leakyIV: Bounding Causal Effects with Leaky Instruments

Introduction

Instrumental variables (IVs) are a popular and powerful tool for estimating causal effects in the presence of unobserved confounding. However, classical methods rely on strong, untestable assumptions such as the exclusion criterion, which states that instrumental effects must be entirely mediated by treatments. In the so-called "leaky" IV setting, candidate instruments are allowed to have some direct influence on outcomes, rendering the average treatment effect (ATE) unidentifiable. But with limits on the amount of information leakage, we may still recover a partial identification interval for the ATE. This package implements methods for ATE bounding in the leaky IV setting with linear structural equations.

We assume data are generated according to the following process: $$X := Z \beta + \epsilon_X$$ $$Y := Z \gamma + X \theta + \epsilon_Y,$$ where the correlation between additive noise terms $\epsilon_X, \epsilon_Y$ is given by unobserved parameter $\rho$. The ATE is denoted by $\theta$. Whereas classical IV models require each $\gamma$ coefficient to be zero, we permit some direct signal from $Z$ to $Y$. Specifically, leakyIV can compute an exact ATE interval when bounding $\lVert \gamma \rVert_p$ or placing separate bounds on each $\gamma$ coefficient. Several algorithms for covariance matrix estimation are implemented, as well as a bootstrapping procedure for uncertainty quantification and a Monte Carlo test for violations of the exclusion criterion.

Installation

The leakyIV package is available on CRAN.

install.packages('leakyIV')

To install the development version from GitHub, using devtools, run:

devtools::install_github('dswatson/leakyIV')

Examples

First, generate data according to the leaky IV model.

set.seed(123)

# Hyperparameters
n <- 200
d_z <- 4
beta <- rep(1, d_z)
gamma <- rep(0.1, d_z)
theta <- 2
rho <- 0.5

# Simulate correlated residuals
S_eps <- matrix(c(1, rho, rho, 1), ncol = 2)
eps <- matrix(rnorm(n * 2), ncol = 2)
eps <- eps %*% chol(S_eps)

# Simulate observables from a leaky IV model
z <- matrix(rnorm(n * d_z), ncol = d_z)
x <- z %*% beta + eps[, 1]
y <- z %*% gamma + x * theta + eps[, 2]
obs <- cbind(x, y, z)

Now we run the leaky IV algorithm to compute lower and upper bounds on the ATE with a leakage threshold of $\tau = 1$.

leakyIV(obs, tau = 1)

To get a bootstrap distribution, we use the n_boot argument.

leakyIV(obs, tau = 1, n_boot = 10)

The function also works with covariance matrix input.

S <- cov(obs)
leakyIV(S, tau = 1)