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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,

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Regression Discontinuity

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Regression discontinuity l.jpg

Regression Discontinuity

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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

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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

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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):

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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

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  • 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

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Some pictures – underlying relationship between y and W is linear

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Now introduce treatment

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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

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  • 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.

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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

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Suppose true relationship is:

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Observed relationship between E(y) and W

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  • 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).

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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.

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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

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The Picture

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The Estimates

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

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