  This paper explores the validity of the two-stage estimation procedure for
sparse linear models in high-dimensional settings with possibly many endogenous
regressors. In particular, the number of endogenous regressors in the main
equation and the instruments in the first-stage equations can grow with and
exceed the sample size n. The analysis concerns the exact sparsity case, i.e.,
the maximum number of non-zero components in the vectors of parameters in the
first-stage equations, k1, and the number of non-zero components in the vector
of parameters in the second-stage equation, k2, are allowed to grow with n but
slowly compared to n. I consider the high-dimensional version of the two-stage
least square estimator where one obtains the fitted regressors from the
first-stage regression by a least square estimator with l_1-regularization (the
Lasso or Dantzig selector) when the first-stage regression concerns a large
number of instruments relative to n, and then construct a similar estimator
using these fitted regressors in the second-stage regression. The main
theoretical results of this paper are non-asymptotic bounds from which I
establish sufficient scaling conditions on the sample size for estimation
consistency in l_2-norm and variable-selection consistency. A technical issue
regarding the so-called "restricted eigenvalue (RE) condition" for estimation
consistency and the "mutual incoherence (MI) condition" for selection
consistency arises in the two-stage estimation from allowing the number of
regressors in the main equation to exceed n and this paper provides analysis to
verify these RE and MI conditions. Depending on the underlying assumptions, the
upper bounds on the l_2-error and the sample size required to obtain these
consistency results differ by factors involving k1 and/or k2. Simulations are
conducted to gain insight on the finite sample performance.
