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- [Instructor] This is the first
part of a three-part series

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on time series econometrics.

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I hope you find this section useful.

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This section on time series
has three main components.

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This is the first one,

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and I'm going to do a little introduction.

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Talk about two common models,

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static and finite distributed lag.

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Talk about very well-behaved
time series data,

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which will remind you a lot

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of the very well-behaved
cross-sectional data

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we did many weeks ago.

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And what we can assume
OLS does based on those

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as far as unbiased and efficient.

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And in the last part, we're
gonna talk about trends

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and seasonal effects and
how to account for those.

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So, so far, most of our time

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has been spent with cross-sectional data

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where we have a single
timeframe, T equals one,

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but many observations and equals many.

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For example, doing a survey
over a specific time period,

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but administering it and gaining data

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from lots of different people.

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We also spoke a bit about panel data,

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where you have, usually
a big N, many respondents

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and more than one time period,

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but usually a fairly small number.

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Say two, three, maybe at most four times.

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Now we're looking at a time series

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where there's one N but many Ts.

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So we're looking at one
set of variables over time,

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such as GDP or sales of some good.

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It could be a sociological thing

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like voting, voting rate,
or many other examples,

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educational attainment in a given place,

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or a biophysical example,

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like phosphorous in the lake or land use.

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But in any case, it's looking at

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the same respondent, in
a sense, over many years.

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And since there's a logical ordering here,

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we have to think about,

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we're not drawing a random
sample of responses.

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We're looking at the same one,

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and there's a clear direction of causality

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that we can in many cases assume

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and we have to account for.

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That the past influences the present

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and the present will influence the future.

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Again, we'll be looking
at a time series process

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that has both deterministic
and stochastic components,

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that there will be part
of our dependent variable,

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which the predicted value is
explained by the regressors.

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And also a stochastic
element where we have

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an error term of that which
was sort of unexpected

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or cannot be measured each time as well.

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Looking at all of the topics

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in this time series

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section again today,

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we're gonna be looking at
very well behaved data.

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What are the assumptions,
which are in some cases

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sort of unrealistic and simplistic
and building from there,

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and then looking at a slightly
less well behaved data

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and transformations that we can make

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to make them better behaved.

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And last, how do we test for
these assumptions being true

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and what do we adjust

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and how do we adjust if
we find that the data

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are not as well behaved as we would like?

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There are two basic kinds of models

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in time series, static
and finite distributed lag

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and I will talk to you about each one.

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This is a static model and it's static

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because you can see in the
equation that starts at YT

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that we're looking at every
thing over the same time.

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So the T would be say
the years of the study.

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So we might have started
collecting data in 2001,

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in which case T equals 1 and continue on

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through the year 2020, when T equals 20.

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Here all of our variables
are contemporaneous,

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that they're happening at the same time,

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that we only assume
that the regressor, ZT,

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had any effect on that year's YT.

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That nothing that happened
in the past has any effect,

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and that is why it's static

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and that it's all
happening in a single time

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and this is very much
like cross sectional data.

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And just like there that we can assume

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with a single regressor
that the change in Y

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will be beta one times the change in Z.

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And assuming that the Z is exogenous

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and it has no effect on you

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that we can do this sort
of simple calculation.

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And the a beta is then the change in Y

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as a result of the change in
Z, the slope of that line.

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We could of course, we could
of course add more regressors.

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Z2, Z3 as many as what
we think are correct.

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And these data are very much in this model

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is very much like cross section,

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except now we're looking across time

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instead of across space.

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We're looking at the different realization

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of a variable over time,

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instead of that variable
across different respondents.

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In contrast here is a
finite distributed lag.

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So lag meaning that
regressors from past years

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are thought to have an
effect on this year's Y.

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So you say YT and ZT,

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but also ZT minus one, last year's Z,

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and ZT minus two, two years ago's Z.

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And since there are two lagged regressors,

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this is known as a finite distributed

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lag model of order two,

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because there are two lags
and we could have more lags

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if we think that that
is the correct model.

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Here are the assumptions

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of well-behaved OLS time series data.

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So the first is, and these
should look pretty familiar,

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the first is that it
is linear in parameters

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that we can model the
data generating process

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as a linear function.

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Next that as before each regressor has

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to contribute some new information.

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That no Z or X can be a constant

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or a linear function of
the other regressors.

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Again, that it has to
provide some new information.

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Third is the exogeneity assumption

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that every error term is uncorrelated

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with each of the regressors.

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If this holds true, a weaker assumption

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that this year's error term

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is uncorrelated with
this year's regressors,

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this is called
contemporaneously exogenous,

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but there's also a stronger assumption

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that this year's error term

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is uncorrelated with any
regressors for any other time.

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And that the regressors are uncorrelated

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with the error term for any other time.

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This again is a stronger assumption

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and when this holds, this
is called strict exogeneity.

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More on the third assumption
that we don't really have

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to worry about this in
cross sectional data,

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that your error term,

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some strange thing that
may have happened to you,

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or some relatively small and minor thing

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that we are not able to measure

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is not effected by my regressors.

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So whether you found a dollar

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or got a flat tire

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is not effected by my income
and my age and my behaviors

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and so forth since it's a random sample.

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But we don't randomly
sample in time series

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because we want a series of data

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and they are all very intimately tied.

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That what happened last year

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might affect what happens to this year,

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because we're looking at the
same phenomenon over time.

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This assumption three fails the
same in cross sectional data

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in cases where there's a measurement error

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or omitted variables, as
well as in time series.

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If there is a lag defect
that this may hold true,

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that this year's error term

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may be effected by last year's regressor.

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And in this case, we want to
put last year's regressors,

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or last two years
regressors into our model.

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So we can specifically
account and control for them

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and to take them out of the error term.

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When these three assumptions hold,

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we can assume that OLS is unbiased.

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And when there is an omitted variable,

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the same kind of omitted variable
bias analysis that we did

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way back in the beginning
of class would hold true.

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The same basic themes and
analyses would hold true.

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The fourth assumption

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in well-behaved time series
data is homoskedasticity.

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That the variance of our error term,

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given any value of X, our
regressor is a constant.

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So before it was that the variance

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of your error term and my error term,

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had a constant variance.

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Now it's that the variance
of this year's error term

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and last year's error
term, going back in time,

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all have a constant variance.

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That's what homoskedasticity
means in this context.

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And we could do the same
basic kinds of tests

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like we did in cross sectional data

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to test for homoskedasticity

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such as the White test and
the Breusch-Pagan test.

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The fifth assumption is
no serial correlation

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that this year's error term

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and last year's error
term are uncorrelated.

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And in fact, this year's
error term is uncorrelated

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with any error term from any past year.

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The violation is that something
abnormally high this year

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is also abnormally high last year,

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that the thing that we
sort of failed to measure,

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the minuscule thing that
we failed to measure

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or didn't account for this year
is that same minuscule thing

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that we didn't account for last year.

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And we will learn a
test on how to do that.

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I think it's a lot like the in panel data,

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the idea that there's
this AI that goes along

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with every individual respondent

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that is very hard to measure
but that holds through

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and here again, I think
that the parts of the model

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that are just very hard to
measure will follow through

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because we're looking at the same thing

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and it's likely that the
errors will be correlated.

