what is the Causal effect of education on wages?
If you got 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 sitting around watching basketball games all the
time, who knows?
Well 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.
So does that mean there is 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 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 got applauded 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, how it directly effects how effective they are at their
job.
And if it makes more productive for some reason.
So it's 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 generated the fake data, which is why
I generated 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 unobserved ability could be driving the observed correlation between education and hourly wage not a true causal effect of education on wage. So what are we gonna do? Well 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 its called the distant to college instrument.
So why would this make sense?
Well if a student lives closer to the university when they are in
high school, then the benefits from going to the university might
be more sailing to them.
They might recognize 'oh look at all this other students who've
gone to university and they having a great time, and they are really
getting a lot of benefits.
I should go too.
The cost might be lower for them to attend because they could have attended university in their home town, they won't have to pay for an apartment, or travel out of state or somewhere else. So this suggests that distant to college has causal effect on whether you attend college or not. So it has a causal effect on how much
education you get overall.
Now 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.
So why would high ability people live close to the universities?
If people were just blown randomly across the countries, some people
have high abilities and some people have low ability then there's
no reason to expect that ability will be related to distance to
college, that was another assumption we needed, and in practice
this is somewhat contentious and usually it is the most contentious
assumption in instrumental variable 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 abilities that they pass on to their children who then live
near a university, and that would mean that actually living close
to a university is correlated with having higher ability, that would
be a problem for using distant to college as an instrument
but putting that issue aside, we're just gonna go with it for now
for the sake of illustration, so the final assumption that we need
for instrumental variable analysis is that there's not a direct effect
of the instrument on outcome variables, so how close you live to
the university doesn't matter whether, it 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 the university or not? They don't
they just care about things like your education and how good you're
in doing the job, so with that said we 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 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, same history what's going on in this block.
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This next before I 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 it's unrelated to other possible com-founders like ability.
So finally, what do we do?
Now that we've justified our instrument, we go and we look at the
coloration between the instrument distance to college and the outcome
variable, the hourly wage.
And we see that there is a correlation here there is a negative relationship
between distant college and early 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
a causal
effect of years of education on hourly wage
of the treatment on the outcome, by looking at the correlation between
the instrument 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 their are potential confounder but in real life we don't
observe this potential com-founders.
But that's okay for instrumental variables analysis.
We don't need to observe the unobserved confounders.
All we need is our instrument our treatment and the outcome variable.
And with those three variables alone
and our assumptions on the instrument.
We can get it the causal effect of treatments and outcomes
or in this case the effect of going to school on wages.
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