Thank you.
Good morning, everybody, and welcome to the lecture on time series econometrics.
I have projected here again a slide with which we started already in the last lecture
because I'm going to repeat a little bit the introduction to univariate non-stationary
processes before we cover new material in this lecture.
But before doing so, let me ask you if there are any questions relating to the material
covered so far.
If so, please raise your hand.
I do not see any questions, so let me first perhaps put into perspective what we will
do today and what we have done in the last lecture when we started discussing univariate
non-stationary processes.
Before dealing with univariate non-stationary processes, we've covered the theory of stationary
and invertible processes basically covered by the class of autoregressive moving average
processes, so ARMA processes, and these were the dominant ways to model economic time series
and not only economic time series, but I'm an economist, so I speak just about economic
time series, but actually the Box Jenkins approach was meant to cover every kind of
time series application in whatever discipline, and it found widespread acceptance to do so
in form of ARMA modeling in the 1970s basically.
Now we leave the class of ARMA models and look at models which are non-stationary,
and there are good reasons, particularly in economics, to do so because many economic
time series which we have are clearly non-stationary because they display a trend, often trend
which is approximately linear, in particular when we work with logarithms, then many economic
time series have a linear trend because the growth rate of economic variables is often
approximately constant over time or fluctuates around a constant mean, and you know from
basic mathematics that processes with a constant growth rate have a linear trend in their logarithms,
which also explains why economists like to work with variables in logs because if this
is a well-behaved process with an approximately constant growth rate over time, growth rate
which fluctuates around a constant mean, then the process is easily depicted by its logarithm
and has then a linear trend. This was well known already in the 70s of course,
and that was the reason why in the years prior to let's say the end of the 70s,
the beginning of the 1980s, many people thought that there is a fairly easy way to transform
non-stationary processes with a linear trend, two processes which can be modeled as ARMA processes
because as we know an ARMA process does not admit a linear trend, but what people thought they could
do was that they just trend adjust the data. Basically they took the raw data which had a
linear trend and then they said okay let us estimate this linear trend by a simple regression,
say we regress the raw data which is trending on a very simple model, let's say just on a constant
and on a linear trend term so that the linear trend is actually captured by the linear trend
term in the regression and everything else is mapped on the resilience of this regression.
And then the estimated trend was subtracted from the raw data so that as a result researchers
obtained trend adjusted series or at least they thought these were trend adjusted series
and the trend adjusted series was of course equal to the regression residual. So to everything in
the data which was not explained by a constant term or a trend term, a linear trend term and
that seems to be, that seemed to be the most natural way to do it. People were usually not
so much interested in constant and trend, but they were interested in the fluctuations of a
series let's say for business cycle analysis. So they removed the constant and the trend and looked
at the residual thinking that in this residual there is the business cycle information or whatever
type of cyclical information they were looking for or they were possibly not looking for but
any kind of cyclical information should be thought in the residuals of this regression
and they called the residual the trend adjusted type series. Now from what we know since the 1980s
what has been done with this approach was mostly completely raw and actually basically all the
economics literature sorry the econometrics literature which has been done prior to say
1979 is completely worthless today you just need to look at it because there have been serious errors
by which serious errors committed by researchers who thought they were doing just the right thing
by a trend adjusting of the time series and today we know that this was not the case
and actually that the research they have published was greatly biased and absolutely misleading
actually so that as I say it is not worth to read econometric studies with type series data prior to
1979. Unfortunately it is not even worth to read many time series studies which have been conducted
after 1979 and this knowledge about this takes which I'm going to teach you today about began
to spread the econometrics community there were still many many people who ignored what you
facts were learned about time series properties by researchers who dealt with non-stationary
processes even today there are studies where a knowledgeable reader looks at it for five minutes
and recognizes immediately that these people still haven't learned their lesson and still commit the
same errors that were perhaps excusable in the 1970s so like 50 years ago but not anymore today
so you should be well aware of these type of errors you should know that you must never commit
them and you should be able to detect them if somebody else presents a paper and is not up
to date in his econometric knowledge about non-stationary processes so I think what you'll
learn today you'll find most illuminating it is in fact highly important it's about the most important
thing you can learn in time series econometrics what we do today and I will give you examples of
the errors that have been committed and that in some studies are still being committed by people
who don't have sufficient knowledge in time series econometrics we used to say in modern time series
econometrics but actually it isn't that modern anymore because the the tools I teach you now
were developed in the 1980s and perhaps in the beginning of the 1990s so this is now also 30
years ago or even more than 30 years and by now everybody should know about it but I still get
papers for refereeing purposes for instance papers submitted to scientific journals where the authors
use methods which are completely outdated and where I just need like 20 minutes looking at the
paper and then I send it back and say this cannot be published and this is just an indication that
that such papers still exist that they are being submitted to journals and unfortunately I have
also seen such papers being published in some journals which apparently don't have good reference
anyway and this is why I want to go again through the first about 20 pages which I covered in
the last lecture to point out the truly important parts which I covered last lecture to
put into perspective what I will do today and as you recall I just started out with an
example of a real world time series an example which we had already had a look on right at the
beginning of the first set of slides which is us real GDP per capita in a certain time period 1955
to 2007 you have the series in the data supplied on stina and you know it looks like this time
series here so it's one of the most central time series we have in our discipline because almost
everybody likes to look at us data for certain questions and GDP is arguably the most important
time series we have about income and production and everything so probably this there's no more
prominent time series than us real GDP per capita which you see here and we very clearly see that
this series has a linear trend on an approximately linear trend clearly this linear trend I mean
there are fluctuations very clearly discernible it there are also periods where you might say
that the slope of the linear trend is perhaps rather steep like for instance here and then
it seems to be a little flatter here and perhaps becomes a little steeper here there but by and
large you might well think if you just take sort of a an average slope you can estimate this average
slope and then it's not the worst approximation to say that would be what eyeball econometrics
teaches you it's not the worst approximation to just take a linear trend which is fitted to this
time series of GDP and this may give you also hints at how GDP will develop over the next
50 or 60 years when you want to extrapolate these type of data and you may then also think about
looking at the deviations of the actual series from the fitted trend saying well everything
which fluctuates around the trend this is somehow business cycle fluctuations or some other types
of fluctuations okay so and the only thing I wanted to say here is that it is quite important
to be able as an econometrician to find an appropriate trend model for GDP
and the point I'm going to make is that there are at least two appropriate trend models
one of which is just fitting a linear trend and saying the fluctuations about this linear trend
are stationary deviations from the trend and the other model would say yes there is something which
is similar to a linear trend we will actually call it a drift but formally it's the same thing as a
linear trend but the deviations from this drift are not stationary and not stationary means
the deviations do not have any tendency to return back to the trend once they have departed
from the trend so while a stationary series as you know always has some tendency to fluctuate
around its mean meaning that some deviations which are greater than others imply that there
is a tendency of the series to return to its mean or non-stationary series as we will see there is
no tendency to return to the mean and therefore the dynamic properties of such a non-stationary
model are completely different from tendencies which the properties which a stationary type
series model has for such a type series okay what we did was if you recall last lecture that we
just tried to fit an ARMA model first an ARMA model without a trend actually to this type of data
to the GDP data here and what we found out when we looked at these results here was that it seems
that the coefficients here imply that the lag polynomial so the adding up of all the coefficients
of the lag polynomial here implies that alpha of one has a zero on that one is a root of the
lag polynomial alpha of l because the coefficients which we have estimated here add up almost exactly
to one as you recall that we pointed out last lecture so one minus the estimated coefficients
here are zero almost zero quite exactly zero and then we found out that this constitutes a problem
because if it is true that alpha of one is equal to zero so if the lag polynomial evaluated at the
point l equal to one is zero then actually these time series would not have a well-defined
expected value since we would divide the constant nu which we have estimated by
by zero which is illegal which we cannot do all right now we did this in various
aspects look at this model here and then came finally to a representation of the time series
perhaps i should go back to this case here we first looked at an AR1 process with constant term
in the form of equation one here yt is equal to a constant nu plus alpha times yt minus one
plus ut so very innocuous exactly the type of AR1 process which we have been looking at in the first
set of slides but now we do not necessarily require that the alpha in here is smaller than
one in absolute value but we just write some alpha and thereby allow for the possibility
that alpha actually equals one or minus one for that matter but empirically alpha equal to one
is the important case and alpha to minus one is analogous but doesn't have as many applications
in real world data so usually we focus on alpha equal to one so what we found out last lecture
was that when we develop this AR1 process here by successive substitutions which we did here
we end up with a representation for yt for the value of the time series in period t
which embodies a linear trend because this representation here which just follows from
