Regression Discontinuity/Event Studies

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Regression Discontinuity/Event Studies
Methods of Economic Investigation Lecture 21

Last Time Non-Stationarity Estimating Causality in Time Series
Orders of Integration Differencing Unit Root Tests Estimating Causality in Time Series A brief introduction to forecasting Impulse Response Functions

Today’s Class Returning to Causal Effects
Brief return to Impulse Response Functions Event Studies/Regression Discontinuity Testing for Structural Breaks

What happens with there’s a “shock”?
Source: Cochrane, QJE (1994)

Impulse Response Function and Causality
Can look, starting at time t was there a change Don’t know if shock (or treatment) was independent. The issue is the counterfactual What would have happened in the future if the shock had not occurred OR What would the past have looked like, in a world where the treatment existed

Back to old selection bias problem: Shock occurs in time t and we observe a change in y Maybe y would have changed anyway at time t E[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 Idea Sometimes something changes sharply with time: e.g
your sentence for a criminal offence is higher if you are above a certain age (an ‘adult’) The interest changes suddenly/surprisingly at after a meeting There is a change in the CEO/manager at a firm Do outcomes also change sharply?

Not just time series Doesn’t 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 Assignment assignment 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 t0 t0 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 discontinuity The basic idea—no reason other outcomes should be discontinuous but for treatment assignment rule

Basics of Estimation Suppose average outcome in absence of treatment conditional on τ is: Suppose average outcome with treatment conditional on t is: Treatment effect conditional on t is g1(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 δ→0 this comes to: i.e. treatment effect at t = t0

Comments Want 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 discontinuity says nothing about the treatment effect away from the discontinuity An important assumption is that underlying effect on t on outcomes is continuous so only reason for discontinuity is treatment effect

Now introduce treatment
E(y│t) E(y│t) β t t0 t0 t World with No Treatment World with Treatment

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 Remember the 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: no good reason to assume relationship is linear this may cause problems Indicator which is 1 if t>t0

Suppose true relationship is:
g0(t) E(y│t) g1(t) t0 t

Observed relationship between E(y) and W
g0(t) E(y│t) g1(t) t0 t

Splines Doesn’t need to be only a level shift
Maybe the parameters all change Can test changes in the slope and intercept with interactions in the usual OLS model Things are trickier in non-linear models Depends 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 Relationship
one 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 t it could be quadratic or something else. Discontinuity may not be in the level, it may be in the underlying function

The trade-offs a value of δ Choose a underlying functional form
Larger means more precision from a larger sample size Risk of misspecification of the underlying outcome function Choose a underlying functional form the cost is 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 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 date
So far, we’ve assumed that we know when the break in the series occurred but sometimes we don’t Suppose we are interested in the relationship between x and y, before and after some date t yt = xt’β1 + εt , t = 1,…,t = xt’β2 + εt , t = t+1,…,T Assume the x’s are stationary and weakly exogenous and the ε’s are serially uncorrelated and homoskedastic. Want to test H0: β1=β2 against β1≠β2 If t is known: this is well defined If t is unknown, and especially if we’re not sure t exists, then the null is not well defined

What to do? (You don’t 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 χ2 not valid here Andrews (1993) showed that under appropriate regularity conditions, the QLR statistic has a “nonstandard limiting distribution.”

Distribution with unknown break (You don’t 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 - 1 Effect of quarterly earnings announcement on Market Returns (MacKinlay, 1997) Outcome: “Abnormal Returns” Testing for a break

Issues How big a window should we choose?
Wider window might allow more volatility which makes it harder to detect jumps Narrower window has few observations reducing our ability to detect a small effect How to model abnormal returns Different ways to model how expectations of returns form This 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, 2008 In Quebec before 1989 childless benefit recipients received higher benefits when they reached their 30th birthday Does this effect Employment rates?

The Picture

The Estimates

Issues What window to choose
Close 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 estimate Underlying function can explain more jumps if it’s got more curvature Splines can also explain a lot.

Next Time Multivariate time series Cointegration State-space form
Multiple/Simultaneous Equation Models