- [Instructor] This is the first part of a three-part series
on time series econometrics.
I hope you find this section useful.
This section on time series has three main components.
This is the first one,
and I'm going to do a little introduction.
Talk about two common models,
static and finite distributed lag.
Talk about very well-behaved time series data,
which will remind you a lot
of the very well-behaved cross-sectional data
we did many weeks ago.
And what we can assume OLS does based on those
as far as unbiased and efficient.
And in the last part, we're gonna talk about trends
and seasonal effects and how to account for those.
So, so far, most of our time
has been spent with cross-sectional data
where we have a single timeframe, T equals one,
but many observations and equals many.
For example, doing a survey over a specific time period,
but administering it and gaining data
from lots of different people.
We also spoke a bit about panel data,
where you have, usually a big N, many respondents
and more than one time period,
but usually a fairly small number.
Say two, three, maybe at most four times.
Now we're looking at a time series
where there's one N but many Ts.
So we're looking at one set of variables over time,
such as GDP or sales of some good.
It could be a sociological thing
like voting, voting rate, or many other examples,
educational attainment in a given place,
or a biophysical example,
like phosphorous in the lake or land use.
But in any case, it's looking at
the same respondent, in a sense, over many years.
And since there's a logical ordering here,
we have to think about,
we're not drawing a random sample of responses.
We're looking at the same one,
and there's a clear direction of causality
that we can in many cases assume
and we have to account for.
That the past influences the present
and the present will influence the future.
Again, we'll be looking at a time series process
that has both deterministic and stochastic components,
that there will be part of our dependent variable,
which the predicted value is explained by the regressors.
And also a stochastic element where we have
an error term of that which was sort of unexpected
or cannot be measured each time as well.
Looking at all of the topics
in this time series
section again today,
we're gonna be looking at very well behaved data.
What are the assumptions, which are in some cases
sort of unrealistic and simplistic and building from there,
and then looking at a slightly less well behaved data
and transformations that we can make
to make them better behaved.
And last, how do we test for these assumptions being true
and what do we adjust
and how do we adjust if we find that the data
are not as well behaved as we would like?
There are two basic kinds of models
in time series, static and finite distributed lag
and I will talk to you about each one.
This is a static model and it's static
because you can see in the equation that starts at YT
that we're looking at every thing over the same time.
So the T would be say the years of the study.
So we might have started collecting data in 2001,
in which case T equals 1 and continue on
through the year 2020, when T equals 20.
Here all of our variables are contemporaneous,
that they're happening at the same time,
that we only assume that the regressor, ZT,
had any effect on that year's YT.
That nothing that happened in the past has any effect,
and that is why it's static
and that it's all happening in a single time
and this is very much like cross sectional data.
And just like there that we can assume
with a single regressor that the change in Y
will be beta one times the change in Z.
And assuming that the Z is exogenous
and it has no effect on you
that we can do this sort of simple calculation.
And the a beta is then the change in Y
as a result of the change in Z, the slope of that line.
We could of course, we could of course add more regressors.
Z2, Z3 as many as what we think are correct.
And these data are very much in this model
is very much like cross section,
except now we're looking across time
instead of across space.
We're looking at the different realization
of a variable over time,
instead of that variable across different respondents.
In contrast here is a finite distributed lag.
So lag meaning that regressors from past years
are thought to have an effect on this year's Y.
So you say YT and ZT,
but also ZT minus one, last year's Z,
and ZT minus two, two years ago's Z.
And since there are two lagged regressors,
this is known as a finite distributed
lag model of order two,
because there are two lags and we could have more lags
if we think that that is the correct model.
Here are the assumptions
of well-behaved OLS time series data.
So the first is, and these should look pretty familiar,
the first is that it is linear in parameters
that we can model the data generating process
as a linear function.
Next that as before each regressor has
to contribute some new information.
That no Z or X can be a constant
or a linear function of the other regressors.
Again, that it has to provide some new information.
Third is the exogeneity assumption
that every error term is uncorrelated
with each of the regressors.
If this holds true, a weaker assumption
that this year's error term
is uncorrelated with this year's regressors,
this is called contemporaneously exogenous,
but there's also a stronger assumption
that this year's error term
is uncorrelated with any regressors for any other time.
And that the regressors are uncorrelated
with the error term for any other time.
