1. EMPIRICAL STUDY AND FORECASTING (II)
2018 Spring
By Elliott Fan
Economics, NTU
Elliott Fan: ESAF 2018 Spring Lecture 1

2. Elliott Fan: ESAF 2018 Spring Lecture 1
A little jumpy here
Elliott Fan: ESAF 2018 Spring Lecture 1

3. Elliott Fan: ESAF 2018 Spring Lecture 1
A little jumpy here
Mortality risk shoots up on and immediately following a twenty-first birthday
This spike adds about 100 deaths to a baseline level of about 150 per day.
The age-21 spike doesn’t seem to be a generic party-hardy birthday effect. If this spike reflects birthday partying alone, we should expect to see deaths shoot up after the twentieth and twenty-second birthdays as well, but that doesn’t happen.
There’s something special about the twenty-first birthday.
Elliott Fan: ESAF 2018 Spring Lecture 1

4. Elliott Fan: ESAF 2018 Spring Lecture 1
A little jumpy here
This figure plots death rates from all causes against age in months. The lines in the figure show fitted values from a regression of death rates on an over-21 dummy and age in months (the vertical dashed line indicates the minimum legal drinking age (MLDA) cutoff).
Elliott Fan: ESAF 2018 Spring Lecture 1

5. Elliott Fan: ESAF 2018 Spring Lecture 1
Define the terms
Forcing variable (or running variable): age
Cutoff point: the 21st birthday
Treatment variable: legal privilege for drinking alcohol
Outcome variable: mortality rate
Elliott Fan: ESAF 2018 Spring Lecture 1

6. Jumpy here and there -- examples
I combine a regression discontinuity design with rich data on academic and labor market outcomes for a large sample of Florida students to estimate the returns to college admission for academically marginal students. Students with grades just above a threshold for admissions eligibility at a large public university in Florida are much more likely to attend any university than below-threshold students. The marginal admission yields earnings gains of 22% between 8 and 14 years after high school completion. These gains outstrip the costs of college attendance, and they are largest for male students and free-lunch recipients.
Elliott Fan: ESAF 2018 Spring Lecture 1

7. Jumpy here and there -- examples
Elliott Fan: ESAF 2018 Spring Lecture 1

8. Jumpy here and there -- examples
Elliott Fan: ESAF 2018 Spring Lecture 1

9. What can we do to use the jump?
Recall that removing selection bias demands random assignment of the treatment
Observations just on the right of the cutoff point and those just on the left of the cutoff point are very similar except for the likelihood of treatment taking up
So the key requirement is that people cannot precisely control the exact position along the forcing variable around the cutoff point
Note that there are different degree of “control”
Elliott Fan: ESAF 2018 Spring Lecture 1

10. Methodologies – the RD design
The RD design utilizes the randomness from incomplete manipulation of a characteristic around the threshold point for the treatment.
From Lee and Lemieux (2009):
Elliott Fan: ESAF 2018 Spring Lecture 1

11. Elliott Fan: ESAF 2018 Spring Lecture 1
What is RDD?
A regression discontinuity design (RDD) is a quasi-experimental design that elicits the causal effects of interventions by assigning a cutoff or threshold above or below which an intervention is assigned. By comparing observations lying closely on either side of the threshold, it is possible to estimate the average treatment effect in environments in which a randomized trial was unfeasible.
Elliott Fan: ESAF 2018 Spring Lecture 1

12. Elliott Fan: ESAF 2018 Spring Lecture 1
Why RDD?
One major challenge of empirical studies is to pin down causality, which demands sources of exogenous variations in the “treatment” variables.
Such sources are rare.
Some variables exhibit a discontinuity, forming a potential source for such exogenous variations in the variable of interest.
Elliott Fan: ESAF 2018 Spring Lecture 1

13. Elliott Fan: ESAF 2018 Spring Lecture 1
Identification
Consider the following setup:
Forcing variable: X
Discontinuity: X=c
We can now define something we might like to know:
Treatment Effect at Discontinuity-TAD
Elliott Fan: ESAF 2018 Spring Lecture 1

