Selecting Valid Instrumental Variables in Linear Models with Multiple Exposure Variables: Adaptive Lasso and the Median-of-Medians Estimator

TL;DR

This paper develops a method combining adaptive Lasso and median-of-medians estimators to select valid instruments in linear models with multiple exposures, improving causal effect estimation accuracy.

Contribution

It extends the median-of-medians estimator to multiple exposures and integrates it with adaptive Lasso for consistent instrument selection with oracle properties.

Findings

The proposed method achieves consistent instrument selection in simulations.

The median-of-medians estimator requires only moderate conditions for validity.

Application to Mendelian Randomization demonstrates practical utility.

Abstract

In a linear instrumental variables (IV) setting for estimating the causal effects of multiple confounded exposure/treatment variables on an outcome, we investigate the adaptive Lasso method for selecting valid instrumental variables from a set of available instruments that may contain invalid ones. An instrument is invalid if it fails the exclusion conditions and enters the model as an explanatory variable. We extend the results as developed in Windmeijer et al. (2019) for the single exposure model to the multiple exposures case. In particular we propose a median-of-medians estimator and show that the conditions on the minimum number of valid instruments under which this estimator is consistent for the causal effects are only moderately stronger than the simple majority rule that applies to the median estimator for the single exposure case. The adaptive Lasso method which uses the initial median-of-medians estimator for the penalty weights achieves consistent selection with oracle properties of the resulting IV estimator. This is confirmed by some Monte Carlo simulation results. We apply the method to estimate the causal effects of educational attainment and cognitive ability on body mass index (BMI) in a Mendelian Randomization setting.