Partial Identifiability in Discrete Data With Measurement Error
This work provides a method for researchers and practitioners to obtain reliable bounds on distributions in discrete data models, even when faced with measurement errors and without relying on strong, often unjustifiable, identification assumptions.
This paper addresses the challenge of measurement errors in discrete data by demonstrating how various modeling assumptions can be formulated as linear constraints. It then uses linear programming to derive sharp bounds for factual and counterfactual distributions, even for scenarios like instrumental variables with measurement error where no bounds were previously available.
When data contains measurement errors, it is necessary to make assumptions relating the observed, erroneous data to the unobserved true phenomena of interest. These assumptions should be justifiable on substantive grounds, but are often motivated by mathematical convenience, for the sake of exactly identifying the target of inference. We adopt the view that it is preferable to present bounds under justifiable assumptions than to pursue exact identification under dubious ones. To that end, we demonstrate how a broad class of modeling assumptions involving discrete variables, including common measurement error and conditional independence assumptions, can be expressed as linear constraints on the parameters of the model. We then use linear programming techniques to produce sharp bounds for factual and counterfactual distributions under measurement error in such models. We additionally propose a procedure for obtaining outer bounds on non-linear models. Our method yields sharp bounds in a number of important settings -- such as the instrumental variable scenario with measurement error -- for which no bounds were previously known.