The Instrumental Variable Model with Categorical Instrument, Exposure, and Outcome: Characterization, Partial Identification, and Statistical Inference

Abstract

Instrumental variable (IV) analysis is a crucial tool in estimating causal relationships that addresses the issue of confounding variables that may lead to bias. Under certain IV assumptions, the causal effect may be partially identified. The binary IV model has been well studied in economics, statistics, and epidemiology, while IV models for general categorical exposure and outcome are less explored. This dissertation studies several aspects of the instrumental variable model with categorical instrument, exposure, and outcome including giving a characterization of the model (Chapters 2, 3 and 5), methods for statistical inference (Chapter 4), and a study of the variation independence properties of the marginal counterfactual distributions (Chapter 6). In Chapter 2, we first give a simple closed-form characterization of the set of joint distributions of the potential outcomes compatible with a given observed probability distribution via a set of inequalities. In Chapter 3, we further derive conditions for the inequalities in Chapter 2 to be non-redundant and construct the minimal set. To handle sampling variability, we provide an algorithm in Chapter 4 to construct confidence regions for any convex functional of the joint counterfactual distribution, such as the average causal effect (ATE), using a finite-sample tail bound for the KL-divergence due to Guo and Richardson [2021]. We also illustrate our methods in Chapters 2 and 4 using data from the Minneapolis Domestic Violence Experiment. In Chapter 5, we study falsification tests for the categorical IV model through simulations. We explore the variation dependence property of the marginal counterfactual distributions and discuss its practical implications in Chapter 6. We conclude with a discussion and directions for future work in Chapter 7.