Two-stage least squares with treatment-covariate interactions for treatment effect heterogeneity

Abstract

Treatment effect heterogeneity with respect to covariates is common in instrumental variable (IV) analyses. An intuitive approach, which we call the interacted two-stage least squares (2sls), is to postulate a working linear model of the outcome on the treatment, covariates, and treatment-covariate interactions, and instrument it using the IV, covariates, and IV-covariate interactions. We clarify the causal interpretation of the interacted 2sls under the local average treatment effect (LATE) framework when the IV is valid conditional on the covariates. Our main findings are threefold. First, we show that the coefficients on the treatment-covariate interactions from the interacted 2sls are consistent for estimating treatment effect heterogeneity with respect to covariates among compliers for any outcome-generating process if and only if the product of the IV propensity score and covariates are linear in the covariates, referred to as the linear IV-covariate interactions condition. Second, assuming that the covariate vector has dimension K and includes a constant term, we show that the linear IV-covariate interactions condition holds only if the IV propensity score takes at most K distinct values. As a result, this condition is difficult to satisfy beyond two special cases: (a) the covariates are categorical with K levels, or (b) the IV is randomly assigned. These results underscore the difficulty of interpreting regression coefficients from specifications with treatment-covariate interactions when the covariates are not saturated and the IV is not unconditionally randomized, absent correct specification of the outcome model. Third, as an application of our theory, we show that the interacted 2sls with demeaned covariates is consistent for estimating the LATE under the linear IV-covariate interactions condition.