Vector-treatment contamination decomposition and testing

Develop a framework for vector treatments that disentangles single-treatment functional-form contamination from the cross-treatment contamination identified by Goldsmith et al. (2024), treating the object of interest as a contamination matrix rather than a scalar.

Background

The paper’s results establish an exact contamination decomposition, diagnostic tests, and a correction for a scalar treatment in linear instrumental variables with covariates. The authors note that extending the framework to multiple treatments is substantively different because functional-form contamination can coexist with the cross-treatment channel studied in prior work, which persists even when the instrument propensity is saturated.

For vector treatments, the unresolved task is to characterize and disentangle these two contamination mechanisms and determine how they should be tested or corrected when the relevant object is matrix-valued rather than scalar.