So in this lecture, we will extend our RHP model in order to obtain Gurche.
So this lecture will cover three main points.
The first point to cover is why do we use Gargash?
Why is there not enough?
The second point to cover is what does Gaja actually look like?
So as with the previous lecture, we'll learn about the equation that is used for the garden process.
This is what will give us the most insight.
And the third point to cover is how does this relate back to Armah?
So at that point, we'll do a little bit of analysis on the model to better understand how it works
and to gain deeper insights about how it behaves.
OK, so let's begin by discussing why do we need Gargash?
So as you recall, the arch model is defined such that the conditional variance depends on past error
terms.
Now, as you know, these error terms are completely random.
So what tends to happen is this Sigma is very erratic with what we call high frequency movements.
High frequency means that the values change very quickly.
What we've seen, however, is that the volatility tends to persist.
So Gurche models tend to be more expressive, allowing Sigma to persist for a longer period of time.
In other words, it allows us to have more flexible modeling capabilities.
OK, so the second thing to consider in this lecture is what does the garden model look like?
Well, one way to write it is this a garden cue is what we get when we take an RFP and we add Kuis previous
sigma terms.
So in other words, RHP is do IRP as P is to Amah P.
Q Note that we use the Greek letter Beita for the Sigma coefficients.
Now one very weird thing about this is that it may seem as if we have P and Q backwards.
In fact, in the literature you may keep you switched around.
It seems to make sense if Sigma is the Time series we're trying to model than the past, Sigma terms
would correspond to the auto regressive component.
Epsilon, as usual, would correspond to the past error component.
So that's just something to be aware of.
Some resources may switch to, however, recall that for us, Epsilon Squared is the Time series, but
in some way this does not seem to make sense since Sigma is not past random noise.
Well, one way to see it is that this is simply the convention.
We'll look at another perspective later in this lecture.
So one characteristic of this model that we can infer from this equation is that it allows Sigma to
persist over time.
As you recall, this was our motivation for using garbage.
So it might help to look at a large one, one which has one passed epsilon term and one passed Sigma
Term.
We can see that although the Epsilon term is random, the Sigma term is not random.
As you recall, the randomness and epsilon comes from the randomness in Z.
The Sigma part is not considered to be random.
It represents the variance of Epsilon, but it is not itself random.
Thus, you can imagine if Beta had a value like zero point nine, it would always take a zero point
nine of whatever the previous variance was, and that influence would taper off slowly over time.
OK, so the next question to answer in this lecture is, how does this relate to.
We've seen that Gaja looks kind of like AAMA, but not quite.
In fact, it's possible to manipulate the guard equation so that it looks even more analogous to AAMA.
So to do this, let's create a new variable entity.
This will be the term that is analogous to noise in Aamot will set it equal to Epsilon T squared, minus
Sigma T squared.
Let's start by considering an arch one.
In this case, we're going to express the model in terms of sigma squared.
Now let's suppose that we had ality to both sides.
Thus we can see that this is precisely an R one in terms of Ypsilanti squared.
Notice that on the left hand side, this is simply equal to Ypsilanti squared as per our original definition
of Áder.
Another way of looking at this is to pretend that Epsilon T is its own time series.
So we simply call it Y of T.
This might make it easier to see that this is just an A1 in terms of Y squared instead of just Y, which
is what we would normally do if we wanted to model the mean instead of the variance.
So here's one thing to consider, which will hopefully make the analogy make more sense.
It may seem like our substitution of a T is arbitrary.
Why should it be defined the way it is?
Well, let's consider what the mean of ADT is.
If we take the expected value of both sides, we know that the expectation of Epsilon squared is just
Sigma squared, which is its variance.
Therefore, we have Sigma squared minus Sigma squared, which is zero.
So this makes sense in terms of AAMA because the model does say that the additive noise has zero, meaning
there is one small difference, however, because we know that the epsilon squares are not uncorrelated.
Therefore the A's do have serial correlation, which makes them a bit different from a typical model.
The next step is to consider a March one one in this case, we're going to make the same substitution
using a T, we're again going to express our model in terms of sigma on the left side.
OK, so the trick this time is to replace the Sigma squared at T minus one on the right side.
As you recall, we defined ADA T to be Ypsilanti squared minus Sigma T squared.
Therefore Sigma squared is just equal to Epsilon squared minus eight.
And note that the subscripts for this is T minus one.
At this point we can combine the two Epsilon terms together.
The next step is the same as before, where we add ality to both sides.
Of course, the left hand side is simply Epsilon Square T, which is what we had before.
Now, you might have to squint your eyes a little bit, but this is an arm, a one one in terms of Epsilon
squared.
We have a bias term, which is Omega.
We have one auto regressive term, which is Alpha one plus beta one.
We have one moving average term, which is minus beta one.
Note that this is multiplied by the previous error term at time T minus one.
And we have the current error term, which is eight a T.
So the symbols are a bit different, but this is in fact in Arma one one.
This should also help you understand why we use P for the past Epsilon terms and Q for the past Sigma
terms, although it also makes sense to switch them around.
So it's really up to you how you want to think of it as an exercise.
You might want to think about how this derivation might be done for a generic Q.