This paper contributes to the literature on high-dimensional sparse M-estimation by allowing the loss function to depend on a functional nuisance parameter, which we estimate by modern machine learning tools. For a class of single-index conditional moment restrictions (CMRs), we explicitly derive the loss function. We first adjust the moment function so that the gradient of the future M-estimator loss is insensitive (formally, Neyman-orthogonal) with respect to the first-stage regularization bias  . We then take the loss function to be an indefinite integral of the adjusted moment function with respect to the single-index . The proposed l1-regularized M-estimator achieves the oracle convergence rate, where the oracle knows the nuisance parameter and solves only the parametric problem. Our framework nests a novel approach to modeling heterogeneous treatment effects with a binary dependent variable. In addition, we apply our results to conditional moment models with missing data and static games of incomplete information. Finally, we generalize our results to generic extremum estimation with a nuisance component .
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 .
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contributes to the literature on
proposes a Lasso-type estimator for a
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M-estimation by allowing the loss function to
parameter identified by a single index conditional moment restriction (CMR). In addition to this parameter, the moment function can also
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functional nuisance parameter,
nuisance function, such as the propensity score or the conditional choice probability,
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For a class of single-index conditional moment restrictions (CMRs), we explicitly derive the loss function.
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M-estimator loss
loss function
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, preserving the single index property
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single-index
single index
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l1-regularized M-estimator achieves the oracle convergence
Lasso estimator converges at the oracle
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parameter
function
R
Our framework nests a novel approach to modeling heterogeneous treatment effects with a binary dependent variable. In addition, we apply our results to conditional moment models with missing data and static games of incomplete information. Finally, we generalize our results to generic extremum estimation with a nuisance component
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