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OK, so in this video, we are going to continue our discussion of Arche, so let's start with the one

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you can think of.

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This is analogous to the R-1 process with some minor differences.

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Specifically, we're going to say Epsilon of T is equal to zero T times the square root of Omega plus

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Alpha one times Epsilon at T minus one.

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So let's break down what each part of this means.

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Of course, the left side, we know this is just the noise time series we're trying to model on the

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right side.

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We start with ZT in March.

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This of T takes on the role of the unpredictable noise will assume it has mean zero and variance one,

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for example, it could be in zero one the standard normal.

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In fact, most of the time we'll just assume that it's in zero one, although it's not necessary.

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Next we have Omega.

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You can think of Omega like the bias term.

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After Omega, we have Alpha one.

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So in March we use Alpha as our symbol for the auto regressive coefficients.

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Since this is a large one, we'll just have one auto regressive coefficient called out for what?

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Finally, we have the previous Noisome and the Time series, which is Epsilon at T minus one.

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Now, one alternative way to look at the arch model is to simply square all the terms when we do it

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this way, it looks a lot more like an air one.

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Basically, instead of building a linear model of Epsilon of T, we're building a linear model of Epsilon

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of T squared.

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Furthermore, the unpredictable noise terms of T is multiplicative rather than additive.

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So at this point, we're going to introduce a new symbol, sigma of tea, as you might expect.

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We want sigma squared of tea to represent the variance of epsilon of tea.

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So to make sure that this relationship is true.

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We're going to say that Epsilon of tea is equal to Z of Tea Times, Sigma of Tea.

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Now, why does this make sense?

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Well, if it doesn't make sense to you right away, I would encourage you to try this experiment in

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Python.

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Firstly, I think we can all agree that if you call NPE not random, not random, then you will get

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a random sample from the standard normal distribution with mean zero and variance one.

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Now, suppose I take this random sample and I multiply it by some number, say three.

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If I do this, the result will be that this new random sample will have variants of three squared or

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nine.

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Now, again, I encourage you to try this yourself, draw a bunch of random samples using and beat out

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random random and multiply all those samples by three.

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You should be able to confirm that the variance of these samples is approximately nine.

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And because of this, we can conclude that if we take Z of T, which is a sample from the standard normal

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and we multiply that by Sigma of T, the result will be a sample from the normal with mean zero and

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variance sigma squared of T.

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So what else can we do with this new symbol?

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Well, let's take the model we had before and let's suppose that we move the zaev down to the left side,

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of course, given the relationship we just learned about.

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We know that Epsilon of T divided by society is just Sigma T..

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So what we really end up with is that sigma squared of T is equal to this linear equation of epsilon

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squared of T minus one.

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So what is this really saying?

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It's saying that the arch model is really a linear model of variance.

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The variance is a linear function of previous squared errors.

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Now, it's important to recognize that Gargash models are not directly analogous to Arima.

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We know that there must be some constraints since this variance must be non-negative.

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In particular, notice how we have a square root.

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Since we want the thing inside the square root to be non-negative, we must have that omega's greater

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than zero and alpha one is greater than or equal to zero.

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In addition, if we want the air to be stationary with finite variance, we need Alpha1 to be strictly

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less than one.

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So you can see that if one were to be greater than or equal to one, the variance could explode.

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OK, so at this point, we've determined how we can model that conditional variants of Epsilon.

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That is what is the variance of Epsilon at time t given the previous values of Epsilon.

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One further question we can ask is what is the unconditional variance of Epsilon?

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That is, if we do not know anything about the other values, what is the variance of Epsilon?

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Another way of thinking about this is what is the variance of epsilon infinitely far into the future?

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Let's call the results simply Sigma Squared without any subscript t in this case, we can simply take

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the unconditional expectation of both sides on the right side.

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Since Zaev in Ypsilanti are independent, we can separate these expectations.

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Furthermore, recall that the variance of ZT is one.

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Finally, we can rearrange and isolate sigma squared.

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We get that sigma squared.

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The unconditional variance of epsilon is omega divided by one minus Alpha one.

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So this is another reason for Alpha one to be less than one.

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If it were not less than one than this result would be nonsensical.

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As a side note, please be aware that I'm leaving out some details.

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When Alpha one is one, the variance is infinite, but you may still get a stationary series.

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In any case, this and most of the math behind Gurche is considered to be outside the scope of this

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cause.

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One thing we'll be able to observe in the code is that if we forecast for a long time into the future,

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our forecasts will actually converge to this value.

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So what this means is that Arche and Gurche are useful for short term forecasts of the variance.

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As with all other financial times, there is models we typically cannot forecast far into the future.

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This makes sense since we can't expect a model like this to predict something like Elon Musk making

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a tweet about Bitcoin or some country banning a cryptocurrency exchange.

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Another obvious example is covid-19.

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Clearly, Gargash is not going to be able to predict these world scale events.

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These events are really what caused large deviations in the variance.

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But of course, without insider trading, you probably wouldn't know about them beforehand.

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OK, so the final thing I want to discuss in this lecture is the RFP.

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Basically, it's a very simple extension of the arch one where we simply let Epsilon of t depend on

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people's legs instead of just one.

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So you can think of this as being analogous to AARP.

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As usual, we can use tools like the AIC to help us determine the best value of P.