发帖
雷达卡
Basic Idea
The regression discontinuity design exploits a jump or discontinuity in the likelihood of being treated at some threshold point. In the RKD design, there is instead a change in slope at the likelihood of being treated at a kink point, resulting in a discontinuity in the first-derivative of the assignment function. These types of kinks arise in a number of government policies.
For example, Simonsen et al. (2015) use a kink in the Danish government’s prescription drug reimbursement schedule: the subsidy is based on the total prescription costs the individual has paid during the year – there is 0% subsidy for the first 500 DKK in expenses, then 50% subsidy once you have paid 500 up until you have paid 1200, then 75% subsidy, and eventually an 80% subsidy for expenses above 2800 DKK. The result is that the share of the price paid out of pocket kinks as shown in Figure 1:
例如,西蒙森et al(2015)使用一个拐点例子说明丹麦政府的处方药还款计划:补贴是基于个人在一年的已有总处方支付费用。对于第一500丹麦克朗的花费，其补贴为0%，而一旦你花费超过500丹麦克朗，而一直没有达到1200丹麦克朗，其补贴为50%，接着是75%的补贴，最终对于超过2800丹麦克朗花费的补贴为80%。支付价格份额的拐点如图1所示
Figure 1: Y-axis is the share of the price paid out of pocket. It falls as one approaches 500 DKK since if you have spent 480 and buy something for 50, you get 0% subsidy on 20 DKK and then 50% subsidy on the 30 DKK that is expenditure past the threshold.
y轴是支付价格的份额。当接近500丹麦克朗时，它就会下降。因为如果你已经花了480丹麦克朗，再花50丹麦克朗，那么，20 丹麦克朗花费的补贴为0% ，超过临界值的30丹麦克朗花费的补贴为50%。
A second example is government unemployment insurance payments, which often cover up to some percentage of your previous wage, up to some maximum, and sometimes with a floor. For example, Figure 2, from a recent paper by Landais, shows the unemployment benefit in Louisiana as a function of the highest quarter earnings:
第二个例子是政府的失业保险金，它通常覆盖你以前工资的一定比例，最高可达一定的上限，有时还有最低下限。例如，Landais最近发表的一篇论文中的图2显示了路易斯安那州的失业救济金作为最高季度收入的函数:
Then the idea is to see whether the outcome of interest also exhibits a kink/change of slope in its relation to the running variable at this same point. For example, Landais is interested in how long people spend in unemployment as function of the amount of benefit they receive. Figure 3 below shows a strong change in the shape of the relationship between unemployment duration and highest quarter earnings at this kink-point.
接下来的思路是观察这样一个令人感兴趣的结果，即在同一点的斜率，以及与其有关的运行其变量之间也显示出一个的拐点/变化。例如，Landais感兴趣的是人们花在失业上的时间与他们得到的福利之间的函数关系。下面的图3显示了失业持续时间和最高季度收入之间的关系在这个拐点上的强烈变化。
Then the causal impact is found by dividing the change in slope for the outcome by the change in slope for the treatment.
When the policy rule is implemented with some error, a fuzzy RKD design can be used (see Card et al. (2015))
然后通过将结果变量的斜率变化除以处理变量的斜率变化得到因果影响。当政策规则实施伴随一些误差时，可以使用模糊RKD设计(参见Card et al. (2015))
Estimation
This can be estimated by using a polynomial regression. Let x be the running variable (e.g. highest quarter previous earnings), Y the outcome variable (e.g. duration of unemployment); k the kink point, and D a dummy variable for being to the right of the kink. Then estimate:
Y = a0 + a1*(x-k) + b1*D*(x-k) + a2*(x-k)^2 + a3*D*(x-k)^2 + …
Then b1 tells you the change in the slope of the outcome at the kink point.
Then run the same regression to get the change in the slope of the treatment variable at the kink point of the running variable, and take the ratio of the two coefficients. The standard error can then be bootstrapped or recovered using the Delta method.Estimation can also be done non-parametrically using local polynomials.
这可以用多项式回归来估计。设x为运行变量(如上季度最高收益)，Y为结果变量(如失业持续时间);k是拐点，D是拐点右边的虚拟变量。然后估计:
Y = a0 + a1*(x-k) + b1*D*(x-k) + a2*(x-k)^2 + a3*D*(x-k)^2 + …
b1告诉你拐点处结果变量斜率的变化。
然后进行同样的回归，得到处理变量在运行变量的拐点处的斜率的变化，并取这两个系数的比值（注意变化前后为两个值，在拐点左右两边，虚拟变量的取值不同....）。然后可以使用Delta方法自助抽样或恢复标准误。还可以使用局部多项式进行非参数估计。
Key assumptions and checks
The key assumptions are similar to those for regression discontinuity and are checked in similar ways:
There should be no change in slope at the kink point for other covariates (e.g. Landais shows no kinks in slope for age, education, capital income, or number of dependents at the kink point)
There is no manipulation of the assignment variable at the kink point – e.g. in the unemployment case, people don’t strategically manipulate their past earnings in expectation of how this will affect their future unemployment benefits. This can be tested by McCrary-type tests that show the distribution of the assignment variable is continuous at the kink point, and also that the first derivative of this pdf is also smooth. Card et al. (2015) show that you can allow agents to do some sorting around the kink points, so long as they make small optimization errors or mistakes so that this sorting is not completely deterministic.
This paper suggests some placebo tests you can do at non-kink points.
Technical details can be found in this Econometrica paper by Card et al. (2015).
Caveats
The RKD design can do pretty poorly in small samples, and typically will require a wider bandwidth or larger sample than you would need with a RDD.
There are methods for optimally choosing the bandwidth, and papers that aren’t able to justify this choice and/or show robustness to different bandwidth choices face criticism. See the comments on this AEJ Economic Policy paper for an example.
1. 其他协变量在拐点的斜率不应该发生变化(例如，Landais研究表明，在拐点处，年龄、教育程度、资本收入或受抚养人数量上的斜率没有变化)。
2.在拐点不存在对赋值变量的操纵——例如，在失业的例子，人们不会战略性地操纵他们过去的收入，以期望这将如何影响他们未来的失业福利。这可以通过McCrary-type检验来测试，该检验显示赋值变量在拐点的分布是连续的，而且该pdf的一阶导数也是平滑的。Card等人(2015)表明，当你允许代理人围绕拐点进行些排序， 只要代理犯很小优化误差或错误，这种排序就不会是完全确定的。
3.这篇论文建议在非拐点处进行一些安慰剂检验。
在Econometrica论文中可以找到Card等人(2015)提供技术细节。
警告：
RKD设计在小样本中做得很差，通常需要比RDD更宽的带宽或更大的样本。
有一些方法可以优化选择带宽，而那些无法证明这种选择的合理性和/或不能显示对不同带宽选择的稳健性的论文将面临批评。参见以这个AEJ的经济政策论文为例的一些评论。
参考文献:
The AEJ paper of Landais has data and replication Stata code available online.
This Stata Journal article by Calonico et al. has some RD code that also covers the RKD case.
美国经济学会网站的四篇参考文献：
https://www.aeaweb.org/search?q=+Regression+Kink+Design
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