Author(s): Bradic, Jelena; Wager, Stefan; Zhu, Yinchu | Abstract: Many popular methods for building confidence intervals on causal effects under high-dimensional confounding require strong "ultra-sparsity" assumptions that may be difficult to validate in practice. To alleviate this difficulty, we here study a new method for average treatment effect estimation that yields asymptotically exact confidence intervals assuming that either the conditional response surface or the conditional probability of treatment allows for an ultra-sparse representation (but not necessarily both). This guarantee allows us to provide valid inference for average treatment effect in high dimensions under considerably more generality than available baselines. In addition, we showcase that our results are semi-parametrically efficient.
@cite_4: We consider inference about a scalar parameter under a non-parametric model based on a one-step estimator computed as a plug in estimator plus the empirical mean of an estimator of the parameter's influence function. We focus on a class of parameters that have influence function which depends on two infinite dimensional nuisance functions and such that the bias of the one-step estimator of the parameter of interest is the expectation of the product of the estimation errors of the two nuisance functions. Our class includes many important treatment effect contrasts of interest in causal inference and econometrics, such as ATE, ATT, an integrated causal contrast with a continuous treatment, and the mean of an outcome missing not at random. We propose estimators of the target parameter that entertain approximately sparse regression models for the nuisance functions allowing for the number of potential confounders to be even larger than the sample size. By employing sample splitting, cross-fitting and @math -regularized regression estimators of the nuisance functions based on objective functions whose directional derivatives agree with those of the parameter's influence function, we obtain estimators of the target parameter with two desirable robustness properties: (1) they are rate doubly-robust in that they are root-n consistent and asymptotically normal when both nuisance functions follow approximately sparse models, even if one function has a very non-sparse regression coefficient, so long as the other has a sufficiently sparse regression coefficient, and (2) they are model doubly-robust in that they are root-n consistent and asymptotically normal even if one of the nuisance functions does not follow an approximately sparse model so long as the other nuisance function follows an approximately sparse model with a sufficiently sparse regression coefficient.
@cite_2: In this article we study a class of parameters with the so-called mixed bias property'. For parameters with this property, the bias of the semiparametric efficient one step estimator is equal to the mean of the product of the estimation errors of two nuisance functions. In non-parametric models, parameters with the mixed bias property admit so-called rate doubly robust estimators, i.e. estimators that are consistent and asymptotically normal when one succeeds in estimating both nuisance functions at sufficiently fast rates, with the possibility of trading off slower rates of convergence for the estimator of one of the nuisance functions with faster rates for the estimator of the other nuisance. We show that the class of parameters with the mixed bias property strictly includes two recently studied classes of parameters which, in turn, include many parameters of interest in causal inference. We characterize the form of parameters with the mixed bias property and of their influence functions. Furthermore, we derive two functional moment equations, each being solved at one of the two nuisance functions, as well as, two functional loss functions, each being minimized at one of the two nuisance functions. These loss functions can be used to derive loss based penalized estimators of the nuisance functions.
We also note two recent papers that consider estimators that resemble ours. consider an estimator that, in the spirit of , first fit a penalized covariate-balancing propensity model, and then re-fit without penalty those coefficients that correspond to features that are relevant to outcome modeling. Meanwhile, augments a penalized covariate-balancing propensity model in an outcome regression; it turns out that his covariate-balancing mechanism designed to address the issue of misspecification is also helpful for relaxing sparsity requirements. Neither paper, however, achieves sparsity double robustness as discussed here; rather, they require both the outcome parameter vector @math and the propensity parameter vector @math to be ultra-sparse---or, if there is misspecification they require the population minimizers of both the outcome and propensity loss functions to be ultra-sparse. Under the framework of @cite_4 @cite_2 , are classified as examples of model double robustness, which means that one of the models (either conditional mean or propensity score) is misspecified. Rate double robustness requires that the product of the @math -norms of the estimation errors in two models is of the order @math .