This again is a stronger assumption
and when this holds, this is called strict exogeneity.
More on the third assumption that we don't really have
to worry about this in cross sectional data,
that your error term,
some strange thing that may have happened to you,
or some relatively small and minor thing
that we are not able to measure
is not effected by my regressors.
So whether you found a dollar
or got a flat tire
is not effected by my income and my age and my behaviors
and so forth since it's a random sample.
But we don't randomly sample in time series
because we want a series of data
and they are all very intimately tied.
That what happened last year
might affect what happens to this year,
because we're looking at the same phenomenon over time.
This assumption three fails the same in cross sectional data
in cases where there's a measurement error
or omitted variables, as well as in time series.
If there is a lag defect that this may hold true,
that this year's error term
may be effected by last year's regressor.
And in this case, we want to put last year's regressors,
or last two years regressors into our model.
So we can specifically account and control for them
and to take them out of the error term.
When these three assumptions hold,
we can assume that OLS is unbiased.
And when there is an omitted variable,
the same kind of omitted variable bias analysis that we did
way back in the beginning of class would hold true.
The same basic themes and analyses would hold true.
The fourth assumption
in well-behaved time series data is homoskedasticity.
That the variance of our error term,
given any value of X, our regressor is a constant.
So before it was that the variance
of your error term and my error term,
had a constant variance.
Now it's that the variance of this year's error term
and last year's error term, going back in time,
all have a constant variance.
That's what homoskedasticity means in this context.
And we could do the same basic kinds of tests
like we did in cross sectional data
to test for homoskedasticity
such as the White test and the Breusch-Pagan test.
The fifth assumption is no serial correlation
that this year's error term
and last year's error term are uncorrelated.
And in fact, this year's error term is uncorrelated
with any error term from any past year.
The violation is that something abnormally high this year
is also abnormally high last year,
that the thing that we sort of failed to measure,
the minuscule thing that we failed to measure
or didn't account for this year is that same minuscule thing
that we didn't account for last year.
And we will learn a test on how to do that.
I think it's a lot like the in panel data,
the idea that there's this AI that goes along
with every individual respondent
that is very hard to measure but that holds through
and here again, I think that the parts of the model
that are just very hard to measure will follow through
because we're looking at the same thing
and it's likely that the errors will be correlated.
And again, we will learn a test
on how to see if this is true later on in this series.
Note that let's not rule out that the Ys are correlated,
that they almost always are.
That last year's GDP
is highly correlated with the year before that
and the year before that, et cetera.
When these five assumptions hold,
the variance of the beta J has the same formula as before.
It is a function of the variance
of the overall error term, sigma squared,
as well as how spread out that XJ is
and how collinear that XJ is
with the other regressors in our model.
And under those assumptions,
those five assumptions, the three unbiasedness
and the sort of spherical errors assumption for time series.
If they all hold, if they are all true, then OLS is blue.
It's the best linear unbiased estimator.
It's the most efficient, unbiased estimator.
Remember that when we started to get into hypothesis tests
that we added a new assumption,
and that is the error term is normally distributed
with mean zero
and variance sigma squared.
And we are adding another slightly stronger assumption
that the Us are not only independent of X as before,
I think at assumption four,
but that they are independently
and identically distributed, IID.
So if the sixth assumption holds
then the OLS estimators also
have predictable statistical properties,
and we can use the kind of inference tests,
T-tests, the F-tests,
and we can create confidence intervals the same,
and they have the same interpretation
as we learned about in cross section.
So just a bit more on IID.
IID stands for independent and identically distributed.
So one way of thinking about this is
the bingo ball wheel that you're drawing the errors from.
It does not change, that it's the exact same distribution.
That's what the identically distributed means.
It's the exact same bell curve probability
of drawing any error term over every year.
And the independent means that it has no memory,
that the ball is replaced each year,
that the error term that you drew this time
has no effect on the error term that you draw next time.
We very often want to use dummy variables in time series
to account for whether
an important event or policy or regime
was in place during that time.
A good example would be if there was
a big recall of romaine, which we've seen a few times,
that the demand for romaine would depend not only sort of
on the supply and the prices and the people's preferences
and things, and their incomes and things like that,
but you would want to have a dummy variable in place
that accounts for there was this huge scare
and there was this huge recall
that happened during that time.
Otherwise you're not really going to get a good account
of what is the effect of income
and price and supply and preferences.
