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So in this lecture, we will extend our RHP model in order to obtain Gurche.

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So this lecture will cover three main points.

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The first point to cover is why do we use Gargash?

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Why is there not enough?

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The second point to cover is what does Gaja actually look like?

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So as with the previous lecture, we'll learn about the equation that is used for the garden process.

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This is what will give us the most insight.

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And the third point to cover is how does this relate back to Armah?

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So at that point, we'll do a little bit of analysis on the model to better understand how it works

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and to gain deeper insights about how it behaves.

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OK, so let's begin by discussing why do we need Gargash?

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So as you recall, the arch model is defined such that the conditional variance depends on past error

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terms.

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Now, as you know, these error terms are completely random.

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So what tends to happen is this Sigma is very erratic with what we call high frequency movements.

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High frequency means that the values change very quickly.

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What we've seen, however, is that the volatility tends to persist.

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So Gurche models tend to be more expressive, allowing Sigma to persist for a longer period of time.

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In other words, it allows us to have more flexible modeling capabilities.

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OK, so the second thing to consider in this lecture is what does the garden model look like?

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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

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sigma terms.

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So in other words, RHP is do IRP as P is to Amah P.

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Q Note that we use the Greek letter Beita for the Sigma coefficients.

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Now one very weird thing about this is that it may seem as if we have P and Q backwards.

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In fact, in the literature you may keep you switched around.

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It seems to make sense if Sigma is the Time series we're trying to model than the past, Sigma terms

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would correspond to the auto regressive component.

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Epsilon, as usual, would correspond to the past error component.

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So that's just something to be aware of.

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Some resources may switch to, however, recall that for us, Epsilon Squared is the Time series, but

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in some way this does not seem to make sense since Sigma is not past random noise.

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Well, one way to see it is that this is simply the convention.

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We'll look at another perspective later in this lecture.

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So one characteristic of this model that we can infer from this equation is that it allows Sigma to

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persist over time.

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As you recall, this was our motivation for using garbage.

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So it might help to look at a large one, one which has one passed epsilon term and one passed Sigma

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Term.

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We can see that although the Epsilon term is random, the Sigma term is not random.

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As you recall, the randomness and epsilon comes from the randomness in Z.

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The Sigma part is not considered to be random.

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It represents the variance of Epsilon, but it is not itself random.

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Thus, you can imagine if Beta had a value like zero point nine, it would always take a zero point

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nine of whatever the previous variance was, and that influence would taper off slowly over time.

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OK, so the next question to answer in this lecture is, how does this relate to.

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We've seen that Gaja looks kind of like AAMA, but not quite.

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In fact, it's possible to manipulate the guard equation so that it looks even more analogous to AAMA.

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So to do this, let's create a new variable entity.

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This will be the term that is analogous to noise in Aamot will set it equal to Epsilon T squared, minus

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Sigma T squared.

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Let's start by considering an arch one.

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In this case, we're going to express the model in terms of sigma squared.

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Now let's suppose that we had ality to both sides.

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Thus we can see that this is precisely an R one in terms of Ypsilanti squared.

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Notice that on the left hand side, this is simply equal to Ypsilanti squared as per our original definition

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of Áder.

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Another way of looking at this is to pretend that Epsilon T is its own time series.

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So we simply call it Y of T.

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This might make it easier to see that this is just an A1 in terms of Y squared instead of just Y, which

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is what we would normally do if we wanted to model the mean instead of the variance.

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So here's one thing to consider, which will hopefully make the analogy make more sense.

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It may seem like our substitution of a T is arbitrary.

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Why should it be defined the way it is?

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Well, let's consider what the mean of ADT is.

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If we take the expected value of both sides, we know that the expectation of Epsilon squared is just

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Sigma squared, which is its variance.

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Therefore, we have Sigma squared minus Sigma squared, which is zero.

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So this makes sense in terms of AAMA because the model does say that the additive noise has zero, meaning

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there is one small difference, however, because we know that the epsilon squares are not uncorrelated.

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Therefore the A's do have serial correlation, which makes them a bit different from a typical model.

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The next step is to consider a March one one in this case, we're going to make the same substitution

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using a T, we're again going to express our model in terms of sigma on the left side.

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OK, so the trick this time is to replace the Sigma squared at T minus one on the right side.

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As you recall, we defined ADA T to be Ypsilanti squared minus Sigma T squared.

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Therefore Sigma squared is just equal to Epsilon squared minus eight.

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And note that the subscripts for this is T minus one.

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At this point we can combine the two Epsilon terms together.

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The next step is the same as before, where we add ality to both sides.

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Of course, the left hand side is simply Epsilon Square T, which is what we had before.

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Now, you might have to squint your eyes a little bit, but this is an arm, a one one in terms of Epsilon

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squared.

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We have a bias term, which is Omega.

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We have one auto regressive term, which is Alpha one plus beta one.

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We have one moving average term, which is minus beta one.

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Note that this is multiplied by the previous error term at time T minus one.

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And we have the current error term, which is eight a T.

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So the symbols are a bit different, but this is in fact in Arma one one.

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This should also help you understand why we use P for the past Epsilon terms and Q for the past Sigma

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terms, although it also makes sense to switch them around.

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So it's really up to you how you want to think of it as an exercise.

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You might want to think about how this derivation might be done for a generic Q.