OK, so in this video, we are going to continue our discussion of Arche, so let's start with the one
you can think of.
This is analogous to the R-1 process with some minor differences.
Specifically, we're going to say Epsilon of T is equal to zero T times the square root of Omega plus
Alpha one times Epsilon at T minus one.
So let's break down what each part of this means.
Of course, the left side, we know this is just the noise time series we're trying to model on the
right side.
We start with ZT in March.
This of T takes on the role of the unpredictable noise will assume it has mean zero and variance one,
for example, it could be in zero one the standard normal.
In fact, most of the time we'll just assume that it's in zero one, although it's not necessary.
Next we have Omega.
You can think of Omega like the bias term.
After Omega, we have Alpha one.
So in March we use Alpha as our symbol for the auto regressive coefficients.
Since this is a large one, we'll just have one auto regressive coefficient called out for what?
Finally, we have the previous Noisome and the Time series, which is Epsilon at T minus one.
Now, one alternative way to look at the arch model is to simply square all the terms when we do it
this way, it looks a lot more like an air one.
Basically, instead of building a linear model of Epsilon of T, we're building a linear model of Epsilon
of T squared.
Furthermore, the unpredictable noise terms of T is multiplicative rather than additive.
So at this point, we're going to introduce a new symbol, sigma of tea, as you might expect.
We want sigma squared of tea to represent the variance of epsilon of tea.
So to make sure that this relationship is true.
We're going to say that Epsilon of tea is equal to Z of Tea Times, Sigma of Tea.
Now, why does this make sense?
Well, if it doesn't make sense to you right away, I would encourage you to try this experiment in
Python.
Firstly, I think we can all agree that if you call NPE not random, not random, then you will get
a random sample from the standard normal distribution with mean zero and variance one.
Now, suppose I take this random sample and I multiply it by some number, say three.
If I do this, the result will be that this new random sample will have variants of three squared or
nine.
Now, again, I encourage you to try this yourself, draw a bunch of random samples using and beat out
random random and multiply all those samples by three.
You should be able to confirm that the variance of these samples is approximately nine.
And because of this, we can conclude that if we take Z of T, which is a sample from the standard normal
and we multiply that by Sigma of T, the result will be a sample from the normal with mean zero and
variance sigma squared of T.
So what else can we do with this new symbol?
Well, let's take the model we had before and let's suppose that we move the zaev down to the left side,
of course, given the relationship we just learned about.
We know that Epsilon of T divided by society is just Sigma T..
So what we really end up with is that sigma squared of T is equal to this linear equation of epsilon
squared of T minus one.
So what is this really saying?
It's saying that the arch model is really a linear model of variance.
The variance is a linear function of previous squared errors.
Now, it's important to recognize that Gargash models are not directly analogous to Arima.
We know that there must be some constraints since this variance must be non-negative.
In particular, notice how we have a square root.
Since we want the thing inside the square root to be non-negative, we must have that omega's greater
than zero and alpha one is greater than or equal to zero.
In addition, if we want the air to be stationary with finite variance, we need Alpha1 to be strictly
less than one.
So you can see that if one were to be greater than or equal to one, the variance could explode.
OK, so at this point, we've determined how we can model that conditional variants of Epsilon.
That is what is the variance of Epsilon at time t given the previous values of Epsilon.
One further question we can ask is what is the unconditional variance of Epsilon?
That is, if we do not know anything about the other values, what is the variance of Epsilon?
Another way of thinking about this is what is the variance of epsilon infinitely far into the future?
Let's call the results simply Sigma Squared without any subscript t in this case, we can simply take
the unconditional expectation of both sides on the right side.
Since Zaev in Ypsilanti are independent, we can separate these expectations.
Furthermore, recall that the variance of ZT is one.
Finally, we can rearrange and isolate sigma squared.
We get that sigma squared.
The unconditional variance of epsilon is omega divided by one minus Alpha one.
So this is another reason for Alpha one to be less than one.
If it were not less than one than this result would be nonsensical.
As a side note, please be aware that I'm leaving out some details.
When Alpha one is one, the variance is infinite, but you may still get a stationary series.
In any case, this and most of the math behind Gurche is considered to be outside the scope of this
cause.
One thing we'll be able to observe in the code is that if we forecast for a long time into the future,
our forecasts will actually converge to this value.
So what this means is that Arche and Gurche are useful for short term forecasts of the variance.
As with all other financial times, there is models we typically cannot forecast far into the future.
This makes sense since we can't expect a model like this to predict something like Elon Musk making
a tweet about Bitcoin or some country banning a cryptocurrency exchange.
Another obvious example is covid-19.
Clearly, Gargash is not going to be able to predict these world scale events.
These events are really what caused large deviations in the variance.
But of course, without insider trading, you probably wouldn't know about them beforehand.
OK, so the final thing I want to discuss in this lecture is the RFP.
Basically, it's a very simple extension of the arch one where we simply let Epsilon of t depend on
people's legs instead of just one.
So you can think of this as being analogous to AARP.
As usual, we can use tools like the AIC to help us determine the best value of P.