Regression discontinuity design

The most common non-parametric method used in the RDD context is a local linear regression. This is of the form:

${\displaystyle Y=\alpha +\tau D+\beta _{1}(X-c)+\beta _{2}D(X-c)+\varepsilon ,}$

where ${\displaystyle c}$ is the treatment cutoff and ${\displaystyle D}$ is a binary variable equal to one if ${\displaystyle X\geq c}$. Letting ${\displaystyle h}$ be the bandwidth of data used, we have ${\displaystyle c-h\leq X\leq c+h}$. Different slopes and intercepts fit data on either side of the cutoff. Typically either a rectangular kernel (no weighting) or a triangular kernel are used. The rectangular kernel has a more straightforward interpretation over sophisticated kernels which yield little efficiency gains.[4]

The major benefit of using non-parametric methods in an RDD is that they provide estimates based on data closer to the cut-off, which is intuitively appealing. This reduces some bias that can result from using data farther away from the cutoff to estimate the discontinuity at the cutoff.[4] More formally, local linear regressions are preferred because they have better bias properties[5] and have better convergence.[6] However, the use of both types of estimation, if feasible, is a useful way to argue that the estimated results do not rely too heavily on the particular approach taken.

An example of a parametric estimation is:

${\displaystyle Y=\alpha +\beta _{1}x_{i}+\beta _{2}c_{i}+\beta _{3}c_{i}^{2}+\beta _{4}c_{i}^{3}+\varepsilon ,}$

where

${\displaystyle x_{i}={\begin{cases}1{\text{ if }}c_{i}\geq {\bar {c}}\\0{\text{ if }}c_{i}<{\bar {c}}\end{cases}}}$

and ${\displaystyle {\bar {c}}}$ is the treatment cutoff. Note that the polynomial part can be shortened or extended according to the needs.

• Policies in which treatment is determined by an age eligibility criterion (e.g. pensions, minimum legal drinking age).[7][8]
• Elections in which one politician wins by a marginal majority.[9][10]
• Placement scores within education that sort students into treatment programs.[11]

It is impossible to definitively test for validity if agents are able to determine their treatment status perfectly. However, some tests can provide evidence that either supports or discounts the validity of the regression discontinuity design.

McCrary (2008)[12] density test on data from Lee, Moretti, and Butler (2004).[13]

McCrary (2008) suggested examining the density of observations of the assignment variable.[12] Suppose there is a discontinuity in the density of the assignment variable at the threshold for treatment. In this case, this may suggest that some agents were able to manipulate their treatment status perfectly.

For example, if several students are able to get a "mercy pass", then there will be more students who just barely passed the exam than who just barely failed. Similarly, if students are allowed to retake the exam until they pass, then there will be a similar result. In both cases, this will likely show up when the density of exam grades is examined. "Gaming the system" in this manner could bias the treatment effect estimate.

Since the validity of the regression discontinuity design relies on those who were just barely treated being the same as those who were just barely not treated, it makes sense to examine if these groups are similarly based on observable variables. For the earlier example, one could test if those who just barely passed have different characteristics (demographics, family income, etc.) than those who just barely failed. Although some variables may differ for the two groups based on random chance, most of these variables should be the same.[13]

Similar to the continuity of observable variables, one would expect there to be continuity in predetermined variables at the treatment cutoff. Since these variables were determined before the treatment decision, treatment status should not affect them. Consider the earlier merit-based scholarship example. If the outcome of interest is future grades, then we would not expect the scholarship to affect previous grades. If a discontinuity in predetermined variables is present at the treatment cutoff, then this puts the validity of the regression discontinuity design into question.

If discontinuities are present at other points of the assignment variable, where these are not expected, then this may make the regression discontinuity design suspect. Consider the example of Carpenter and Dobkin (2011) who studied the effect of legal access to alcohol in the United States.[8] As the access to alcohol increases at age 21, this leads to changes in various outcomes, such as mortality rates and morbidity rates. If mortality and morbidity rates also increase discontinuously at other ages, then it throws the interpretation of the discontinuity at age 21 into question.

If parameter estimates are sensitive to removing or adding covariates to the model, then this may cast doubt on the validity of the regression discontinuity design. A significant change may suggest that those who just barely got treatment to differ in these covariates from those who just barely did not get treatment. Including covariates would remove some of this bias. If a large amount of bias is present, and the covariates explain a significant amount of this, then their inclusion or exclusion would significantly change the parameter estimate.[4]

Recent work has shown how to add covariates, under what conditions doing so is valid, and the potential for increased precision.[14]

The identification of causal effects hinges on the crucial assumption that there is indeed a sharp cut-off, around which there is a discontinuity in the probability of assignment from 0 to 1. In reality, however, cutoffs are often not strictly implemented (e.g. exercised discretion for students who just fell short of passing the threshold) and the estimates will hence be biased.

In contrast to the sharp regression discontinuity design, a fuzzy regression discontinuity design (FRDD) does not require a sharp discontinuity in the probability of assignment. Still, it is applicable as long as the probability of assignment is different. The intuition behind it is related to the instrumental variable strategy and intention to treat. Fuzzy RDD does not provide an unbiased estimate when the quantity of interest is the proportional effect (e.g. vaccine effectiveness), but extensions exist that do.[17]

When the assignment variable is continuous (e.g. student aid) and depends predictably on another observed variable (e.g. family income), one can identify treatment effects using sharp changes in the slope of the treatment function. This technique was coined regression kink design by Nielsen, Sørensen, and Taber (2010), though they cite similar earlier analyses.[18] They write, "This approach resembles the regression discontinuity idea. Instead of a discontinuity of in the level of the stipend-income function, we have a discontinuity in the slope of the function." Rigorous theoretical foundations were provided by Card et al. (2012)[19] and an empirical application by Bockerman et al. (2018).[20]

Note that regression kinks (or kinked regression) can also mean a type of segmented regression, which is a different type of analysis.

Final Considerations

The RD design takes the shape of a quasi-experimental research design with a clear structure that is devoid of randomized experimental features. Several aspects deny the RD designs an allowance for a status quo. For instance, the designs often involve serious issues that do not offer room for random experiments. Besides, the design of the experiments depends on the accuracy of the modelling process and the relationship between inputs and outputs.