Regression Kink Design

Regression kink (RK) design is useful for retrieving causal estimates when a mostly differentiably continuous policy variable (unemployment insurance payments, earned-income tax credits) has kinks, or discontinuities in the first derivative.

Where the RD estimates the causal relationship from a jump in the outcome associated with a jump in the treatment, the RK estimates the causal relationship from a kink in the outcome associated with a kink in the treatment.

The regression analysis is run first by retrieving B1 in this equation:

Y = A0 + A1(v-k) + B1D*(v-k) + A2(v-k)^2 + A2D*(v-k)^2 + ...

Where Y is the outcome variable of interest; k is the kink point; v is the running variable, and D allows the slopes to change at the kink.

The B1 retrieves the change in slope in outcomes at the kink point in the treatment variable.

The causal impact per unit is found by dividing B1 by the difference in derivatives around the kink which is calculated mathematically rather than econometrically. This step may not be intuitive. Think about it a little bit. The kink in outcomes needs to be scaled by the kink in treatment. Consider an outcome where there's a modest kink. Perhaps the kink in treatment is also modest, or perhaps it is fairly striking. We account for the degree of the kink by dividing the outcome slope change by the treatment slope change.

Here's a nice graph that may shed some more light on RK designs:

This is from Camille Landais' paper on unemployment insurance. Here, benefits increase with highest quarterly earnings until some maximum level. If you earn more than a certain amount, you get the same level of unemployment benefits regardless of how much more you earn. Camille exploits this. He finds this kink in the duration of unemployment when plotted on the highest quarterly earnings of the individual (this is the running variable). He then divides the slope change observed above by the slope change in the benefits level at that kink.

The early papers with RK from Card and Lee don't explain how to run the analysis. There is an unreadable paper by Ying Ying Dong on the subject and here is a job market paper from Cornell that explains the practical part of RK estimation.

There is a small wrinkle: sometimes a the policy that makes a treatment kink is not always followed. For instance, we use a kink in the benefit formula in Missouri's unemployment insurance program to estimate the effect of benefit generosity on labor supply. Some individuals were awarded benefit amounts that are not consistent with the formula and thus the kink variation does not apply to them. To address this, practitioners can use a fuzzy RKD design just as a fuzzy RDD can work in contexts were not everyone's treatment status is determined by their placement around the threshold. As in a fuzzy RD, the first stage kink needs to be estimated and used to scale the reduced form effect to get the causal effect of treatment (Card et al. 2015)

Colleagues at Harvard University, Peter Ganong and Simon Jager, provide a useful placebo test using a permutation test that can be implemented with RK. Card et al. (2015) point out that while this test has an intuitive appeal, practitioners need use caution; the test may not be informative if the curvature of the conditional expectation function changes. Ganong and Jager are using as placebo estimates the estimated kinks at other points in the running variable. If the curvature changes, then the distribution of empirical estimates does not reflect the true distribution of apples-to-apples estimates.

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