Local linear tie-breaker designs

Tie-breaker experimental designs are hybrids of Randomized Control Trials (RCTs) and Regression Discontinuity Designs (RDDs) in which subjects with moderate scores are placed in an RCT while subjects with extreme scores are deterministically assigned to the treatment or control group. The design maintains the benefits of randomization for causal estimation while avoiding the possibility of excluding the most deserving recipients from the treatment group. The causal effect estimator for a tie-breaker design can be estimated by fitting local linear regressions for both the treatment and control group, as is typically done for RDDs. We study the statistical efficiency of such local linear regression-based causal estimators as a function of $\Delta$, the radius of the interval in which treatment randomization occurs. In particular, we determine the efficiency of the estimator as a function of $\Delta$ for a fixed, arbitrary bandwidth under the assumption of a uniform assignment variable. To generalize beyond uniform assignment variables and asymptotic regimes, we also demonstrate on the Angrist and Lavy (1999) classroom size dataset that prior to conducting an experiment, an experimental designer can estimate the efficiency for various experimental radii choices by using Monte Carlo as long as they have access to the distribution of the assignment variable. For both uniform and triangular kernels, we show that increasing the radius of randomized experiment interval will increase the efficiency until the radius is the size of the local-linear regression bandwidth, after which no additional efficiency benefits are conferred.