Post-selection and post-regularization inference in linear models with many controls and instruments

In this note, we offer an approach to estimating structural parameters in the presence of many instruments and controls based on methods for estimating sparse high-dimensional models. We use these high-dimensional methods to select both which instruments and which control variables to use. The approach we take extends Belloni et al. (2012), which covers selection of instruments for IV models with a small number of controls, and extends Belloni, Chernozhukov and Hansen (2014), which covers selection of controls in models where the variable of interest is exogenous conditional on observables, to accommodate both a large number of controls and a large number of instruments. We illustrate the approach with a simulation and an empirical example. Technical supporting material is available in a supplementary appendix.