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

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.

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About Frank Windmeijer

Frank obtained his PhD in Econometrics in 1992 from the University of Amsterdam under the supervision of Heinz Neudecker and Piet Groeneboom. He subsequently held positions at the Dept of Statistics at the ANU in Canberra, the Dept of Economics at UCL and the Institute for Fiscal Studies in London, where he was co-director of the ESRC Centre for Microdata Methods and Practice, and the University of Bristol. He currently is a professorial research fellow the Dept of Statistics and Nuffield College at the University of Oxford. His main research focus early on was on dynamic panel/longitudinal data estimation methods. He is currently working on causal inference using instrumental variables methods, dealing with weak and/or invalid instruments, with applications in Mendelian randomisation.