the successive substitutions implies that yt is equal to a constant which in this case is
just the first observation on y which we have available so y by zero plus nu times the number
of time series which have passed since we observed by zero so nu times t is a linear trend term here
plus something which is not an error term anymore in the sense in which we usually had it but
accumulation of error terms where all the error terms which have occurred since period zero
are being added up right so that's the simple mathematics of it if alpha is equal to one
then we find that yt equals this representation constant plus linear trend plus sum over
accumulated errors note this is only true of course for the case alpha equal to one we have
done this kind of successive substitutions in the first set of slides for the case of alpha
smaller than one and then we knew that we obtained a completely different representation
here because these sums converge when time went back to infinity minus infinity we did not get
anything with a linear trend but the quality of the process the type of the process changes
decisively and actually discontinuously when we move from an alpha which is just a tiny little
bit smaller than one to an alpha which is exactly equal to one and in the case of alpha equal to be
an equal to one we get this type of representation here which suddenly involves a linear trend even
though in equation one we didn't see any when if you set alpha equal to one in equation one
you see yt is equal to nu plus yt minus one plus ut so essentially this tells us yt is the same
value as the previous period plus some constant term nu plus some white noise right and and
perhaps at first sight you just don't realize that this representation with alpha equal to one
does indeed induce a linear trend to the time series yt if you think about it a little bit
more thoroughly you see it immediately though already from this representation that there is
a linear trend in it because note yt is equal set alpha equal to one right yt is equal to yt minus
one plus nu so equation one basically tells you yt is greater than the previous value by nu
we forget about the ut because ut has an expected value of of zero so sometimes there's a little
bit more and sometimes there's a little bit less of an increase in yt but sort of on average it
is true that yt value of the observed series in period t is greater by nu than the value of the
previous period and if this is true for all periods then you see that yt increases by nu every period
and if yt increases by a constant nu every period period then clearly yt as a linear trend
and this is made explicit in this highlighted expression here yt is the value of y in period
zero plus nu plus t times the increase the average increase of nu which this time series has period
by period plus of course the effect of all the white noise arochops okay so in this representation
here we clearly see that such a unit root process as a linear trend now we convinced ourselves
in the last lecture that the same property holds true if we consider a general arp
process with unit root so suppose that alpha of l is now a p-dimensional polynomial polynomial of
order p so provides for at most p legs of the endogenous variable and still has the property
that it has a root of one so that when i replace the leg operator l in the leg polynomial alpha
then alpha of one is equal to zero so when we look at such an arp process written in this
form here alpha of l yt is nu plus ut and we rearrange this a little bit then eventually we
arrive at the representation which i have down here where we see yt is after the successive
substitutions and rearrangements and this kind of thing is equal to some type of constant y zero
multiplied by some transformed leg polynomial alpha tilde to the power of t
plus a trend term nu times t as we had it in the case of the ar1 plus some type of accumulation
of the use of the white noise error terms which occurred between period zero and period t
so this just generalizes the setup of the ar1 process to the setup of an arp process
and shows us that the basic phenomenon which we detected in the ar1 context for alpha equal to one
also holds true for the arp process for the arp setting when the leg polynomial alpha
has a root of one so that alpha of one is equal to zero
now what i concluded from this is that when we have some type of ar
process whose leg polynomial has this unit root property and alpha of one is equal to zero
then this implies that the data which we may explain by such a process must be data which
have a linear trend or this is true when we assume that nu is different from zero okay
so we will look at the case of nu equal to zero separately but for the time being it is easier to
look at the case of nu different from zero because it is very clear when we have an ar
leg polynomial with a unit root and the constant of this process is different from zero
then such an autoregressive process can only explain data which have a linear trend
because the linear trend is clearly being built up in this process as representation
for here makes clear now when we when i say that such a unit root process can only explain
data with a linear trend then this does not yet imply that data with a linear trend
do require a unit root process right the unit root process is only
a unit root process can only explain data with a linear trend but clearly it is possible that
data with a linear trend can be explained not only by a unit root process but also by some other
trend model and this other trend model is basically the traditional trend model which
research researchers have used up to the end of the 1970s by just assuming that the data are
being made up by some type of deterministic component consisting of constant and linear trend
term plus some type of stationary fluctuations fluctuations which fluctuate around the linear
trend and that was the first trend model we have been looking at in the last lecture as i said
already in the beginning of this lecture what we will do today is that we look at two rivaling
models to explain linear trends in time series data the big error that i mentioned in the
beginning of this lecture by many researchers was that they thought there's only one trend model
namely the trend model which i have defined here as trend stationarity where i defined the time
series yt to be trend stationary if the expectation of yt is a trending function of time
and the difference between yt and its expectation is stationary so the difference between yt and its
trend is something which is stationary that was the dominant trend model in the 1970s and
before and most people probably thought that this is the only reasonable trend model which we can
fit to the data now we in this lecture know already a little more we know by the analysis
of last lecture that there is a different possibility to model a linear trend namely
supplying an ar like polynomial with a root of unity so-called unit root model
because this also implies the existence of a linear trend and therefore we have to ask the
question do trending data actually require such a unit root model and the answer is as we have
pointed out last lecture no they don't it is possible that they can be explained by a trend
stationary model now and we looked at this case and found out where the difference between yt and
its trend has a voltage representation if yt is stationary and this voltage representation is
advertible so we can write the process as alpha of l times yt minus its trend is equal to ut and ut
is white noise and it's stationary and with this representation we could then show that a
trend stationary process is covariance stationary and and we computed this and proved this last
lecture and this of course implies that also the variance of the process is constant over time
one way to model a trend stationary process and probably the dominant way the way we will
use almost all of the time in this lecture is that we say well we can think of a trend stationary
process as being a process with some lag polynomial alpha of l which does not have a unit root
so where all the roots of alpha of l have modulus greater than one
it's such a lag polynomial multiplied by yt is equal to a constant plus a linear trend term plus
some white noise error so that would be a trend stationary process where i assume that the theta
parameter here which governs the linear trend is different from zero so that indeed we have a
non-trivial and non-trivial linear trend in this type of data okay i think this is basically what
we have done we don't need to go through this formula here and now finally we are left off
last lecture namely at pointing out that there is a second possibility to model trending data
which would be to have have a a r model alpha of l times yt is equal to just constant plus ut so
unlike the uh representation i just showed you there is no term on the right hand side of this
equation seven of the form theta times t so this is missing right go back here this is basically
the same equation but theta times t is in here and theta was different from zero now what i
claim is or what we have already sensed in last lecture if theta is equal to zero and this term
here drops out then we can still provide for a linear trend if on the other hand we assume
that the alpha of l has a root of unity here we have excluded roots of unity because we have said
alpha of l have modulus greater than one plus we have said theta is different from zero now if we
say theta is different from zero and also relax this assumption here so basically we strengthen
our assumptions on theta saying that theta is equal to zero but we relax the assumption here
by allowing for a modulus of the root of alpha of l of being equal to one and we can also depict a
linear trend when we are here right so this model is completely different model even though
it looks quite related it's a completely different model because the linear trend here is missing
but we allow that alpha of one is equal to zero so that the leg polynomial has a unit root
what we do next is that we study the consequences of this change set of assumptions here
the first consequence is that when alpha of l has a unit root then clearly this leg polynomial alpha
of l is not invertible right we know that invertibility requires that all the roots of the
leg polynomial are greater than one in modulus moreover we know that while the trend component
is not explicit in this representation seven and we just see constant and we see white noise
error term but we know that we have analyzed this process already and have by successive
substitutions rewritten it in this form here this was equation four where we do see the linear trend
yt is equal to t times nu plus some type of constant plus q related white noise here with
certain weights alpha tilde of l with the power of i it's not so important what weights these
exactly are the important thing is that suddenly we do see a linear trend here and we do see
something that resembles a constant namely y zero okay so let us treat the starting value of our
time series y zero just as a given constant then we know that the expectation of this process yt
is equal to the trend term t times nu right this is deterministic and we suppose y zero is a given
constant so its expectation is also y zero so the expectation of this thing here is alpha tilde l to
the power of t times y zero so it's basically just the first two components of the sum here
well and the expectation of this summary of the accumulation of error terms the expectation of
this is of course zero because the expectation of u is zero in any period so this term here
vanishes when we take the expectation of y so the expectation of y and therefore the trend
component of y is well nu times t plus alpha tilde of l to the power of t times y zero so
basically we may say constant plus nu times t where actually the constant is just the y zero
and this thing here is as you see not constant because it's being raised to the power of t and
therefore is being changed in every period but for linguistic simplicity i want to sometimes
just refer to this part here as the concept now we may ask is this representation seven
