Testability of Instrumental Variables in Additive Nonlinear, Non-Constant Effects Models

We address the issue of the testability of instrumental variables derived from observational data. Most existing testable implications are centered on scenarios where the treatment is a discrete variable, e.g., instrumental inequality (Pearl, 1995), or where the effect is assumed to be constant, e.g., instrumental variables condition based on the principle of independent mechanisms (Burauel, 2023). However, treatments can often be continuous variables, such as drug dosages or nutritional content levels, and non-constant effects may occur in many real-world scenarios. In this paper, we consider an additive nonlinear, non-constant effects model with unmeasured confounders, in which treatments can be either discrete or continuous, and propose an Auxiliary-based Independence Test (AIT) condition to test whether a variable is a valid instrument. We first show that, under the completeness condition, if the candidate instrument is valid, then the AIT condition holds. Moreover, we illustrate the implications of the AIT condition and demonstrate that, under certain additional conditions, the AIT condition is necessary and sufficient to detect all invalid IVs. We also extend the AIT condition to include covariates and introduce a practical testing algorithm. Experimental results on both synthetic and three different real-world datasets show the effectiveness of our proposed condition.

💡 Research Summary

This paper tackles the long‑standing problem of testing the validity of instrumental variables (IVs) when the treatment variable may be continuous and the causal effect is allowed to be nonlinear and non‑constant. Existing testable implications such as Pearl’s instrumental inequality or the recent IV‑PIM condition are limited to discrete treatments or constant‑effect models. To fill this gap, the authors introduce the Additive Nonlinear, Non‑Constant Effects (ANINCE) model, a non‑parametric additive framework in which
(X = g(Z) + \phi_X(U) + \varepsilon_X,)
(Y = f(X) + \phi_Y(U) + \varepsilon_Y,)
with an unobserved confounder (U) and independent noise terms. The key identification assumption is completeness: for any integrable function (\psi(X)), (E