So far all of our analysis has used data on a bunch
of units at a single point and time, that's called cross sectional
data.
Often, we actually observe units at multiple points in time, that's
called a panel data set or longitudinal data set.
So in the next few modules we're
gonna look at how we can use panel data to get at new ways
of learning about causality.
So, to start we're gonna look at an applied example, a very controversial
question, does the death penalty have a deterrent effect?
So, because we have the death penalty in the US or some states do, maybe someone
whom was gonna commit a crime, wants to murder somebody is going
to say, eh, never mind I am not gonna do that because I am worried
about getting executed.
That's what the deterring effect is.
So, some people propose that this is one reason we should have the
death penalty because it has a deterring effect, but is that claim
empirically true.
Well to answer
that, let's look at some data.
In 1972 the US Supreme Court case Furman versus Georgia imposed
a de facto ban on the death penalty. So from 1972 on, the death
penalty was illegal in every single state.
In 1976 however, the case Gregg versus Georgia over turned
the previous case, and at that point the death penalty became legal
in 32 states and illegal in the rest of the states.
So, here's the data we've got. We are looking at our outcome variable
is homicide rates per units, per 100,000 residents.
Our treatment invariable is whether a state has the death penalty
or not,
so here our units of analysis are states, so in 1975 the death penalty
was illegal in every single state.
In 1977 it was legal in some states and not others.
So the untreated
states, these are the states where
was illegal in both years.
The treated states are where death penalties was legal in 1977 and
illegal in 1975.
So, for the treated states we see that in 1975 the homicide rate
was 10.3 and in 1977 it was 9.7 and in the untreated states, the homicide rate
in 1975 was 8 and in 1977 it was 6.9.
Now there are three different kinds of analysis we can do to use
this data to learn about the causal effect of the death penalty.
So the first approach is to ignore the fact that we actually have
panel data and just look at the 1977 data and do a contemporaneous
comparison.
Just compare this number, the homicide rate for the treaty group,
states that have the death penalty with this number the homicide
rate for the states that don't have the death penalty.
So if we do that we see that we get the homicide rate for the treatment
group minus the homicide rate for our control group, and that's gonna
be positive 2.8.
So this suggests that the death penalty actually increases homicides if we just do a contemporaneous
comparison. So this right here is the average outcome in the treatment
group and this right here is the average outcome in the control
group, the difference is an estimate of the average treatment effect
of the death penalty, but hold on a second, this number if we're
gonna believe that is a true causal effect what assumptions
do we need? Well we need that
the death penalty is randomly assigned to states.
That there's no systematic difference between states that have the
death penalty and states that don't, but that assumption probably
isn't true, probably the states that have a death penalty are different
in some
systematic way from the states that don't. For example maybe these
states passed the death penalty precisely because they thought that
they were gonna have high homicide rates or they were having problems
with this and they passed the death penalty to try to lower it,
that would cause a correlation just like this where we see that
states with the death penalty have higher homicide rates whereas
states without the death penalty have lower homicide rates. So
this contemporaneous comparison probably we don't believe it, probably
this is not the actual causal effect of the death penalty. The
second approach is to do it before and after comparison we just
look at the states who had the death penalty in 1977, and compare
their outcome when they had the death penalty with their outcome
when they didn't.
So when we do that, we see that when they had the death penalty in 1977,
their outcome was 9.7, and when they didn't, their outcome was 10.3
with a difference of minus 0.6.
So we see that in 75, they didn't have the death penalty, and they had
10.3 homicide rate.
Then in 77, they had the death penalty and homicides went down by
0.6.
So this suggests that the death penalty actually lowered homicide
rates.
Right?
Because they didn't have the death penalty then they had it, and
homicides went down.
So that's one way we can use the panel dimension of our data to
learn a causal effect.
But do we really believe this number?
Is this really a good estimate of the causal effect of the death
penalty?
Well, what do we need for this assumption for this to be a good
estimate?
We need that nothing changes over time except the treatment status.
That because we wanna conclude that this difference over time is
due solely to the death penalty and cannot be explained by
anything else that's changing in these states over time.
Well that assumption's probably not correct, because for example,
if you look over here at the untreated states, their treatment status
doesn't change over time, so if nothing else was changing over time,
then we should see 8.0, 8.0 but actually their homicide rate also
changed overtime, even though their treatment status didn't.
The death penalty's illegal in both years.
So, this assumption that nothing else is changing overtime, that
would be enough to let us interpret this as a causal effect doesn't
seem to hold.
The problem with the before and after comparison is that other
things are changing over time, not just the treatment status, so
what are we gonna do?
Well, we're gonna introduce the common trends assumption we're gonna
assume that anything that's changing over time that's affecting 118 00:06:57,030 --> 00:00:00,000 our outcome variable, is affecting the treatment group and the control group
identically, exactly the same, that the trend in these unobserved
things that are changing overtime is the same.
It's common to both groups, the common trends assumption.
Now why is this helpful ?
Well, that means that if we look at our control group, here, and
we look at the difference in their outcome overtime so that's 6.9
minus 8.0, that gives us the common trend minus 1.1 so there's a
trend, a downward trend in homicides in general overtime in all states
of minus 1.1, now let's look at the treatment group. Over here in our
before and after comparison we saw that homicides went down in this
group by minus 0.6 but by the common trends assumption we know that
homicides in general were going down by -1.1, so here homicide actually
went down by
there's less than they did in the control group, so the only explanation
here, because of our common trends assumption is the fact that these
people were treated, so the reason that homicides went down by last here
is because of the death penalty so let's right that one out in algebra.
So here, we've got the difference in outcomes over time in the treatment
group and then I'm going to subtract off the common trends assumption.
Subtract off the common trend which I can do by the common trend
assumption and that's gonna give me positive 0.5. So this is my estimate
of the causal effect of death penalty on the homicide rate.
It suggests that having the death penalty actually increases your
homicide rate by 0.5 and why is that?
Well, if these states hadn't have had the death penalty they would have been
exactly like these states just by the common trends assumptions
and what happened in these states?
Well homicides went down by -1.1 but over here they didn't go down
by quite that much.
What explains the difference?
Treatment, and treatment is the death penalty so that's this analysis.
The difference between the treatment group and untreated
group is this 0.5, the death penalty actually increases the homicide
rate.
That's our conclusion, so if you were to go over these states that
are untreated and implement the death penalty if we believe this
number that would say that homicide rates would have actually increased.
Now, this idea of assuming common trends and then removing it
from before and after comparisons, is the main idea behind all the
ways we're gonna get at causal effects by using panel data.
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