Motivated by recent findings in the field of consumer science, this paper evaluates the causal effect of debit cards on household consumption using population-based data from the Italy Survey on Household Income and Wealth (SHIW). Within the Rubin Causal Model, we focus on the estimand of population average treatment effect for the treated (PATT). We consider three existing estimators, based on regression, mixed matching and regression, propensity score weighting, and propose a new doubly-robust estimator. Semiparametric specification based on power series for the potential outcomes and the propensity score is adopted. Cross-validation is used to select the order of the power series. We conduct a simulation study to compare the performance of the estimators. The key assumptions, overlap and unconfoundedness, are systematically assessed and validated in the application. Our empirical results suggest statistically significant positive effects of debit cards on the monthly household spending in Italy.

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Expanding a lower-dimensional problem to a higher-dimensional space and then projecting back is often beneficial. This article rigorously investigates this perspective in the context of finite mixture models, specifically how to improve inference for mixture models by using auxiliary variables. Despite the large literature in mixture models and several empirical examples, there is no previous work that gives general theoretical justification for including auxiliary variables in mixture models, even for special cases. We provide a theoretical basis for comparing inference for mixture multivariate models with the corresponding inference for marginal univariate mixture models. Analytical results for several special cases are established. We show that the probability of correctly allocating mixture memberships and the information number for the means of the primary outcome in a bivariate model with two Gaussian mixtures are generally larger than those in each univariate model. Simulations under a range of scenarios, including mis-specified models, are conducted to examine the improvement. The method is illustrated by two real applications in ecology and causal inference.

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The exclusion restriction is usually assumed for identifying causal effects in true or only natural randomized experiments with noncompliance. It requires that the assignment to treatment does not have a direct causal effect on the outcome. Despite its importance, the restriction can often be unrealistic, especially in situations of natural experiments. It is shown that, without the exclusion restriction, the parametric model is identified if the outcome distributions of various compliance statuses are in the same parametric class and that class is a linearly independent set over the field of real numbers. However, the relaxation of the exclusion restriction yields a parametric model that is characterized by the presence of mixtures of distributions. This scenario complicates the likelihood-based estimation procedures because it implies more than one maximum likelihood point. A two-step estimation procedure based on detecting the root that is closest to the method of moments estimate of the parameter vector is then proposed and analyzed in detail, under normally distributed outcomes. An economic example with real data concerning returns to schooling concludes the paper.