And I was thinking as well,
that when we model time series GDP
and other economic variables for right now
that we will probably need to have a dummy variable
for the coronavirus that, you know,
the intercept shift of what happened
just as a result of that holding all else equal.
It's also very common in time series
to acknowledge that the data
have natural trends and seasons.
So trends take into account that there's,
especially in economics, that there is a time trend,
that our GDP increases over time,
worker productivity increases over time
sort of independent of many, many other factors.
That it adds this sort of a natural
or recurring time trend built into it.
And it's important that we account for that
because if productivity and GDP are both trending upward,
that failing to account for that trend in both of them
would make the relationship between them
seem stronger than it actually is
and we would get a biased estimate.
That it would be in a sense,
an omitted variable bias by failing
to account for that trend.
Here is a very simple example
of a Y that has a time trend.
So T would just be the actual year
that the data were collected.
So again, if we have 2001 to 2020 data,
T equals 1 in year 2001, and so on,
T equals 20 in year 2020,
and in this model, holding all else equal,
Y will increase or shrink by the rate of A1 each year.
So if A1 is greater than zero,
the predicted value of Y will increase by A1 each year.
We very often model things with
a number of regressors and a time trend.
So you see here, there's two regressors,
X1, X2, as well as this time trend, T.
And again, that this accounts for the trend
that may be in either Y or in X1 or X2,
and by leaving out this trend by not controlling for it,
it can cause omitted variable bias.
And note that you should include a time trend,
even if the Xs trend and the Ys don't that you want
to take out the effect of time over these Xs
and holding time equal, how does X affect Y.
The R squared of a time series model tends to be quite high,
that these kind of aggregated data tend to explain
the relationships and result in a higher R squared
than we get with something
like doing a survey and how do the regressors
across individual people affect the dependent.
So, one way to get a better sense
of how much of the variation in Y is explained by X,
that is R squared independent of time
is to detrend our data.
And you do that
by regressing YT only on T
and saving those residuals,
which I call this weird Y umlaut.
And then R squared is this formula
where SSR is that of regressing,
YT on X1, X2 and T.
And it gives you a better idea of taking away time trend,
how well do the Xs explain our Y.
There's also many economic phenomena
that have a clear seasonality,
such as most Christmas trees
are sold in November and December,
hay is sold in the summer when it's fresh.
Another example is nursery flower transplants
are much more often sold in the spring.
So you can include dummy variables
for specific months or quarters or seasons
where you think that sales will be particularly high or low,
model it with them and you can use a T or an F-test
to determine whether that has
a significant effect on our model
and much like we did before, we can deseasonalize our Ys,
where we regress our YTs
only on our seasonal dummy variables, save those residual,
and then regress our Xs
on those Y umlauts again,
and use R squared as another measure
of the goodness of fit of the model.
Basically, again, that many of the assumptions
are unrealistic, too restrictive,
doesn't really meet the real world.
So we're gonna look at those models
and look at some common ones and some principles
that we have to abide by
for those assumptions to fit and last,
a transformation that you're already familiar with
that can deal with it.
So two pieces of vocabulary that we're gonna deal with,
dependence and stationarity.
And both of them
cause issues with inference,
that it can mess up our hypothesis tests, our DNF tests
that we would like to run on our data.
We think about dependence and how much
does the past affect the present and the future.
And we say that a time series process
is a weakly dependent,
that as the distance between time periods
gets larger, the relationship
between those variables gets small.
So basically we want weak dependence
so that this year's
realization of X and future years,
that relationship gets very small, very fast.
You can think of independent is that like
when we're drawing a random sample
where the coin has no memory at all,
that last year that you and I as respondents
don't have any thing to do with each other.
Dependence really, where they are very intimately related
and last year and past years effect this year
in a very strong way, or thinking about
that the coin remembers every flip
really messes up inference and it makes it impossible.
Where with weak dependence, this memory fades over time.
Stationarity means that the variables
have the same probability distribution over time.
So much as, we assumed that they have
this normal IID distribution over time,
we need to make assumptions
and know that the probability distribution
of our error terms and of our dependent variables
have a predictable, the same distribution
in order for us to be able to make any kind of inference
and use T-tests and F-tests
and those tests that we have learned about.
Basically weak dependence is needed
for the law of large numbers
and the central limit theorem to hold.
