Regression
Discontinuity/Event Studies
Methods of Economic
Investigation
Lecture 21
Last Time

Non-Stationarity




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

Impulse Response Function:



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

Return to Selection Bias

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:
E[Y| t , T  0]  g 0(t )
 Suppose average outcome with treatment
conditional on t is:
E[Y| t , T  1]  g 1(t )



Treatment effect conditional on t is
g 1 (t) – g 0 (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:
E[Y| t 0    t  t 0]  E[Y| t 0  t  t 0   ]


As δ→0 this comes to:
g1 (t 0 )  g 0 (t 0 )
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
t
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
t0
World with No Treatment
t
t0
World with Treatment
t
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:
y   0  1T   2t  
Indicator which is 1 if t>t0
no good reason to assume relationship is
linear
 this may cause problems

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



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
QLRT 

max
t{t min ,..., t max }
FT (t )
When t is unknown: the standard
assumptions used to show that the LRstatistic is asymptotically χ2 not valid here

Andrews (1993) showed that under appropriate
regularity conditions, the QLR statistic has a
“nonstandard limiting distribution.”
Bk (r )' Bk (r )
QLRT  sup
D r[ r , r ]
r (1  r )
min max
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