  This paper proposes a Lasso-type estimator for a high-dimensional sparse
parameter identified by a single index conditional moment restriction (CMR). In
addition to this parameter, the moment function can also depend on a nuisance
function, such as the propensity score or the conditional choice probability,
which we estimate by modern machine learning tools. We first adjust the moment
function so that the gradient of the future loss function is insensitive
(formally, Neyman-orthogonal) with respect to the first-stage regularization
bias, preserving the single index property. We then take the loss function to
be an indefinite integral of the adjusted moment function with respect to the
single index. The proposed Lasso estimator converges at the oracle rate, where
the oracle knows the nuisance function and solves only the parametric problem.
We demonstrate our method by estimating the short-term heterogeneous impact of
Connecticut's Jobs First welfare reform experiment on women's welfare
participation decision.
