Locally Robust Semiparametric Estimation (1608.00033v4)

Abstract: Many economic and causal parameters depend on nonparametric or high dimensional first steps. We give a general construction of locally robust/orthogonal moment functions for GMM, where moment conditions have zero derivative with respect to first steps. We show that orthogonal moment functions can be constructed by adding to identifying moments the nonparametric influence function for the effect of the first step on identifying moments. Orthogonal moments reduce model selection and regularization bias, as is very important in many applications, especially for machine learning first steps. We give debiased machine learning estimators of functionals of high dimensional conditional quantiles and of dynamic discrete choice parameters with high dimensional state variables. We show that adding to identifying moments the nonparametric influence function provides a general construction of orthogonal moments, including regularity conditions, and show that the nonparametric influence function is robust to additional unknown functions on which it depends. We give a general approach to estimating the unknown functions in the nonparametric influence function and use it to automatically debias estimators of functionals of high dimensional conditional location learners. We give a variety of new doubly robust moment equations and characterize double robustness. We give general and simple regularity conditions and apply these for asymptotic inference on functionals of high dimensional regression quantiles and dynamic discrete choice parameters with high dimensional state variables.

Essay on "Locally Robust Semiparametric Estimation"

The paper "Locally Robust Semiparametric Estimation" by Chernozhukov et al. addresses the problem of estimating economic and causal parameters that rely on nonparametric or high-dimensional first steps. The research offers a comprehensive methodology for constructing locally robust, or orthogonal, moment functions for Generalized Method of Moments (GMM) estimators. These moment conditions are characterized by having zero derivative with respect to the first-stage estimation steps, thus significantly reducing bias associated with model selection and regularization, often encountered in machine learning applications.

Core Contributions

One of the paper's primary contributions is the delineation and application of orthogonal moment functions. These functions augment traditional identifying moments by incorporating the nonparametric influence function, which captures the effect of the first step on identifying moments. This approach enables the construction of debiased machine learning estimators for several econometric contexts, including functionals of high-dimensional conditional quantiles and dynamic discrete choice models with complex state spaces.

The research articulates a general debiasing technique, notably enhancing the asymptotic properties of the estimators. The result is a suite of novel, doubly robust moment equations characterized by the independence of asymptotic distribution from first-order biases in the initial estimation. The paper further demonstrates how cross-fitting—a sample-splitting technique—contributes to these improvements by mitigating biases related to own-observation dependencies and circumventing the need for Donsker conditions, a constraint not satisfied by many machine learning methods.

Theoretical Implications

This work has considerable implications for semiparametric estimation, particularly within the high-dimensional landscapes typified by modern econometrics and data science. By ensuring the orthogonality of moment conditions, the authors provide a robust framework that not only addresses first-step bias but also maintains valid confidence intervals in scenarios involving local alternatives and regularized initial estimates. The authors extend the literature on functional estimation, mapping their methodologies to broader contexts where existing approaches either falter or are overly restrictive.

Practical Applications

Practitioners employing machine learning algorithms such as Lasso, neural networks, and boosting would benefit significantly from the debiased GMM estimator outlined in this paper. The estimator's resilience against model selection and regularization biases make it particularly suited for applications in econometrics where high-dimensional covariates are at play. Moreover, the general approach to automating the estimation of unknown functions within the nonparametric influence function further eases the use of these techniques in empirical research.

Future Developments

The paper's approach suggests multiple avenues for future research, particularly in extending the robustness properties to broader classes of functions and exploring automatic estimation techniques for non-standard settings beyond conditional quantiles and dynamic choice. Another potential development would be the integration of advanced machine learning algorithms, examining not only their first-step applications but also their potential in further reducing estimator bias and enhancing inference accuracy.

In summary, "Locally Robust Semiparametric Estimation" provides an invaluable contribution to the econometric literature, offering practical tools and theoretical insights that address the challenges of high-dimensional, nonparametric first steps critical to many economic and causal parameters. The methodologies proposed not only advance the state-of-the-art in semiparametric estimation but also furnish a vital toolkit for econometricians and data scientists working with complex models in high-dimensional settings.