  In genetical genomics studies, it is important to jointly analyze gene
expression data and genetic variants in exploring their associations with
complex traits, where the dimensionality of gene expressions and genetic
variants can both be much larger than the sample size. Motivated by such modern
applications, we consider the problem of variable selection and estimation in
high-dimensional sparse instrumental variables models. To overcome the
difficulty of high dimensionality and unknown optimal instruments, we propose a
two-stage regularization framework for identifying and estimating important
covariate effects while selecting and estimating optimal instruments. The
methodology extends the classical two-stage least squares estimator to high
dimensions by exploiting sparsity using sparsity-inducing penalty functions in
both stages. The resulting procedure is efficiently implemented by coordinate
descent optimization. For the representative $L_1$ regularization and a class
of concave regularization methods, we establish estimation, prediction, and
model selection properties of the two-stage regularized estimators in the
high-dimensional setting where the dimensionality of covariates and instruments
are both allowed to grow exponentially with the sample size. The practical
performance of the proposed method is evaluated by simulation studies and its
usefulness is illustrated by an analysis of mouse obesity data. Supplementary
materials for this article are available online.
