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J Am Stat Assoc. 2018 Nov 13;114(527):1339-1350. doi: 10.1080/01621459.2018.1498346. eCollection 2019.

On the Use of the Lasso for Instrumental Variables Estimation with Some Invalid Instruments.

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Department of Economics, University of Bristol, Bristol, United Kingdom.
MRC Integrative Epidemiology Unit, Bristol, United Kingdom.
Center for the Economics of Aging, Max Planck Society Munich, Germany.
School of Social and Community Medicine, University of Bristol, Bristol, United Kingdom.


We investigate the behavior of the Lasso for selecting invalid instruments in linear instrumental variables models for estimating causal effects of exposures on outcomes, as proposed recently by Kang et al. Invalid instruments are such that they fail the exclusion restriction and enter the model as explanatory variables. We show that for this setup, the Lasso may not consistently select the invalid instruments if these are relatively strong. We propose a median estimator that is consistent when less than 50% of the instruments are invalid, and its consistency does not depend on the relative strength of the instruments, or their correlation structure. We show that this estimator can be used for adaptive Lasso estimation, with the resulting estimator having oracle properties. The methods are applied to a Mendelian randomization study to estimate the causal effect of body mass index (BMI) on diastolic blood pressure, using data on individuals from the UK Biobank, with 96 single nucleotide polymorphisms as potential instruments for BMI. Supplementary materials for this article are available online.


Causal inference; Instrumental variables estimation; Invalid instruments; Lasso; Mendelian randomization.

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