  Consider the problem of estimating the local average treatment effect with an
instrument variable, where the instrument unconfoundedness holds after
adjusting for a set of measured covariates. Several unknown functions of the
covariates need to be estimated through regression models, such as instrument
propensity score and treatment and outcome regression models. We develop a
computationally tractable method in high-dimensional settings where the numbers
of regression terms are close to or larger than the sample size. Our method
exploits regularized calibrated estimation, which involves Lasso penalties but
carefully chosen loss functions for estimating coefficient vectors in these
regression models, and then employs a doubly robust estimator for the treatment
parameter through augmented inverse probability weighting. We provide rigorous
theoretical analysis to show that the resulting Wald confidence intervals are
valid for the treatment parameter under suitable sparsity conditions if the
instrument propensity score model is correctly specified, but the treatment and
outcome regression models may be misspecified. For existing high-dimensional
methods, valid confidence intervals are obtained for the treatment parameter if
all three models are correctly specified. We evaluate the proposed methods via
extensive simulation studies and an empirical application to estimate the
returns to education.
