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Instrumental variables have been widely used for estimating the causal effectbetween exposure and outcome. Conventional estimation methods require completeknowledge about all the instruments' validity; a valid instrument must not havea direct effect on the outcome and not be related to unmeasured confounders.Often, this is impractical as highlighted by Mendelian randomization studieswhere genetic markers are used as instruments and complete knowledge aboutinstruments' validity is equivalent to complete knowledge about the involvedgenes' functions.  In this paper, we propose a method for estimation of causal effects when thiscomplete knowledge is absent. It is shown that causal effects are identifiedand can be estimated as long as less than $50$% of instruments are invalid,without knowing which of the instruments are invalid. We also introduceconditions for identification when the 50% threshold is violated. A fastpenalized $\ell_1$ estimation method, called sisVIVE, is introduced forestimating the causal effect without knowing which instruments are valid, withtheoretical guarantees on its performance. The proposed method is demonstratedon simulated data and a real Mendelian randomization study concerning theeffect of body mass index on health-related quality of life index. An R package\emph{sisVIVE} is available online.

Instrumental Variables Estimation with Some Invalid Instruments and its Application to Mendelian Randomization