So, Covariance stationary time series,
common structure over time, nothing
changes.
Now, we often talk about non-stationary
processes.
So what's the definition of a
non-stationary process?
Well, it's something that's not stationary
[laugh] right?
And, and you know, may, may sound, you
know, it's, it is, it's tautological.
The reason why I say it that way is
nonstationarity can come about in many,
many different kinds of ways.
Remember stationarity means, you have a
common structure over time.
If something's nonstationary, then
something is not constant over time.
Something.
Maybe the mean's not constant over time.
Maybe the variance is not constant over
time.
Maybe the time dependence is not constant
over time.
So, nonstationarity, by itself, is not
specific.
Right?
So it's like saying you know, something is
non-white.
Well what is it then?
Well it's not white.
Right?
[laugh] So, in terms of nonstationery we
want anytime we say nonstationery we want
to be specific about where the non
stationery is coming from.
Alright so here's a example.
This is what's called a determinant, de,
deterministically trending process.
So one way that a time series maybe
nonstationary, if the mean changes over
time.
One way to make the mean change over time
is to explicitly make the mean a function
of time.
So if we let Yt be beta not, plus beta one
times time, plus epsilon t where epsilon t
is a random variable that's white noise.
So epsilon is an uncorrelated time series
with mean zero and constant variance.
Alright?
So what are the properties of this time
series?
Well, if we take the expected value of Yt,
the expected value of Yt is beta zero plus
beta one, times time, plus the expected
value of epsilon.
But epsilon has mean zero.
So this is beta not, plus beta one times
t.
So, this is the mean of Yt.
Depends upon time.
So, it's not stationary.
Okay.
Here's a plot of a deterministically
trending process, okay?
So, it's obviously non stationary because
it's mean is an increasing function of
time.
Any time series that has a trend, any
discernible trend, by definition, is not
stationary, okay?
So this is, we often will say this is
exhibiting non stationarity in the mean,
because the mean is a function of time.
Okay?
Now, very often when we have a
nonstationary time series, there is, a
simple transformation of the time series
that makes it stationary.
So, for example, if we subtract off the
deterministic trend, so we define Xt as Yt
minus beta not, minus beta 1<i>t< /i> well
that's the light noise air term, which is</i>
stationary.
So, de-trending a deterministically
trending process creates a stationary time
series.
If asset returns on average have mean
zero, then the expected gain is zero by
investing in doing that, right?
So you know, and here the question is, is
a mean zero a expected return a reasonable
assumption for a expected return?
And, and here the answer depends upon the
investment Horizon.
So, say for example your investment
horizion is one day.
What is your expected daily return on any
investment?
And in that case you know, essentially the
growth rate per day is typically extremely
small and the approximation zero could
actually be very good.
I mean you can take that even further.
Suppose you have an investment for just
one hour.
What's the expected return on that
investment?
You know, again, what is the expected
price growth over any one hour period?
It's probably very, very, very small.
On the other hand, if your investment
horizon is a year, then, you know, the
expected return is, is most likely going
to be non zero, right?
I mean, T bills give you at least, you
know, one percent per year and if you're
going to invest in stocks, you would
expect to get at least better than that.
So if your investment horizon is a longer
period of time, then you typically view
the expected return as being positive.
Whereas if we're, the return horizon is
quite small, then you actually might find
expected return being equal to zero to be,
be quite reasonable.
And we'll see this very clearly in the
data as well.
Alright.
So let me continue on with a nonstationary
time series.
So another kind of non stationary time
series is, is the random walk.
And the random walk has a, a very long
literature and holds a important, it has
an important role in finance as well.
And so, the random walk process is a
nonstationary process where the
nonstationarity is actually in the
variance.
So again, something that's not stationary,
something about the time series is
changing over time, and in the random walk
process, we'll see what's changing over
time is the volatility of the process.
Right?
So, what is a random walk?
A random walk is a time series process
such that the random variable at time t is
the random variable at time t - one plus a
white noise air term.
So the error term has mean zero and a
constant volatility.
Alright?
So, anybody know why a random walk is
called a random walk?
The, the term the random walk has its
origins in describing the behavior of a
drunken sailor leaving a bar.
And the idea is that the drunken sailor,
cuz he's drunk, you know?
The idea's the random walk is trying to
tell you, where's the drunken sailor going
to be at any point in time?
And the drunken sailor, and so you think
of epsilon as being equal you know, if
it's positive, then the, the drunken
sailor veers off to the right.
And if epsilon is zero, then the drunken
sailor goes forward.
And if epsilon is negative, then the
drunken sailor veers to the left.
And the idea is that, you know, these air
terms, which represent the direction that
the sailor is walking are unpredictable.
And so the, the drunk leaves the bar, and
can wander into the canal or, you know,
whatever.
And so the idea's the, the evolution of
this thing is this, this random walk over
time.
And, and we think of stock prices, you
know, the idea is this would be the stock
price today.
What, what's the stock price today?
Well, it's what it was yesterday plus an
unpredictable component, you know, that we
don't know.
And, and later on, we'll give an
interpretation in the context of stock
prices.
This could be like the random news that
hits the market.
We don't, today we don't know what the
news is going to be tomorrow.
It could be good news, so the stock price
goes up.
It could be neutral, so that the stock
price stays exactly where it is or it
could be bad news, the stock price could
go down.
But the idea is we can't predict what the
news is going to be.
So this is a, stochastic process, and so
let's study this a little bit and see why
it's nonstationery.
So the thing about the random walk is that
we can do a little bit of manipulation
with it.
So let's look at the random walk.
So Yt = Yt - one + epsilon t.
Epsilon t is white noise.
Okay, Now we need to do recursive
substitution.
