  Identification in a regression discontinuity (RD) research design hinges on
the discontinuity in the probability of treatment when a covariate (assignment
variable) exceeds a known threshold. When the assignment variable is measured
with error, however, the discontinuity in the relationship between the
probability of treatment and the observed mismeasured assignment variable may
disappear. Therefore, the presence of measurement error in the assignment
variable poses a direct challenge to treatment effect identification. This
paper provides sufficient conditions to identify the RD treatment effect using
the mismeasured assignment variable, the treatment status and the outcome
variable. We prove identification separately for discrete and continuous
assignment variables and study the properties of various estimation procedures.
We illustrate the proposed methods in an empirical application, where we
estimate Medicaid takeup and its crowdout effect on private health insurance
coverage.
