Hey guys, in this video I'm going to do this
time series problem. It essentially is giving you a
list of different time series or stochastic processes
and it's asking you to determine which of them,
if any, are stationary. This question
comes from the Brockwell and Davis
time series book, where they define stationarity
as weak stationarity.
There's technically a definition of weak stationarity
and also strict stationarity
but for time series, usually we want to just meet
the definition of weak stationarity.
Weak stationarity is a less strict stationarity
assumption and it's based on two parts.
It's based on essentially
looking for stationarity in the first moment
and the second moment of your stochastic process.
This definition is also from the Brockwell
and Davis book and it says that a time series
is weakly stationary if the first moment
does not depend on time t, in other words
that the mean is constant
for your time series and that the autocovariance
function for t and t
plus a lag h is independent of time t.
This is saying that the autocovariance
of X(t) is also constant.
Essentially if you're checking for stationarity of a time series,
what you want to do is get the expected value
of your time series and make sure it's a constant
and doesn't depend on the time, and also check for
the autocovariance between
time t and time t plus any lag
and make sure that also is constant
and if your first two moments are constant,
then your time series meets the definition of weak stationarity.
For this video,
I'm going to do the first two problems. I'm going
to do A and B. So let's go ahead and get started.
I'm going to start with problem A and for
every problem I'm going to first check that the mean
is constant and then check the autocovariance function.
The expected value or the mean
of X(t) is just going to be plugging in the
expected value of the righthand side, which is A
plus B times Zt
and we're given in the problem that Zts are
distributed with mean 0 and variance sigma squared.
We're going to use that information later. Plus,
c times Zt-2.
And we're also given that a, b, and c
are constants. The expected value of a constant
is just a constant. We can simplify this as
the expected value of a is just a
plus b times the expected value
of Zt plus c
times the expected value of Zt-2.
We're given in the problem
that the expected values of the Zts are
zero. This is going to be equal to a
plus 0--or b times 0,
which is just 0. Plus c times 0,
which is just 0. The expected value
of X(t) is equal to a, which does not depend on t
and so we've met the first
assumption for weak stationarity.
Our mean is stationary and now we have to check
the second moment, or the autocovariance function of A
to make sure that it is weakly stationary.
For the second part, I'm going to check the
covariance between Xt+h--
so, Xt plus a lag h--
and just regular Xt
and again I'm going to plug in Xt
the righthand side and Xt+h, the righthand side
for this part.
That's going to be equal to the autocovariance between
a plus b times Zt+h
plus c
times Zt-2+h and
Xt, which is just a plus b
times Zt plus
c times Zt minus 2.
Then for the next step, I'm essentially going to
expand this single covariance
function into the covariance between
each part of the expressions by
comparing each part of the first expression to each part
of the second expression. I'm going to start
with the first part, which is a. The covariance between
a and a---this first part, second part,
and third part is going to be 0 because a
is just a constant. The covariance between
any constant and any other constant
or random variable is just going to be 0.
I'm finished with comparing a.
For this second part of the expression,
again the covariance between a random variable
and a constant is just zero
so that's zero. Then the covariance
between this first part bZt
plus h and this second part
bZt is equal to the
covariance between bZt+h
and bZt plus
the covariance between this second part and the last part
is going to be equal to the covariance between
bZt+h and c
times Zt-2.
I've essentially just expanded the
expression using covariance properties
and I'm going to do the same thing for the last part.
The covariance between this third part
and a constant is just 0. Then the
covariance between this part
and bZt is going to be expanded out.
And I'm going to do the same thing with the last
two parts of the expressions.
I'm going to have
this as my expanded covariance function.
And now as a final part I have to determine the stationarity of this autocovariance
function for any lag h.
When h is 0, we're going
to have the covariance between
bZt plus 0
and bZt. That's just going to be b squared
and the covariance between
two random variables Zt.
The covariance between Zt and Zt
will just give you the variance of Zt,
which we're given is sigma squared.
That's what the expression will be for this first part.
Here Zt plus 0
and Zt-2,
the covariance between that will be 0. The same thing
will happen for this expression. That will just be 0.
And then here if we plug in 0, we'll get
that this is equal to c squared
times the variance of
Zt-2, which is just sigma squared.
When we have that the
lag h is equal to 0, this will be our
autocovariance function. Then we essentially have to worry about
when these two values in the middle
will not be zero. And we can see that's going to be
when h is equal to plus or minus 2.
When h is equal to minus 2,
that's going to give us the covariance between
bc
and Z-2 and Z-2, which is
the variance of Zt-2
or sigma squared. The same thing is going to
happen when we have h is equal to positive 2.
If we plug in positive 2 for h
here, we're going to get Zt and Zt.
Then the covariance between c times Zt
and b times Zt is going to be
bc and the variance of Zt, which is sigma squared.
Then, the autocovariance is going to be 0
for any other lags.
The autocovariance is going to be 0 for any other lag
that's not 0 or plus or minus 2.
This is our final autocovariance function
expression. And as we can see for any
lag h, the autocovariance expression does not
depend on time t. So we have
that the first moment or the mean
of the stochastic process does not depend on time t
and we have that the autocovariance function
does not depend on time t for any lag h.
Thus for problem A, we do have
weak stationarity.
And I think I'm going to end the video here
instead of doing part B
because it's already getting a little bit long.
Maybe in the future, I'll make a second video where I can do some of these
other problems. But for now I'll end the video here
and thank you guys for watching!
*chiptune music*