# Regression Discontinuity - PowerPoint PPT Presentation

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Regression Discontinuity. Basic Idea. 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,

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Regression Discontinuity

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## Regression Discontinuity

### Basic Idea

• 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

### More formally..

• 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

### Basics of RDD Estimator

• 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):

### How can we estimate this?

• 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

### The procedure in practice

• 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

### Dangers

• 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 underlying relationship linear

• 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

### Observed relationship between E(y) and 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).

### In practice

• 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.

### An example

• 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

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