  We discuss the fundamental issue of identification in linear instrumental
variable (IV) models with unknown IV validity. With the assumption of the
"sparsest rule", which is equivalent to the plurality rule but becomes
operational in computation algorithms, we investigate and prove the advantages
of non-convex penalized approaches over other IV estimators based on two-step
selections, in terms of selection consistency and accommodation for
individually weak IVs. Furthermore, we propose a surrogate sparsest penalty
that aligns with the identification condition and provides oracle sparse
structure simultaneously. Desirable theoretical properties are derived for the
proposed estimator with weaker IV strength conditions compared to the previous
literature. Finite sample properties are demonstrated using simulations and the
selection and estimation method is applied to an empirical study concerning the
effect of BMI on diastolic blood pressure.
