Matching Estimation for Causal Effect on Compositional Outcomes

Kenichi Ariga   (University of Toronto)

Abstract: Compositional outcomes are not unusual in political science research. Political scientists have analyzed the factors influencing the proportions across multiple categories, such as multiparty vote shares (Katz and King 1999) and budget allocation (Lipsmeyer, Philips, Rutherford and Whitten 2019). In conventional regression methods, the compositional feature of such outcomes creates a challenge of how to model them, and the models based on the multiple logistic transformation of outcomes have been proposed to address these issues (e.g., Katz and King 1999; Mebane and Sekhon 2004; Phillips, Rutherford, Whitten 2015). A matching estimation can be applied to compositional outcomes without such transformation, as the average treatment effect can be non-parametrically estimated, and the results could be more valid than the log-ratio based regression models, as the matching estimation does not rely on the distributional and functional form assumptions needed in the latter models. Despite potential benefits, however, the applications of matching estimation of compositional outcomes are virtually absent. A couple of reasons may be suggested. First, it may be unclear for many applied researchers how a matching method can be used to estimate the causal impact on the distribution of compositional outcomes (e.g., how running an incumbent candidate would change the distribution of multiparty vote shares in a district). Second, the currently available inferential methods are not sufficient for addressing some important questions for compositional outcomes. For example, if a certain causal variable increases one component of the outcome, it should reduce other components, and we may be interested in which components a particular gain comes from (e.g., if there is an electoral advantage from running an incumbent candidate for the Conservative party in Canada, which parties does the Conservative’s vote gain come from? Does it come solely or mostly from one of the major parties? Or equally from all other major parties?). In this context, we may want to compare the impact on one component to that on another (e.g., a hypothesis testing or confidence interval for the difference in the vote loss between Liberals and NDP due to the Conservative incumbency). However, the currently available estimators for the variance of a matching estimator (Abadie and Imbens 2006; Hanson and Sunderam 2012) are not applicable in this context, as they don’t address the particular clustering and dependent structures of compositional outcomes. This paper addresses these two issues. First, it delineates how we can apply a matching estimation to compositional outcomes. It serves as an easy guide for applied researchers who examine compositional outcomes. Second, it derives a consistent estimator for the variance of a non-parametric matching estimator for compositional outcomes. In addition, it also discusses how a sensitivity analysis may be tailored to the compositional outcomes. The paper uses an incumbency advantage in Canadian Federal elections as an example to illustrate how the proposed method can be applied.

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