How much should we trust regression-kink-design estimates?

In a regression kink (RK) design with a finite sample, a confounding smooth nonlinear relationship between an assignment variable and an outcome variable around a threshold can be spuriously picked up as a kink and results in a biased estimate. In order to investigate how well RK designs handle such confounding nonlinearity, I firstly implement Monte Carlo (MC) simulations and then study the effect of fiscal equalization grants on local expenditure in Japan using an RK design. Results in both the MC simulations and the empirical application suggest that RK estimation without covariates can be easily biased, and this problem can be mitigated by adding observed covariates to the regressors. On the other hand, a smaller bandwidth or a higher-order polynomial, even a quadratic polynomial, tends to result in imprecise estimates although they may be able to reduce estimation bias. In sum, RK estimation with confounding nonlinearity often suffers from bias or imprecision, and estimates are credible only when relevant covariates are controlled for. I also examine how placebo RK estimation can effectively address these issues.