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Regression Discontinuity

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Regression Discontinuity

- Sometimes whether something happens to you or not depends on your ‘score’ on a particular variable e.g
- You get a scholarship if you get above a certain mark in an exam,
- you get given remedial education if you get below a certain level,
- a policy is implemented if it gets more than 50% of the vote in a ballot,
- your sentence for a criminal offence is higher if you are above a certain age (an ‘adult’)

- All these are potential applications of the ‘regression discontinuity’ design

- assignment to treatment depends in a discontinuous way on some observable variable W
- simplest form has assignment to treatment being based on W being above some critical value w0- the discontinuity
- method of assignment to treatment is the very opposite to that in random assignment – it is a deterministic function of some observable variable.
- But, assignment to treatment is as ‘good as random’ in the neighbourhood of the discontinuity – this is hard to grasp but I hope to explain it

- Suppose average outcome in absence of treatment conditional on W is:
- Suppose average outcome with treatment conditional on W is:
- This is ‘full outcomes’ approach.
- Treatment effect conditional on W is g1(W)-g0(W):

- Basic idea is to compare outcomes just to the left and right of discontinuity i.e. to compare:
- As δ→0 this comes to:
- i.e. treatment effect at W=w0

- 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
- says nothing about the treatment effect away from the discontinuity - this is a limitation of the RDD effect.
- An important assumption is that underlying effect on W on outcomes is continuous so only reason for discontinuity is treatment effect

E(y│W)

w0

W

E(y│W)

β

w0

W

- If take process described above literally should choose a value of δ that is very small
- This will result in a small number of observations
- Estimate may be consistent but precision will be low
- desire to increase the sample size leads one to choose a larger value of δ

- If δ is not very small then may not estimate just treatment effect – look at picture
- As one increases δ the measure of the treatment effect will get larger. This is spurious so what should one do about it?
- The basic idea is that one should control for the underlying outcome functions.

- If the linear relationship is the correct specification then one could estimate the ATE simply by estimating the regression:
- But no good reason to assume relationship is linear and this may cause problems

g0(W)

E(y│W)

g1(W)

w0

W

g0(W)

E(y│W)

g1(W)

w0

W

- one would want to control for a different relationship between y and W for the treatment and control groups
- Another problem is that the outcome functions might not be linear in W – it could be quadratic or something else.
- The researcher then typically faces a trade-off:
- a large value of δ to get more precision from a larger sample size but run the risk of a misspecification of the underlying outcome function.
- Choose a flexible underlying functional form at the cost of some precision (intuitively a flexible functional form can get closer to approximating a discontinuity in the outcomes).

- it is usual for the researcher to summarize all the data in the graph of the outcome against W to get some idea of the appropriate functional forms and how wide a window should be chosen.
- But its always a good idea to investigate the sensitivity of estimates to alternative specifications.

- Lemieux and Milligan “Incentive Effects of Social Assistance: A regression discontinuity approach”, Journal of Econometrics, 2008
- In Quebec before 1989 childless benefit recipients received higher benefits when they reached their 30th birthday

- Note that the more flexible is the underlying relationship between employment rate and age, the less precise is the estimate