Estimand of ordinary least squares and instrumental variable estimator for any outcome variable and any effect heterogeneity

Ordinary least squares (OLS) applied to a linear regression that controls covariates linearly is widely used in applied work. However, such linear specifications are generally misspecified when treatment effects are heterogeneous or when the outcome is noncontinuous. Instrumental variable estimator (IVE) is also often used for similar linear specifications with covariates entering additively. The important question is therefore what OLS and IVE actually estimate under misspecifications. There is now a fairly well-established literature addressing this issue. This paper reviews the literature and derives a new decomposition of the linear IV estimand into a weighted component of conditional local average treatment effects and a bias term. In an application to 401(k) plans, estimated effects on net financial assets differ substantially across specifications, with standard OLS and IV producing smaller estimates than methods that account more flexibly for nonlinear covariate relationships.