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

Abstract: Instrumental variable (IV) analysis is a crucial tool in estimating causal relationships by addressing the issue of confounding variables that may bias the results. Among other work on IV models with binary exposure and outcomes, Richardson and Robins (2014) studied the instrumental variable model with binary exposure (X) and binary outcome (Y) with an instrument (Z) that takes Q states where Q>=2. However, IV models beyond binary X and Y have been less explored. In this work, we consider the instrumental variable model with categorical X, Y, Z taking values in {1, ..., K}, {1, ..., M}, and {1, ..., Q} respectively. We first give a simple closed-form characterization of the set of joint distributions of the potential outcomes P(Y(x=1), ..., Y(x=K)) compatible with a given observed probability distribution P(X, Y | Z). We further show the bounds we derived are necessary, sufficient, and non-redundant, and they hold under various versions of the independence assumptions that have been discussed in the literature. We also provide how a confidence region of any convex function of the joint counterfactual probability including the average causal effect (ATE) can be computed using an algorithm proposed by Guo and Richardson (2021) which is based on a new tail bound for the KL-divergence. We implement our bounds and provide practical recommendations through a real data example of a cash-incentive smoking cessation program.