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And again, we will learn a test

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on how to see if this is
true later on in this series.

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Note that let's not rule out
that the Ys are correlated,

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that they almost always are.

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That last year's GDP

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is highly correlated
with the year before that

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and the year before that, et cetera.

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When these five assumptions hold,

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the variance of the beta J has
the same formula as before.

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It is a function of the variance

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of the overall error term, sigma squared,

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as well as how spread out that XJ is

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and how collinear that XJ is

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with the other regressors in our model.

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And under those assumptions,

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those five assumptions,
the three unbiasedness

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and the sort of spherical errors
assumption for time series.

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If they all hold, if they are
all true, then OLS is blue.

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It's the best linear unbiased estimator.

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It's the most efficient,
unbiased estimator.

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Remember that when we started
to get into hypothesis tests

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that we added a new assumption,

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and that is the error term
is normally distributed

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with mean zero

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and variance sigma squared.

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And we are adding another
slightly stronger assumption

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that the Us are not only
independent of X as before,

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I think at assumption four,

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but that they are independently

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and identically distributed, IID.

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So if the sixth assumption holds

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then the OLS estimators also

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have predictable statistical properties,

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and we can use the kind
of inference tests,

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T-tests, the F-tests,

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and we can create confidence
intervals the same,

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and they have the same interpretation

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as we learned about in cross section.

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So just a bit more on IID.

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IID stands for independent
and identically distributed.

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So one way of thinking about this is

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the bingo ball wheel that
you're drawing the errors from.

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It does not change, that it's
the exact same distribution.

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That's what the identically
distributed means.

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It's the exact same bell curve probability

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of drawing any error term over every year.

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And the independent means
that it has no memory,

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that the ball is replaced each year,

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that the error term
that you drew this time

257
00:18:34,980 --> 00:18:39,273
has no effect on the error
term that you draw next time.

258
00:18:47,150 --> 00:18:51,560
We very often want to use
dummy variables in time series

259
00:18:51,560 --> 00:18:53,740
to account for whether

260
00:18:53,740 --> 00:18:58,740
an important event or policy or regime

261
00:18:59,360 --> 00:19:01,653
was in place during that time.

262
00:19:03,190 --> 00:19:07,590
A good example would be if there was

263
00:19:07,590 --> 00:19:12,590
a big recall of romaine,
which we've seen a few times,

264
00:19:13,940 --> 00:19:18,940
that the demand for romaine
would depend not only sort of

265
00:19:19,540 --> 00:19:24,540
on the supply and the prices
and the people's preferences

266
00:19:26,410 --> 00:19:30,410
and things, and their
incomes and things like that,

267
00:19:30,410 --> 00:19:35,140
but you would want to have
a dummy variable in place

268
00:19:35,140 --> 00:19:37,670
that accounts for there
was this huge scare

269
00:19:37,670 --> 00:19:39,690
and there was this huge recall

270
00:19:41,300 --> 00:19:43,873
that happened during that time.

271
00:19:45,920 --> 00:19:49,770
Otherwise you're not really
going to get a good account

272
00:19:49,770 --> 00:19:52,610
of what is the effect of income

273
00:19:52,610 --> 00:19:57,270
and price and supply and preferences.

274
00:19:57,270 --> 00:19:59,580
And I was thinking as well,

275
00:19:59,580 --> 00:20:03,670
that when we model time series GDP

276
00:20:03,670 --> 00:20:08,610
and other economic variables for right now

277
00:20:08,610 --> 00:20:12,940
that we will probably need
to have a dummy variable

278
00:20:12,940 --> 00:20:17,940
for the coronavirus that, you know,

279
00:20:19,200 --> 00:20:21,290
the intercept shift of what happened

280
00:20:21,290 --> 00:20:24,353
just as a result of that
holding all else equal.

281
00:20:27,820 --> 00:20:31,460
It's also very common in time series

282
00:20:31,460 --> 00:20:33,620
to acknowledge that the data

283
00:20:33,620 --> 00:20:36,433
have natural trends and seasons.

284
00:20:37,420 --> 00:20:42,420
So trends take into account that there's,

285
00:20:43,560 --> 00:20:47,420
especially in economics,
that there is a time trend,

286
00:20:47,420 --> 00:20:51,210
that our GDP increases over time,

287
00:20:51,210 --> 00:20:55,050
worker productivity increases over time

288
00:20:55,050 --> 00:20:59,540
sort of independent of
many, many other factors.

289
00:20:59,540 --> 00:21:02,000
That it adds this sort of a natural

290
00:21:02,000 --> 00:21:04,903
or recurring time trend built into it.

291
00:21:06,440 --> 00:21:10,390
And it's important that
we account for that

292
00:21:10,390 --> 00:21:15,390
because if productivity and
GDP are both trending upward,

293
00:21:18,250 --> 00:21:22,730
that failing to account for
that trend in both of them

294
00:21:22,730 --> 00:21:25,380
would make the relationship between them

295
00:21:25,380 --> 00:21:27,690
seem stronger than it actually is

296
00:21:27,690 --> 00:21:29,390
and we would get a biased estimate.

297
00:21:29,390 --> 00:21:31,690
That it would be in a sense,

298
00:21:31,690 --> 00:21:34,600
an omitted variable bias by failing

299
00:21:34,600 --> 00:21:37,183
to account for that trend.

300
00:21:40,710 --> 00:21:44,440
Here is a very simple example

301
00:21:47,890 --> 00:21:50,460
of a Y that has a time trend.

302
00:21:50,460 --> 00:21:54,070
So T would just be the actual year

303
00:21:54,070 --> 00:21:57,630
that the data were collected.

304
00:21:57,630 --> 00:22:02,630
So again, if we have 2001 to 2020 data,

305
00:22:03,270 --> 00:22:07,730
T equals 1 in year 2001, and so on,

306
00:22:07,730 --> 00:22:11,530
T equals 20 in year 2020,

307
00:22:13,760 --> 00:22:17,680
and in this model, holding all else equal,

308
00:22:17,680 --> 00:22:22,680
Y will increase or shrink
by the rate of A1 each year.

309
00:22:23,430 --> 00:22:26,550
So if A1 is greater than zero,

310
00:22:26,550 --> 00:22:31,550
the predicted value of Y will
increase by A1 each year.

311
00:22:39,290 --> 00:22:42,230
We very often model things with

312
00:22:42,230 --> 00:22:45,860
a number of regressors and a time trend.

313
00:22:45,860 --> 00:22:48,457
So you see here, there's two regressors,

314
00:22:48,457 --> 00:22:52,870
X1, X2, as well as this time trend, T.

315
00:22:52,870 --> 00:22:57,170
And again, that this
accounts for the trend

316
00:22:57,170 --> 00:23:02,170
that may be in either Y or in X1 or X2,

317
00:23:03,570 --> 00:23:08,570
and by leaving out this trend
by not controlling for it,

318
00:23:08,830 --> 00:23:12,520
it can cause omitted variable bias.

