  This paper considers generalized linear models in the presence of many
controls. We lay out a general methodology to estimate an effect of interest
based on the construction of an instrument that immunize against model
selection mistakes and apply it to the case of logistic binary choice model.
More specifically we propose new methods for estimating and constructing
confidence regions for a regression parameter of primary interest $\alpha_0$, a
parameter in front of the regressor of interest, such as the treatment variable
or a policy variable. These methods allow to estimate $\alpha_0$ at the
root-$n$ rate when the total number $p$ of other regressors, called controls,
potentially exceed the sample size $n$ using sparsity assumptions. The sparsity
assumption means that there is a subset of $s<n$ controls which suffices to
accurately approximate the nuisance part of the regression function.
Importantly, the estimators and these resulting confidence regions are valid
uniformly over $s$-sparse models satisfying $s^2\log^2 p = o(n)$ and other
technical conditions. These procedures do not rely on traditional consistent
model selection arguments for their validity. In fact, they are robust with
respect to moderate model selection mistakes in variable selection. Under
suitable conditions, the estimators are semi-parametrically efficient in the
sense of attaining the semi-parametric efficiency bounds for the class of
models in this paper.
