There are actually two different kinds of regression discontinuity
designs.
The ones we've talked about so far are called Sharp RDDs, because
if you are one side of the cut off, you always get treated and if
your on the other side of the cut off, you never get treated.
So which side of the cut off your on, completely determines your
treatment.
The other kind is called a Fuzzy RDD and that's when on one side
of the cut off you're very likely to get treated but you might not
and on the other side of the cut off you're very likely not to get
treated but you might actually actually get treated. So which
side of the cut off your on does not completely determine whether
you get treated or not.
Now there's actually a real nice analogy between regular RDD or
sharp RDD and regression and fuzzy RDD, and instrumental variables
analysis.
So basically with sharp RDD
we saw that all we had to do was compare average outcomes
of people on the treated side of the cut off with average outcomes
of people on the untreated side of the cut off.
Just like in a randomized experiment.
So that's also what you would learn in just doing a regression analysis
of the correlation between which side of the cut off you are.
So with fuzzy RDD however, you can't just compare the average outcomes
of people on one side of the cut off with average outcomes on the
other because on the treated side of the cut off we've actually
got some other people who didn't get treated and on the other side
of the cut off where you're not likely to get treated some people
actually do, so there's a problem, your groups are sort of mixed
up, so you can't just do that average comparison.
That's actually exactly the case that happened in the non-compliance
situation.
There, we assign people to be treated but some of them didn't comply.
They actually didn't get treated and
conversely we could assign some people not to get treated and they
may actually get treated.
So that's the non compliance problem.
So how do we solve the non compliance problem?
One way was by using instrumental variables and that's exactly how
we can solve the fuzzy RDD problem.
We can use which side of the cut off you're on as an instrument for
whether you get or not, so that's because if you are on one side
of the cut off, that makes you a lot more likely get treated although
it doesn't completely determine whether you're treated or not, so
that's how we'll ultimately use instrumental variable analysis to
analyze fuzzy RDD situations.
So let's just look at an example of this.
So in this example, we're gonna go all the way back to 1520 in Switzerland.
One region in Switzerland broke out into war, and after the war,
in 1520, one region became Protestant and was ruled by Protestant
leaders and the other region remained Catholic. So there was this
split in religion and one side of the region was Protestant and
the others were Catholic.
So the researchers today can have used this split to do an RDD analysis
of the effect of religion on those things like preference for leisure
or government redistribution. So let's look at how that works.
Here in this chart, I've got the share of Protestants living in
a region in 1980, so I'm gonna measure that share between one and
zero, and my units of analysis are gonna be municipalities and towns
and villages in Switzerland.
Here at zero I'm drawing the border okay, this is people, if you're
town here zero you're living on that 15, 20 border where the
region was split into two halves.
The Protestants side and the Catholic side. 66 00:03:56,912 --> 00:00:00,000 So then on the horizontal axis I have distance to the border over here it's got
I have distance to the border over here a town right here is minus 20
kilometers from the border.
And over here we've got 20 kilometers from the border and the other
direction okay.
So what we see here is that in this side of the border, a few people
are Protestants only somewhere around maybe 10% are people maybe
of people are Protestants.
Over here however on this side of the border it jumps dramatically
just going a few kilometers across the border the percentage of
Protestants in towns and villages jumps all the way up to something.
Okay it doesn't go to one so that's why we know we have
a fuzzy RDD design because this is not one and this right here is
not zero that's what you would see if you have a sharp RDD, it be
zero and then right as soon as you cross the border it's one.
So here is not quite but but it's pretty close.
So that says that the border really does
That says that the border really does for the share of Protestants
living in villages today.
Remember this is the 1520 border and our data here is from 1980.
Okay so that is what we cal the first stage in instrumental variable
analysis.
Which side of the border you are on affects your treatment, affects the percentage of Protestants
living in that village okay.
So now let's think about our outcome variables.
Well Switzerland is nice because they have a form of direct democracy
where their citizens vote directly on many different referendum.
So what the authors did is they the
data on all the referendum that the Swiss voted on from the 1980
on wards that was about 44 different referendums, and they categorized
that into three group.
The first group is referendum about leisure.
So this is things like should the official retirement age be lowered or not.
The second group was government re-distribution, so should the old
age and disability insurance be expanded so if you say if a village
votes yes mostly then we say that you have a preference for re-distribution
from young workers to old age retired people or people with disabilities. The third
group is about your preferences for government intervention so things
like do you think the government should be regulating prices or
not, so we've gone down on each municipality and how they voted on
this referendum, and then we can just compare how this municipalities voted
for the ones This side of the border and for the ones who are on
this side of the border, and with the ones who are on that side
of the border, so let's do that.
So here I've got their group of referenda preferences for leisure
so they've taken all the referendum that are like should the official
retirement age be lowered or not.
Okay, so over here, on the catholic
Okay so over here on the catholic side we see that if the town is
really close to the border, it about 48% of the time votes for this
leisure based referendum and then over here on the provinces side
however it drops 39% so we see that there is about a nine percent
difference in their preferences for leisure just by where you are
which side of the board you are.
And remember a whole idea of RDD is that because the border basically
determines your treatment with your products as a catholic this
different Catholic this difference here is solely explained by the
fact that this misfalities/g are protestant and these are catholic.
So this is a cause of that okay.
Now we have to be careful because this is a Fuzzy RDD so this differences,
actually something like an intention to treat number it's because
these municipalities they have a few Catholics living in them
and these municipalities have a few protestants living in them.
So if you think that actually being Protestant does have a true
colossal effect on lowering your preference for leisure as is suggested
here, then actually the true colossal effects is going to be even
bigger because if we got rid of Compliance the Catholics over here
and we got rid of the Protestants who were none compliance over
here then we see the gap increase.
And we'll see that happen a little bit later.
But right now we're just looking at this effect the intention to
treat effect.
So that's the outcome for leisure let's look at the next outcome
Over here we have the
redistribution and we see an effect that is not quite
as large.
On the Catholic side, it's about 43% and then on the Protestant
side, it's about 39% so something about 4% before we found 9% so
that's still pretty big but maybe it's a little
maybe is a little bit of an effect, about 4%, being on the protestant
side lowers your preferences for you distribution, so you're as
inclined to distribute in the old age or the disabled, let's look
at our third and final all come variable intervention things like
regulation of prizes, so over here again we see the Protestants have
lower preferences for intervention than the Catholics do, it's by
about 53% to about 48%,so that's about a 5% in difference, and outcomes
for the preference for intervention, so that's your first main result
from this paper, we see that the causal effect, the intention to
treat effect of being protestant versus catholic lowers your preferences
for government intervention, lowers your preferences for redistribution,
lowers your preferences for leisure, in the next module we'll see
those exact same numbers in
table form and do the instrumental analysis through solve this fuzzy
RDD problem or this new non-compliance problem
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