Abstract:
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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. 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 exceed the sample size $n$, using sparsity assumptions. The estimators and these resulting confidence regions are valid uniformly over $s$-sparse models. Moreover, these procedures are robust with respect to ``moderate" model selection mistakes in variable selection steps. Moreover, the estimators are semi-parametrically efficient in the sense of attaining the semi-parametric efficiency bounds for the class of models in this paper.
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