1. The Regression Discontinuity Design
Héctor Lamadrid-Figueroa
Center for Population Health Research
National Institute of Public Health, Mexico
Our last evaluation design is somewhat special as it works only in a particular set of programs, but in thse cases it might be very seful, particularly in settings when control groups are not available. Let’s see what it is about.

2. Basic Idea All these are potential applications of the
Sometimes an outcome it depends on you being
BELOW or ABOVE a threshold in a certain
“score”...
You get a scholarship if you get above a certain mark in an
exam
You get remedial education if you get below a certain
academic performance level
You get enrolled in a PROGRAMME if you are below a “poverty
threshold”
All these are potential applications of the
‘regression discontinuity’ design
On occasions, something like getting a scholarship, getting admitted at a school, getting a credit in a bank; all happen depending on reaching certain threshold in a continuous score. Whenever that occasion arises, it is potentially a good candidate to be analyzed by RDA.

3. Regression Discontunuity
Assignment to treatment depends in a discontinuous way on some observable variable S
Assignment to treatment being based on S being above some critical value s0- the discontinuity
Method of assignment to treatment is NOT random but totally deterministic!
However.... assignment to treatment is as good as random in very near the discontinuity....
What do we mean by this?
One if the key things here is that the Score ariable has to be OBSERVED. Another key thing is that in this cases assignment to treatment is completey deterministic, HOWEVER it can be as good as random in an interesting way.

4. Regression Discontinuity - Baseline
Suppose this graph represents the relationship between a continuous variable (such as income), and a “desirable” outcome (think of nutrition status). When people are making more money they tend to have better nutrition in average. This is the natural state of affairs, in the absence of interventions.
This is what happens “naturally”…. That is in the absence of any programme

5. Regression Discontinuity - Baseline
Threshold at 4.5
Now suppose you want to devise a program that will help poor people. And you decide that those below a threshold will get the intervention… If we were to ask you if you think that people to the left and the right f the discontinuity are compatable, you would probably quickly say NO, as the rich people are too the right and the poor to the left. HOWEVER… let’s look what happens if we “zoom in” to look at people very near the threshold.
4.5

6. …very near the threshold
Do you see any pattern here? These are the exact same datapoints, but zooming in near the threshold. They look quite similar now…. Let’s understand why: imagine the line represents the line of poverty which is living on 1.25 USD per day; do you think it would make much difference to live with 1.20 dollars a day rather than 1.30 dollars a day? Probably NOT… however, someone who makes 1.20 dollar will get the intervention wheras someoone with 1.30 dollars will not. This means that we can use the 2nd person as a control!
Subjects above and below the threshold are quite similar! Perhaps a good proxy of the counterfactual…

7. Regression Discontinuity design
We can compare subjects just above and just below the discontinuity in order to estimate the treatment effect
Where d defines a “window” near the threshold
As δ→0 this becomes:
.... Which is the treatment effect at S=s0
This formulas tell us that we need to define a window in which we can presume that the treated and untreated groups are quite similar. The difference in the functions will then be the impact effect…let’s explain.

8. RDD estimator
The RDD estimator compares the outcome of people who are just on both sides of the discontinuity - difference in means between these two groups is an estimate of the treatment effect at the discontinuity
Local treatment effect: says little about the treatment effect far from the discontinuity
A crucial assumption is that the underlying functional relationship of S and the outcome is linear
The difference between the two functions is called the RDD estimator. This estimator has two very important nuances though, the first is that we should bear in mind it is what we call a Local estimator, meaning that it probably only pertains to units near the threshold. The other thing is that the functional form of the relationship is linear or otherwise we have to correctly specify it.

9. Model specification
This is the classical way of getting the RDD estimator:
Eligibility Score
Covariates
Dummy variable for the cutoff point
This is what the model to get the RDD estimator looks like in general. Two very important things to notice: adummy variable to be above or below the threshold has to be included, the beta coefficient of this variable will be the RDD estimator; another important characteristic: we have to add the eligibility score as a continuous variable. We can add covariates if we like.

10. The regression line for those who received the program is…
The regression line for those who DID NOT receive the program is…
Therefore the impact of the program is …

11. Regression Discontinuity - Baseline
Threshold at 4.5
Not Eligible
Going back to our original example… those belo the threshold are deemed eligible. Please remember that this is NOTdone by the evaluator but this happens because that’s the way the program is actually implemented.
Eligible
4.5

12. Regression Discontinuity - Baseline
Group without Program
Group with Program
Not Eligible
Eligible
4.5

13. Regression Discontinuity: Baseline (Regression lines)
Group without Program
Group with Program
Not Eligible
If we fit the least squares line, we can see that there is a clear naturally occurring linear relationship betweent the score and the outcome.
Eligible
4.5

14. Regression Discontinuity : Post-Intervention
Group without Program
Group with Program
After the implementation fo the program, those who received the treatment move up in the outcome scale.. We now have two lines, which differ in the intercept. For those not treated, the intercept is b0, for the treated the intercept is b1… b1 is the RDD estimate.

