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So the real interesting thing about Gargash is not just plugging in some data into the code, but it's

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really understanding why this is a good model.

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So in this lecture, we're going to look at a few more properties of the arch one.

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And this should give you more confidence that everything makes sense.

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Now, as mentioned, that this section won't be focused on deriving these properties.

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Instead, it's the properties themselves which are of interest.

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In other words, simply looking at them and understanding what they mean is useful.

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So the first property we want to look at is that the conditional meaning of Epsilon is zero.

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That is given the past epsilon's the expected value of the next steps.

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The line is always zero.

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This makes sense since we usually expect to have zero centered noise despite knowing the past values

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of the noise.

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Another similar but different fact is that the unconditional mean of Epsilon is also zero.

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That is to say, if we do not know anything about the past, the expected value of epsilon is still

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zero.

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This also makes sense.

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This means that no matter what, whether we know the past epsilons or not, the expected value of the

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next epsilon is always zero.

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The next interesting property is that the covariance between any epsilon and any Epsilon at some other

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point in time is zero.

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So here are the first Epsilon is at time T and the other Epsilon is at time T minus S.

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So as long as S is not zero, then these are two different epsilon's at two different points in time.

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When this is the case, there are covariance is zero.

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And as you recall, correlation is just covariance divided by some constant and thus the correlation

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between any two epsilon's is also zero.

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This makes sense since as you recall, when we look at the akef of log returns, they are in fact zero.

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This model, therefore, seems to be accurate for modeling log returns, at least in this respect.

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Now, one important fact to remember from your study of probability is this it's that absence of correlation

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does not imply independence.

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As you recall, if two random variables are independent, then they are uncorrelated, but the reverse

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is not true.

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So you might wonder, does this fact have any practical use?

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And the answer is yes, because we're seeing an example at this moment.

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The garden model is one example of where the epsilons are uncorrelated, but they are not independent.

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And of course, we know that they are not independent because we have an equation relating past Epsilon's

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to future epsilons.

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OK, so as you recall, one way we can show that the epsilons are not independent is to plot the akef

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of the squares.

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When we do this, we see that there is, in fact, a relationship between Epsilon's at different times.

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Now it's possible to derive the theoretical akef, but this is not necessary to appreciate the results.

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The result is that for simple cases like Arche one, you'll see a geometric decay in the theoretical

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HCF.

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One way to make use of this fact is that if we see something similar in the sample akef, then we know

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that the arch of the Gurche might be a good model.

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OK, so considering these properties, does using Arch to model stock returns makes sense?

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I think the answer is yes.

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Let's consider the fact that the unconditional Menagh zero.

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This makes sense because as we've seen, when we take the mean of stock returns, they are usually very

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close to zero.

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Of course, if they weren't, it would still be subtracted when we fit in a Arima or any other model

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of the mean return.

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How about the fact that the conditional means zero?

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This also makes sense because as we've seen, even despite knowing past returns, the best guess for

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the current return is still typically very close to zero.

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OK, so how about the fact that the Akef shows no correlation with the legs?

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This makes sense because when we look at the act of stock returns, this is precisely what we see.

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In fact, this is why we can think of stock prices as following a random walk.

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And how about the fact that the AKF does show correlation when we look at the squared returns?

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Again, this makes sense.

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We've seen volatility clustering where the magnitude of the return does, in fact depend on past return

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magnitudes.

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We haven't yet had a chance to show this in code, but we will very soon.

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OK, so considering all these facts, I hope you can agree that Arche seems to be a pretty nice fit

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for financial returns.