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

Xichen Guo, Zheng Li, Biwei Huang, Yan Zeng, Zhi Geng, Feng Xie

Abstract

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 if the candidate instrument is valid, then the AIT condition holds. Moreover, we illustrate the implications of the AIT condition and demonstrate that, in certain conditions, AIT conditions are 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.

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

Abstract

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 if the candidate instrument is valid, then the AIT condition holds. Moreover, we illustrate the implications of the AIT condition and demonstrate that, in certain conditions, AIT conditions are 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.

Paper Structure

This paper contains 33 sections, 13 theorems, 46 equations, 6 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Additive Non-Parametric Instrumental Variable Model
  5. Additive Nonlinear, Non-Constant Effects Model
  6. AIT Condition and Its Implications in ANINCE Models
  7. AIT Condition
  8. Implications of AIT Condition in Additive Linear, Constant Effects Models
  9. Implications of AIT Condition in Additive Nonlinear, Non-Constant Effects Models
  10. Practical Implementation of AIT Condition
  11. AIT Condition with Covariates
  12. AIT Condition with Finite Data
  13. Experiments
  14. Synthetic Data
  15. Theoretical Validation of Proposed Method
  16. ...and 18 more sections

Key Result

Theorem 1

Let $X$, $Y$, and $Z$ be the treatment, outcome, and candidate IV in an ANINCE model, respectively. Suppose that $X$, $Y$, and $Z$ are correlated and that the sample size $n \to \infty$ holds. Further, suppose that the probability densities $p(\boldsymbol{\varepsilon_U})$ and $p(\varepsilon_Z)$ are

Figures (6)

  • Figure 1: Graphical illustration of IV models, where $\mathbf{U}$ is the set of unmeasured confounders. (a) $Z$ is a valid IV. (b) $Z$ is an invalid IV due to the edge $\mathbf{U} \to Z$ (Violate $\mathcal{C}2$). (c) $Z$ is an invalid IV due to the edge $Z \to Y$ (Violate $\mathcal{C}3$).
  • Figure 2: Scatter plots of Candidate IV $Z$ and Auxiliary-variable $\mathcal{A}$ under the linear models. (a) All noise terms follow Gaussian distributions. (b) Some noise terms follow non-Gaussian distributions.
  • Figure 3: Scatter plot of Candidate IV $Z$ and Auxiliary-variable $\mathcal{A}$ when all noise terms follow Gaussian distribution in the partially non-linear invalid IV model.
  • Figure 4: Graphical illustration of an IV model for estimating the causal effect of schooling ($S_{ch}$) on returns of education ($R_{e}$) card1993using.
  • Figure 5: Graphical illustration of an IV model for estimating the causal effect of institutions ($I_{ns}$) on economic development ($E_{d}$) acemoglu2001colonial.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2: AIT Condition
  • Theorem 1: Necessary Condition for IV
  • Proposition 1: Non-testability in Linear Gaussian Models
  • Proposition 2: Testability of Exogeneity in Linear Models
  • Remark 3
  • Example 1
  • Proposition 3: Non-testability of Exclusion Restriction in Linear Models
  • ...and 12 more