On the use of the Lasso for instrumental variables estimation with some invalid instruments

We investigate the behaviour of the Lasso for selecting invalid instruments in linear instrumental variables models for estimating causal effects of exposures on outcome, as proposed recently by Kang, Zhang, Cai and Small (2016, Journal of the American Statistical Association). 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 select all invalid instruments in large samples if they are relatively strong. Consistent selection also depends on the correlation structure of the instruments. We propose a median estimator that is consistent when less than 50% of the instruments are invalid, but its consistency does not depend on the strength of the instruments or their correlation structure. This estimator can therefore be used for adaptive Lasso estimation. The methods are applied to a Mendelian randomisation study to estimate the causal effect of BMI on diastolic blood pressure using data on individuals from the UK Biobank, with 96 single nucleotide polymorphisms as potential instruments for BMI.