trend stationary right so this thing here is this auto trend stationary representation we have
looked at trend stationary representation so you might suspect now well this is just another way
to write a process but it is also trend stationary what would we need to do to check if this process
here is or is not trend stationary where we would have to subtract the trend from the variable y t
and after we've done this we have to check whatever remains whether this is stationary so
the trend component of y t is exactly the expected value of y t this is nu times t
plus alpha tilde of l to the power of t times y zero so we subtract this okay
so let's look at the variance of y t first and the rest you'll get as an exercise because this
is then an analysis the variance of y t given that we have the unit root now in the process
and no explicit trend term is equal to the expectation of the squared deviation of y t
minus its trend right y t minus e of y t is y t minus its trend component and this thing
squared this taking the expectation of this square here is the variance of y t
well we know y t minus e of y t squared is just equal to this sum here over the ut minus i's
because if you go back to this equation here and you subtract the expected value of y t which is
this one here from the y t then of course what is left over is just the sum here all right so we
have the sum here and we need to square the sum and then take the expectation of it squaring the
sum resides in a double sum i have index i here and index j here both indices run up to t minus
one right and then of course i have to double the factors here i have alpha tilde of l to the power
of i and alpha tilde of l to the power of j times ut minus i times ut minus j so it's a standard
and i have to take the expectation of that well i can put the expectation operator in each and
every term of this sum here and then no ut minus i times ut minus j has an expectation of
zero because the ut's are white noise and they are uncorrelated over time unless i is the same
thing as j right so all other terms where i is different from j are have an expectation of zero
and therefore the terms drop out we only have to bother about the terms where ut minus i
or where this expectation here refers to ut minus i squared because j is equal to i
so this would be this case here this double sum here simplifies to a simple sum i runs from zero
to t minus one again and then i have here the expectation of ut minus i squared and here i
have of course alpha tilde l to the power of two i because this would be alpha tilde of l to the
power of i times alpha tilde of l again to the power of i j is being equal to i so this thing
here is being squared now we know that e or the expectation of ut minus i squared here
is equal to sigma squared u right so this is just a constant and we know also that a lag polynomial
applied to a constant is just the lag polynomial evaluated at l equal to one so i have here alpha
tilde of one to the power of two i right but alpha tilde of one is equal to one because we
know that alpha of one is equal to zero and alpha tilde is of course just the lag polynomial
without the first component without the one so this thing here is equal to one
now when this term here is equal to one and i sum this from zero to t minus one i have apparently
t terms which i sum so the value of the variance variance is just t times sigma square u
now observe what we have found here very much unlike what we have done so far in the first
set of slides the variance of an ar process which we study here is not time constant anymore
but rather it increases when time increases the variance of y t becomes greater and greater
and actually increases linearly with t so the variance is a function of time and therefore
we know already that y t is not stationary and then of course y t can cannot be trans stationary
right because this is here this is the the variance also of y t minus the trend component
and we know that a trans stationary process would require that this variance of y t minus
the trend component is stationary but the variance is not stationary and therefore we see that y t
according to equation seven so a process of type equation seven is not a trend stationary process
but other trend process and this is the alternative to trend stationarity and this is where the error
comes from which i announced early this morning so we have learned the trend stationary model
which was equation five and the unit root model which was equation seven are very different
models for trending data because the trend stationary model implies a constant variance
for y t whereas the unit root model equation seven implies a linearly increasing variance
for y t linear in time right so these are different models to explain the time series
and when you now look back to the unit to the GDP data which i showed you at the beginning of this
lecture then the key question is does this GDP question have a good representation by model five
or by model seven basically the question is which of the two models equation five or equation seven
seven is a better model to represent a trending economic time series like GDP right so researchers
prior to the 1980s almost mechanically usually applied model five right and they say
trend stationary but they didn't actually test this whether it is trend stationary and we come
to tests of this question which model applies five or seven later i will explain to you that
these tests are not trivial you can't do this in a standard way actually but mostly people just
haven't tested it at all and they just assumed that the data is trend stationary often they
didn't even make this explicit this assumption or they just took it for granted that you could deal
with the data in such a way as if the data were trend stationary but later on when all these
instruments about unit root processes were developed people found out that the assumption
that the time series are trend stationary the economic time series are trends
trend stationary this assumption is mostly wrong right and mostly a model of type seven something
similar applies to two time series not everywhere and you have to test and we do have the tests now
for 30 years or so but you need to know that you have to test and and then that you have to model
or take the modeling choice according to the results of the tests we could also now go into
the analysis of the covariance of yt yt minus tau in the case of model seven but i give this to you
now as an exercise because actually we don't need to do it anymore we we do know already that the
unit root process is not trend stationary we could conclude this already from our analysis
of its variance so it will be no surprise to you to read this result here which you should show in
the exercise that the covariance of yt and yt minus tau in the case of model seven is also not
independent of time but rather it is t minus tau times sigma square u so the covariance function
not only depends on tau but it also depends on t if it were to depend only on the time lag tau
between t and t minus tau then we would have a stationary process we know that the covariance
is the covariance of a stationary process if the covariance just depends on tau but in this case
where we have the unit root model seven it also depends on t as we see here and again this is in
some way a linear process just with a constant term tau times sigma square u but otherwise the
covariance increases with t right and of course the same thing then carries over to the autocorrelation
structure if you want to look at the coefficients of correlation for this unit root case of alpha of
one being equal to zero now i mentioned the term already several times here's the formal definition
a process of type seven which is written as alpha of l yt plus constant is sorry alpha of l times yt
is equal to a constant plus white noise ut with a property that alpha has a unit root that one
unity is the root of the leg polynomial alpha makes the leg polynomial equal to zero such a process
is simply called a unit root process and what we have seen is that these unit root
processes cannot be rendered stationary by subtracting their mean so by subtracting a linear
trend they just don't become stationary we will look at how can we render a unit root process
which by itself is not stationary and which cannot be rendered stationary by subtracting
linear trend how can we render such a process stationary or better how can we transform such
a process to stationarity so that the tools we have developed in the first set of slides can be
applied well it turns out this is very easy actually namely differencing right if we have
if you have a unit root process then you can say take the first difference of this unit root process
and for many unit root processes this will already be sufficient and you'll find that the
first difference of a unit root process of many unit root processes is stationary not of all of
them as i would tell you but the most relevant imperative cases are usually rendered stationary
after differencing just once all right so how can we see this well we know that alpha of one is equal
to zero and we know that alpha of l is a polynomial so we can factor this polynomial so we can write
it as one minus lambda one l times one minus lambda two l times times times times one minus
lambda p minus one l times one minus l we know there is one factor one minus l in here because
we have a unit root right we don't now know what numbers the lambda one and the lambda two and the
lambda p minus one are actually some of these could also be unity but let us first discuss the
case where lambda one and lambda two and lambda p minus one are all smaller than one so that the
root l is actually greater than one and in modulus right if lambda one smaller than one in modulus
than l or the value plugged in for l which makes this term here equal to zero and there were the
whole polynomial equal to zero this value would be greater than one in modulus so this would be a
root which is greater than one in modulus here's this assumption let us first assume that all those
lambdas here are smaller than one or that the inverses are greater than one in absolute value
in this case we know that the lag polynomial alpha tilde of l is invertible and therefore we can
rewrite our unit root model seven as well one minus l times yt is equal to nu divided by the
polynomial alpha tilde of one plus alpha tilde of l inverse c times ut so basically what i do
is i know i cannot invert the whole lag polynomial alpha of l because alpha of l has a unit root
but i know there is just one unit root all the other factors here can be inverted
because all the lambdas here are smaller than one or lambda inverses are greater than one
right so i can invert this factor and i can invert this factor and i can invert all the other
factors except one minus l now all the other factors except this last one i have defined to
be alpha tilde of l so essentially i'm saying here i can't invert alpha tilde of l but i know
i cannot invert one minus l all right so this i use here i leave the one minus l on the left hand
side of the equation but i invert the alpha tilde of l um applying this to a constant as we know
would transform the l to just one because the constant being shifted in time still remains the
same uh constant so the new constant for one minus l y is equal to nu over alpha tilde of one
so that's just a constant term plus here now the inverted lag polynomial alpha tilde of l
inverse times ut so this here is essentially a volt decomposition all right we can show that
it is a volt decomposition and i now do not show you explicitly but we've done this already that
the alpha tilde of l involves coefficients psi i i which are square summable so take this for
granted um so what we see here is that the first difference of yt has a volt representation and
therefore all volt representations are stationary so therefore it is stationary one minus l times
yt is just delta yt right as we know so that's the first difference of yt so we see that taking
the first difference of our unit group process gives us a stationary process so we know now how
