Marginal and Nested Structural Models Using Instrumental Variables

The objective of many scientific studies is to evaluate the effect of a treatment on an outcome of interest ceteris paribus. Instrumental variables (IVs) serve as an experimental handle, independent of potential outcomes and potential treatment status and affecting potential outcomes only through potential treatment status. We propose marginal and nested structural models using IVs, in the spirit of marginal and nested structural models under no unmeasured confounding. A marginal structural IV model parameterizes the expectations of two potential outcomes under an active treatment and the null treatment respectively, for those in a covariate-specific subpopulation who would take the active treatment if the instrument were externally set to each specific level. A nested structural IV model parameterizes the difference between the two expectations after transformed by a link function and hence the average treatment effect on the treated at each instrument level. We develop IV outcome regression, IV propensity score weighting, and doubly robust methods for estimation, in parallel to those for structural models under no unmeasured confounding. The regression method requires correctly specified models for the treatment propensity score and the outcome regression function. The weighting method requires a correctly specified model for the instrument propensity score. The doubly robust estimators depend on the two sets of models and remain consistent if either set of models are correctly specified. We apply our methods to study returns to education using data from the National Longitudinal Survey of Young Men.