  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.