That because we do not random sample,
if the realizations are too closely related,
we can't do inference
and strong dependence totally messes it up.
So we are going to look at models
that have strong and weak dependence
that we may encounter in time series data.
So what we're finding is these autoregressive models
where last year's Y
or past year's Y appear as regressors
with this year's Y on the left side.
And then two examples of that,
the so-called unit root process and the random walk.
And we'll go through each one over in the next few slides.
Here's an example
of an autoregressive process of order one, an AR1.
It's autoregressive, because last year's Y,
YT minus one, appears as a regressor.
And it's of order one because only
one lagged dependent variable is a regressor in this model.
The properties of this model
is that Y starts at some sort
of starting point, at time zero.
Our error term is again, IID zero sigma squared,
and that the ETs are independent of Y naught.
And just for simplicity, we often assume
that the expected value of Y naught equals zero,
or we could subtract out that value each time
to make the math easier.
So we're gonna look at the value
of this coefficient P
or I believe in the Wooldridge book it's rho,
but I just made it a P
because I have P on my keyboard and not rho.
So it's important that for any kind of...
So when rho is less than one,
then we see that with the expected value of YT plus H,
given our starting point YT, is rho to the H times YT,
for any number of years greater than one.
And we see that the influence of our starting point,
YT gets smaller and smaller over time,
and it approaches zero overall as H gets bigger and bigger.
And this is what we want, we have weak dependence.
When rho equals one,
the best prediction of YT plus H is always YT.
We call this a unit root process when rho equals one,
and this leads to strong dependence,
and we need to deal with this
in order to do estimation and inference.
So again, the unit root process
is when this P or rho equals one.
One very common unit root process is the
so-called random walk where Y,
this year, equals last year's Y plus an error term.
And we assume that the error term, ET,
is as a mean of zero and constant variance
and so Y starts at the previous year's value
and adds an error term.
And every year just adds that error term
to last year's value so that there's a random walk.
The random part is the error,
and it might be high, it might be low,
but it'll just sort of walk around that mean.
And if we assume, Y naught, just for mathematical
again equals zero,
and the expected value of Y in any year
equals this starting point.
So you can imagine the line sort of
meandering all around over time, going up and down,
but basically the expected value of any year
is that starting point.
So the value of Y
in year T plus H
is just the sum of the error terms over all those years,
plus YT, the starting point.
So the value of any year is the sum of all those error terms
plus the starting points.
So if we assume a starting point of zero, again,
it'll just be a line that meanders around
the zero line, the X axis.
And that for any value of YT,
that the expected value of YT plus H equals YT.
And you can see in the first equation
that if we have a big enough H
that if we add them all up, that they will,
since the expected value is zero, that they are going
to eventually converge and sum to zero.
And so the expected value of YT plus H
will always equal our starting point, YT.
So another example is the so-called random walk with drift.
So this is again, a unit root process
because our rho, or P, equals one.
And in this case, there's a drift
and that drift is A naught.
So every this year is Y is last year's Y
plus this drift plus this A naught.
So this has both persistence and trend.
It's persistent because it builds on the last year's value
and the expected value is
of what will be carried over.
And the trend is that every year
it adds this factor A naught.
So that let's us talk
a bit more about persistence and trend.
So we talked a bunch about trend last time,
and persistence is again,
when something that happens
is felt over time.
So if we don't deal with them, they'll mess up analysis.
We learned last time, how trend can be
an important omitted, variable bias,
and the persistence is when some shock
continues to be felt into the future,
or that this year's realization
is very closely related to past years,
that all of last year's and previous years values
are sort of remembered over time.
And it's interesting to note
that interest rates are persistent,
that this quarters interest rate
is going to be very closely related to last quarters,
but they don't really trend.
That they're not steadily upward or downward over time
whereas GDP is likely
both persistent and trending.
That, you know, in most years here in the US,
you'll get in a normal sort of growth
that you'll get only about 2% annual growth.
So last year's GDP is very,
plays a big factor on what this year's is,
but there's this sort of overall trend over time.
Now, the huge contraction of our economy
that we're experiencing with the virus
sort of throws it off, but that's hopefully
a one in a lifetime event.
So we can see that the random walk with drift
has both persistence and trend
and so the A naught is the trend part
and the fact that it's a unit root process
makes it highly persistent.