15. Regression Discontinuity: Post-Intervention
Group without Program
Group with Program
Let’s look closely…

16. Regression Discontinuity I Post-Intervention
Impact
You can see there is a discontinuity right at the threshold, which is captured by our b1 estimate.

17. in the whole sample would be adequate
In theory: if the model is well specified an the underlying relationshp between the score and the outcome conditional on covariates is is TRULY a straight line, fitting …
in the whole sample would be adequate
HOWEVER, this is usually considered to be too risky!
Therefore we normally want to focus on those who are near the treshold
Fitting the model in the whole sample to get b1 is considered too risky as it is crucial to get the correct functional form, the chance that the function is too complicated is directly proportional to the range of the score that is included in the calculation. In other words, the chance thhat the functional form is too complicated is smaller if we look very close to the threshold. Choosing an incorrect functional form can have dire consequences in terms of bias , as we’ll see later.

18. Two possible estimation strategies:
Give more weight to observations near the threshold: It might be difficult to come up with a good formula to generate the weights or simply…
Define one, or preferably several, “windows” (define d).
Optimal bandwith procedures based on minimizing the Mean Squared Error are described in the literature :
Ludwig, J., and D. Miller. (2005). Does Head Start Improve Children’s Life Chances? Evidence from a Regression Discontinuity Design. NBER Working Paper Cambridge, MA: National Bureau of Economic Research.
Imbens, G. W., Kalyanaraman K. (2009). Optimal Bandwidth Choice for the Regression. Discontinuity Estimator. NBER Working Paper No
There are actually two ways to estimate the impact using RDD, one is using the window around the thresold approach, which is the most common and straightforward. The other approach is using weights that decrease proprtionally to the distance from the threshold, but in this case there is the problem of how to define weights. Windows can be defined based on “expert knowledge” or accoring to statistical criteria.

19. Regression Discontinuity (Window) - Baseline
Group without Program
Group with Program
Going back to our original example, imagine we select a window of 0.5 units from the threshold.
4.5
“Window”

20. Regression Discontinuity (Window) - Baseline
Group without Program
Group with Program
4.5
“Window”
We will use only those observations in the Window

21. Regression Discontinuity Post-Intervention
Group without Program
Group with Program
We can apply the described strategy to the data points within the window.
“Window”

22. RD: Looking at the window only- Post-intervention
Zooming in near the discontinuity… the impact estimate is the difference between both lines at the discontinuity
4.5

23. RD: Looking at the window only- Post-intervention
Impact
4.5

24. When RDD does not work… Case 1: Treatment effect is heterogeneous
Case 2: Underlying functional form is not a straight line
Now let’s see why RDD would NOT work correctly…

25. When RDA does not work: Heterogeneous treatment effect
Group with Program
Group without Program
Baseline
Going back to our example… imagine the following scenario…

26. When RDA does not work: Heterogeneous treatment effect
Group with Program
Group without Program
Post-intervention
Baseline
What happened here? You can see now that there was a heterogeneous effect: the “poorest of the poor” had a large positive impact, whereas those near the discontinuity had only a modest positive impact. If we only look at those units near the discontinuity we will underestimate the true impact.

27. When RDA does not work: Heterogeneous treatment effect
Group with Program
Group without Program
Post-intervention
Baseline
“Window”

28. When RDA does not work: True functional form not linear
One other very important problem for RDD is when the underlying function is misspecified. Let’s look at a hypothetical sitution in which the TRUE underlying functional form is a curve…

29. Discontinuity! (Program Impact Estimate)
Fitted RDA Model:
If we fit a “regular” simple model to this data, we will se a false “discontinuity” ath the threshold which is nothing but an artifact induced by the model misspecification.

30. This doesn’t solve the problem!
Fitted RDA Model:
Some might try to let the slopes of the linear functions at both sides of the thresholds vary, by introducing an interaction between the score and the treatment dummy variable, in order to minimize the bias, but we see that the problem does not entirely disappear. The only thing than can help us here is to specifiy the correct functional form. A practical advice is: try to model polynomials of the score starting with a high degree, look at the highest degree coefficient checking or significance, if insignificant remove said term and check the immediatley lower degree term..and repeat until reaching a statistically significant coefficient or reaching a 1st order polynomial (which corresponds to a linear model).

31. In summary…
RDD is a very useful strategy when a programme is offered to those below or above a threshold in a continuous variable
Identifying the correct functional form is crucial
To avoid this we look at observations near the threshold
We must remember that the impact estimate is LOCAL… we may miss heterogeneous effects
We have discussed the «sharp» design of RDD, «fuzzy» designs in which there is missclasification around the threshold combine RDD with instrumental variable methods.
… as a final comment, fuzzy designs of RDD combine instrumental variables with the methods we described in this presentation. In that case the instrument is the dummy variable for being above or below the threshold, and the instrumented variable is the actual treatment status.