Regression Discontinuity

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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 d?0 this comes to
• i.e. treatment effect at Ww0

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Comments

• 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 d 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 d

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Dangers

• If d is not very small then may not estimate just
  treatment effect look at picture
• As one increases d 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 d 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

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