we can transform data which are derived by or which are represented by a unit group process
how we can transform these data to stationarity namely first differencing and we know how we can
render trans stationary series stationary namely by subtracting the linear trend
but these are two completely different ways of handling the data whether we estimate and subtract
a linear trend which would be appropriate for model five or whether we take the first difference
of the data which would be approximately which would be appropriate for model seven
okay any questions so far
if this is not the case then we can just generalize a little bit what we have done
so far because here in this representation i have assumed there's just one unit group
one minus l and then i have assumed on the next slide i had assumed that the lambda p minus one
and all the other lambdas here are smaller than one in modules greater that the inverse is greater
than one right same thing now obviously it could be that some of those lambdas are also equal
to one if this were the case let's say the lambda one is also equal to one then we would have one
minus l here and one minus l here and let's suppose all the other other lambdas are smaller
than one absolute value well then i would take all the other factors here and invert them this
would be my alpha tilde and i would have one minus l squared here for the first factor and
the last factor so this would tell us then that differencing twice renders the time series which
initially had unit roots stationary and therefore our definition is unit root process which can be
rendered stationary by differencing d times is called integrated of order d and this is denoted
as i of d or sometimes we just say id right that's an id process a process which is integrated of order
d basically integration is the the opposite of differencing right integrating means
cumulating whereas differencing means de-cumulating so when d is equal to one
then we would call this process i one meaning that i one process needs to be
different one time and then it is stationary so the the unit root process which we have
considered here where the lambda i inverses are all greater than one except for just one of those
things that would be an i one process and empirically in fact these two models either i one
or i zero are the most important i zero means the process is stationary or trans stationary so
doesn't need to be different at all i one means differencing one is necessary to another process
stationary it has turned out in years of experience now that these two are the most important
processes we encounter in economics there are some example of i two processes for some countries and
limited ranges of time periods of time it's particularly the inflation rate which is sometimes
i two but in general this is a rare event and we find a time series of economic meaningful
economic time series i two or even integrated up to a higher degree i three same nevertheless it
happens in particular let's say in the work of students phd students master students or whatever
that they run the tests for unit root properties of their data and they come up proud with some
results saying my time series is i three in this case um usually you've made an error you usually
you've done something something wrong uh i three processes i i can't recall of any
time series in economics i've ever seen which turned out to be i three so be careful this
happens to you um in particular when you're a beginning beginner in these type of things
sometimes you make misjudgments sometimes you specify things around me if you come up with
the results of i two or i three be very very careful whether they are true because most
processes are either i one or they are i zero or i not
if we look for instance at a random walk uh then this is i one very clearly and its difference is
of course stationary because the difference of a stationary of a random walk as you know
is just white noise and white noise is the most trivial stationary process so it is i not in this
case as an exercise look at the first difference of a current stationary process and study whether
this process is also stationary okay let us look at the example of a random walk um
um oh i haven't defined it yet so so here's the definition an a one process with alpha equal to
one and nu equal to zero as the simple form yt is equal to yt minus one plus some white noise term
so basically uh the new value is the old value plus some disturbance which is white noise which
has no structure no auto correlation over time which has a mean of zero such a process is called
a random walk what i just told you i'm sorry i didn't recall that i hadn't defined it yet
the first difference of a random walk is of course just ut because if on both sides of this equation
10 we subtract yt minus one then on the left hand side of this equation we will have yt minus yt
minus one which is the first difference of yt and on the right hand side of the equation we will
just have ut so on the right hand side of this equation we'll just have white noise the first
difference of a random walk process is white noise and thereby it is stationary that it was what i
just explained on the previous page there is a slightly more complicated form um and slightly
more relevant perhaps also or at least very relevant and empirical work um of an of a random
walk process which is characterized by alpha is equal to one as before but now the constant is
admitted with non-zero value so yt is equal to nu plus yt minus one plus ut right such a random
walk process is called a random walk with drift and nu is called the drift term so we've seen that
already this nu here by successive substitutions will develop into a linear trend right even though
we don't see the trend property directly in uh representation 11 and we think that the nu is a
constant the really confusing thing about a unit root econometrics is that we have to be very careful
with constants and with linear trends because sometimes symbols change their meaning and what
seems to be a constant like this nu here is in fact the slope coefficient of a linear trend
factor and the constant which also is um actually existing in this type of process
can't even be seen because the constant that turns out will be the y zero with the starting
value of the process and this is not yet visible in this representation here so if we have these
two um very simple models of a random walk and a random walk with drift then we have the number
of implications first for the random walk without drift so just yt is equal to yt minus one plus
ut we immediately see as i have already pointed out that the first difference of this random walk
is just white noise delta yt is equal to yt minus yt minus one and that is white noise
this means that the change in the random walk process cannot be forecasted because we never
can forecast white noise right it's uncorrelated over time it's just a completely random disturbance
which we have here so there is no way how in period t minus one we could forecast the change
in this variable yt the delta yt cannot be forecasted it is similar for random walk with drift
the growth rate or the first difference of the variable delta of the variable yt so delta yt
would then be equal to nu the drift parameter nu plus white noise so basically this equation 13
tells us that the first difference of a random walk with drift is something which cannot be
forecasted plus plus plus this drift parameter so we we have a positive value right for the first
difference on average plus something which cannot be forecasted now this property that the random
walk cannot be forecasted at all or that a random walk with drift cannot be forecasted except for
some type of constant forecast all the time the same forecast each period that we knew right not
so much a very interesting forecast this property which seems boring is actually very interesting
from an economic point of view because it makes the random walk model a suitable model for efficient
markets efficient markets are markets where people think that all available information
has already influenced the current market price is already somehow taken into account by market
participants to form the current market press and therefore if all information which is available
today has already been built into the price which we observe today then apparently the price of
tomorrow cannot be forecasted because everything which we might need for a forecast is already
embodied in in today's price so the price change will occur only because of new information which
enters the market because of information which we don't know yet of therefore the price change
cannot be forecasted in a market which is efficient in the sense that all relevant information
is already been taken into account it is reflected in today's price usually we think of the stock
market being one of those efficient markets where forecasting is virtually impossible
and there are a number of amusing experiments which economists have done to show the efficient
market hypothesis of stock markets so to show that in the price of stocks or any other kinds
of assets being traded on capital markets all relevant information is already included
is already embodied so that there is no way to forecast price changes one of those amusing tests
was for instance a test done by the San Francisco Chronicle and a daily newspaper in San Francisco
who studied the market decisions of a number of investment funds I think there were like
12 investment funds right so these funds had a certain portfolio at the beginning of a year
and the San Francisco Chronicle then recorded how the fund managers decided how they invested
into certain stocks obviously the obligation of fund managers and the way and the reason why they
are very well paid is that people think they have some knowledge about the development of
stocks or other assets in the future so that they invest in the right assets and thereby maximize
the returns to the investment fund and to their investors this is what the investors believe or
yeah what they are made to be believed by the marketing of the investment funds that the fund
managers can somehow forecast at least on average with a better rate of success than
other market participants the development of of asset prices and what the San Francisco Chronicle
had done was that they tracked I think as I said the development of 12 such investment funds
and also in collaboration with the San Francisco Zoo they presented the
asset prices recorded in the wall street journal so the asset price page of the wall street
journal on each day of the year to an ape for grand ape to an Orangutan actually and they
had trained the Orangutan to point to some well for some asset price on the page of the wall
street journal there received some kind of reward for doing that obviously the Orangutan had no
knowledge whatsoever about what he was pointing at and what kind of price expectations there
there were and how assets developed on the stock market but the San Francisco Chronicle recorded
what the Orangutan pointed at and then they had a fictitious portfolio of assets which they
developed over time following the advice of the Orangutan what to buy and what to sell right
and then at the end of the year they evaluated whether the portfolios managed by true
investment fund brokers had better return than the portfolio recommended by the Orangutan
and it turned out that was not the case right the Orangutan was by no means significantly better
or significantly worse than the decisions of the investment brokers which is a very nice
illustration to the hypothesis that the returns to investment on efficient markets markets like
the capital markets are not forecastable and actually the investment brokers cannot do any
better than than a grand ape can okay that's just an anecdote aside which illustrates the
idea of the efficient market hypothesis and illustrates that random walks may be a very
good model to model the behavior of stocks and just don't believe all those people who tell you
that they know how the stock market will develop tomorrow because if the stock market is efficient
then there is no way how you can forecast stock markets.
