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So in this lecture, we will summarize everything we learned in this section and also discuss some ways

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has been extended in case you wish to further your studies.

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So this section focused on a technique called Gurche.

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Unlike all of the other time series methods we learned in this course, Gaja is not focused on learning

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the mean of a Time series.

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Instead, it's focused on learning about the volatility of the errors.

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This makes it completely unique compared to everything else we studied in this course.

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It turns out that Gurche is a very popular model for Financial Times series because volatility is one

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of the few characteristics of Financial Times series that have predictable structure.

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Specifically, we learn that although stock prices look like a random walk, when you plot the action

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of the returns, they do not look like a random walk when you plot the action of the squared returns.

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This is one factor that differentiates stock prices from random walks.

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What we notice when we look at a time series of stock returns is that they tend to exhibit volatility

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clustering.

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This means that large returns tend to be surrounded by other large returns.

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Small returns tend to be surrounded by other small returns.

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And of course, this is exactly the kind of behavior that garden is designed for.

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It's one of the main properties of garbage.

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In this section, we studied the arch one in depth, and we learn that it has many of the same characteristics

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of stock returns which make them an appropriate model for Financial Times series.

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We then learned about Gurche, which extends the arch by improving its ability to have the variants

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persist after looking at the theory behind Arch and Gargash.

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We then looked at how to use Gartland Python.

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This was hopefully a bit surprising because it's very much unlike the other models we've studied in

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this course.

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We noted that the forecast isn't really a direct forecast of a time series, but instead a forecast

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of variance.

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What's interesting about this is we don't really have a target.

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That is, there's nothing to compare it to, to check if our model predictions are correct.

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So we didn't follow the usual process of training a model and making a forecast and then checking some

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metric like the May.

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The next thing we did in this section was to look at how garbage can be fitted if you were to do it

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yourself from scratch.

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This was mostly conceptual, but of course, there's nothing stopping you from actually implementing

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what you learned.

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One thing we observed was that this opens the door for us to build much more complex models, including

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those that use deep neural networks.

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So as long as you know how to build a custom lost function in your favorite deep learning library.

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This is something you can try.

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The final topic of this lecture is to explore some extensions of garbage, although garbage is already

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very powerful.

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There are some ways it can be improved depending on what you are trying to model.

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One such model is called a park.

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In fact, this model introduces two new improvements.

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The first improvement is that it model something called the leverage effect.

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Essentially, what this means is that large negative returns tend to increase volatility compared to

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large positive returns of the same magnitude.

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This can not be modeled with a simple gurche because garbage can only depend on past the squared error

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terms and the square function is symmetric.

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That is Epsilon at T minus one has the same influence, whether it is positive or negative.

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The way that a park handles this is it introduces this new function where we take the absolute value

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of the old error and subtract a parameter gamma multiplied by the old error.

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Now this is probably confusing, so it helps to simply look at a picture.

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In this picture.

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We can see what happens for different values of gamma when gamma equals zero, we can see that both

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positive and negative values are symmetric.

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When Gamma equals zero point nine, we can see that on the negative side, the influence of these values

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are much higher compared to the positive side.

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Thus, with a positive gamma negative, returns will have a stronger influence on Sigma.

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The second obvious thing to notice about this is that we are not squaring the terms.

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Instead, we raise them to the power delta.

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So this allows for more flexibility by giving you another parameter to tune.

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Yet another way to extend Gargash is to use the multivariate garbage, that is to consider multiple

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asset returns at once.

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This makes sense since it's very possible that returns of one asset could affect the returns of another.

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For example, if there's a sudden movement in Bitcoin, this could transfer over to either as well.

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However, this is challenging for several reasons.

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One reason is that it gives us many more values to estimate.

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As you recall, in multiple dimensions, it's not just the variance we need to know, but the covariance

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matrix.

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If we have the assets, then we need to estimate the Times D plus one over two values in the covariance

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matrix.

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This is not quite squared since this matrix is symmetric, but it still does grow quite dramatically.

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Another issue is that, as you recall, the volatility is unobserved, so we're basically trying to

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estimate a quadratic number of new unobserved variables with only a linear amount of new data.

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Now, it's not just a matter of adding more data in the past, since, of course, the stock market

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is dynamic, the relationships between assets are always changing.

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So the correlation and returns 20 years ago is not the same as it is today.

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Finally, we also have to account for the fact that covariance matrices need to be positive, definite.

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This is the matrix equivalent of saying that the variance needs to be bigger than zero.

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So we can't just pick any numbers for the covariance matrix, but they need to have a special structure.