  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.
