A Permutation Test for the Regression Kink Design

The Regression Kink (RK) design is an increasingly popular empirical method for causal inference. Analogous to the Regression Discontinuity design, which evaluates discontinuous changes in the level of an outcome variable with respect to the running variable at a point at which the level of a policy changes, the RK design evaluates discontinuous changes in the slope of an outcome variable with respect to the running variable at a kink point at which the slope of a policy with respect to the running variable changes. We document empirically that RK estimates are highly sensitive to nonlinearity in the underlying relationship between the outcome and the assignment variable. As an alternative to standard inference, we propose that researchers construct a distribution of placebo estimates in regions with and without a policy kink and use this distribution to gauge statistical significance. Under the assumption that the location of the kink point is random, this permutation test has exact size in finite samples for testing a sharp null hypothesis of no effect of the policy on the outcome. In simulation studies with policy kinks, we find that statistical significance based on conventional standard errors may be spurious. In contrast, our permutation test has exact size even in the presence of non-linearity.