319
00:23:12,520 --> 00:23:17,480
And note that you should
include a time trend,

320
00:23:17,480 --> 00:23:22,010
even if the Xs trend and
the Ys don't that you want

321
00:23:22,010 --> 00:23:26,380
to take out the effect
of time over these Xs

322
00:23:26,380 --> 00:23:30,733
and holding time equal,
how does X affect Y.

323
00:23:37,530 --> 00:23:42,530
The R squared of a time series
model tends to be quite high,

324
00:23:43,970 --> 00:23:48,890
that these kind of aggregated
data tend to explain

325
00:23:48,890 --> 00:23:52,910
the relationships and
result in a higher R squared

326
00:23:52,910 --> 00:23:55,130
than we get with something

327
00:23:55,130 --> 00:23:59,170
like doing a survey and
how do the regressors

328
00:23:59,170 --> 00:24:04,170
across individual people
affect the dependent.

329
00:24:04,220 --> 00:24:06,680
So, one way to get a better sense

330
00:24:06,680 --> 00:24:11,680
of how much of the variation
in Y is explained by X,

331
00:24:13,260 --> 00:24:16,680
that is R squared independent of time

332
00:24:16,680 --> 00:24:19,860
is to detrend our data.

333
00:24:19,860 --> 00:24:20,840
And you do that

334
00:24:22,505 --> 00:24:25,083
by regressing YT only on T

335
00:24:27,310 --> 00:24:29,700
and saving those residuals,

336
00:24:29,700 --> 00:24:32,567
which I call this weird Y umlaut.

337
00:24:33,478 --> 00:24:37,283
And then R squared is this formula

338
00:24:38,930 --> 00:24:42,960
where SSR is that of regressing,

339
00:24:44,000 --> 00:24:48,530
YT on X1, X2 and T.

340
00:24:48,530 --> 00:24:53,530
And it gives you a better idea
of taking away time trend,

341
00:24:54,810 --> 00:24:57,633
how well do the Xs explain our Y.

342
00:25:02,640 --> 00:25:06,940
There's also many economic phenomena

343
00:25:06,940 --> 00:25:08,970
that have a clear seasonality,

344
00:25:08,970 --> 00:25:11,450
such as most Christmas trees

345
00:25:11,450 --> 00:25:14,540
are sold in November and December,

346
00:25:14,540 --> 00:25:19,540
hay is sold in the summer when it's fresh.

347
00:25:22,290 --> 00:25:27,230
Another example is
nursery flower transplants

348
00:25:28,540 --> 00:25:32,200
are much more often sold in the spring.

349
00:25:32,200 --> 00:25:35,610
So you can include dummy variables

350
00:25:35,610 --> 00:25:39,520
for specific months or quarters or seasons

351
00:25:39,520 --> 00:25:44,520
where you think that sales will
be particularly high or low,

352
00:25:45,720 --> 00:25:50,720
model it with them and you
can use a T or an F-test

353
00:25:51,280 --> 00:25:53,900
to determine whether that has

354
00:25:53,900 --> 00:25:56,920
a significant effect on our model

355
00:25:56,920 --> 00:26:01,920
and much like we did before,
we can deseasonalize our Ys,

356
00:26:03,200 --> 00:26:06,380
where we regress our YTs

357
00:26:06,380 --> 00:26:11,380
only on our seasonal dummy
variables, save those residual,

358
00:26:14,390 --> 00:26:19,390
and then regress our Xs

359
00:26:19,560 --> 00:26:22,900
on those Y umlauts again,

360
00:26:22,900 --> 00:26:26,030
and use R squared as another measure

361
00:26:26,030 --> 00:26:28,203
of the goodness of fit of the model.

362
00:26:30,880 --> 00:26:34,050
Basically, again, that
many of the assumptions

363
00:26:34,050 --> 00:26:37,700
are unrealistic, too restrictive,

364
00:26:37,700 --> 00:26:40,000
doesn't really meet the real world.

365
00:26:40,000 --> 00:26:43,100
So we're gonna look at those models

366
00:26:43,100 --> 00:26:48,100
and look at some common
ones and some principles

367
00:26:48,220 --> 00:26:50,300
that we have to abide by

368
00:26:50,300 --> 00:26:52,970
for those assumptions to fit and last,

369
00:26:52,970 --> 00:26:56,750
a transformation that
you're already familiar with

370
00:26:56,750 --> 00:26:57,850
that can deal with it.

371
00:27:02,270 --> 00:27:06,430
So two pieces of vocabulary
that we're gonna deal with,

372
00:27:06,430 --> 00:27:08,860
dependence and stationarity.

373
00:27:08,860 --> 00:27:10,500
And both of them

374
00:27:10,500 --> 00:27:13,023
cause issues with inference,

375
00:27:16,240 --> 00:27:20,700
that it can mess up our
hypothesis tests, our DNF tests

376
00:27:20,700 --> 00:27:22,823
that we would like to run on our data.

377
00:27:28,470 --> 00:27:32,400
We think about dependence and how much

378
00:27:32,400 --> 00:27:37,400
does the past affect the
present and the future.

379
00:27:39,150 --> 00:27:42,603
And we say that a time series process

380
00:27:46,150 --> 00:27:48,500
is a weakly dependent,

381
00:27:48,500 --> 00:27:53,113
that as the distance between time periods

382
00:27:55,380 --> 00:27:58,360
gets larger, the relationship

383
00:27:58,360 --> 00:28:01,390
between those variables gets small.

384
00:28:01,390 --> 00:28:06,390
So basically we want weak dependence

385
00:28:06,690 --> 00:28:10,620
so that this year's

386
00:28:13,310 --> 00:28:18,310
realization of X and future years,

387
00:28:18,520 --> 00:28:23,033
that relationship gets
very small, very fast.

388
00:28:24,190 --> 00:28:28,290
You can think of independent is that like

389
00:28:28,290 --> 00:28:31,070
when we're drawing a random sample

390
00:28:31,070 --> 00:28:34,270
where the coin has no memory at all,

391
00:28:34,270 --> 00:28:38,100
that last year that you
and I as respondents

392
00:28:38,100 --> 00:28:42,713
don't have any thing
to do with each other.

393
00:28:45,030 --> 00:28:50,030
Dependence really, where they
are very intimately related

394
00:28:53,500 --> 00:28:57,360
and last year and past
years effect this year

395
00:28:57,360 --> 00:29:00,740
in a very strong way, or thinking about

396
00:29:00,740 --> 00:29:04,250
that the coin remembers every flip

397
00:29:06,912 --> 00:29:11,507
really messes up inference
and it makes it impossible.

398
00:29:13,170 --> 00:29:17,793
Where with weak dependence,
this memory fades over time.

399
00:29:18,960 --> 00:29:23,960
Stationarity means that the variables

400
00:29:23,960 --> 00:29:28,960
have the same probability
distribution over time.

