  When an exposure of interest is confounded by unmeasured factors, an
instrumental variable (IV) can be used to identify and estimate certain causal
contrasts. Identification of the marginal average treatment effect (ATE) from
IVs relies on strong untestable structural assumptions. When one is unwilling
to assert such structure, IVs can nonetheless be used to construct bounds on
the ATE. Famously, Balke and Pearl (1997) proved tight bounds on the ATE for a
binary outcome, in a randomized trial with noncompliance and no covariate
information. We demonstrate how these bounds remain useful in observational
settings with baseline confounders of the IV, as well as randomized trials with
measured baseline covariates. The resulting bounds on the ATE are non-smooth
functionals, and thus standard nonparametric efficiency theory is not
immediately applicable. To remedy this, we propose (1) under a novel margin
condition, influence function-based estimators of the bounds that can attain
parametric convergence rates when the nuisance functions are modeled flexibly,
and (2) estimators of smooth approximations of these bounds. We propose
extensions to continuous outcomes, explore finite sample properties in
simulations, and illustrate the proposed estimators in an observational study
targeting the effect of higher education on wages.
