Adaptive Estimation of Quadratic Functionals in Nonparametric Instrumental Variable Models

Long before the recent popularity of instrumental variables in modern machine learning causal inference and biostatistics, the instrumental variables technique has been widely used in economics. For instance, instrumental variables regressions are frequently used to account for omitted variables, mis-measured regressors, endogeneity in simultaneous equations and other complex situations in observational data. In economics and other social sciences, as well as in medical research, it is very difficult to estimate causal effects using observational data sets alone. When treatment assignment is not randomized, it is generally impossible to discern between the causal effect of treatments and spurious correlations that are induced by unobserved factors. Instrumental variables are commonly used to provide exogenous variation that is associated with the treatment status, but not with the outcome variable (beyond its direct effect on the treatments). To avoid mis-specification of parametric functional forms, the nonparametric instrumental variables regressions (NPIV) have gained popularity in econometrics and modern causal inference in statistics and machine learning.