401
00:29:29,630 --> 00:29:32,970
So much as, we assumed that they have

402
00:29:32,970 --> 00:29:37,800
this normal IID distribution over time,

403
00:29:42,110 --> 00:29:45,100
we need to make assumptions

404
00:29:45,100 --> 00:29:49,143
and know that the probability distribution

405
00:29:50,397 --> 00:29:55,360
of our error terms and of
our dependent variables

406
00:29:55,360 --> 00:29:59,373
have a predictable, the same distribution

407
00:30:00,920 --> 00:30:05,350
in order for us to be able
to make any kind of inference

408
00:30:05,350 --> 00:30:07,410
and use T-tests and F-tests

409
00:30:07,410 --> 00:30:10,173
and those tests that
we have learned about.

410
00:30:13,180 --> 00:30:15,820
Basically weak dependence is needed

411
00:30:15,820 --> 00:30:18,280
for the law of large numbers

412
00:30:18,280 --> 00:30:22,100
and the central limit theorem to hold.

413
00:30:22,100 --> 00:30:25,570
That because we do not random sample,

414
00:30:25,570 --> 00:30:30,570
if the realizations are
too closely related,

415
00:30:31,600 --> 00:30:33,340
we can't do inference

416
00:30:33,340 --> 00:30:38,340
and strong dependence
totally messes it up.

417
00:30:38,640 --> 00:30:41,460
So we are going to look at models

418
00:30:41,460 --> 00:30:45,910
that have strong and weak dependence

419
00:30:45,910 --> 00:30:50,090
that we may encounter in time series data.

420
00:30:54,070 --> 00:30:57,920
So what we're finding is
these autoregressive models

421
00:30:57,920 --> 00:31:02,300
where last year's Y

422
00:31:02,300 --> 00:31:06,530
or past year's Y appear as regressors

423
00:31:06,530 --> 00:31:11,530
with this year's Y on the left side.

424
00:31:12,018 --> 00:31:13,800
And then two examples of that,

425
00:31:13,800 --> 00:31:18,800
the so-called unit root
process and the random walk.

426
00:31:19,660 --> 00:31:23,323
And we'll go through each one
over in the next few slides.

427
00:31:27,750 --> 00:31:28,940
Here's an example

428
00:31:28,940 --> 00:31:33,803
of an autoregressive process
of order one, an AR1.

429
00:31:34,700 --> 00:31:39,620
It's autoregressive,
because last year's Y,

430
00:31:39,620 --> 00:31:43,323
YT minus one, appears as a regressor.

431
00:31:44,210 --> 00:31:46,870
And it's of order one because only

432
00:31:46,870 --> 00:31:51,870
one lagged dependent variable
is a regressor in this model.

433
00:31:57,470 --> 00:32:01,690
The properties of this model

434
00:32:01,690 --> 00:32:04,770
is that Y starts at some sort

435
00:32:04,770 --> 00:32:08,030
of starting point, at time zero.

436
00:32:08,030 --> 00:32:13,030
Our error term is again,
IID zero sigma squared,

437
00:32:14,060 --> 00:32:19,060
and that the ETs are
independent of Y naught.

438
00:32:19,700 --> 00:32:23,240
And just for simplicity, we often assume

439
00:32:23,240 --> 00:32:27,003
that the expected value
of Y naught equals zero,

440
00:32:28,050 --> 00:32:33,050
or we could subtract
out that value each time

441
00:32:33,270 --> 00:32:37,100
to make the math easier.

442
00:32:37,100 --> 00:32:39,720
So we're gonna look at the value

443
00:32:39,720 --> 00:32:43,620
of this coefficient P

444
00:32:43,620 --> 00:32:47,230
or I believe in the
Wooldridge book it's rho,

445
00:32:47,230 --> 00:32:50,350
but I just made it a P

446
00:32:50,350 --> 00:32:55,203
because I have P on my
keyboard and not rho.

447
00:32:57,180 --> 00:33:01,550
So it's important that for any kind of...

448
00:33:08,490 --> 00:33:12,100
So when rho is less than one,

449
00:33:12,100 --> 00:33:17,100
then we see that with the
expected value of YT plus H,

450
00:33:17,710 --> 00:33:22,133
given our starting point YT,
is rho to the H times YT,

451
00:33:23,360 --> 00:33:27,180
for any number of years greater than one.

452
00:33:27,180 --> 00:33:31,750
And we see that the influence
of our starting point,

453
00:33:31,750 --> 00:33:34,090
YT gets smaller and smaller over time,

454
00:33:34,090 --> 00:33:37,770
and it approaches zero overall
as H gets bigger and bigger.

455
00:33:37,770 --> 00:33:41,470
And this is what we want,
we have weak dependence.

456
00:33:41,470 --> 00:33:44,580
When rho equals one,

457
00:33:44,580 --> 00:33:48,390
the best prediction of
YT plus H is always YT.

458
00:33:52,050 --> 00:33:56,550
We call this a unit root
process when rho equals one,

459
00:33:56,550 --> 00:33:59,400
and this leads to strong dependence,

460
00:33:59,400 --> 00:34:03,180
and we need to deal with this

461
00:34:03,180 --> 00:34:08,180
in order to do estimation and inference.

462
00:34:10,200 --> 00:34:14,120
So again, the unit root process

463
00:34:14,120 --> 00:34:18,263
is when this P or rho equals one.

464
00:34:22,400 --> 00:34:27,400
One very common unit root process is the

465
00:34:28,790 --> 00:34:33,153
so-called random walk where Y,

466
00:34:34,830 --> 00:34:39,830
this year, equals last
year's Y plus an error term.

467
00:34:40,520 --> 00:34:43,100
And we assume that the error term, ET,

468
00:34:43,100 --> 00:34:47,860
is as a mean of zero and constant variance

469
00:34:47,860 --> 00:34:52,860
and so Y starts at the
previous year's value

470
00:34:52,970 --> 00:34:54,830
and adds an error term.

471
00:34:54,830 --> 00:34:59,830
And every year just adds that error term

472
00:35:01,440 --> 00:35:05,783
to last year's value so
that there's a random walk.

473
00:35:06,810 --> 00:35:09,830
The random part is the error,

474
00:35:09,830 --> 00:35:12,640
and it might be high, it might be low,

475
00:35:12,640 --> 00:35:17,010
but it'll just sort of
walk around that mean.

476
00:35:17,010 --> 00:35:22,010
And if we assume, Y naught,
just for mathematical

477
00:35:22,360 --> 00:35:24,533
again equals zero,

478
00:35:27,030 --> 00:35:31,630
and the expected value of Y in any year

479
00:35:34,240 --> 00:35:36,980
equals this starting point.

480
00:35:36,980 --> 00:35:39,930
So you can imagine the line sort of

481
00:35:39,930 --> 00:35:44,930
meandering all around over
time, going up and down,

482
00:35:45,840 --> 00:35:50,840
but basically the
expected value of any year

483
00:35:50,900 --> 00:35:53,143
is that starting point.

484
00:35:56,930 --> 00:36:01,460
So the value of Y

485
00:36:03,140 --> 00:36:05,710
in year T plus H

486
00:36:05,710 --> 00:36:10,710
is just the sum of the error
terms over all those years,

487
00:36:11,150 --> 00:36:14,490
plus YT, the starting point.

488
00:36:14,490 --> 00:36:19,490
So the value of any year is the
sum of all those error terms

489
00:36:21,060 --> 00:36:26,060
plus the starting points.

