Inferring causal effects with unmeasured confounder is a main challenge in causal inference. Many researchers impose parametric assumptions on the distribution of unmeasured confounder. However, due to the unobservable nature of the unmeasured confounder, it is more reasonable to leave its distribution unrestricted. Another key challenge in causal inference is the involvement of invalid instrumental variables, which may lead to biased inference and potentially deceptive results. To this end, we employ a flexible semiparametric model that allows for possibly invalid instruments without specifying the distribution of unmeasured confounder in this work. A penalized semiparametric estimator for causal effects is constructed and its oracle and asymptotic properties are well established for statistical inference. We evaluate the performance of the estimator through simulation studies, revealing that our proposed estimator exhibits asymptotic unbiasedness and robustness in estimating causal effects, along with consistent selection of invalid instruments. We also demonstrate its application using Atherosclerosis Risk in Communities Study data set, which further validates its robustness in the presence of invalid instruments.
Cao et al. (Sun,) studied this question.