So in this lecture, we will summarize everything we learned in this section and also discuss some ways
has been extended in case you wish to further your studies.
So this section focused on a technique called Gurche.
Unlike all of the other time series methods we learned in this course, Gaja is not focused on learning
the mean of a Time series.
Instead, it's focused on learning about the volatility of the errors.
This makes it completely unique compared to everything else we studied in this course.
It turns out that Gurche is a very popular model for Financial Times series because volatility is one
of the few characteristics of Financial Times series that have predictable structure.
Specifically, we learn that although stock prices look like a random walk, when you plot the action
of the returns, they do not look like a random walk when you plot the action of the squared returns.
This is one factor that differentiates stock prices from random walks.
What we notice when we look at a time series of stock returns is that they tend to exhibit volatility
clustering.
This means that large returns tend to be surrounded by other large returns.
Small returns tend to be surrounded by other small returns.
And of course, this is exactly the kind of behavior that garden is designed for.
It's one of the main properties of garbage.
In this section, we studied the arch one in depth, and we learn that it has many of the same characteristics
of stock returns which make them an appropriate model for Financial Times series.
We then learned about Gurche, which extends the arch by improving its ability to have the variants
persist after looking at the theory behind Arch and Gargash.
We then looked at how to use Gartland Python.
This was hopefully a bit surprising because it's very much unlike the other models we've studied in
this course.
We noted that the forecast isn't really a direct forecast of a time series, but instead a forecast
of variance.
What's interesting about this is we don't really have a target.
That is, there's nothing to compare it to, to check if our model predictions are correct.
So we didn't follow the usual process of training a model and making a forecast and then checking some
metric like the May.
The next thing we did in this section was to look at how garbage can be fitted if you were to do it
yourself from scratch.
This was mostly conceptual, but of course, there's nothing stopping you from actually implementing
what you learned.
One thing we observed was that this opens the door for us to build much more complex models, including
those that use deep neural networks.
So as long as you know how to build a custom lost function in your favorite deep learning library.
This is something you can try.
The final topic of this lecture is to explore some extensions of garbage, although garbage is already
very powerful.
There are some ways it can be improved depending on what you are trying to model.
One such model is called a park.
In fact, this model introduces two new improvements.
The first improvement is that it model something called the leverage effect.
Essentially, what this means is that large negative returns tend to increase volatility compared to
large positive returns of the same magnitude.
This can not be modeled with a simple gurche because garbage can only depend on past the squared error
terms and the square function is symmetric.
That is Epsilon at T minus one has the same influence, whether it is positive or negative.
The way that a park handles this is it introduces this new function where we take the absolute value
of the old error and subtract a parameter gamma multiplied by the old error.
Now this is probably confusing, so it helps to simply look at a picture.
In this picture.
We can see what happens for different values of gamma when gamma equals zero, we can see that both
positive and negative values are symmetric.
When Gamma equals zero point nine, we can see that on the negative side, the influence of these values
are much higher compared to the positive side.
Thus, with a positive gamma negative, returns will have a stronger influence on Sigma.
The second obvious thing to notice about this is that we are not squaring the terms.
Instead, we raise them to the power delta.
So this allows for more flexibility by giving you another parameter to tune.
Yet another way to extend Gargash is to use the multivariate garbage, that is to consider multiple
asset returns at once.
This makes sense since it's very possible that returns of one asset could affect the returns of another.
For example, if there's a sudden movement in Bitcoin, this could transfer over to either as well.
However, this is challenging for several reasons.
One reason is that it gives us many more values to estimate.
As you recall, in multiple dimensions, it's not just the variance we need to know, but the covariance
matrix.
If we have the assets, then we need to estimate the Times D plus one over two values in the covariance
matrix.
This is not quite squared since this matrix is symmetric, but it still does grow quite dramatically.
Another issue is that, as you recall, the volatility is unobserved, so we're basically trying to
estimate a quadratic number of new unobserved variables with only a linear amount of new data.
Now, it's not just a matter of adding more data in the past, since, of course, the stock market
is dynamic, the relationships between assets are always changing.
So the correlation and returns 20 years ago is not the same as it is today.
Finally, we also have to account for the fact that covariance matrices need to be positive, definite.
This is the matrix equivalent of saying that the variance needs to be bigger than zero.
So we can't just pick any numbers for the covariance matrix, but they need to have a special structure.