490
00:36:26,274 --> 00:36:31,274
So if we assume a starting
point of zero, again,

491
00:36:31,700 --> 00:36:34,910
it'll just be a line that meanders around

492
00:36:36,140 --> 00:36:40,793
the zero line, the X axis.

493
00:36:42,060 --> 00:36:45,523
And that for any value of YT,

494
00:36:47,900 --> 00:36:51,517
that the expected value
of YT plus H equals YT.

495
00:36:54,780 --> 00:36:58,150
And you can see in the first equation

496
00:36:58,150 --> 00:37:01,440
that if we have a big enough H

497
00:37:01,440 --> 00:37:05,050
that if we add them
all up, that they will,

498
00:37:05,050 --> 00:37:08,450
since the expected value is
zero, that they are going

499
00:37:08,450 --> 00:37:12,150
to eventually converge and sum to zero.

500
00:37:12,150 --> 00:37:17,150
And so the expected value of YT plus H

501
00:37:18,390 --> 00:37:22,743
will always equal our starting point, YT.

502
00:37:27,720 --> 00:37:32,720
So another example is the
so-called random walk with drift.

503
00:37:32,900 --> 00:37:37,470
So this is again, a unit root process

504
00:37:37,470 --> 00:37:40,483
because our rho, or P, equals one.

505
00:37:44,640 --> 00:37:47,587
And in this case, there's a drift

506
00:37:47,587 --> 00:37:50,720
and that drift is A naught.

507
00:37:50,720 --> 00:37:55,720
So every this year is Y is last year's Y

508
00:37:55,910 --> 00:38:00,030
plus this drift plus this A naught.

509
00:38:00,030 --> 00:38:03,903
So this has both persistence and trend.

510
00:38:05,630 --> 00:38:10,630
It's persistent because it
builds on the last year's value

511
00:38:12,621 --> 00:38:14,850
and the expected value is

512
00:38:14,850 --> 00:38:18,167
of what will be carried over.

513
00:38:21,600 --> 00:38:25,170
And the trend is that every year

514
00:38:25,170 --> 00:38:27,463
it adds this factor A naught.

515
00:38:30,730 --> 00:38:32,780
So that let's us talk

516
00:38:32,780 --> 00:38:36,510
a bit more about persistence and trend.

517
00:38:36,510 --> 00:38:39,350
So we talked a bunch
about trend last time,

518
00:38:39,350 --> 00:38:43,890
and persistence is again,

519
00:38:43,890 --> 00:38:47,830
when something that happens

520
00:38:47,830 --> 00:38:51,453
is felt over time.

521
00:38:54,920 --> 00:38:58,420
So if we don't deal with them,
they'll mess up analysis.

522
00:38:58,420 --> 00:39:02,440
We learned last time, how trend can be

523
00:39:02,440 --> 00:39:05,800
an important omitted, variable bias,

524
00:39:05,800 --> 00:39:10,800
and the persistence is when some shock

525
00:39:11,400 --> 00:39:16,400
continues to be felt into the future,

526
00:39:16,890 --> 00:39:21,620
or that this year's realization

527
00:39:21,620 --> 00:39:25,150
is very closely related to past years,

528
00:39:25,150 --> 00:39:29,980
that all of last year's
and previous years values

529
00:39:29,980 --> 00:39:33,720
are sort of remembered over time.

530
00:39:33,720 --> 00:39:36,380
And it's interesting to note

531
00:39:36,380 --> 00:39:39,350
that interest rates are persistent,

532
00:39:39,350 --> 00:39:44,350
that this quarters interest rate

533
00:39:44,700 --> 00:39:48,920
is going to be very closely
related to last quarters,

534
00:39:48,920 --> 00:39:51,930
but they don't really trend.

535
00:39:51,930 --> 00:39:56,930
That they're not steadily
upward or downward over time

536
00:39:57,250 --> 00:39:59,930
whereas GDP is likely

537
00:39:59,930 --> 00:40:02,563
both persistent and trending.

538
00:40:05,320 --> 00:40:10,320
That, you know, in most
years here in the US,

539
00:40:11,890 --> 00:40:14,690
you'll get in a normal sort of growth

540
00:40:14,690 --> 00:40:19,680
that you'll get only
about 2% annual growth.

541
00:40:19,680 --> 00:40:23,053
So last year's GDP is very,

542
00:40:24,780 --> 00:40:28,090
plays a big factor on what this year's is,

543
00:40:28,090 --> 00:40:33,090
but there's this sort of
overall trend over time.

544
00:40:33,160 --> 00:40:37,660
Now, the huge contraction of our economy

545
00:40:40,430 --> 00:40:43,520
that we're experiencing with the virus

546
00:40:43,520 --> 00:40:46,880
sort of throws it off,
but that's hopefully

547
00:40:46,880 --> 00:40:51,773
a one in a lifetime event.

548
00:40:57,000 --> 00:41:02,000
So we can see that the
random walk with drift

549
00:41:02,200 --> 00:41:05,840
has both persistence and trend

550
00:41:05,840 --> 00:41:10,840
and so the A naught is the trend part

551
00:41:13,040 --> 00:41:16,740
and the fact that it's a unit root process

552
00:41:16,740 --> 00:41:19,263
makes it highly persistent.

553
00:41:20,820 --> 00:41:25,820
So for any year, T,

554
00:41:25,860 --> 00:41:27,003
that the YT

555
00:41:33,200 --> 00:41:36,940
equals this A naught added on T time.

556
00:41:36,940 --> 00:41:41,510
So A naught times T plus
all of the error terms,

557
00:41:41,510 --> 00:41:44,010
plus Y naught, our starting point.

558
00:41:44,010 --> 00:41:46,870
And if we, again, assume that
our starting point is zero,

559
00:41:46,870 --> 00:41:51,870
then the expected value
of YT is just A naught T.

560
00:41:52,510 --> 00:41:55,163
And if A naught is greater than zero,

561
00:41:56,160 --> 00:41:58,470
the expected value will grow over time.

562
00:41:58,470 --> 00:42:00,963
And if it's less, it
will shrink over time.

563
00:42:05,030 --> 00:42:10,030
So again, this is a
highly persistent process.

564
00:42:10,530 --> 00:42:15,050
It's a unit root,

565
00:42:15,050 --> 00:42:19,080
it's trending and persistent,

566
00:42:19,080 --> 00:42:23,650
and we need to be able
to transform the data

567
00:42:23,650 --> 00:42:28,650
so that it becomes weakly dependent

568
00:42:29,130 --> 00:42:32,053
and we can work with these data.

569
00:42:35,260 --> 00:42:38,150
Not surprisingly the most

570
00:42:38,150 --> 00:42:42,360
common transformation is differencing.

571
00:42:42,360 --> 00:42:46,640
That we saw that much like
we did with panel data

572
00:42:46,640 --> 00:42:48,940
where we subtract

573
00:42:50,940 --> 00:42:53,700
this years minus last years

574
00:42:53,700 --> 00:42:57,780
and that year's minus the
year before and so on.

575
00:42:57,780 --> 00:43:02,750
And in this way, it gets
rid of both the persistence

576
00:43:02,750 --> 00:43:07,330
because it's only measuring
the change over time.

