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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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basic idea
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
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
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
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
the procedure in practice
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
  • 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 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
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
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
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 that the more flexible is the underlying relationship between employment rate and age, the less precise is the estimate