[MUSIC] What is the causal effect of education on wages?
If you go to school for an extra year, will you make $1,000 more?
$10,000 more or $100,000 more?
Or maybe you're going to lose money, maybe you'll make less money, because you're just
sitting around watching basketball games all the time.
Who knows?
To answer that question, a lot of people have used something called Instrumental Variables
Analysis.
And to see how that works, let's look at an example where I've generated some fake data.
So in my fake data, here I've got the relationship between years of education and how much money
you make.
We see in the data that there is a positive relationship, people with more education tend
to make more money.
Does that mean there's a causal effect of education on wages?
No, not necessarily because education is not randomly assigned, people choose how much
education they get.
So, what could be going on?
Well, this positive relationship might be just be picking up an unobserved confounder.
What's one possible unobserved confounder?
Well, maybe your ability, your innate ability, deep down inside you.
So, over here I've plotted ability versus years of education, so people with higher
ability tend to get more education.
Seems plausible.
Maybe people of higher ability also tend to make more money, because their ability, independent
of how much school they have, directly effects how effective they are at their job and makes
some more productive for some reason.
So, if in your data you don't observe this variable, this innate ability, then you can't
draw these plots.
I can draw them because I've generated the fake data, which is why I generated the fake
data.
But in real life you often don't see the unobserved confounders-that's what makes them unobserved,
so you don't know if this kind of thing is going on behind the scenes.
So, with this situation the unobserved ability could be driving the observed correlation
between education, hourly wage, not a true causal effect of education on wage.
So, what are we gonna do?
That's where the Instrument comes in.
One Instrument that has been proposed is how far a student's high school home is from the
nearest college or university.
So it's called the Distance to College Instrument.
So why would this make sense?
Well, if a student lives closer to university when they're in high school then the benefits
from going to university might be more salient to them, they might recognize, oh, look at
all these students who've gone to university and they're having a great time and they're
really getting a lot of benefits, I should go too.
The cost might be lower for them to attend because they could attend university in their
home town, they wouldn't have to pay for an apartment or travel out of state or somewhere
else.
So, this suggests that distance to college has a causal effect on whether you actually
go to college or not.
So, it has a causal effect on how much education you get overall, that was one of our assumptions
from before.
Another thing you might think about this instrument is that it's not necessarily related to ability.
Okay, so why would high-ability people live closer to universities?
If people were just born randomly across the country, some people have high-ability, some
people have low ability then there's no reason to expect that ability will be related to
distance to college.
Now, that was another assumption we needed, and in practice this is somewhat contentious,
and usually is the most contentious assumption in an instrumental variables analysis.
Because some people have argued that, well, universities often have university faculty
or other staff members living near them and those people may have some innate ability
that they pass on to their children who then live near a university, and that would mean
that actually living close to the university is correlated with having high ability, and
that would be a problem for using distance to college as an instrument.
But putting that issue aside, we're just going to go with it for now for the sake of illustration.
So the final assumption that we need for instrumental variables analysis is that there is not a
direct effect of the instrument on outcome variables, so how close you live to a university
doesn't affect your wages at all.
So, this kind of makes sense because why would an employer care about where you lived when
you were in high school?
Why would they care if you lived close to a university or not?
They don't, they just care about things like your education and how good you are at doing
the job.
So, with that said we've sort of justified that this one instrument can satisfy all of
the three assumptions we needed, so let's see what it looks like in the data.
So over here, I've plotted distance to college versus years of education.
So people who live closer to a college over here tend to get more education, so that was
our first assumption, that there is a causal effect of distance to college on years of
education.
Seems to be what's going on in this plot.
This next plot plotted ability versus distance to college.
And you can see there isn't any relationship, it's just a big cloud of points.
So that was our next assumption, that distance to college is randomly assigned to people
that's unrelated to other possible confounder like ability.
So finally what do we do?
Now that we've justified our instrument, we go and we look at the correlation between
the instrument, distance to college, and the outcome variable, the hourly wage, and we
see that there is a correlation.
Here, there's a negative relationship between distance of college and hourly wage.
People who live closer tend to make more money.
So, because we've said that this instrument satisfies those three assumptions, we can
conclude from this plot, that this represents this correlation represents a causal effect
of years of education on hourly wage of the treatment on the outcome by looking at the
correlation between the instruments and the outcome.
That's the main idea of instrumental variables analysis, and one important thing to take
away from this is that in my fake data I had ability there are potential confounder, but
in real life we don't observe these potential confounder, but that's okay for instrumental
variable analysis.
We don't need to observe the unobserved confounder.
All we need is our instrument, our treatment and the outcome variable, and with those instruments
alone and our assumptions on the instrument, we can get at casual effect of treatments
on outcomes, or in this case the effect of going to school on wages.
[MUSIC].