577
00:43:07,330 --> 00:43:12,013
And the trend, because the A naught trend

578
00:43:13,080 --> 00:43:15,260
is subtracted out every time.

579
00:43:15,260 --> 00:43:20,260
And in that way, when we
transform with the difference,

580
00:43:20,730 --> 00:43:24,400
a random walk, then the change in Y

581
00:43:24,400 --> 00:43:28,070
then only becomes the
change in an error term.

582
00:43:28,070 --> 00:43:30,580
And if we have other regressors

583
00:43:30,580 --> 00:43:33,590
much like we saw with panel data,

584
00:43:33,590 --> 00:43:37,530
we can regress how the change in Y

585
00:43:37,530 --> 00:43:40,740
is a function of the change
in Xs just like we did

586
00:43:40,740 --> 00:43:44,633
in the first differencing
model in panel data.

587
00:43:49,090 --> 00:43:53,810
So by way of overview in the first section

588
00:43:53,810 --> 00:43:58,810
we looked at well behaved data
with restrictive assumptions

589
00:43:59,330 --> 00:44:01,540
and what are their properties.

590
00:44:01,540 --> 00:44:03,460
Here we're looking at times

591
00:44:03,460 --> 00:44:07,010
when the data are less well behaved,

592
00:44:07,010 --> 00:44:09,610
but with more realistic symptoms

593
00:44:09,610 --> 00:44:13,660
and what we can do to transform it.

594
00:44:13,660 --> 00:44:17,160
And then last, in the next one,

595
00:44:17,160 --> 00:44:21,957
we begin to look at
what happens if we have

596
00:44:22,990 --> 00:44:27,513
either serial correlation
or heteroskedasticity.

597
00:44:30,600 --> 00:44:34,070
Now we're gonna get into section three

598
00:44:34,070 --> 00:44:36,630
and look at what happens

599
00:44:36,630 --> 00:44:41,200
when the last few assumptions don't hold.

600
00:44:41,200 --> 00:44:44,990
And specifically when the assumption five

601
00:44:44,990 --> 00:44:48,980
of a serial correlation

602
00:44:49,860 --> 00:44:52,470
is a problem.

603
00:44:52,470 --> 00:44:57,120
So again, under serial correlation,

604
00:44:57,120 --> 00:44:59,760
our assumption does not hold,

605
00:44:59,760 --> 00:45:04,760
and as we'll see much
like heteroskedasticity,

606
00:45:06,390 --> 00:45:09,770
it leads to faulty inference

607
00:45:09,770 --> 00:45:12,350
and biased errors and things like that.

608
00:45:12,350 --> 00:45:15,970
So, and also inefficient models.

609
00:45:15,970 --> 00:45:18,320
So we're gonna learn how to test for it

610
00:45:18,320 --> 00:45:19,900
and how to deal with it

611
00:45:21,380 --> 00:45:25,513
and we will also revisit
heteroskedasticity briefly.

612
00:45:28,570 --> 00:45:31,723
Much like we learned with
cross-sectional data,

613
00:45:33,437 --> 00:45:37,187
heteroskedasticity and
autocorrelated errors,

614
00:45:39,070 --> 00:45:44,070
give us a faulty variance
estimator of the OLS estimators,

615
00:45:46,330 --> 00:45:48,910
and therefore we need to deal with them

616
00:45:48,910 --> 00:45:52,790
and they tend to be fairly
common in time series,

617
00:45:52,790 --> 00:45:55,900
especially autocorrelated errors,

618
00:45:55,900 --> 00:45:59,570
because we are not
random sampling each time

619
00:46:02,780 --> 00:46:05,420
parts of the model that we don't estimate

620
00:46:05,420 --> 00:46:10,420
that we can't measure,
that are seemingly random,

621
00:46:10,950 --> 00:46:15,950
recur over time, and we
need to deal with it.

622
00:46:16,530 --> 00:46:19,460
So, as we've learned before,

623
00:46:19,460 --> 00:46:23,523
when we have autocorrelated errors,

624
00:46:24,520 --> 00:46:29,170
the good news is that there is no bias,

625
00:46:29,170 --> 00:46:32,367
but as we learned with heteroskedasticity,

626
00:46:34,500 --> 00:46:38,170
this model will yield a higher variance

627
00:46:38,170 --> 00:46:39,710
so it's no longer blue.

628
00:46:39,710 --> 00:46:41,940
It's no longer the most
efficient estimator

629
00:46:41,940 --> 00:46:45,350
and the T-test and F-tests don't work

630
00:46:45,350 --> 00:46:48,373
due to the biased estimate

631
00:46:51,980 --> 00:46:54,930
of the variance.

632
00:46:54,930 --> 00:46:58,570
So the good news is that if

633
00:46:58,570 --> 00:47:02,840
the dynamics are properly
specified, there is no problems.

634
00:47:02,840 --> 00:47:05,560
So if we have put the right lags in there

635
00:47:05,560 --> 00:47:08,670
and every thing else we can deal with it.

636
00:47:08,670 --> 00:47:10,740
And there's also techniques

637
00:47:10,740 --> 00:47:13,370
that we will learn about
now to test for it.

638
00:47:13,370 --> 00:47:17,283
And in some cases to transform the data.

639
00:47:23,600 --> 00:47:28,120
Again, the problem with serial correlation

640
00:47:30,070 --> 00:47:33,180
is that OLS is no longer blue

641
00:47:33,180 --> 00:47:37,170
and all of the tests that
we use no longer work.

642
00:47:37,170 --> 00:47:39,940
Very much like heteroskedasticity

643
00:47:39,940 --> 00:47:41,390
that we've dealt with before.

644
00:47:43,780 --> 00:47:45,930
So we learned last time about

645
00:47:49,451 --> 00:47:50,901
the autoregressive model one.

646
00:47:52,000 --> 00:47:54,830
And we can assume in some cases

647
00:47:54,830 --> 00:47:59,520
that the errors follow a similar pattern.

648
00:47:59,520 --> 00:48:03,853
That this year's error is a
function of last year's error,

649
00:48:05,080 --> 00:48:09,380
plus some new E, some new
sort of random information.

650
00:48:09,380 --> 00:48:13,883
So when P here, rho, equals zero,

651
00:48:16,120 --> 00:48:19,500
then the errors are not correlated

652
00:48:19,500 --> 00:48:21,340
and we don't have a problem.

653
00:48:21,340 --> 00:48:23,573
When it's greater than zero,

654
00:48:24,790 --> 00:48:27,950
OLS underestimates the variance,

655
00:48:27,950 --> 00:48:31,990
when P is less than zero it overestimates,

656
00:48:31,990 --> 00:48:36,990
and our T and F and
Lagrange multiplier test

657
00:48:38,270 --> 00:48:40,623
and all of those no longer work.

658
00:48:44,800 --> 00:48:48,690
So when we have strictly
exogenous regressors,

659
00:48:48,690 --> 00:48:51,130
where none of the regressors

660
00:48:51,130 --> 00:48:54,590
or the error terms over
time are correlated,

661
00:48:54,590 --> 00:48:56,800
it makes it a little bit easier.

