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.