that missing data are missing at random (MAR, Rubin, 1976; Little and Rubin, 1987). The main result in the paper makes precise the extent to which refreshment samples are informative about the attrition process; a class of non-ignorable missing data models can be identified without making strong distributional or functional form assumptions if refreshment samples are available.

Abstract:
This paper develops semiparametric Bayesian methods for inference in dynamic linear panel data models, and applies them to longitudinal data on labor earnings from the Panel Study of Income Dynamics. We focus on characterizing not only parameters related to conditional means and variances, but the entire joint distribution of earnings. Full distributional inference in semiparametric panel data models must solve a nonparametric deconvolution problem arising from the existence of unobserved additive individual-specific terms. A computational Bayesian approach based on variable augmentation can deal with such latent-variable models effectively, and can allow considerable flexibility in distributional assumptions. We use relatively simple, low-order linear models with individual heterogeneity, but estimate the distribution of the disturbances without requiring that it belong to a restrictive parametric class, such as the class of normal distributions. A prior distribution on the space of probability distributions is used to impose smoothness on an otherwise general specification for the disturbance terms. This nonparametric prior is constructed using a countable mixture of normals representation, where the mixing distribution is given a Dirichlet process prior.