So for any year, T,
that the YT
equals this A naught added on T time.
So A naught times T plus all of the error terms,
plus Y naught, our starting point.
And if we, again, assume that our starting point is zero,
then the expected value of YT is just A naught T.
And if A naught is greater than zero,
the expected value will grow over time.
And if it's less, it will shrink over time.
So again, this is a highly persistent process.
It's a unit root,
it's trending and persistent,
and we need to be able to transform the data
so that it becomes weakly dependent
and we can work with these data.
Not surprisingly the most
common transformation is differencing.
That we saw that much like we did with panel data
where we subtract
this years minus last years
and that year's minus the year before and so on.
And in this way, it gets rid of both the persistence
because it's only measuring the change over time.
And the trend, because the A naught trend
is subtracted out every time.
And in that way, when we transform with the difference,
a random walk, then the change in Y
then only becomes the change in an error term.
And if we have other regressors
much like we saw with panel data,
we can regress how the change in Y
is a function of the change in Xs just like we did
in the first differencing model in panel data.
So by way of overview in the first section
we looked at well behaved data with restrictive assumptions
and what are their properties.
Here we're looking at times
when the data are less well behaved,
but with more realistic symptoms
and what we can do to transform it.
And then last, in the next one,
we begin to look at what happens if we have
either serial correlation or heteroskedasticity.
Now we're gonna get into section three
and look at what happens
when the last few assumptions don't hold.
And specifically when the assumption five
of a serial correlation
is a problem.
So again, under serial correlation,
our assumption does not hold,
and as we'll see much like heteroskedasticity,
it leads to faulty inference
and biased errors and things like that.
So, and also inefficient models.
So we're gonna learn how to test for it
and how to deal with it
and we will also revisit heteroskedasticity briefly.
Much like we learned with cross-sectional data,
heteroskedasticity and autocorrelated errors,
give us a faulty variance estimator of the OLS estimators,
and therefore we need to deal with them
and they tend to be fairly common in time series,
especially autocorrelated errors,
because we are not random sampling each time
parts of the model that we don't estimate
that we can't measure, that are seemingly random,
recur over time, and we need to deal with it.
So, as we've learned before,
when we have autocorrelated errors,
the good news is that there is no bias,
but as we learned with heteroskedasticity,
this model will yield a higher variance
so it's no longer blue.
It's no longer the most efficient estimator
and the T-test and F-tests don't work
due to the biased estimate
of the variance.
So the good news is that if
the dynamics are properly specified, there is no problems.
So if we have put the right lags in there
and every thing else we can deal with it.
And there's also techniques
that we will learn about now to test for it.
And in some cases to transform the data.
Again, the problem with serial correlation
is that OLS is no longer blue
and all of the tests that we use no longer work.
Very much like heteroskedasticity
that we've dealt with before.
So we learned last time about
the autoregressive model one.
And we can assume in some cases
that the errors follow a similar pattern.
That this year's error is a function of last year's error,
plus some new E, some new sort of random information.
So when P here, rho, equals zero,
then the errors are not correlated
and we don't have a problem.
When it's greater than zero,
OLS underestimates the variance,
when P is less than zero it overestimates,
and our T and F and Lagrange multiplier test
and all of those no longer work.
So when we have strictly exogenous regressors,
where none of the regressors
or the error terms over time are correlated,
it makes it a little bit easier.
And let's assume that we have
this AR1 model for serial correlation
that we saw last slide,
where this year's error is a function of last year's error.
So you could simply run a regression
where you put this year's error,
basically error T on the left side
and the year before that, T minus one on the right side.
Run a regression, our null hypothesis is that our P,
the coefficient with UT minus one equals zero.
And in this case,
we hope that we get a small test stat,
and we can fail to to reject our null
and assume that we do not have serial correlation.
But if we do, if we reject our null,
then we have to deal with it.
So here's how you would do it.
Again, First regress YT
on each of your regressors, save the residuals,
then regress each UT on the previous year.
So you would have to sort of go in in Excel
and cut and paste,
and sort of just knock it down one row,
and run a T-test on this coefficient.
And hopefully you can
fail to reject the null.
And note that this only tests
if the adjacent errors are correlated,
since it's an AR1,
it's only looking at the previous year's error.
So if UT and UT minus two
are correlated, this test wouldn't pick that up.