14. Elliott Fan: ESAF 2018 Spring Lecture 1

15. Elliott Fan: ESAF 2018 Spring Lecture 1
Identification
In general, we cannot estimate TAD in a statistically reliable way. What can we estimate in the data? Let’s try an RD estimator. As before, the game will be to figure out what assumptions are necessary for the RD estimator (which we can see) to deliver the TAD (which we cannot see).
Elliott Fan: ESAF 2018 Spring Lecture 1

16. Elliott Fan: ESAF 2018 Spring Lecture 1
Identification
We will start with what we can estimate, then add and subtract three terms. After manipulation, we can clarify the assumptions necessary to get the TAD.
Now add and subtract three things:
Elliott Fan: ESAF 2018 Spring Lecture 1

17. Elliott Fan: ESAF 2018 Spring Lecture 1
Identification
Rearrange the equation, we obtain:
The four terms refer to:
TAD: this is our object of interest.
Selection bias: Captures any difference in Y0 between treatment and control. If assignment at discontinuity is as good as random, can assume this is zero.
Limit gap from above: Captures any jump in Y1 at c. Can assume zero if Y1 continuous at c.
Limit gap from below: Captures any jump in Y0 at c. Can assume zero if Y0 continuous at c.
Elliott Fan: ESAF 2018 Spring Lecture 1

18. Two types of designs -- Sharp and Fuzzy
Sharp design: When treatment taking is deterministically relied on eligibility. That is, everybody complies.
Fuzzy design: When treatment taking is NOT deterministically relied on eligibility. That is, there are non-compliers
Elliott Fan: ESAF 2018 Spring Lecture 1

19. Sharp and Fuzzy Designs
Imbens and Lemieux (2008)
Elliott Fan: ESAF 2018 Spring Lecture 1

20. Elliott Fan: ESAF 2018 Spring Lecture 1
An example of sharp RDD
Go back to the case of legal access to alcohol on death rates.
The treatment variable in this case can be written Da , where Da = 1 indicates legal drinking and is 0 otherwise. Da is a function of age a : the MLDA transforms 21-year-olds from underage minors to legal alcohol consumers.
We capture this transformation in mathematical notation by writing:
In sharp RD designs, treatment switches cleanly off or on as the running variable passes a cutoff.
Elliott Fan: ESAF 2018 Spring Lecture 1

21. Elliott Fan: ESAF 2018 Spring Lecture 1
Running a regression
Ma is the death rate in month a (again, month is defined as a 30-day interval counting from the twenty-first birthday).
Da, the treatment dummy, is defined as in the previous slide
a is a linear control for age in months.
Elliott Fan: ESAF 2018 Spring Lecture 1

22. Elliott Fan: ESAF 2018 Spring Lecture 1
Linear or non-linear?
Elliott Fan: ESAF 2018 Spring Lecture 1

23. Elliott Fan: ESAF 2018 Spring Lecture 1
Linear or non-linear?
Elliott Fan: ESAF 2018 Spring Lecture 1

24. Elliott Fan: ESAF 2018 Spring Lecture 1
Linear or non-linear?
Elliott Fan: ESAF 2018 Spring Lecture 1

25. Elliott Fan: ESAF 2018 Spring Lecture 1
Linear or non-linear?
Elliott Fan: ESAF 2018 Spring Lecture 1

26. Running a regression – normalized form
Ma is the death rate in month a (again, month is defined as a 30-day interval counting from the twenty-first birthday).
Da, the treatment dummy, is defined as in the previous slide
a is a linear control for age in months
a0 refers to the cutoff point
Elliott Fan: ESAF 2018 Spring Lecture 1

27. Running a regression – making it more flexible
Now, adding 𝛿[(𝑎− 𝑎 0 ) 𝐷 𝑎 ] allows the slope to be different before and after the cutoff point
You should be careful interpreting the main coefficients
Elliott Fan: ESAF 2018 Spring Lecture 1

28. Elliott Fan: ESAF 2018 Spring Lecture 1
Compare the results
Elliott Fan: ESAF 2018 Spring Lecture 1

29. Which function form is best?
There are no general rules here, and no substitute for a thoughtful look at the data.
When different forms provide varied estimates, caution is needed.
Elliott Fan: ESAF 2018 Spring Lecture 1

30. The next question: how do you know it is MLDA?
Elliott Fan: ESAF 2018 Spring Lecture 1

31. The next question: how do you know it is MLDA?
Elliott Fan: ESAF 2018 Spring Lecture 1