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[Instructor] Hello, and
welcome to the SPS exercise

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for panel data two for CDAE
359 Applied Econometrics.

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So I did a bunch of the
data transformation already.

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What I'm going to show you,

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and I made this claim in class

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is that when you only
have two time periods

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that the fixed effects time demeaning

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and the first differencing
data transformations

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give us the same betas.

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So you can see that this is
data from 64 cities on rent,

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and it looks at rent, which is our y

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as a function of population
of the average income,

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the log of average income

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and the percentage of student population.

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And there are 64 cities.

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And it looks at,

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well, what is the effect of
these three regressors on rent?

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And it has data from 1980 and 1990.

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And so the first thing I did
was to transform the data.

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And I actually did this in
Excel and then copy pasted.

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So it shows it here as one and zeros

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because it's such a small number,

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but basically for each city.

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So you can see that it's arranged here,

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cities one all the way down
year, year, year, year.

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And for each of these, I just subtracted.

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So this is the change in
the log of population,

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subtracting zero from 1990,

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from 1980 same way.

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But this is our dependent and et cetera.

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And in a minute, I will show you

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how I transformed the
data for fixed effects.

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But this was the data transformation.

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And this C in front of it here
means that it's the change.

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So it's the 1990 value
minus the 1980 value

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for each of these cities.

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So let's run a regression,
and it's a little linear.

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And our dependent is the
change in the log of rent.

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Just wanna make, go back to the sheet

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and make sure that I'm doing this.

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Clpop, Cl average income
and C percentage student.

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So clpop, boom, cl average income

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and change in percentage student

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Nope, we do not put our
dummy variable for 1990 here.

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That kind of wouldn't make sense to

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because it's only gonna, it would be,

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since we subtract out, it would be a,

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it would be one or zero for every row.

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And that would be a,

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you can't have a constant as a regressor

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because it is perfectly
colinear with the intercept.

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So let's just run this now.

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Boom, and here are the results.

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And I want you to look at the beta.

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So there's two things significant here.

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Or three, actually, sorry.

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The only one that's not is
the change in population,

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but both the average income

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and the percentage of
students has a positive sign,

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and is significant.

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So what what that means is

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as the average income
changes in a given city,

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and as that the percentage
of students changes

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in a given city that results in on average

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a change in the average
rent in that same city.

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So let's look at the
other way to look at it.

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Now, I'll make this small.

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And the other thing I did
was did a fixed effects.

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So here I'm putting,

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I took the mean here,
and I just subtracted it.

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So fixed effects, L rent,

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fixed effects Log of population,

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fixed effects average income

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and fixed effects percentage mean.

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Note this pattern here,

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and this would make sense
that in a given city

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that these will just be the
same number in opposite sign.

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So it's the mean would be
halfway in between these two.

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So since there's only two, this
is minus 2.8 under the mean.

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And then in year, the second year,

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it's that same amount over.

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You can see this pattern

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for all of the fixed
effects data transformation.

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And that makes sense.

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I hope it makes sense
that it would be that way

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for this T plus two.

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So let's do a regression again.

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And we do, again a
simple linear regression.

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We can reset it.

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And now it is, our dependent
is this fixed effects,

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L rent as a function of fixed effects.

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L pop, fixed effects.

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Log average income, fixed effects,

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percentage student,

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and to control for the year

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because there's one observation per year,

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and it makes sense to include it,

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we include the dummy variable for year 90,

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which is one, if it's a 1990 observation.

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Zero, if it's a 1980,
and then we hit okay,

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and we look at the betas and
note that they're the same.

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Let's go up and see and
note that these betas

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for pop average income and
percentage students are the same.

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And note that now the
year in the 90 is now

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the constant.

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That's the same figure.

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But the key here is that these
three betas are the same.

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And the point of this is to
show again, when T equals two,

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it doesn't matter if you first difference

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or time demean fixed effects.

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It doesn't matter which data
transformation that you use.

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The betas are the same.

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So thank you for watching.

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Bye.