A very common test is a so-called Durbin-Watson test,
where we also use UT or the UT hat.
So we save the residuals,
our null hypothesis again, is that P equals zero.
And this stat sums up on the denominator,
UT minus, UT minus one, and squares that,
and in the denominator,
just the sum of squared residuals as we've come to know.
And then we look at the stats,
so this stat will be approximately equal
to two times the quantity one minus P.
So if P equals zero,
and then we can fail to reject the null
and that's a good thing that we don't have it.
But if the number is far less than zero,
so you're multiplying by a number almost one.
So one minus P is a very small number,
multiplied by two is still a very small number
and implies that P is significantly different than zero
and we have serial correlation.
When we don't have strict exogeneity,
and this will happen most often
when we have lagged regressors,
when we not only put XT into the model,
this year's regressor, but also put last years in.
This is more common,
and this could be valid for any number of regressors.
So this is the more sort of general case.
The test is much the same,
except now in the second step, we include our regressors.
So we start with regressing YT just on our regressors
and save the residuals.
And then we re regress the residuals
on the original regressors and on UT minus one,
and run the T-test on P, the coefficient of UT minus one.
And by including the regressors,
we are controlling for any correlations
between our Xs and our error terms.
So we don't need to make this assumption,
but we might have to calculate, use robust errors
if we suspect that heteroskedasticity is present.
We can also test if there's
an autoregressive process of order two
where we regress UT on our regressors
plus each of the past two years
and do a joint F-test on P1 and P2,
because here our null is that P2 and P2
are jointly equal to zero.
And you could do it for any number of regressors
and some general number Q,
where you regress the UT on Xs, save the residuals,
and then regress them on Q lagged residuals,
and use a joint F-test
that all of these rows P1, P2, da-da-da,
up to PQ are equal to zero.
And again, we want to fail to reject that null,
hoping that we don't have serial correlation.
If these tests determined
that you do have serial correlation,
if the coefficient on last year's errors
are significant on this year error,
then we have to do some sort of transformation.
One of the things that we learned before
was to difference and subtract
this year minus last year and last year,
minus the year before that and so on.
But you lose an observation
and this should remind you a little bit of panel data,
where differencing cost us an observation
and therefore cost of efficiency.
So there is a transformation,
very similar to random effects models,
where you can do a transformation.
You need to know the value of this P or rho, but if you do,
then you can do a transformation like this,
and I'm gonna show you.
And I just used a
slide from the textbook slides
'cause they make them look much more neat.
So here, if you have strictly exogenous regressors
that you can do this kind of transformation
where it's sort of like differencing,
but you only subtract rho times the past year.
So if you look at about the third equation down
with the red boxes all around it,
you instead of of subtracting out the full value
of last year's Y, that you only subtract out rho
and rho being some number between zero and one,
only subtract out that.
The problem is that you need
a good estimator for this value of rho.
And you can do the same thing
if you have an AR2 or an ARQ process,
and most of these are available
in more advanced stats packages.
I don't believe SPSS has an easy way to do this,
but, you know, again,
if you're working with time series data,
you're probably gonna want
to get and work with a more sort of econometrics specific
statistical software package like SAS.
So again, it becomes a trade off here
that if you have a static model like this,
and you think that the UT follows this AR1 process,
that then you can just subtract it out.
And then it basically fixes a lot of our efforts
because UT then, if UT as a random walk,
then subtracting it out and taking the difference
makes it a zero mean constantly various variant error term.
And we can use OLS and we can use our F and T-tests
and everything's good.
But again, we lose an observation.
There is also specific transformations.
The so-called Newey-West corrections,
most software that works specifically
with time series will have this.
It does require that this is another FGLS transformation,
and it does require strict exogeneity
and ARQ correlation,
but it can fix and give you
the correct standard errors,
which will allow you to do better T-tests.
It does need a large sample.
And as before, if you really have severe serial correlation,
it might be the best idea to simply
take the difference, and use that model instead.
There are also tests for heteroskedasticity
that you can use the ones that we are familiar with,
but we have to first rule out
serial correlation that these tests will sort
of give you a false positive
that if there is serial correlation,
it might make you think that there is heteroskedasticity.
So you want to do the serial correlation tests first
and only if that's ruled out,
then do the Breusch-Pagan or White tests that we've learned
and then we can use things
like weighted least squares as before.