662
00:48:56,800 --> 00:48:59,210
And let's assume that we have

663
00:49:00,400 --> 00:49:05,400
this AR1 model for serial correlation

664
00:49:05,470 --> 00:49:07,370
that we saw last slide,

665
00:49:07,370 --> 00:49:11,870
where this year's error is a
function of last year's error.

666
00:49:11,870 --> 00:49:15,280
So you could simply run a regression

667
00:49:15,280 --> 00:49:18,023
where you put this year's error,

668
00:49:19,180 --> 00:49:22,860
basically error T on the left side

669
00:49:22,860 --> 00:49:27,860
and the year before that, T
minus one on the right side.

670
00:49:28,650 --> 00:49:33,373
Run a regression, our null
hypothesis is that our P,

671
00:49:34,370 --> 00:49:38,447
the coefficient with UT
minus one equals zero.

672
00:49:42,840 --> 00:49:46,440
And in this case,

673
00:49:46,440 --> 00:49:51,170
we hope that we get a small test stat,

674
00:49:51,170 --> 00:49:54,450
and we can fail to to reject our null

675
00:49:54,450 --> 00:49:59,450
and assume that we do not
have serial correlation.

676
00:49:59,610 --> 00:50:02,250
But if we do, if we reject our null,

677
00:50:02,250 --> 00:50:03,700
then we have to deal with it.

678
00:50:06,520 --> 00:50:08,440
So here's how you would do it.

679
00:50:08,440 --> 00:50:12,730
Again, First regress YT

680
00:50:12,730 --> 00:50:16,410
on each of your regressors,
save the residuals,

681
00:50:16,410 --> 00:50:21,180
then regress each UT on the previous year.

682
00:50:21,180 --> 00:50:23,740
So you would have to
sort of go in in Excel

683
00:50:23,740 --> 00:50:25,070
and cut and paste,

684
00:50:25,070 --> 00:50:29,500
and sort of just knock it down one row,

685
00:50:29,500 --> 00:50:34,500
and run a T-test on this coefficient.

686
00:50:34,524 --> 00:50:39,524
And hopefully you can

687
00:50:39,680 --> 00:50:42,720
fail to reject the null.

688
00:50:42,720 --> 00:50:44,850
And note that this only tests

689
00:50:44,850 --> 00:50:49,610
if the adjacent errors are correlated,

690
00:50:49,610 --> 00:50:51,910
since it's an AR1,

691
00:50:51,910 --> 00:50:55,780
it's only looking at the
previous year's error.

692
00:50:55,780 --> 00:51:00,510
So if UT and UT minus two

693
00:51:00,510 --> 00:51:05,510
are correlated, this test
wouldn't pick that up.

694
00:51:09,630 --> 00:51:13,250
A very common test is a
so-called Durbin-Watson test,

695
00:51:13,250 --> 00:51:17,700
where we also use UT or the UT hat.

696
00:51:17,700 --> 00:51:19,970
So we save the residuals,

697
00:51:19,970 --> 00:51:24,530
our null hypothesis again,
is that P equals zero.

698
00:51:24,530 --> 00:51:29,530
And this stat sums up on the denominator,

699
00:51:30,220 --> 00:51:34,230
UT minus, UT minus one, and squares that,

700
00:51:34,230 --> 00:51:35,900
and in the denominator,

701
00:51:35,900 --> 00:51:40,830
just the sum of squared
residuals as we've come to know.

702
00:51:40,830 --> 00:51:43,220
And then we look at the stats,

703
00:51:43,220 --> 00:51:47,700
so this stat will be approximately equal

704
00:51:47,700 --> 00:51:52,700
to two times the quantity one minus P.

705
00:51:53,490 --> 00:51:58,283
So if P equals zero,

706
00:52:00,370 --> 00:52:04,730
and then we can fail to reject the null

707
00:52:04,730 --> 00:52:08,560
and that's a good thing
that we don't have it.

708
00:52:08,560 --> 00:52:12,050
But if the number is far less than zero,

709
00:52:12,050 --> 00:52:16,340
so you're multiplying
by a number almost one.

710
00:52:16,340 --> 00:52:19,070
So one minus P is a very small number,

711
00:52:19,070 --> 00:52:22,610
multiplied by two is
still a very small number

712
00:52:22,610 --> 00:52:27,180
and implies that P is
significantly different than zero

713
00:52:27,180 --> 00:52:30,153
and we have serial correlation.

714
00:52:32,730 --> 00:52:36,060
When we don't have strict exogeneity,

715
00:52:37,177 --> 00:52:39,700
and this will happen most often

716
00:52:39,700 --> 00:52:42,510
when we have lagged regressors,

717
00:52:42,510 --> 00:52:46,030
when we not only put XT into the model,

718
00:52:46,030 --> 00:52:49,333
this year's regressor, but
also put last years in.

719
00:52:50,560 --> 00:52:52,840
This is more common,

720
00:52:52,840 --> 00:52:57,840
and this could be valid for
any number of regressors.

721
00:52:58,070 --> 00:53:01,393
So this is the more sort of general case.

722
00:53:06,350 --> 00:53:09,190
The test is much the same,

723
00:53:09,190 --> 00:53:14,190
except now in the second step,
we include our regressors.

724
00:53:15,690 --> 00:53:20,690
So we start with regressing
YT just on our regressors

725
00:53:20,980 --> 00:53:23,050
and save the residuals.

726
00:53:23,050 --> 00:53:26,560
And then we re regress the residuals

727
00:53:26,560 --> 00:53:31,560
on the original regressors
and on UT minus one,

728
00:53:31,640 --> 00:53:36,640
and run the T-test on P, the
coefficient of UT minus one.

729
00:53:38,600 --> 00:53:41,710
And by including the regressors,

730
00:53:41,710 --> 00:53:46,710
we are controlling for any correlations

731
00:53:47,590 --> 00:53:52,550
between our Xs and our error terms.

732
00:53:52,550 --> 00:53:57,490
So we don't need to make this assumption,

733
00:53:57,490 --> 00:54:02,407
but we might have to
calculate, use robust errors

734
00:54:03,410 --> 00:54:07,693
if we suspect that
heteroskedasticity is present.

735
00:54:14,410 --> 00:54:17,580
We can also test if there's

736
00:54:17,580 --> 00:54:21,520
an autoregressive process of order two

737
00:54:21,520 --> 00:54:26,520
where we regress UT on our regressors

738
00:54:26,700 --> 00:54:29,320
plus each of the past two years

739
00:54:29,320 --> 00:54:32,500
and do a joint F-test on P1 and P2,

740
00:54:32,500 --> 00:54:35,710
because here our null is that P2 and P2

741
00:54:35,710 --> 00:54:38,373
are jointly equal to zero.

742
00:54:43,130 --> 00:54:47,700
And you could do it for
any number of regressors

743
00:54:47,700 --> 00:54:50,440
and some general number Q,

744
00:54:50,440 --> 00:54:54,910
where you regress the UT
on Xs, save the residuals,

745
00:54:54,910 --> 00:54:59,910
and then regress them
on Q lagged residuals,

746
00:55:00,760 --> 00:55:02,610
and use a joint F-test

747
00:55:02,610 --> 00:55:05,983
that all of these rows P1, P2, da-da-da,

748
00:55:07,330 --> 00:55:11,410
up to PQ are equal to zero.

