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.