On the Predictive Distributions of Outcome Gains in the Presence of an Unidentified Parameter
在标准潜变量选择模型框架下，展示了如何获得结果增益的完整预测分布，而不仅仅是平均处理效应，并利用协方差矩阵的正定性约束从未识别参数中学习信息，应用于1903年新泽西童工识字对周薪影响的预测。
In this article we describe methods for obtaining the predictive distributions of outcome gains in the framework of a standard latent variable selection model. Although most previous work has focused on estimation of mean treatment parameters as the method for characterizing outcome gains from program participation, we show how the entire distributions associated with these gains can be obtained in certain situations. Although the out-of-sample outcome gain distributions depend on an unidentified parameter, we use the results of Koop and Poirier to show that learning can take place about this parameter through information contained in the identified parameters via a positive definiteness restriction on the covariance matrix. In cases where this type of learning is not highly informative, the spread of the predictive distributions depends more critically on the prior. We show both theoretically and in extensive generated data experiments how learning occurs, and delineate the sensitivity of our results to the prior specifications. We relate our analysis to three treatment parameters widely used in the evaluation literature—the average treatment effect, the effect of treatment on the treated, and the local average treatment effect—and show how one might approach estimation of the predictive distributions associated with these outcome gains rather than simply the estimation of mean effects. We apply these techniques to predict the effect of literacy on the weekly wages of a sample of New Jersey child laborers in 1903.