And yes perhaps one more warning about that if your your your bank advisor recommends you
certain investment funds he would typically show you some graphs which show you that the
value of the investment fund which he recommends has increased more substantially and better
than let's say the the average of the market says let's say some some kind of average performance
index has that is of course very easily explainable by the strategy of setting up many many different
investment funds by the bank and dropping those which have performed less well than the market
and continuing those which have coincidentally developed better than the market in the past
there will be always some which have developed better and others which have developed less well
and so those which have developed less well are buried very silently and those which have developed
better are marketed to the consumers but of course and they typically write this even in
their market information there's no guarantee that a fund which has coincidentally developed
better than the market in the past will also develop better in the future.
Quite likely it will not develop better in the future but rather perhaps be sort of on the same
track as the market possibly be versed possibly better we don't know that's mere chance mere
coincidence the fund will be dropped silently if it under performs for a substantial period of
of time and the new funds are being developed all the time and some of them will be marketed
so just don't believe all the stuff which banks value when they try to convince you to invest
your money into certain funds. You can equally well do this just yourself you don't even need
a grand egg for that just mere chance on on on the course sheet of The Wall Street Journal
or they will do exactly the same thing on long-run averages all right
enough of that we will first deal with the general case of a random walk with drift
it seems to be somewhat more complicated than the random walk without drift but you'll see
that actually perhaps the results for the random walk with drift are more instructive
um then the results are more intuitive let's say more intuitive or easier to understand
than the results for a random walk without drift pure random walk but obviously this case here
does involve the pure random walk as a special case when we said nu equal to zero what we know
in this case is that the lag polynomial alpha of l is just one minus l right move the y t minus one
here over to the other side and drive this lag polynomial way then you have alpha of l equal to
one minus l so alpha of one is equal to zero one minus one is equal to zero so the random walk
process with or without drift is just a huge root process what we know is that the first difference
is stationary so the random walk again with or without drift is difference stationary
if we do the successive substitution stuff again well then that is easy and we have essentially
already done that then we see that y t is equal to t times nu of type of trend component plus
starting value by zero serves as a constant plus the accumulation of white noise arochamps so the
usual stuff here it is again this type of representation of a random walk with drift
if nu is different from zero now i have written it in such a way that i could most easily argue
well that's a constant or we take y zero as a constant and nu t is nu times t is the linear
trend term which we actually in this case in the case of a unit root called drift right so nu is
the slope of the drift term and this thing here is not so intuitively called a stochastic trend
so y t consists now of a constant or actually just the starting value plus drift plus what is called
a stochastic trend accumulation of the ut's here why is this called a stochastic trend well
actually a non-stochastic trend or drift in this term is the accumulation of a constant namely nu
nu is accumulated t times and here a non-constant namely white noise disturbance is accumulated in
exactly the same way right so this is the reason why this thing is called a trend and this is a
called trend but the first type of trend is a deterministic trend and the second type of trend
is a stochastic uh trend it is important to note that the stochastic trend has nothing at all to
do with the vault representation because the vault representations a vault representation which
has an upper limit here of infinity but for y t we have the upper limit t minus one and the
upper limit of the sum here keeps changing as the uh as the process moves in time so for the next
period y t plus one the upper limit will be t and so forth so that the upper limit is time variable
this representation here where is the vault representation for all possible y t's y t plus
one y t plus two we have as an upper limit here infinity so it has nothing to do with a vault
representation quite to the contrary we know in the vault representation the coefficients
psi i must be square summable so if this thing here goes to infinity we would have lots of
coefficients of one here and of course the ones are not square summable for an infinite sum of
ones because the sum would be equal to infinity
now i come now to the practical question and i will give you the first impression of what's
going on here before we go to the break because i want to illustrate to you now why it is so
tremendously important to distinguish between two types of models for trending data
as i have pointed out already at the beginning of today's lecture prior to the 1980s it was
common practice to regress the trending time series on a linear trend so on a constant and
a linear trend actually and then just look at the residues and fit an armor model to the residues
now let us suppose that the data generating mechanism for some time series is a random walk
with drift if it is a random walk with drift so if it is a model of this type here
then we would think that there should be no business cycle component in the series right
or more generally there should be no cyclical component in this series because yt is just
nu which is not cyclical well yt minus one is the same thing as yt just the previous
value of the same time series right there's no evidence of cyclicality in there if yt is equal
to nu plus yt minus one and then ut is also not cyclical because this is just white noise
so in the unit root series with drift we should not see any kind of cycle but we know we should
see a trend in this series now suppose further and finally that we believe that the time series
is trend stationary and we remove the trend by ols regression so what i will what i will do with
you now is that i generate a random walk with drift and i apply the wrong trend adjustment
procedure to the series because i do the same type of error which researchers prior to the 1980s
usually committed by assuming that this model is trend stationary okay what we know already the
unit root process the random walk is not trend stationary what i want to show you is if i do this
then i will spuriously generate cycles in my time series which actually are not my time series
which cannot be my time series as we have just convinced ourselves by looking again at the
law of motion for a random walk with drift here is the demonstration of this really stunning
phenomenon we just take a computer and generate a random walk with drift with artificial data
so i provided my artificial data to you on steener random walk one right so basically
you have the random walk there already but you can do the same and i would encourage you to do it
do the same experiment with a random data which you generated for yourself in some type of program
you know xo generates random numbers and many other programs also do this just generate for
yourself a random walk with drift and repeat what i'm doing here with as many observations as you like
i chose just 100 observations but you can easily do it with thousands or 10,000 observations and
see whether you get similar results i have 100 observations here and i took a new of 0.01 so
basically if these are logs then this would be a one percent growth right and then i draw random
numbers from a normal distribution with a constant variance sigma square u the variance are also set
at some value in this case it is by the standard deviation is 0.01 and the variance would be the
square that's so much smaller than that important is to note that the ut's which i generate with
my random number generator are white noise so they are stochastically independent they are not
autocorrelated over time they do not display any kind of cycle and the series looks like this one
here right now when you just have a cursory view look a cursory look at this time series here
random walk one then it looks very much actually like the time series we had in the beginning of
the lecture right quite similar it's of course completely different time series right but i mean
if i if i had given you this time series here without telling you that's GDP but i had given you
where is it this time series here without telling you that's artificially generated data
and then i had just asked you well tell me which of those series is us real GDP per capita i'm not
sure that many of you would have found the right uh the correct solution and so it seems quite a
natural economic time series or could quite naturally be an economic time series which we observe
in reality i have here 100 observations so i pretend this is let's say GDP from 1870
1872 1969 assuming that this is a researcher working in the 1970s and he has no clue about
unit group econometric thing okay what does this research had do the researcher thinks that the
process has a linear trend that this linear trend should be removed because he wants to fit some
arama model so what he is doing is he runs a regression where the random walk random walk one
is regressed on a constant or under linear time trend and here are the results
the constant is estimated with a value of negative 0.008 and a statistic which is not
quite significant but but doesn't really play a role 1.3 essentially and trend component and
the trend component is 0.008 and this is highly significant right not surprisingly because
this trend is in the data highly significant with a t statistic of 78 so beyond any doubt
does the researcher say there is a trend in this type of data and obviously it's discernible by
just eyeball econometrics there is this trend and now he subtracts this trend from the data
so he just looks at the resilience of this regression which essentially means it takes
the data random walk number one this type of data subtracts this constant well he has
estimated a constant it's not significant or bother about it he just subtracts a constant
nce subtracts 0.008 time trend and looks what is left right and what do we expect to find
we would naively at least naively we would expect to find that we now find cumulated white noise
cumulated white noise because this is what the uh the model tells us i would not expect cycles
but actually the resilience of this regression look like this right and that should be a surprise
because what we see here actually looks like a very long cycle this way going up again here
going down again here not clear how it continues there a very long cycle combined with shorter
cycles going up here going down here this is already 15 years could be almost a business cycle
again an upswing an upswing and a downswing not quite clear if this is an upswing then down again
upswing downswing upswing downswing upswing downswing upswing not quite clear what happens here
downswing upswing downswing upswing so if this year is a hundred years you would perhaps say as a
researcher very nice what i observe here is one let's say two three four five six seven
five seven or eight business seconds something like that in a hundred years time this makes well
the length of a business cycle a little longer than 10 years right class says the researcher look
how interesting my results are i found a conductive cycle not sure if you know what
a contractive cycle is but the the idea of a contractive cycle was that these are very long
