In this lecture we will talk about invertibility and stationarity conditions.
Objectives is to articulate invertibility condition for
moving average processes with the order q.
Discover stationarity condition for auto-regressive processes with order p.
And then we will relate at moving average processes and
auto-regressive processes, through duality.
So, let's start with MA(q) process.
This MA(q) process is basically Xt is a linear combination
of innovations q lags back.
And if you write B to use backward shift operator,
and write this in terms of Xt is equal to beta B Zt and
beta B is going to be our polynomial operator.
And what we are trained to do, as we did in the last lecture for
MA(1) process, we tried to find inverse operator for beta B and
write this as alpha zero + alpha one B and so forth.
The thing is finding this alpha zero, alpha one, alpha two is not that easy, but
at least you would like to be able to find inverse of this polynomal operator.
It throws up that we have a condition.
And we're going to use this condition as a black box.
I will not give the proof.
A proof for MA(1) process will be in this optional lecture.
MA(q) process is in invertal.
In other words, we can write Zt in terms of Xts as an infinite sum.
If the roots of this polynomial, which might be a real R complex,
lies outside of a unit circle, where we regard B as
a complex number, not as an operator.
So for example let's look at the MA(1) process, this is our Xt, Zt + beta Zt- 1.
Our operator, polynomial operator is 1 + beta B, the only route,
in this case there's only one real route, is negative 1 over beta.
Which has to have a magnitude with greater than 1 so
that it's outside of the unit circle that would tell us that magnitude of, or
absolute value in this case, of beta is actually less than one.
Then we can work Zt as infinite sum of Xts.
Let's look at MA(2) process.
Now this is MA(2) process.
And polynomial operator is 1 + pi over 6B + 1 over 6B squared.
If you would like to check if this process is invertible, then all we
have to do according to this invertibility condition is to check whether or
not all routes of this polynomial lie outside of the unit circle.
In terms of complex numbers this is our quadratic equation.
And it turns out that in this case we still have real roots, which is 2 and
3, both of which are lines outside of the unit circle, they're on its axis.
But both of them are on one side of it, they're above 1.
So we can try and find inverse of the polynomial operator,
usually you would not be able to find it.
But in this case we can actually do it, we write this 1 over our polynomial and
factorize our polynomial, and use partial fractions.
And write this into two fractions, if you actually find common denominator here,
you will see you that you will get back to this rational function.
There B is regarded as a complex number.
Now, we can expand each fraction as a geometric series.
There, we start at 3 and our R is going to be half B, or going to be one third B,
and if we combine these 2 geometric series, we'll have a expression for
inverse of this polynomial operator, and this is going to be the inverse.
In other words, Zt can be written as this operator acting on an Xt,
which will give us this pi k, and pi ks are basically these coefficients.
So, as we can see, MA(2) process can be inverted into AR(infinity) process.
If invertible, the condition is satisfied.
Now remember MA(q) processes are always stationary.
But it is not the case for AR(p) processes.
It turns out that there's a counterpart for
AR(p) processes and similar condition which hold it for
immutability for MA(q) processes holds for stationarity of AR(p) processes.
So what is the condition?
We have AR(p)process where Xt is regressed from the previous
p lags with some random noise.
We write this using backwards shift operator, our operator polynomial is going
to look like 1- phi 1B- phi 2B square- phi pB to the P, because order is P.
And all we have to check, in this case, is again, to make sure the polynomial,
this polynomial, has all of its complex rules outside of the unit circle.
So in case of AR (1) process.
Our polynomial operator is 1- phi 1B, which has only one real root].
Which is 1 over phi 1.
You have to make sure that this slides outside of the unit circle.
In other words, the magnitude.
In this case, the absolute value has to be greater than 1.
Which means that phi1 must be less than 1.
In other words, if phi1 has absolute value less than 1 then
AR(1) process is going to be stationary.
And we can actually invert it.
And when I say invert, we can find inverse of this operator acting on Zt.
It's going to be 1 + phi 1B + phi 1B squared.
This is basically geometric expansion, and it's acting on Zt, and
we write Xt as infinite sum of Zts, right?
So in other words, we actually express the AR(P) process as MA(infinity) process.
We'll come back to that, but let's just take another look at phi 1.
If you actually take the variance of Xt, variance of infinite sum,
assuming that infinite sum is convergent in some sense which is again
the optional video, it's in mean square sense.
Then, we can basically distribute the variance because all of the Zs
are uncorrelated, and variance of Zs are sigma squared,
become pull off sigma squared, we'll get this infinite sum of numbers.
But this is a geometric series, right?
This is our usual geometric series.
R is phi 1 squared, so it has to have magnitude less than 1, so
that this infinite sum is convergent, so that the variance do exists.
In other words, you want phi 1 to be less than 1.
So you just found that,
if I want my AR(1) process to be stationary in two different ways,
we actually proved that absolute value of phi 1 must be less than 1.
So, AR(p) processes can be written as MA(infinity) process,
if we have this stationarity condition that that holds.
So that gives us duality, so we have a duality between AR,
other aggressive processes, and MA processes, moving other processes.
Under invertibility condition MA(q) processes can be written
as AR(infinity) process.
Under stationarity condition,
AR(p) process can be written as MA(infinity) process.
So what have we learned?
We have learned invertibility conditions for MA(q) processes.
We have learned stationarity condition for AR(p) processes and we have learned
duality between moving average processes and other regressive processes.