That is, lets start, lets look at the
process at time t=1.
We know that Y1 = Y0 + E1.
Then we're going to look at the process at
time two where Y2 = Y1 + E2.
But notice that Y1 here is equal to Y0 +
E1.
So this process is equal to Y0 + E1 + E2.
Okay?
And then, similarly, we go down at time T
equal, you know, capital T.
Y at time capital T, is Y at time capital
T - one + epsilon T, and then by this
recursive substitution, we know that this
is Y0 plus E1, plus E2, plus Et.
So again, this is the, where the drunken
sailor is.
This is where he is when he leaves the
bar.
Okay?
And so where he is his first step out of
the bar is the bar door, plus the random
direction of, of where it is.
At time two, where is the drunken sailor?
Well, it's where he was after, at time
one, plus the random step, but where he
was at time one was leaving the bar, the
first random step, the second random step.
And then finally, the drunken sailor at
time capital T.
Well, it's where he was at time T - one
plus the random step.
Well that's where he left the bar, plus
the accumulation of all the random steps
until that point.
So, going back to this recursive
substitution, we, in a random walk model
we can write Y and find capital T as Y0 +
e1 + e2 up to et.
So now we can, we can study the properties
of this process.
So what's the expected value of Yt.
Well, it's equal to the expected value of
y0 + e1 + e2 + et.
So that's equal to y0 plus the expected
value of e1 plus the expected value of e2
plus the expected value of et.
But because all the epsilons are white
noise, those are all zero, so this is
equal to y0.
So again this is assumed to be fixed.
So we do the random walk, we assume, we
start the random walk somewhere, right?
This is the, the entrance to the bar,
right and so this is saying like where do
we expect the drunken sailor to be, right
we expect the drunken sailor essentially
because we don't, can't predict which way
he's going, we expect him to be
essentially on a straight line outside of
the bar door right but he could, you know,
and, and, and, so, so that's not average
you know, where the drunken sailor is.
But you know, again, you think of, well
what does it mean to be on average where
he is?
Think of, like putting the drunken sailor
out of the bar, you know, 500 times, and
you let him walk, right?
So sometimes he's going to veer off to the
left, sometimes he's going to veer off to
the right?
What kind of on average he's going to go
straight.
So that, that's essentially what, what
this means.
So, lets look at the variance of this
process.
The variance of Yy, is equal to the
variance of, Y0 + e1 + e2 + eT.
Now, If we assume Y0 is fixed, this is
like a constant, so this is, so this
doesn't have any variance, you know, this
is the, this is the entrance, the entrance
to the bar.
And then these are all the random steps.
Now, in a white noise process these are
all uncorrelated.
Remember the variance of the sum is the
sum of the variances plus two times all
the covariances.
The covariances between all the air terms
are equal to zero because we have a white
noise process.
That is the, the, the step the drunken
sailor takes, you know, on the T-step is
independent, is uncorrelated with the next
step and so on.
So this is equal to the variance of e1
plus the variance of e2 plus the variance
of eT and this is because all covariances
are equal to zero, okay?
Now, in a white noise process, each one of
these air terms have the same variance,
sigma epsilon^2.
So this is equal to sigma epsilon^2 +
sigma epsilon^2 + sigma epsilon^2.
So that's T sigma epsilon^2.
So notice the, the variance.
This is, where is the sailor after capital
T steps?
Well, the uncertainty about where the
variance is, is equal to the number of
steps times the, the volatility of each
step.
So the idea here is, the more steps the
sailor takes, the less certainty we know
where the sailor is going to be, right?
So, the volatility after one step is just
sigma epsalon squared.
The volatility after two steps is two
sigma epsalon squared.
The volatility after capital T steps is
capital T times sigma epsilon squared.
So this is just saying, the further the
sailor walks, the less certainly we know
about where the sailor is.
So that means the volatility depends upon
time, it depends upon you know, what point
in the sequence we have it.
So this is a process that's nonstationary
in the variance because the variance of Yt
is t times sigma epsilon squared, it
depends upon time.
So, and that's, that derives this
relationship right here.
And again, maybe a picture would help.
So if we think of Yt and t, and here is
Y0.
And if we're doing a computer simulation.
Say, here's one process of the random
walk.
Okay?
And then, we simulate this thing again,
and we get other random components down
over here.
And then you know, we do it again in, in
as many different colors as, as, you know,
we can and so on and so forth.
Now, the idea behind this, so if we look
at some point in time capital T.
The volatility of the process at time t is
essentially the spread in the possible
values of this path.
So, when t = one, you know, this process
can't go very far, right?
But when t is a hundred then the drunken
sailor can be, you know, way up here, or
it can be way down there.
So the uncertainty about where he is
spreads out as time goes along.
That's non stationary invariance.
So just like in a deterministically
trending process, we can create a
stationary process by a simple
transformation.
So in this random walk model, if we take
the first difference of Yt, so if you look
at Yt, - Yt - one, by definition that's
equal to epsilon and that's stationary.
So if we have a random walk process, you
take the first difference of the process
and we create a stationary process and so
the question is, you know, how important
is where you start?
Partly it depends upon the volatility of
the steps, right?
So if the sailor just takes tiny, tiny
little steps then you can't wander too far
away from where he starts.
But if his volatility term is relatively
large that the sailor can take big steps,
then eventually when you get far out in
sequence, you're going to be pretty, you
know, mostly you're going to wander away
from where the starting point is.
So what's true about the random walk is
that you know, if we simulate this process
out say to infinity, it diverges to plus
or minus infinity.
So, eventually, you know, your, your away
from where you started with probability
one.
And, and you've wondered off yo know, who
knows where.