Regression Discontinuity/Event StudiesMethods of Economic InvestigationLecture 21

Last TimeNon-StationarityOrders of IntegrationDifferencingUnit Root Tests

Estimating Causality in Time SeriesA brief introduction to forecasting Impulse Response Functions

Todays ClassReturning to Causal EffectsBrief return to Impulse Response FunctionsEvent Studies/Regression DiscontinuityTesting for Structural Breaks

What happens with theres a shock?Source: Cochrane, QJE (1994)

Impulse Response Function and CausalityImpulse Response Function:Can look, starting at time t was there a changeDont know if shock (or treatment) was independent.

The issue is the counterfactualWhat would have happened in the future if the shock had not occurredORWhat would the past have looked like, in a world where the treatment existed

Return to Selection BiasBack to old selection bias problem:Shock occurs in time t and we observe a change in yMaybe y would have changed anyway at time tE[Yt | shock = 1] Et-1[Yt | shock = 0]} =E[Yt | shock = 1] E[Yt | shock = 0] + {E[Yt | shock = 0] Et-1 [Yt | shock = 0]}

The issue is that shocks/treatments are not randomly assigned to a time period

Basic IdeaSometimes something changes sharply with time: e.gyour sentence for a criminal offence is higher if you are above a certain age (an adult)The interest changes suddenly/surprisingly at after a meetingThere is a change in the CEO/manager at a firm

Do outcomes also change sharply?

Not just time seriesDoesnt only have to be time could be some other dimension with a discontinuous change 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,

All these are potential applications of the regression discontinuity design

Treatment Assignmentassignment to treatment (T) depends in a discontinuous way on some observable variable t simplest form has assignment to treatment being based on t being above some critical value t0t0 is the discontinuity or break date

method of assignment to treatment is the very opposite to that in random assignment it is a deterministic function of some observable variable.assignment to treatment is as good as random in the neighbourhood of the discontinuityThe basic ideano reason other outcomes should be discontinuous but for treatment assignment rule

Basics of EstimationSuppose average outcome in absence of treatment conditional on is:

Suppose average outcome with treatment conditional on t is:

Treatment effect conditional on t isg1(t) g0(t)This is full outcomes approach

How can we estimate this?Basic idea is to compare outcomes just to the left and right of discontinuity i.e. to compare:

As 0 this comes to:

i.e. treatment effect at t = t0

CommentsWant to compare the outcome that are just on both sides of the discontinuity difference in means between these two groups is an estimate of the treatment effect at the discontinuitysays nothing about the treatment effect away from the discontinuityAn important assumption is that underlying effect on t on outcomes is continuous so only reason for discontinuity is treatment effect

Now introduce treatmentE(yt)t0tE(yt)t0tWorld with No TreatmentWorld with Treatment

The procedure in practiceIf take process described above literally should choose a value of that is very smallThis will result in a small number of observationsEstimate may be consistent but precision will be lowdesire to increase the sample size leads one to choose a larger value of

DangersIf is not very small then may not estimate just treatment effect Remember the pictureAs one increases 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 linearIf the linear relationship is the correct specification then one could estimate the ATE simply by estimating the regression:

no good reason to assume relationship is linear this may cause problemsIndicator which is 1 if t>t0

Suppose true relationship is:g0(t)g1(t)E(yt)t0t

Observed relationship between E(y) and Wg0(t)g1(t)E(yt)t0t

SplinesDoesnt need to be only a level shiftMaybe the parameters all changeCan test changes in the slope and intercept with interactions in the usual OLS modelThings are trickier in non-linear modelsDepends heavily on the correct specification of the underlying function

Splines allow you to choose a certain type of function (e.g. linear, quadratic) and then test if the parameter in the model changed at the break date t0

Non-Linear Relationshipone would want to control for a different relationship between y and t for the treatment and control groups

Another problem is that the outcome functions might not be linear in tit could be quadratic or something else.Discontinuity may not be in the level, it may be in the underlying function

The trade-offsa value of Larger means more precision from a larger sample size Risk of misspecification of the underlying outcome function

Choose a underlying functional formthe cost is some precisionintuitively a flexible functional form can get closer to approximating a discontinuity in the outcomes

In practiceit is usual for the researcher to summarize all the data in a graph Should be able to see a change outcome at t0 get some idea of the appropriate functional forms and how wide a window should be chosen. It is always a good idea to investigate the sensitivity of estimates to alternative specifications.

Breaks at an unknown dateSo far, weve assumed that we know when the break in the series occurred but sometimes we dont

Suppose we are interested in the relationship between x and y, before and after some date tyt = xt1 + t , t = 1,,t= xt2 + t , t = t+1,,TAssume the xs are stationary and weakly exogenous and the s are serially uncorrelated and homoskedastic.

Want to test H0: 1=2 against 12If t is known: this is well definedIf t is unknown, and especially if were not sure t exists, then the null is not well defined

What to do? (You dont need to know this for the exam)In the case where t is unknown, use LR statistic

When t is unknown: the standard assumptions used to show that the LR-statistic is asymptotically 2 not valid hereAndrews (1993) showed that under appropriate regularity conditions, the QLR statistic has a nonstandard limiting distribution.

Distribution with unknown break (You dont need to know this for the exam)Distribution is a Brownian Bridge and distribution values are calculated as a function of rmin and rmax

The applied researcher has to choose rmin and rmax without much guidance. Think of rmin as the minimum proportion of the sample that can be in the first subsample Think of 1 rmax as the minimum proportion of the sample that can be in the second subsample.

An example - 1Effect of quarterly earnings announcement on Market Returns (MacKinlay, 1997)Outcome: Abnormal Returns

Testing for a break

IssuesHow big a window should we choose?Wider window might allow more volatility which makes it harder to detect jumpsNarrower window has few observations reducing our ability to detect a small effect

How to model abnormal returnsDifferent ways to model how expectations of returns formThis is akin to considering functional form

An Example 2 (Micro Example)Lemieux and Milligan Incentive Effects of Social Assistance: A regression discontinuity approach, Journal of Econometrics, 2008In Quebec before 1989 childless benefit recipients received higher benefits when they reached their 30th birthdayDoes this effect Employment rates?

The Picture

The Estimates

IssuesWhat window to chooseClose to 30 years old? Not many people on social assistance

Note that the more flexible is the underlying relationship between employment rate and age, the less precise is the estimateUnderlying function can explain more jumps if its got more curvatureSplines can also explain a lot.

Next TimeMultivariate time seriesCointegrationState-space formMultiple/Simultaneous Equation Models

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