cycles which go along with major innovations major progress major technological progress
the the basic idea of contractive cycles was and is there's still people researching
conductive cycles and there may actually be something about the idea of contractive cycles
the major idea of contractive cycles was that there are some inventions which are so important
that they actually lead to big expansions of economic activity for a long period of time
some of those big inventions were let's say the steam engine in the end of the 18th century or
mechanical weaving in the first half of the 19th century or railroads in the second half of the
19th century or electricity end of the 19th beginning of the 20th century or chemical
engineering in the first part of the 20th century or any kind of mechanical engineering
or construction activity may then be the next general purpose technology in the middle of the
20th century certainly at the end of the 20th century would say the internet digitalization
these could be the type of major economic advances which give rise to very long lasting cycles
the root of thumb for contractive cycles was always to have cycles of say 50 years time or
something like this and look what this researcher who is being fooled by my random walk data would
possibly find in this time series would find a nice long contractive cycle right downswing
upswing downswing again could well be a 50-year cycle which seems to see seems to find in this
data so the point i'm making here is we generate what i call spurious cycles in time series analysis
if we do the wrong type of trend adjustment and what i have done here is that i use completely
random data which have actually no cyclical property whatsoever just a random walk with drift
right this random walk with drift then applied the trend adjustment which researchers have
typically applied to trending data prior to 1970 and many people have done this long after 1970
actually do this type of trend adjustment get the resilience from this trend adjustment
and then be enthusiastic about the cycles we find right where this random walk here would give you
perhaps a contractive cycle perhaps a business cycle quite likely both of it right or at least
the researcher could easily now publish a paper saying he found a contractive cycle and he found
a business cycle and it's all garbage what he what he publishes and has nothing to do with
the reality random walk data generated in my computer and actually the cycle is completely
spurious after the break i will break now we will try to understand why this happens
but before the break let me express to you my sincere conviction
that what i showed you here with data which i generated in my computer is not an exception
which i generated with some pain some effort perhaps some kind of exception where i generated
many many random walks before i found this type of random walk here which after trend adjustment
generates spurious cycles but rather that was the first random walk which came out of my computer i
just applied this method and got this result and many people have found similar results so it is
very easy to reproduce i mean if you don't believe me check it in your own computer generate for
yourself a random walk with drift apply this type of trend adjustment by regressing on a constant
and linear trend which is very easily done and then look at the data you get and you'll see not
exactly the same picture by far not the trend sorry the cycle which you may see in your data then may
look completely different from that but i'm very confident you'll see a cycle and possibly a long
cycle and several smaller cycles which you could easily mistake for economic cycles in the sense of
quantitative cycles in business cycles so completely spurious completely misleading cycles
because you have applied method which was wrong because you have not understood or in this case
you just pretend that not to understand time series models well because you have committed an
error which almost everybody has committed prior to 1970 many people have still committed it after
1970 so far for now that was 90 minutes of lecturing for this purpose but i think it is
very important here and we uh i continued the lecture in five minutes and then we would try
to understand better what happens
okay i hope you're all back so let us continue
as i said um when we apply the wrong um way of trend adjustment to the data then
we find long cycle and also shorter cycles in data which had no cycle absolutely no cycle by
construction so there is what is called spurious cycle in the data and there is a famous paper
actually which you find the reference by Peter Phillips understanding spurious regressions
which basically explains on a very high mathematical level why these type of spurious
cycles are generated in unit root time series if the wrong way of trend adjustment is applied
i will try to give you a more simpler explanation of that is it's not it is not
completely exact explanation of what happens because the correct interpretation is in fact
beyond the mathematical scope of this lecture here it involves certain
techniques vina processes and Brownian motions which i haven't introduced to you and which are
quite demanding but i think you can get at least some type of more than only intuitive
understanding of what happens by the way i would will explain here now what we know is our unit
root process our random walk with drift can be represented in the following form yt is equal to
y not plus the drift term nu times t plus accumulated white noise which we have here
now call this accumulated white noise just wt if you define this as wt then it looks like a well
almost like a trend stationary process where you would have yt is equal to a constant plus a trend
term plus some error term wt right wt has an expected value of zero like a true error term
should have an expected value of zero so the important thing is what kind of second moments
does this wt process have so we have to look at the variances and covariances and the best
thing to do this is to do it in a standpoint by looking at the autocorrelation function
of wt now this autocorrelation function of wt or here i have written it as the
estimated value of wt because the true wt we cannot observe the autocorrelation function of the
estimated wt should asymptotically be equal to one at all legs right because basically in the
long run this term here and the same term shifted by some legs will still have many many overlaps
and therefore the autocorrelation function should actually be equal to one at all legs
but if we look at the autocorrelation function which we have actually estimated from the data
then we see that slowly decaying namely the autocorrelation function of
this now says u1 here but then it's meant to be w right the autocorrelation function is slowly
decaying very gradually only right but it is decaying you see that even the 12th autocorrelation
here is still significant right it's still beyond the two sigma band which i have
shown here in dashed lines so it seems that w hat t is estimated to be stationary rather
than non-stationary because the autocorrelations decay but in truth we know this is just some type
of estimation problem and some type of finite sample problem the reason why the autocorrelation
seems to be decaying or why we seem to find a stationary process for the ws here
is that the least squares estimated tries to minimize the residual variance
and this is always the case this is by construction least squares means make the residuals in their
squared form as small as possible which basically means make the variance as small as possible
now the problem is that for a non-stationary process like our unit root process here the
unit root with drift the variance is asymptotically infinite recall we showed this um this lecture
oh where was it not here not here um here the variance of some type of unit root process
you're not even just for the random walk but for any unit root process um this variance is t times
sigma square u so it increases with t and becomes infinite asymptotically so that's
quite a challenge for the least squares process which tries to minimize the residual variance
but how can you minimize a residual variance um if the variance is actually asymptotically infinite
well the only way is that the least squares procedure tries by all means to estimate
a stationary process rather than an in stationary the unit root process the least squares procedure
in some way is biased towards finding a stationary process rather than the unit root process which
is actually at the root of the data the underlying problem of that is that a random walk process
and a unit root process in general is not an ergodic process remember we define this concept
of ergodicity in slides one of this lecture and it basically means that the concept of ergodicity
if you don't recall it precisely basically means that the limit in mean squares of an estimator
is the true value of say the mean or the variance so basically ergodicity said with standard
estimators we can asymptotically correctly estimate the mean and the variance right
this is not the case for a random walk with drift or a random walk without drift and we show you
that this ergodicity property does not hold for a unit root process by looking just at the simple
example of a random walk without a drift and for simplicity to make things really easy even let's
suppose that the starting value is zero so that yt is just cumulated white noise right no starting
value no why not no drift term because we look at the process without drift so nu is equal to zero
we can ask the question whether this thing here is mean ergodic
so can we estimate the mean of this process asymptotically correct
in the lim sense that the limit mean is the true value so does it hold that lim u hat or t
is equal to mu and therefore smaller than infinity well clearly this mu here would be equal to zero
right because we don't have a constant term here and we don't have drift term it's just
u's so just just cumulated white noise what would be mu hat t that's the standard estimator that's
one over t times a sample of the y's so sometimes some of the y's over a sample of t periods
right so that's the standard estimator for the mean and now applying our knowledge that yt is
equal to the cumulation of the ut's here this is equal to this term here so we have a double sum
here t runs from one to capital t and tau runs from zero to t minus one if you analyze this
double sum here a little bit more closely then you see that each u in this double sum here is counted
differently many times and you can actually count how often is each single u being counted
how often does it appear the sum and you realize that the sum can be written as one over t times
the sum just the single sum now from small t going from one to capital t over capital t plus
one minus small t times u t so the expectation of the estimator for the mean is zero and that's
quite all right but LIM convergence requires that the variance of mu hat t converges to zero
so LIM convergence if you recall requires that both the bias goes to zero asymptotically
and the variance of the estimator goes to zero asymptotically here we see the bias is fine that
goes to zero or actually there's never a bias we always have expected value of mu hat t being equal
to zero but how about the variance well the variance of mu hat t is the expectation of mu
hat t squared since the expectation of mu hat t is equal to zero so what do we have to do we have to
look at this estimator here squared which would be one over t squared a factor which we can pull
out of the expectation operator times this simple sum here with weights t plus one minus
small t times u t being squared well what is this obviously uh squaring this thing results in
the variance of u sigma square u whenever u t is multiplied by by itself in this case the
parentheses here are also multiplied by themselves so the variance it's the square of t plus plus
one minus small t squared and that's summed t times now the rest is just a little algebra
we can multiply out this parentheses here it's t plus one squared minus two times t plus one
