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2457096_part1Slidesforwebsite9-regressiondiscontinuitydesign.pptxEC338: regression discontinuity design
RDD exploits the rules governing entitlement to treatment to create a treatment and control group either side of a cut-off
e.g. date of birth or income and entitlement to benefits
Idea is that treatment is close to randomly allocated around discontinuity, and hence that comparison gives causal effect
The further from the discontinuity, the less likely this is to hold, but can account for selection by controlling for running variable
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
To undertake RDD:
z and z0 (the assignment rule) must be observable to the researcher
In addition, the following assumptions must hold:
Assumption 1: individuals cannot perfectly manipulate z
Assumption 2: and are continuous in z at z0
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Assumption 1: individuals cannot perfectly manipulate z
Imagine z is a test score. Individuals scoring at or above a given threshold are given a scholarship; those scoring below are not
We want to estimate the impact of scholarships on attainment
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
z is clearly not entirely randomly determined
Smarter, more motivated students are likely get higher test scores
But can they ensure that they score just above rather than just below the threshold? Perhaps not . . .
There could be positive or negative shocks to performance (e.g. illness on exam day) or marking (e.g. stricter or less strict exam graders)
The influence of such factors is what we rely on for treatment to be as good as randomly determined in the vicinity of the cut-off
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Assumption 2: and are continuous in
Effectively means that all observed and unobserved factors determining the outcome are continuous at the threshold
i.e. that there are no jumps in other covariates that could be correlated with any observed change in the outcomes at z0
Treatment is the only thing that changes discontinuously at the cut-off; hence comparing outcomes of those just above and just below should give us a causal estimate of the treatment
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Some things to note . . .
A key assumption for OLS (which we saw explicitly in the context of matching) is that there is “common support”
i.e. that we observe individuals with the same (observed) characteristics who do and do not receive treatment, so that we can compare their outcomes to estimate the causal treatment effect
But with RDD, there is no value of z at which we observe individuals receiving and not receiving treatment
We must therefore rely on extrapolation to identify what would have happened to the relationship between z and y in the absence of the discontinuity
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
This means that functional form – specifically the relationship between z (or c) and y – matters a lot
e.g. if the relationship is non-linear and we constrain it to be linear, then we will get a biased estimate of the effect at the discontinuity
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
RDD only estimates treatment effect around cut-off
i.e. it cannot tell us how much very low- or very high- scoring students might benefit from a scholarship (unless treatment effects are homogeneous)
Generalisability of this treatment effect is uncertain
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
An empirical example:
Lee (2008), Randomized experiments from non-random selection in US House elections, Journal of Econometrics, Vol. 142, pp. 675-697
Incumbent candidates or parties (those who won the last election) are more likely to win the next election
Is this because incumbents are inherently more electable?
Or because incumbency gives them an advantage?
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
To identify incumbency advantage, we’d like to rule out that those who were elected before are “better” candidates and hence more likely to be elected again
i.e. we’d like to randomise chances of winning election and compare outcomes of those who won vs. those who lost
Can proxy such a situation using RDD . . .
There is a clear threshold for victory:
You need more votes than your opponent
If you reach this threshold, you are almost certainly “treated” (i.e. almost certainly take up office)
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
What is y?
Election outcome or winning margin in election at time t+1
What is D?
Whether candidate/party is incumbent, i.e. won last election
What is z?
Vote share at last election (at time t)
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Is this design likely to meet the assumptions required?
Assumption 1:
Vote share cannot be perfectly manipulated
Credible? May be circumstances on the day that sway turn-out in an unpredictable way (e.g. weather)
Assumption 2:
All other factors determining election outcome/vote share at t+1 are continuous around discontinuity
Credible? Yes if believe variation like that described above is random (i.e. not related to characteristics of winners/losers)
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Discontinuities in other characteristics around cut-off?
Placebo test: effect of vote share at time t on election result at t-1
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
In this example, treatment was basically “deterministic”
i.e. once we know z, we know D with certainty
e.g. once we knew vote share, we knew who was in office
Or if kids with low test scores are forced to attend remediation class
This is known as a “sharp” RDD
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
But often only the probability of treatment varies at the cut-off
e.g. if kids scoring below a threshold had the option to attend the class
Or parents of 3 year olds have the option to send their child to nursery
Or those with income below a cut-off have the option to receive benefits
This is known as a “fuzzy” RDD
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Under sharp RDD:
Under fuzzy RDD:
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Can we still use a regression discontinuity design if only the probability of treatment varies?
i.e. if the discontinuity is “fuzzy” rather than “sharp”?
Yes – but how?
Combine it with IV methods . . .
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Let’s go back to our returns to schooling example . . .
We were interested in the effect of schooling on wages:
But were concerned that S might be endogenous
So we looked for sources of variation in S that we thought would be unrelated to unobservable characteristics
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
One way to generate exogenous variation in schooling is date of birth discontinuities in starting or leaving ages . . .
In England kids start school in September after they turn four → treatment varies discontinuously around 1 Sept
Sample of kids born in August or September: we give them a test on 1 September after their 5th birthday
Those born in August have 1 year of schooling by this point while those born in September have none
Use this to estimate causal effect of one year of schooling
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: instrumental variables
If admissions rule was implemented perfectly, then schooling is a deterministic function of age and we have a sharp RDD
If we observe (and can control for) age, we can fully account for the selection of individuals into different amounts of schooling
We can estimate the causal effect of S on y using this approach
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: instrumental variables
But what if the admissions rule wasn’t deterministic?
What if children could start school in the September after they turn four, but didn’t have to start until they turned five?
We are re-introducing some choice into the admissions process
We can no longer perfectly observe (and hence no longer fully control for) the selection process
So a regression of y on S may not produce a causal estimate
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
What do we do in this case?
There are two options . . .
If we observe date of birth (but not when kids actually started school), then use an intention to treat or reduced form approach
Uses assigned treatment status rather than observed treatment status, thus avoiding potentially endogenous variation in actual schooling choices
Still enables us to answer a policy relevant question, e.g. what is the impact of starting school in an area in which the policy is X rather than Y?
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
If we observe both date of birth and school start date, then can use assigned treatment status (based on date of birth) as an instrument for observed treatment status (actual school start)
Is this likely to be a valid instrument (relevant and exogenous)?
Relevant: unless there are a lot of non-compliers, assigned treatment status should be highly correlated with observed treatment status
Exogenous:
Relies on assigned treatment status being (conditionally) random and observed treatment status being the only route through which initial treatment status matters
Danger here would be if we think parents choose to have children on one side of the discontinuity or the other: could potentially violate both if selection is unobserved
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Our basic sharp RDD model would be:
Simplest fuzzy RDD model instruments S using D where
First stage:
Second stage:
Handout 9 for EC338 2017-18: Claire Crawford
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EC338: regression discontinuity design
Summary of key learning points:
RDD exploits discontinuous changes in treatment from assignment rules based on observed characteristics
Generates something close to random assignment of treatment near the discontinuity, assuming that:
Individuals cannot perfectly manipulate the running variable
and are continuous in z at z0
And hence enables us to estimate causal effect of treatment – but only in the vicinity of cut-off
Handout 9 for EC338 2017-18: Claire Crawford
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