Chapter 4: Aggregated Designs: RD Design Theory and Design

Including Additional Baseline Covariates

The inclusion in the RD models of additional covariates—measured at baseline—can improve the precision of the impact estimates. Similar to experimental designs, covariates can increase power by explaining some of the variance in the outcome measures across units (that is, by increasing regression R² values). The use of covariates (such as pretests) is especially important for improving precision in groupbased designs where statistical power is often a major concern (Bloom et al. 1999; Schochet 2008).

Conditional on the assignment scores, the covariates will be asymptotically uncorrelated with treatment status if (1) the outcome-score relationship is modeled correctly, and (2) the covariates are a continuous function of the scores (at the cutoff value). Thus, under a well-designed RD study, the inclusion of additional covariates in the RD model should have little effect on the impact estimates (and if they do, model specification error may be present). The situation is analogous to the use of covariates in experimental designs which are asymptotically uncorrelated with treatment status due to random assignment.

When additional covariates are included in the RD model, the asymptotic OLS variance estimator for α̂₁ can be expressed as follows:

where R_{RD_X}² is the asymptotic regression R² value when y_i^RD is regressed on T_i^RD, Score_i, and the vector of covariates X_i (which could include strata indicator variables). The analogous variance expression for the RA design is:

Importantly, the numerators in (12) and (13) are the same. Thus, the RD design effect in (11) applies also when additional covariates are included in the estimation models.