[MUSIC] In many applications of differences-in-differences, you'll often see a plot like this.
On the horizontal axis, we've got time.
On the vertical axis, we've got our outcome variable like employment in the Mariel Boatlift
case.
Then we've got two lines, one for the treatment group like Miami, one for the control group
like the Comparison Cities.
Now suppose that the boatlift happens some other time in the future, that for all of
this time periods there isn't massive immigration in either Miami or the comparison cities.
Then the Common Trends Assumption says that although the levels of employment can be different
in these two groups, the trends have to be the same.
So if Miami increases by some amount, the comparison cities have to also increase by
that amount.
If Miami decreases by some amount, the comparison cities have to decrease by the same amount.
But the two lines don't overlap.
They don't have to overlap.
They just have to have the same trend.
They have to move together.
Now let's suppose that the treatment, the immigration boatlift, actually happens at
this day right here.
So before this date, we don't have immigration in either Miami or the comparison cities,
and after this date we have immigration in Miami, but not the comparison cities.
Well, let's think about the first case, where immigration doesn't have any effect whatsoever
on employment.
Then what would we see?
Well, before immigration happens, we will see exactly this: employment just following
whatever its common trend is in both groups.
After the treatment, we would see the control group would continue with some trend.
It's not affected by treatment at all, because there isn't immigration there.
But in the treatment group, Miami, what would happen there?
Well, supposing that immigration doesn't have the any effect, then we know that the trend
in the control group is going to continue exactly as in the treatment group.
So we're going to see in the control group it didn't change employment, so in Miami,
employment's also not going to change, because of common trends.
So this number here is what we should see if there was not any effect of treatment,
because it's the same thing we saw in the comparison cities.
So that's by the Common Trends Assumption.
Well, now suppose that there actually was a positive treatment effect of immigration
on employment, so immigration increased employment, then what would we see?
Well we wouldn't see this line here.
Instead we would see some higher number, like this.
And so we would see this trend line.
So, what does that mean?
Well, that means that there's now a gap between what we see and what we would have seen had
the common trend from the comparison group continued.
That gap is precisely the treatment effect of immigration on employment.
It's the gap between what we see and what would what would've happened had the common
trend continued.
So this is precisely the number we compute with the differences-in-differences estimate.
So that's what's going on.
Let's take a look at the real data.
So over here, I've got the same chart.
I've got Time on the horizontal axis, and Employment on the vertical axis.
I've got two lines.
One for Miami, in solid bold and another in shaded grey for the Comparison Cities.
And here I've drawn a vertical line showing when the Mariel Boatlift happened.
So before this, both Miami and the comparison cities didn't have immigration; after this,
Miami had this massive immigration and the comparison cities did not.
But so what do we see?
Well, if we look right around when the immigration happened, we see that there isn't much difference
between the trends in the control group - the comparison cities - and the trends in Miami,
which suggests that there is not a treatment effect of immigration on employment, because
that's what we would see if there wasn’t a treatment effect.
So that's the first finding from this.
It suggests there isn't a treatment effect.
Second, if we look before in this “pre” group, before immigration, we can compare
to trends in Miami with the trends in the comparison group, and we see that the trends
are basically the same.
So this suggests that the common trend assumption for these dates probably holds.
So this is often given as evidence for the common trends assumptions in paper by looking
at the trends of many years before the treatment actually occurred.
[MUSIC].