On the Existence and Information of Orthogonal Moments

Locally Robust (LR)/Orthogonal/Debiased moments have proven useful with machine learning first steps, but their existence has not been investigated for general parameters. In this paper, we provide a necessary and sufficient condition, referred to as Restricted Local Non-surjectivity (RLN), for the existence of such orthogonal moments to conduct robust inference on general parameters of interest in regular semiparametric models. Importantly, RLN does not require either identification of the parameters of interest or the nuisance parameters. However, for orthogonal moments to be informative, the efficient Fisher Information matrix for the parameter must be non-zero (though possibly singular). Thus, orthogonal moments exist and are informative under more general conditions than previously recognized. We demonstrate the utility of our general results by characterizing orthogonal moments in a class of models with Unobserved Heterogeneity (UH). For this class of models our method delivers functional differencing as a special case. Orthogonality for general smooth functionals of the distribution of UH is also characterized. As a second major application, we investigate the existence of orthogonal moments and their relevance for models defined by moment restrictions with possibly different conditioning variables. We find orthogonal moments for the fully saturated two stage least squares, for heterogeneous parameters in treatment effects, for sample selection models, and for popular models of demand for differentiated products. We apply our results to the Oregon Health Experiment to study heterogeneous treatment effects of Medicaid on different health outcomes.