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 w 0 - 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 g 1 (W)-g 0 (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=w 0

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

Some pictures – underlying relationship between y and W is linear E(y│W) w0w0 W

Now introduce treatment E(y│W) w0w0 W β

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

Suppose true relationship is: E(y│W) w0w0 W g 0 (W) g 1 (W)

Observed relationship between E(y) and W E(y│W) w0w0 W g 0 (W) g 1 (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 30 th birthday

The Picture

The Estimates

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