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Bounding Causal Effects with Leaky Instruments

David S. Watson, Jordan Penn, Lee M. Gunderson, Gecia Bravo-Hermsdorff, Afsaneh Mastouri, Ricardo Silva

Abstract

Instrumental variables (IVs) are a popular and powerful tool for estimating causal effects in the presence of unobserved confounding. However, classical approaches rely on strong assumptions such as the $\textit{exclusion criterion}$, which states that instrumental effects must be entirely mediated by treatments. This assumption often fails in practice. When IV methods are improperly applied to data that do not meet the exclusion criterion, estimated causal effects may be badly biased. In this work, we propose a novel solution that provides $\textit{partial}$ identification in linear systems given a set of $\textit{leaky instruments}$, which are allowed to violate the exclusion criterion to some limited degree. We derive a convex optimization objective that provides provably sharp bounds on the average treatment effect under some common forms of information leakage, and implement inference procedures to quantify the uncertainty of resulting estimates. We demonstrate our method in a set of experiments with simulated data, where it performs favorably against the state of the art. An accompanying $\texttt{R}$ package, $\texttt{leakyIV}$, is available from $\texttt{CRAN}$.

Bounding Causal Effects with Leaky Instruments

Abstract

Instrumental variables (IVs) are a popular and powerful tool for estimating causal effects in the presence of unobserved confounding. However, classical approaches rely on strong assumptions such as the , which states that instrumental effects must be entirely mediated by treatments. This assumption often fails in practice. When IV methods are improperly applied to data that do not meet the exclusion criterion, estimated causal effects may be badly biased. In this work, we propose a novel solution that provides identification in linear systems given a set of , which are allowed to violate the exclusion criterion to some limited degree. We derive a convex optimization objective that provides provably sharp bounds on the average treatment effect under some common forms of information leakage, and implement inference procedures to quantify the uncertainty of resulting estimates. We demonstrate our method in a set of experiments with simulated data, where it performs favorably against the state of the art. An accompanying package, , is available from .
Paper Structure (31 sections, 10 theorems, 56 equations, 9 figures, 1 table)

This paper contains 31 sections, 10 theorems, 56 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Notation.
  4. The Leaky IV Setting.
  5. Theory
  6. Scalar $\tau$-exclusion
  7. Vector $\tau$-exclusion
  8. Inference
  9. Experiments
  10. Point Estimators.
  11. Bayesian Methods.
  12. Power.
  13. Coverage.
  14. Related Work
  15. Discussion
  16. ...and 16 more sections

Key Result

Lemma 1

There is a bijective, strictly decreasing function $f: [-1, 1] \mapsto \mathbb{R}$ that maps values of the confounding coefficient $\rho$ to the ATE $\theta$:

Figures (9)

  • Figure 1: Causal diagram with treatment $X$, outcome $Y$, unobserved confounder $U$ (shaded), and candidate instruments $Z_1, \dots, Z_{d_{\boldsymbol{Z}}}$. Dashed edges suggest possible violations of the exclusion criterion. Edges among $\boldsymbol{Z}$ are allowed, but omitted for simplicity.
  • Figure 2: Causal diagram of the SEM described by Eqs. \ref{['eq:scmx']}-\ref{['eq:scmSigma']}. Edge weights correspond to linear coefficients, while unobserved confounding effects are represented by the bidirected edge connecting $\epsilon_x$ and $\epsilon_y$. The dashed edge from $\boldsymbol{Z}$ to $Y$ denotes possible violations of (A3).
  • Figure 3: Example curves illustrating the relationships between parameters in the leaky IV model. (A) A $\rho$-$\theta$ curve maps the relationship between latent confounding and causal effects. (B) A $\theta$-$\lVert \boldsymbol{\gamma} \rVert_2$ curve maps the relationship between causal effects and information leakage. Shading represents 95% confidence intervals estimated via the bootstrap.
  • Figure 4: Minimum and oracle leakage values impose a three-partition of the threshold space. Below $\check{\tau}_2$, we have the infeasible region (grey striped area), where no configuration of latent parameters satisfies our structural constraints. Between $\check{\tau}_2$ and $\tau^*_2$, we have the error region (red area), where bounds are identifiable but invalid. Above $\tau^*_2$, we have the valid region (rest of the plot), where bounds are guaranteed to contain the true ATE $\theta^*$, represented by the vertical blue line.
  • Figure 5: Comparison against various methods at a range of values for the confounding coefficient $\rho$, SNR for $Y$, and number of candidate instruments $d_{\boldsymbol{Z}}$. The horizontal black line at $\theta = 1$ represents the true ATE $\theta^*$.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Lemma 1: ATE as a function of confounding
  • Lemma 2: Leakage as a function of ATE
  • Lemma 3: Minimum leakage as a function of ATE
  • Lemma 4: Minimum leakage as a function of confounding
  • Theorem 1: Identifiability
  • Corollary 1.1
  • Theorem 2: ATE bounds
  • Corollary 2.1
  • Theorem 3: Exclusion test
  • Theorem 4: Coverage