In this lecture, we will talk about backward shift operator.
Objective is the following.
We're going to define and utilize backward shift operator.
So, let's start.
We have a stochastic process, X1, X2, X3, and so forth.
Each one of these guys are random variables.
And we define backward shift operator as a follow up.
B is our backward shift operator, and we'll take Xt, and
we'll take it back one step, which will take it back one lambda.
So, BXt will go to Xt-1.
In other words, if you have B squared Xt, B squared is defined
B applying backward shift operator twice, so we have BBXt,
BXt is going to be Xt-1, and if take one step back you going to go to Xt-2.
In other words B to the k Xt, Im sorry, B to the k Xt going to the Xt-k.
With the Xt, I would take it back k steps and let's use it.
So, this is the definition of the backward shift operator, so
how can we utilize this?
We're going to do the following.
Think of the random walk right this is one of our examples we have talked about it.
Basically you start from the previous step,
and you add some random disturbance to it right.
Zt is our random disturbance, or you know bayesian, or white noise,
and then we go to the Xt, in other words we can write actually Xt as sum of
the equations of the whole past innovations, all disturbances.
Xt-1 can be expressed as BXt, that's backwards shift operator.
And we take the Xt to the left-hand side, and pull out Xt, so
that we get another operator 1-B is going to be our polynomial operator.
1 is basic identity 1Xt, Xt, or 1-B will give you
(1-B)Xt will give us Xt-Xt-1.
We define (1- B) by phi B.
And then phi (B)Xt becomes Zt.
So, by using back work shift operator we rewrite random walk in this format.
This is a MA(2) process.
One example of MA(2) process is it goes back two steps.
You see this is, the basically, sum weighted sum,
of the three innovations,
right, the present random noise, the previous random noise, and
then the random noise in lack two, and they all add it up to give the Xt stack.
Today, that's Xt.
So, how come you neutralize backward chief operator here.
Instead of Zt-1, we can write BZt, and instead of Zt-2 we can write B squared Zt.
Now, everybody has Zt in it.
We can kind of pull out Zt, and all polynomial operator,
which we call beta B is going to be 1+0.2B.
Plus 0.04B squared.
Now, we write this MA(2) process in this format, Xt = Beta(B)Zt,
where Beta(B) is actually a polynomial operator.
We can utilize backwards shift operator in AR(2) processes.
So, this is our Xt, a regressed on two previous
lags with some random disturbance, innovation, or white noise at that step.
And instead of Xt minus 1, we can write BXt,
Xt minus 2 to write B square Xt take these two terms to the left outside,
pull out Xt, we have polynomial operator we call this phi(B).
So, we write this as phi(B)Xt = Zt.
But what is our phi(B)?
Phi(B) is basically polynomial operator in terms of the backward shift operator.
It is 1- 0.2B- 0.3B squared.
In general, if you look at MA(q) processes, moving average processes, for
their q, assessing there's some drift.
Mu is our drift, our expectation of Xt, in other words.
And Xt is equal to, it's drift plus
basically the linear combination of the random noises.
Q steps back, q lags back instead of Xt, I write Zt instead of Zt -1, we write BZt.
Instead of Zt minus q, we can write this BqZt.
We can pull out Zt, put the drift to the left-hand side.
I have Xt minus mu is equal to polynomial operator acting on Zt.
And this polynomial operator is basically phi 0 + phi 1 B + phi q B q.
Okay, so let me just mention that these phis are actually not phis,
this is a typo, it is our basically Beta 0, Beta 1, Beta q to the end.
Okay, If you have AR(p) processes, basically Xt is
regressed on the previous P steps of the same stochastic process,
plus some random noise, we can use, backwards shifht and
put all these terms to the left, and come up with the format, phi(B)Xt equals Zt.
Where phi(B) is the backwards shift, I'm sorry, it's a polynomial for each ratio.
The back ration is 1-phi 1b and so forth.
So, what have we learned?
We have learned the definition of backward shift operator,
which we will use a lot later on as well.
And we used how to utilize backward shift operator to write MA(q) processes and
AR(p) processes.