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In this paper we address the problem of bias in machine learning of parameters following covariate shifts. Covariate shift occurs when the distribution of input features change between the training and deployment stages. Regularization and model selection associated with machine learning biases many parameter estimates. In this paper, we propose an automatic debiased machine learning approach to correct for this bias under covariate shifts. The proposed approach leverages state-of-the-art techniques in debiased machine learning to debias estimators of policy and causal parameters when covariate shift is present. The debiasing is automatic in only relying on the parameter of interest and not requiring the form of the form of the bias. We show that our estimator is asymptotically normal as the sample size grows. Finally, we demonstrate the proposed method on a regression problem using a Monte-Carlo simulation.

Bias correction can often improve the finite sample performance of estimators. We show that the choice of bias correction method has no effect on the higher-order variance of semiparametrically efficient parametric estimators, so long as the estimate of the bias is asymptotically linear. It is also shown that bootstrap, jackknife, and analytical bias estimates are asymptotically linear for estimators with higher-order expansions of a standard form. In particular, we find that for a variety of estimators the straightforward bootstrap bias correction gives the same higher-order variance as more complicated analytical or jackknife bias corrections. In contrast, bias corrections that do not estimate the bias at the parametric rate, such as the split-sample jackknife, result in larger higher-order variances in the i.i.d. setting we focus on. For both a cross-sectional MLE and a panel model with individual fixed effects, we show that the split-sample jackknife has a higher-order variance term that is twice as large as that of the `leave-one-out' jackknife.

Many causal and policy effects of interest are defined by linear functionals of high-dimensional or non-parametric regression functions. $\sqrt{n}$-consistent and asymptotically normal estimation of the object of interest requires debiasing to reduce the effects of regularization and/or model selection on the object of interest. Debiasing is typically achieved by adding a correction term to the plug-in estimator of the functional, which leads to properties such as semi-parametric efficiency, double robustness, and Neyman orthogonality. We implement an automatic debiasing procedure based on automatically learning the Riesz representation of the linear functional using Neural Nets and Random Forests. Our method only relies on black-box evaluation oracle access to the linear functional and does not require knowledge of its analytic form. We propose a multitasking Neural Net debiasing method with stochastic gradient descent minimization of a combined Riesz representer and regression loss, while sharing representation layers for the two functions. We also propose a Random Forest method which learns a locally linear representation of the Riesz function. Even though our method applies to arbitrary functionals, we experimentally find that it performs well compared to the state of art neural net based algorithm of Shi et al. (2019) for the case of the average treatment effect functional. We also evaluate our method on the problem of estimating average marginal effects with continuous treatments, using semi-synthetic data of gasoline price changes on gasoline demand., Comment: Accepted for a long presentation at the ICML. Code available at https://github.com/victor5as/RieszLearning

Debiased machine learning is a meta algorithm based on bias correction and sample splitting to calculate confidence intervals for functionals, i.e. scalar summaries, of machine learning algorithms. For example, an analyst may desire the confidence interval for a treatment effect estimated with a neural network. We provide a nonasymptotic debiased machine learning theorem that encompasses any global or local functional of any machine learning algorithm that satisfies a few simple, interpretable conditions. Formally, we prove consistency, Gaussian approximation, and semiparametric efficiency by finite sample arguments. The rate of convergence is $n^{-1/2}$ for global functionals, and it degrades gracefully for local functionals. Our results culminate in a simple set of conditions that an analyst can use to translate modern learning theory rates into traditional statistical inference. The conditions reveal a general double robustness property for ill posed inverse problems., Comment: Biometrika 2022

A variety of interesting parameters may depend on high dimensional regressions. Machine learning can be used to estimate such parameters. However estimators based on machine learners can be severely biased by regularization and/or model selection. Debiased machine learning uses Neyman orthogonal estimating equations to reduce such biases. Debiased machine learning generally requires estimation of unknown Riesz representers. A primary innovation of this paper is to provide Riesz regression estimators of Riesz representers that depend on the parameter of interest, rather than explicit formulae, and that can employ any machine learner, including neural nets and random forests. End-to-end algorithms emerge where the researcher chooses the parameter of interest and the machine learner and the debiasing follows automatically. Another innovation here is debiased machine learners of parameters depending on generalized regressions, including high-dimensional generalized linear models. An empirical example of automatic debiased machine learning using neural nets is given. We find in Monte Carlo examples that automatic debiasing sometimes performs better than debiasing via inverse propensity scores and never worse. Finite sample mean square error bounds for Riesz regression estimators and asymptotic theory are also given., Comment: arXiv admin note: text overlap with arXiv:1809.05224