times small t plus small t squared and then i have to take the sum over these three components
going from one to capital t well this is then we see capital t plus one squared is independent of
the index of the sum of the small t so here we have just capital t times t plus one squared
which is this expression here capital t times t plus one squared similarly t plus one is a
constant factor here which can be pulled out of the sum just like this two and we have to sum
the small t from small t going from one to capital t well the small t being summed is then capital t
times t plus one over two and this being multiplied by t plus one and two so the two
cancels and the t plus one here is squared so essentially this thing and second term cancel
and then we also have to sum the small t squared capital t times well look at some
collections of formula if you don't know this by heart which you don't need to know
this is t times t plus one times two t plus one over six and very easy if you divide this thing
then by a t here you get sigma squared u divided by t times t plus one times two t plus one so we
have a capital t squared here in the numerator and we just have t in the denominator so the
variance converges to infinity and therefore mu hat t does not convert to zero or the variance
of mu hat t does not converge to zero and therefore we see that even for this simple case of a random
walk without drift and with starting value zero the series is not agotic
i'll give you here an exercise where you can reproduce basically what i've done so far with
with the computer and i think it's very useful to do it so write some computer routine which
gives you a random walk with drift for some drift term nu which you may specify and for vector of
white noise random numbers which you have generated in your computer and then you can generate a few
random walks without drift let's say r w u of zero check if you find evidence for trends in the data
and try to explain your result intuitively by supposing that the random walks that the random
shots you hear can be either plus one or minus one and nothing else right um i think this will aid
your intuitive understanding what is at stake here and what we analyze here if you do this exercise
b and then write a simple program in your computer which for instance for a thousand times does the
following kind of loop which simulates certain results say you generate 200 observations on a
random walk without drift and then you regress the random walk on a constant and a linear trend
and compute the t statistic of the trend term in the regression and then you collect the results
on the t statistics over the thousand runs now check how often would you find a significant
t statistic and how often do you actually find a significant t statistic so often we expect this
and how often do we actually find a significant t statistic is this in line with each other and
plot the histograms of the t statistic
before i go on with spurious regressions um may i ask you if you have any questions
and please uh do take the opportunity to take questions if there are any
because this is really important what we do here
raise your hand if you have any questions no nothing okay then we come to spurious regressions
spurious regressions and spurious cycles are very closely related and as i already told you the
famous paper by peter phillips is called understanding spurious regressions
what we now do is that we also look at the interaction of two random walks so look at two
random walks which i have generated which are completely independent of each other so random
walk one and random walk two you find both data on steeden so these both of them were generated
by computer but this is the only thing they have in common they were completely different random
data used for random walk one and used for random walk two so there is no relationship
between random walk one and random walk two but suppose we don't know that suppose we don't know
that that the random walks have been generated by computer but you're just giving the data
like i gave them to you on steeden and you think that perhaps these data have something to do with
each other or you want to check right so what you do in a simple type of more or less coordination
analysis is that you regress random walk two on random walk one or you could also in the
first way regress random walk one on random walk two the model you estimate is then random walk
two is equal to 0.09 plus 1.05 times random walk one and the t values are 7.8 for the constant
and 42 for the slope coefficient of random walk one so your result in this case would be that
random walk one is very closely related to random walk two look the r squared is 95 percent
very high r squared which you get here and the t statistic for the slope coefficient of random
walk one is 42 and very rarely with true data you find t statistics as high as that
right so you would say this beyond any doubt beyond any reasonable doubt that random walk one
is somehow related to random walk two but as we know that's not true the two things have been
are generated completely independent of each other and they have nothing to do
with each other right actually is perhaps you would even say random walk one is causal for
random walk two if you confuse causality with correlation but that of course is completely
untrue because if you reverse this regression and you would regress random walk one on random walk
two you would also get a highly significant t statistic here so it has nothing to do with
causality but you may be easily misled to say well there is a significant correlation so these two
random walks must have something to do with each other high r squared very high t value for random
walk one in the regression of random walk two on a constant and on random walk one
why is this so well let us first look at the time series plots of these two random walks and
then you'll see already one of the reasons why this is so actually this should teach you as I
have already taught you in slides one you should always first look at the time series look plots
and then start regressing things on each other always first have a graphic look at the data and
then use geometric methods we've done it here in the wrong way this is why I write look rather late
at the time series plots of this data and they look like this okay so green is random walk one
and the dashed line blue is random walk two this is interesting because we see part of the problem
or can explain parts of the results already we see two things actually in this time series but
namely both random walks have an upward trend right they have a drift and random walks with
drift and there is well a positive slope for both random walks right coincidentally I used the same
drift parameter here which is why I estimated a coefficient here close to one basically saying
the trend term is the same right the trend the slope of the trend is the same or approximately
the same for both random walks however this would be the same result if I use different
if I had used different values of nu if I had used say nu of one for green process and nu of minus
two negative two for the dashed blue process then my regression would have resulted in
some type of negative coefficient around negative zero point five probably and it would have been
highly significant right so it is not the result does not hinge on the fact that I have used the
same nu coefficient here and you can easily test this by generating your own random walk with drift
and regressing them one on the other so very clearly the two series have both of them have
an upward trend but also quite clearly the fluctuations which they do about their trends
are not so closely related actually when you look at the green line here you see that the fluctuations
of the green line about its trend seem to have nothing to do with the fluctuations the blue line
has about its own trend so in terms of cycles actually it seems that there is no correlation
and therefore this result here is highly misleading because you will have expected
and many researchers interpreted their results this way that not only the same trend between
the two series generates the significant result here but also the same type of cyclical behavior
but this need not be true if you have trending series then it is very likely that you find
significant statistics in regression such as this just because the trend components
explain or quotation marks explain each other seem to explain each other because there is the same
type of correlation either positive or negative which is pronounced and which gives rise to this
coefficient here but not more than some type of commonality in trend is explained and as I already
pointed out even the trend need not be common they can completely different new
values and the coefficients can actually be negative or whatever what we estimate here if
the trends are not common but go into opposite directions for instance or have opposite slopes
then that would be corrected by the size of this coefficient here and the key statistic would still
be highly significant so our hypothesis is that random walk one is significant only because it
also has a drift and as I said it's in fact the same drift so our hypothesis is that the
coefficient the estimated regression coefficient on random walk one is significant only because
one trend seems to be explained by the other trend now that's a nice hypothesis and this
could rather easily explain our results so let us make a second experiment now by looking at two
random walks which do not have a drift so two random walks without drift for better comparison
I take the same shocks which I use to generate random walk one and random walk two right random
walk one random walk two use certain random numbers and they supplied some non-zero drift
a term new now I use the same random numbers which I use to construct random walk one
to construct a random walk without drift and I call it rw1 and I use the same random walk numbers
the same random numbers as in random walk two to construct random walk so rw2 here a random walk
without drift so in this case cannot be the case anymore that the two trends explain each other or
seem to explain each other and this is why rw1 or random walk one is significant in the regression
of random walk two on random walk one so it has now regressed rw2 the random walk without drift
on rw1 a regressor without drift here are the two series right I see the blue series
which is rw2 you see the green series which is rw1 and actually I think just by eyeball
econometrics you would say well these series don't really have much to do with each other
now when I regress rw2 on rw1 in much exactly the same way as before then I estimate a coefficient
a constant here of 0.04 with a t value of 5 so highly significant and I estimate a slope parameter
for rw1 of 0.24 with a t statistic of 3.2 so also significant at the 1% level actually
right applying standard inference which actually is wrong here right so actually significance is
much more complicated in the setting than in the standard case but if you apply your standard
knowledge you would say anything which has a t statistic greater than two is significant
t statistic greater than two means significant at the 5% level t statistic greater than three
means standard normally distributed tested statistics significant at the 1% level so you
would if you have no knowledge of unit root econometrics think well apparently rw1 is really
important for explaining rw2 or the other way around the correlation is significant at the 1%
level okay the r squared is much lower than before it's not just 10% but we find some positive r
squared and we find positive significant coefficient or we find significant coefficients that's the
important thing significant coefficients for our simple regression with t values of 5.3 and 3.2
why the t statistics are much lower than before before because this trend component has
disappeared we still have seemingly spuriously spuriously significant
regression coefficients in this regression and that was the big surprise which peter phillips
