We propose an l1-regularized M-estimator for a high-dimensional sparse parameter that is identified by a class of semiparametric conditional moment restrictions ( CMR). We estimate the nonparametric nuisance parameter by modern machine learning methods. Plugging the first-stage estimate into the CMR, we construct the M-estimator loss functionfor the target parameter so that its gradient is insensitive (formally, Neyman-orthogonal) with respect to the first-stage regularization bias. As a result, the estimator achieves oracle convergence rate \sqrt{k \log p/n oracle knows the true first stage and solves only a parametric problem. We apply our results to conditional moment models with missing data , games of incomplete information and treatment effects in regression models with non-linear link functions .
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 .
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We propose an l1-regularized M-estimator for a
This paper contributes to the literature on
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parameter that is identified by
M-estimation by allowing the loss function to depend on a functional nuisance parameter, which we estimate by modern machine learning tools. For
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semiparametric
single-index
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CMR). We estimate the nonparametric nuisance parameter by modern machine learning methods. Plugging the first-stage estimate into the CMR, we construct the M-estimator loss functionfor the target parameter so that its gradient
CMRs), we explicitly derive the loss function. We first adjust the moment function so that the gradient of the future M-estimator loss
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As a result, the estimator achieves
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
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\sqrt{k \log p/n
, where the
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true first stage
nuisance parameter
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the
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Our framework nests a novel approach to modeling heterogeneous treatment effects with a binary dependent variable. In addition, we
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and static
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and treatment effects in regression models with non-linear link functions
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