Multidimensional heterogeneity and endogeneity are important features of models with multiple treatments. We consider a heterogeneous coefficients model where the outcome is a linear combination of dummy treatment variables, with each variable representing a different kind of treatment. We use control variables to give necessary and sufficient conditions for identification of average treatment effects. With mutually exclusive treatments we find that, provided the heterogeneous coefficients are mean independent from treatments given the controls, a simple identification condition is that the generalized propensity scores (Imbens, 2000) be bounded away from zero and that their sum be bounded away from one, with probability one. Our analysis extends to distributional and quantile treatment effects, as well as corresponding treatment effects on the treated. These results generalize the classical identification result of Rosenbaum and Rubin (1983) for binary treatments., Comment: 19 pages, 1 figure. This is the scientifically accepted version

This paper is about the feasibility and means of root-n consistently estimating linear, mean-square continuous functionals of a high dimensional, approximately sparse regression. Such objects include a wide variety of interesting parameters such as regression coefficients, average derivatives, and the average treatment effect. We give lower bounds on the convergence rate of estimators of a regression slope and an average derivative and find that these bounds are substantially larger than in a low dimensional, semiparametric setting. We also give debiased machine learners that are root-n consistent under either a minimal approximate sparsity condition or rate double robustness. These estimators improve on existing estimators in being root-n consistent under more general conditions that previously known.

The drift diffusion model (DDM) is a model of sequential sampling with diffusion (Brownian) signals, where the decision maker accumulates evidence until the process hits a stopping boundary, and then stops and chooses the alternative that corresponds to that boundary. This model has been widely used in psychology, neuroeconomics, and neuroscience to explain the observed patterns of choice and response times in a range of binary choice decision problems. This paper provides a statistical test for DDM's with general boundaries. We first prove a characterization theorem: we find a condition on choice probabilities that is satisfied if and only if the choice probabilities are generated by some DDM. Moreover, we show that the drift and the boundary are uniquely identified. We then use our condition to nonparametrically estimate the drift and the boundary and construct a test statistic.

Multidimensional heterogeneity and endogeneity are important features of a wide class of econometric models. We consider heterogenous coefficients models where the outcome is a linear combination of known functions of treatment and heterogenous coefficients. We use control variables to obtain identification results for average treatment effects. With discrete instruments in a triangular model we find that average treatment effects cannot be identified when the number of support points is less than or equal to the number of coefficients. A sufficient condition for identification is that the second moment matrix of the treatment functions given the control is nonsingular with probability one. We relate this condition to identification of average treatment effects with multiple treatments., Comment: 15 pages

Many causal and structural effects depend on regressions. Examples include policy effects, average derivatives, regression decompositions, average treatment effects, causal mediation, and parameters of economic structural models. The regressions may be high dimensional, making machine learning useful. Plugging machine learners into identifying equations can lead to poor inference due to bias from regularization and/or model selection. This paper gives automatic debiasing for linear and nonlinear functions of regressions. The debiasing is automatic in using Lasso and the function of interest without the full form of the bias correction. The debiasing can be applied to any regression learner, including neural nets, random forests, Lasso, boosting, and other high dimensional methods. In addition to providing the bias correction we give standard errors that are robust to misspecification, convergence rates for the bias correction, and primitive conditions for asymptotic inference for estimators of a variety of estimators of structural and causal effects. The automatic debiased machine learning is used to estimate the average treatment effect on the treated for the NSW job training data and to estimate demand elasticities from Nielsen scanner data while allowing preferences to be correlated with prices and income., Comment: Econometrica 2022

There are many interesting and widely used estimators of a functional with finite semiparametric variance bound that depend on nonparametric estimators of nuisance functions. We use cross-fitting (i.e. sample splitting) to construct novel estimators with fast remainder rates. We give cross-fit doubly robust estimators that use separate subsamples to estimate different nuisance functions. We obtain general, precise results for regression spline estimation of average linear functionals of conditional expectations with a finite semiparametric variance bound. We show that a cross-fit doubly robust spline regression estimator of the expected conditional covariance is semiparametric efficient under minimal conditions. Cross-fit doubly robust estimators of other average linear functionals of a conditional expectation are shown to have the fastest known remainder rates for the Haar basis or under certain smoothness conditions. Surprisingly, the cross-fit plug-in estimator also has nearly the fastest known remainder rate, but the remainder converges to zero slower than the cross-fit doubly robust estimator. As specific examples we consider the expected conditional covariance, mean with randomly missing data, and a weighted average derivative.

