Functional Generalized Empirical Likelihood Estimation for Conditional Moment Restrictions

Moment restrictions identify a parameter of interest by restricting the expectation value of so-called moment functions, which depend on the parameter and random variables representing the underlying noisy data generating process. Important problems in causal inference, economics, and generally robust machine learning can be cast in this form [Newey, 1993, Ai and Chen, 2003, Bennett and Kallus, 2020b, Dikkala et al., 2020]. Particularly challenging are problems formulated as conditional moment restrictions (CMR), which constrain the conditional expectation of the moment function. Such problems appear, e.g., in instrumental variable (IV) regression [Newey and Powell, 2003, Angrist and Pischke, 2008], where the expectation of the residual of the prediction conditioned on so-called instruments is restricted to be zero. Other applications are policy learning [Bennett and Kallus, 2020a] and off-policy evaluation in reinforcement learning [Kallus and Uehara, 2020, Bennett et al., 2021, Chen et al., 2021] and double/debiased machine learning [Chernozhukov et al., 2016, 2017, 2018]. As conditional moment restrictions are difficult to handle directly, a common approach is to transform them into an infinite number of corresponding unconditional moment restrictions Bierens [1982]. Generalizing the corresponding estimation methods from the finite dimensional case to the infinite case is an active area of research [Carrasco and Florens, 2000, Carrasco et al., 2007, Chaussé, 2012, Carrasco and Kotchoni, 2017, Muandet et al., 2020, Bennett and Kallus, 2020b, Zhang et al., 2021]. One of the most popular approaches to learning with moment restrictions is Hansen's celebrated generalized method of moments (GMM) [Hansen, 1982]. In order to improve the small sample properties of GMM estimators, alternative methods have been proposed and are generally known as generalized empirical likelihood (GEL) estimators [Smith, 1997, 2005, Newey and Smith, 2004].