arXiv2025
Non-binary instrumental variables, especially continuous ones, are common in practice. A binary recoding induces a Wald ratio but may discard useful variation and reduce efficiency. Although fully nonparametric approaches can in principle use the entire instrument, they often require high-dimensional nuisance estimation which can be unstable with rich covariates. We address this problem by developing a generalized Wald estimand for binary treatments that uses the full variation in a non-binary instrument. Under standard instrumental-variable assumptions and a homogeneity condition, the estimand yields a common identification formula for categorical and continuous instruments. We further develop its semiparametric efficiency theory and construct a locally efficient debiased estimator using risk-minimization reparameterizations and double cross-fitting to accommodate flexible machine learning while improving numerical stability. The central technical challenge is that the many Wald ratios generated by a non-binary instrument must agree, thereby imposing overidentifying restrictions on the observed-data law. In this setting, characterizing the tangent space is nonstandard: it requires a second-order parametric submodel, a construction that, to our knowledge, has not been standard in semiparametric efficiency theory. Simulations show stable performance across sample sizes and greater efficiency than estimators based on dichotomized instruments. In an application to the Princess Margaret Cancer Centre lung cancer cohort, associational analyses link excess body weight to lower two-year mortality, a seemingly protective pattern often called the obesity paradox. The proposed instrumental-variable analysis instead suggests increased mortality, pointing to residual confounding behind this paradox.