Robins et al. (2008, 2017) applied the theory of higher order influence functions (HOIFs) to derive an estimator of the mean $\psi$ of an outcome Y in a missing data model with Y missing at random conditional on a vector X of continuous covariates; their estimator, in contrast to previous estimators, is semiparametric efficient under the minimal conditions of Robins et al. (2009b), together with an additional (non-minimal) smoothness condition on the density g of X, because the Robins et al. (2008, 2017) estimator depends on a nonparametric estimate of g. In this paper, we introduce a new HOIF estimator that has the same asymptotic properties as the original one, but does not impose any smoothness requirement on g. This is important for two reasons. First, one rarely has the knowledge about the properties of g. Second, even when g is smooth, if the dimension of X is even moderate, accurate nonparametric estimation of its density is not feasible at the sample sizes often encountered in applications. In fact, to the best of our knowledge, this new HOIF estimator remains the only semiparametric efficient estimator of $\psi$ under minimal conditions, despite the rapidly growing literature on causal effect estimation. We also show that our estimator can be generalized to the entire class of functionals considered by Robins et al. (2008) which include the average effect of a treatment on a response Y when a vector X suffices to control confounding and the expected conditional variance of a response Y given a vector X. Simulation experiments are also conducted, which demonstrate that our new estimator outperforms those of Robins et al. (2008, 2017) in finite samples, when g is not very smooth., Comment: 42 pages

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.

Shape restrictions have played a central role in economics as both testable implications of theory and sufficient conditions for obtaining informative counterfactual predictions. In this paper we provide a general procedure for inference under shape restrictions in identified and partially identified models defined by conditional moment restrictions. Our test statistics and proposed inference methods are based on the minimum of the generalized method of moments (GMM) objective function with and without shape restrictions. Uniformly valid critical values are obtained through a bootstrap procedure that approximates a subset of the true local parameter space. In an empirical analysis of the effect of childbearing on female labor supply, we show that employing shape restrictions in linear instrumental variables (IV) models can lead to shorter confidence regions for both local and average treatment effects. Other applications we discuss include inference for the variability of quantile IV treatment effects and for bounds on average equivalent variation in a demand model with general heterogeneity. We find in Monte Carlo examples that the critical values are conservatively accurate and that tests about objects of interest have good power relative to unrestricted GMM.

Many useful parameters depend on nonparametric first steps. Examples include games, dynamic discrete choice, average exact consumer surplus, and treatment effects. Often estimators of these parameters are asymptotically equivalent to a sample average of an object referred to as the influence function. The influence function is useful in local policy analysis, in evaluating local sensitivity of estimators, constructing debiased machine learning estimators, in efficiency comparisons, and in formulating primitive regularity conditions for asymptotic normality, We show that the influence function is a Gateaux derivative with respect to a smooth deviation evaluated at a point mass. This result generalizes the classic Von Mises (1947) and Hampel (1974) calculation to estimators that depend on smooth nonparametric first steps. We give explicit influence functions for first steps that satisfy exogenous or endogenous orthogonality conditions. We use these results to generalize the omitted variable bias formula for regression to policy analysis for and sensitivity to structural changes. We apply this analysis and find no sensitivity to endogeneity of average equivalent variation estimates in a gasoline demand application.

The linear regression model is widely used in empirical work in Economics, Statistics, and many other disciplines. Researchers often include many covariates in their linear model specification in an attempt to control for confounders. We give inference methods that allow for many covariates and heteroskedasticity. Our results are obtained using high-dimensional approximations, where the number of included covariates are allowed to grow as fast as the sample size. We find that all of the usual versions of Eicker-White heteroskedasticity consistent standard error estimators for linear models are inconsistent under this asymptotics. We then propose a new heteroskedasticity consistent standard error formula that is fully automatic and robust to both (conditional)\ heteroskedasticity of unknown form and the inclusion of possibly many covariates. We apply our findings to three settings: parametric linear models with many covariates, linear panel models with many fixed effects, and semiparametric semi-linear models with many technical regressors. Simulation evidence consistent with our theoretical results is also provided. The proposed methods are also illustrated with an empirical application.

Non-standard distributional approximations have received considerable attention in recent years. They often provide more accurate approximations in small samples, and theoretical improvements in some cases. This paper shows that the seemingly unrelated "many instruments asymptotics" and "small bandwidth asymptotics" share a common structure, where the object determining the limiting distribution is a V-statistic with a remainder that is an asymptotically normal degenerate U-statistic. We illustrate how this general structure can be used to derive new results by obtaining a new asymptotic distribution of a series estimator of the partially linear model when the number of terms in the series approximation possibly grows as fast as the sample size, which we call "many terms asymptotics".

In parametric, nonlinear structural models a classical sufficient condition for local identification, like Fisher (1966) and Rothenberg (1971), is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix. We derive an analogous result for the nonparametric, nonlinear structural models, establishing conditions under which an infinite-dimensional analog of the full rank condition is sufficient for local identification. Importantly, we show that additional conditions are often needed in nonlinear, nonparametric models to avoid nonlinearities overwhelming linear effects. We give restrictions on a neighborhood of the true value that are sufficient for local identification. We apply these results to obtain new, primitive identification conditions in several important models, including nonseparable quantile instrumental variable (IV) models, single-index IV models, and semiparametric consumption-based asset pricing models.