Deploying Differential Distance as an Instrumental Variable: Alternative Forms, Estimators, and Specifications.

Despite well-established econometric theory, less attention is paid to the type of treatment effects being estimated using alternate instrumental variable (IV) approaches and the support for IV in the health literature. We illustrate this case using a commonly used IV-differential distance (DD). We summarize the literature and find that DD was used as an IV in various forms and approaches in the literature, leading to the estimation of different identified parameters, which were not always explained. We illustrate the sources of these differences using theoretical reasoning and a case study to evaluate the causal effects of going to a for-profit (FP) hospital versus a not-for-profit (NFP) hospital on the total cost of psychiatric inpatient stay. We find that estimates of treatment effects differ considerably when using two-stage least squares with binary versus continuous DD. In contrast, two-stage residual inclusion (2SRI) approaches using binary or continuous DD yield similar estimates of the treatment effects when we adequately model the control function. Both the 2SRI estimates are close to the average treatment effect estimate generated by local IV approaches, which can illustrate the extent of selection into FP versus NFP hospitals through marginal treatment effect heterogeneity.