749
00:55:11,410 --> 00:55:15,560
And again, we want to
fail to reject that null,

750
00:55:15,560 --> 00:55:18,793
hoping that we don't
have serial correlation.

751
00:55:25,600 --> 00:55:27,930
If these tests determined

752
00:55:27,930 --> 00:55:31,440
that you do have serial correlation,

753
00:55:31,440 --> 00:55:36,440
if the coefficient on last year's errors

754
00:55:37,180 --> 00:55:39,980
are significant on this year error,

755
00:55:39,980 --> 00:55:44,740
then we have to do some
sort of transformation.

756
00:55:44,740 --> 00:55:46,920
One of the things that we learned before

757
00:55:46,920 --> 00:55:51,670
was to difference and subtract

758
00:55:53,280 --> 00:55:56,070
this year minus last year and last year,

759
00:55:56,070 --> 00:55:58,893
minus the year before that and so on.

760
00:55:59,930 --> 00:56:02,060
But you lose an observation

761
00:56:02,060 --> 00:56:06,660
and this should remind you
a little bit of panel data,

762
00:56:06,660 --> 00:56:11,170
where differencing cost us an observation

763
00:56:11,170 --> 00:56:13,680
and therefore cost of efficiency.

764
00:56:13,680 --> 00:56:16,750
So there is a transformation,

765
00:56:16,750 --> 00:56:21,750
very similar to random effects models,

766
00:56:23,300 --> 00:56:27,330
where you can do a transformation.

767
00:56:27,330 --> 00:56:32,330
You need to know the value of
this P or rho, but if you do,

768
00:56:33,910 --> 00:56:38,280
then you can do a
transformation like this,

769
00:56:38,280 --> 00:56:39,483
and I'm gonna show you.

770
00:56:40,510 --> 00:56:43,190
And I just used a

771
00:56:45,580 --> 00:56:47,510
slide from the textbook slides

772
00:56:47,510 --> 00:56:50,140
'cause they make them look much more neat.

773
00:56:50,140 --> 00:56:55,140
So here, if you have
strictly exogenous regressors

774
00:56:55,660 --> 00:56:58,600
that you can do this
kind of transformation

775
00:56:58,600 --> 00:57:03,090
where it's sort of like differencing,

776
00:57:03,090 --> 00:57:08,090
but you only subtract
rho times the past year.

777
00:57:09,430 --> 00:57:13,350
So if you look at about
the third equation down

778
00:57:13,350 --> 00:57:15,590
with the red boxes all around it,

779
00:57:15,590 --> 00:57:20,590
you instead of of subtracting
out the full value

780
00:57:22,240 --> 00:57:26,730
of last year's Y, that
you only subtract out rho

781
00:57:26,730 --> 00:57:31,730
and rho being some number
between zero and one,

782
00:57:32,640 --> 00:57:34,713
only subtract out that.

783
00:57:35,620 --> 00:57:37,721
The problem is that you need

784
00:57:37,721 --> 00:57:42,283
a good estimator for this value of rho.

785
00:57:49,640 --> 00:57:52,930
And you can do the same thing

786
00:57:52,930 --> 00:57:57,930
if you have an AR2 or an ARQ process,

787
00:57:59,740 --> 00:58:03,640
and most of these are available

788
00:58:03,640 --> 00:58:07,960
in more advanced stats packages.

789
00:58:07,960 --> 00:58:12,550
I don't believe SPSS has
an easy way to do this,

790
00:58:12,550 --> 00:58:14,040
but, you know, again,

791
00:58:14,040 --> 00:58:17,470
if you're working with time series data,

792
00:58:17,470 --> 00:58:18,550
you're probably gonna want

793
00:58:18,550 --> 00:58:23,550
to get and work with a more
sort of econometrics specific

794
00:58:25,180 --> 00:58:28,833
statistical software package like SAS.

795
00:58:35,080 --> 00:58:40,080
So again, it becomes a trade off here

796
00:58:42,910 --> 00:58:47,830
that if you have a static model like this,

797
00:58:47,830 --> 00:58:52,830
and you think that the UT
follows this AR1 process,

798
00:58:54,430 --> 00:58:58,010
that then you can just subtract it out.

799
00:58:58,010 --> 00:59:03,010
And then it basically
fixes a lot of our efforts

800
00:59:04,990 --> 00:59:09,840
because UT then, if UT as a random walk,

801
00:59:09,840 --> 00:59:14,220
then subtracting it out
and taking the difference

802
00:59:14,220 --> 00:59:19,220
makes it a zero mean constantly
various variant error term.

803
00:59:22,120 --> 00:59:26,720
And we can use OLS and we
can use our F and T-tests

804
00:59:26,720 --> 00:59:27,810
and everything's good.

805
00:59:27,810 --> 00:59:29,933
But again, we lose an observation.

806
00:59:35,610 --> 00:59:40,610
There is also specific transformations.

807
00:59:42,200 --> 00:59:46,440
The so-called Newey-West corrections,

808
00:59:46,440 --> 00:59:51,160
most software that works specifically

809
00:59:51,160 --> 00:59:54,090
with time series will have this.

810
00:59:54,090 --> 00:59:58,600
It does require that this is
another FGLS transformation,

811
01:00:00,750 --> 01:00:04,197
and it does require strict exogeneity

812
01:00:05,092 --> 01:00:07,103
and ARQ correlation,

813
01:00:08,510 --> 01:00:13,510
but it can fix and give you

814
01:00:13,610 --> 01:00:15,750
the correct standard errors,

815
01:00:15,750 --> 01:00:19,150
which will allow you to do better T-tests.

816
01:00:19,150 --> 01:00:22,740
It does need a large sample.

817
01:00:22,740 --> 01:00:27,740
And as before, if you really
have severe serial correlation,

818
01:00:30,130 --> 01:00:34,740
it might be the best idea to simply

819
01:00:37,030 --> 01:00:41,073
take the difference, and
use that model instead.

820
01:00:43,130 --> 01:00:47,040
There are also tests
for heteroskedasticity

821
01:00:47,040 --> 01:00:51,940
that you can use the ones
that we are familiar with,

822
01:00:51,940 --> 01:00:56,940
but we have to first rule out

823
01:00:57,780 --> 01:01:02,780
serial correlation that
these tests will sort

824
01:01:03,120 --> 01:01:04,950
of give you a false positive

825
01:01:04,950 --> 01:01:07,640
that if there is serial correlation,

826
01:01:07,640 --> 01:01:11,080
it might make you think that
there is heteroskedasticity.

827
01:01:11,080 --> 01:01:15,710
So you want to do the serial
correlation tests first

828
01:01:15,710 --> 01:01:17,600
and only if that's ruled out,

829
01:01:17,600 --> 01:01:22,500
then do the Breusch-Pagan or
White tests that we've learned

830
01:01:22,500 --> 01:01:24,470
and then we can use things

831
01:01:24,470 --> 01:01:26,983
like weighted least squares as before.