Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Assumptions and Bounds in the Instrumental Variable Model

Thomas S. Richardson, James M. Robins

TL;DR

This work establishes sharp, model-wide constraints for instrumental-variable graphs with binary treatment and outcome and a $K$-state instrument $Z$. It proves that, under any of four independence regimes, the joint distribution of the counterfactual pair $(Y(x_0),Y(x_1))$ is fully characterized by $4K$ inequality constraints, via a constructive mapping $\,\phi$ showing equivalence between model-implied and observed distributions. The authors derive universal, variation-independent bounds on $P(Y(x_0))$, $P(Y(x_1))$, and the average causal effect $ACE(X\rightarrow Y)$, expressed through a function $g(i,j)$; for binary $Z$ these reduce to the Balke–Pearl bounds and relate to Dawid and Pearl’s formulations. The paper also clarifies redundancy in the bound set and situates the results within SWIG/FFRCISTG frameworks, highlighting connections to classical causal-bounds literature and providing tools for sharp inference in IV settings with multi-level instruments.

Abstract

In this note we give proofs for results relating to the Instrumental Variable (IV) model with binary response $Y$ and binary treatment $X$, but with an instrument $Z$ with $K$ states. These results were originally stated in Richardson & Robins (2014), "ACE Bounds; SEMS with Equilibrium Conditions," arXiv:1410.0470.

Assumptions and Bounds in the Instrumental Variable Model

TL;DR

This work establishes sharp, model-wide constraints for instrumental-variable graphs with binary treatment and outcome and a -state instrument . It proves that, under any of four independence regimes, the joint distribution of the counterfactual pair is fully characterized by inequality constraints, via a constructive mapping showing equivalence between model-implied and observed distributions. The authors derive universal, variation-independent bounds on , , and the average causal effect , expressed through a function ; for binary these reduce to the Balke–Pearl bounds and relate to Dawid and Pearl’s formulations. The paper also clarifies redundancy in the bound set and situates the results within SWIG/FFRCISTG frameworks, highlighting connections to classical causal-bounds literature and providing tools for sharp inference in IV settings with multi-level instruments.

Abstract

In this note we give proofs for results relating to the Instrumental Variable (IV) model with binary response and binary treatment , but with an instrument with states. These results were originally stated in Richardson & Robins (2014), "ACE Bounds; SEMS with Equilibrium Conditions," arXiv:1410.0470.
Paper Structure (8 sections, 6 theorems, 56 equations, 1 figure, 3 tables)

This paper contains 8 sections, 6 theorems, 56 equations, 1 figure, 3 tables.

Table of Contents

  1. Instrumental Variable Models
  2. Characterization of the IV Models
  3. Bounds on $P(Y(x_0))$, $P(Y(x_1))$ and $ACE(X\!\rightarrow\! Y)$
  4. Bounds on the ACE$(X\rightarrow Y)$, when $X$ is binary
  5. Detailed derivation of inequalities relating $P(Y(x_0),Y(x_1))$ and $P(X,Y\mid Z=k)$
  6. Relation to the ACE$(X\rightarrow Y)$ bounds of Balke & Pearl when $Z$ has two levels
  7. Upper Bounds on $ACE (X\rightarrow Y)$
  8. Lower Bounds on $ACE (X\rightarrow Y)$

Key Result

Theorem 1

Under any of the assumptions (i), (ii), (iii), or (iv) the joint distribution $P(Y(x_0), Y(x_1))$ is characterized by the $4K$ pairs of inequalities: with $y, \tilde{y}, i \in \{0,1\}$ and $z\in \{1,\ldots ,K\}$.

Figures (1)

  • Figure 1: (a) IV model with no confounding between $Z$ and $X$; (b) SWIG representing $P(Z, X(z),Y(x),U)$; (c) IV model with confounding between $Z$ and $X$; (d) SWIG representing $P(Z, X,Y(x),U,U^*)$.

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 2