in his famous paper explained which people had not understood before 1987 when peter phillips
paper appeared right and it is indeed very surprising that two random series which have
nothing to do with each other and where any child can tell you that these things don't
have anything to do with each other in a very simple static regression seems to be significant
and again this is a typical result which we you will find many many examples which are
constructed along the same lines as i have constructed my example here i did not have
to look for two random walks which finally displayed the significant result do it in your
own computer it's easy to do easy to reproduce i bet you very easily find results similar as
these ones here when you just regress two random walks which are independently generated which
have no drift on each other now let's try to understand this or at least partially understand
this suppose we have two random walks without drift x and y and the true data generating
process of random walk x is x t is x t minus one plus u one t and y's process is y t is y t minus
one plus u two t so very clearly two random walks without drift u one t and u two t are completely
independent of each other and they are simply white noise so what would you expect when you
regress x t on no when you regress y t on x t i think if you know that this is the data generating
process and you know y t has nothing to do with x t and the error terms in y t have nothing to do
with the error terms in u t and there's no influence of y t on x t and there's no influence
of x t on y t nothing at all you would at first think when we regress y on x then we would expect
to estimate a coefficient on x of zero right so when we estimate this regression here y t is equal
to beta naught plus beta one x t plus u t i think your expectation would be that beta one is on
average at least estimated at zero and whatever we find here is an insignificant coefficient
that's my phone sorry for that um okay however there's one problem with this regression here
and this problem is that the true model 15 and 16 is not nested in equation 17 here
right so whatever value beta one takes on in this regression here
the resulting model is always false right so if beta one is equal to zero
then this regression equation here does not give us the true model it is actually false it would
say y t is equal to beta zero plus u t u t would we would think is stationary right stationary
white noise so it would basically tell us y t is a constant plus white noise but that's wrong because
y t is a random walk plus white noise or is a random walk is y t is y t minus one plus white noise
so model 16 is not nested in model 17 for beta one equal to zero but model 16 is also not necessary
nested in 17 for beta one different from zero because if beta one were different from zero
then this would tell us y t is equal to beta zero plus some influence of x e plus white noise
but that's also wrong because as we see from equation 16 there is no influence of x t
so whatever our least squares procedure does in estimating beta one it will always estimate
something which is wrong okay so that i have explained here on this flat suppose now that
we have estimated beta one hat to be exactly equal to zero never happens that we estimate
something exactly to be equal to zero but suppose it is then we would have y hat t y t is equal to
beta naught hat so the estimated constant plus u hat t for all t's right if this were the case that
we estimate the slope coefficient of the x t as equal to zero which was perhaps your initial idea
that it should happen now all s estimation always aims at minimizing the sum of squares and
the sum of squares is u hat prime times u hat right so this is the sum of the squared resilience
right the sum of the u hat t squares well in this case we would have that u hat prime times
u hat is equal to y tilde prime times y tilde if you define y tilde to be the mean adjusted
value of the y t so to be y t minus beta zero hat right or in this case y tilde is equal to y
minus y bar times yota and yota is just a vector of once and essentially y bar times yota is as
you know from basic econometrics is just beta zero hat so in this case we would therefore find
that u hat prime u hat is equal to the estimated variance of or t times the estimated variance
of y tilde here this means the estimated variance of the residuals one over t u hat prime u hat
would be the same as the estimated variance of the ys which would be one over t utilde prime
utilde and therefore in terms of ols estimation it is absolutely unattractive for the ols procedure
to find some beta one hat of zero because this would not make the residuals as small as possible
quite to the contrary it would make the residuals as large as explained as the observed variation
basically it would imply an r squared of zero so this thing will never happen in such a regression
that we estimate beta one hat of zero and it would actually not even be desirable because as i pointed
out beta one equal to zero is the wrong model therefore what we now know is ols will always
estimate the beta one with a positive or negative coefficient but not with a zero coefficient
and actually it would typically try to find a pronounced non negative non zero coefficient
so let's look at the case that beta one hat is different from zero in this case u hat would be
y the observed value minus the constant the estimated constant minus beta one hat times
the x right what you may show as an exercise please is that this implies that y bar so the
mean of the y's is equal to one over t times jota prime y is equal to beta zero hat plus
beta one hat times x bar where x bar is just the mean over the x's
if we subtract the means from the y's then this yields u hat namely u hat minus
u hat is equal to y minus y bar jota minus beta one hat x minus x bar jota right this basically
is the mean adjustment which needs to take place to get uh the u hats here and i denote this y by
y tilde so y tilde minus beta one hat x tilde is equal to u hat so we find that u hats tilde
u hat prime u hat is equal to those y tildes prime y tilde minus two beta one hat y tilde
prime x tilde plus beta one square x tilde prime x tilde so what does OLS estimation do
OLS estimation tries to minimize the sum of the squared residuals and we now know that the sum
of the squared residuals is equal to the right hand side of this equation here so OLS tries to
find beta one hat such that this expression here becomes minimal how is this done well this
expression here on the right hand side of the equation is independent in its first term of
beta one hat so we can ignore the y tilde prime y tilde here so basically we need to make this
thing here the second two terms as small as possible oops excuse me
so we will find beta one hat different from zero
if it is true that negative two beta one hat y tilde prime x tilde plus beta one hat squared
x tilde prime x tilde is smaller than zero that will be the case whenever uh the uh y tilde is
not exactly orthogonal to the x tilde so this is a very weak condition which we have here
so that this thing here is smaller than zero is almost we can almost take take for granted
unless y tilde is completely orthogonal for x tilde
now when we minimize this term here over beta one hat then we just get the normal equations
which you know from basic econometrics so two beta one hat x tilde prime x tilde is equal to two
y tilde prime x tilde so the OLS estimate of beta one hat is as usual the estimated covariance
between x and y and the estimated divided by the estimated variance of x
so the value which we estimate for the beta one hat will tend to be very large
and therefore will tend to have a t value greater than two when the covariance of x and y is large
relative to the estimated variance of x so when the covariance between x and y is great relative
to the variance of x then this thing here will be relatively great uh therefore will deviate much
from zero and then it is likely to find a t value which is greater than two
now is it possible that the estimated covariance between x and y is large relative to the variance
of x we know that in theory x and y have high autocorrelations so the autocorrelation function
is equal to one at all x therefore it may easily be the case that we find correlation between x
and y because each of these has high autocorrelation right so the autocorrelation for
each series which are completely independent the autocorrelation is high for both of these
series so since the pattern of autocorrelating with its own past is similar you may erroneously
find high autocorrelations between the series which have nothing to do with each other
and that is the clue to understanding the spurious regression results since both of the artificially
generated random walks have high autocorrelation just with reference to their own legs to their
own legs and their own disturbances but because the pattern is so extended we may seem to find
or we may find evidence of cross correlation between them so we may find evidence of high
covariance suppose for instance in series x we have some very positive
error term at the beginning then we know this positive error extends very long over time in
the autocorrelation function so we find we seem to find positive residuals for the x series of an
extended period of time suppose for y we either have a positive or negative shock in the initial
period due to a high autocorrelation we will find either a positive or negative autocorrelation
also sorry we find a positive autocorrelation therefore also either positive shocks or
negative shocks at all other legs or many other legs so this seems as if the resilience which
we find of the x process then stand in some relationship to the resilience of the y process
which gives rise to the result of a high estimated covariance between the two processes
yep this is what i explained to you in this example here suppose that at time t x t is high
relative to other x values then many neighboring values of this one observation will also likely
be high and suppose that at time t say y t is low relative to other y values then many neighboring
values will also likely be rather low and in this case if we have high values here over an extended
period of time and low values and y over an extended period of time then we will find
negative correlation between x and y even though they have nothing to do with each other
or if the y correlation was also high we find positive correlation but this correlation is
completely spurious since x and y are independent processes we've generated them
this way and we've written down them in the data generating process
but the spurious correlation gives rise to spuriously high t values in linear regression
i conclude here again with an exercise where you can reproduce this result on your own computer
so generate two independent random walks compute the correlation between them do this
a high number of times say a thousand times and look at the estimate of
row which where the true value is zero and the standard error is one over square root of t
and see how often do you falsely reject the null right and then plot the distribution of the rows
and you can also do this rather than a thousand times ten thousand times to just check whether
the problem vanishes asymptotically and you'll find it does not vanish
okay so much for today i hope you have got a good idea of what kind of problems we have in
time series econometrics when we as we should take into account that there are different possibilities
for linear trends for the way how linear trends can be generated and that stochastic trends
also come into play even if we don't have deterministic trade trends which correlate
with each other and that the correlation of stochastic trends which have nothing to do with
each other can also lead to erroneous findings of spurious correlation and therefore
forward results in time series analysis
unless you have questions if you do then please raise your hand i will stop the lecture here
i do not see any questions so thanks for your attention and see you next week