So the real interesting thing about Gargash is not just plugging in some data into the code, but it's
really understanding why this is a good model.
So in this lecture, we're going to look at a few more properties of the arch one.
And this should give you more confidence that everything makes sense.
Now, as mentioned, that this section won't be focused on deriving these properties.
Instead, it's the properties themselves which are of interest.
In other words, simply looking at them and understanding what they mean is useful.
So the first property we want to look at is that the conditional meaning of Epsilon is zero.
That is given the past epsilon's the expected value of the next steps.
The line is always zero.
This makes sense since we usually expect to have zero centered noise despite knowing the past values
of the noise.
Another similar but different fact is that the unconditional mean of Epsilon is also zero.
That is to say, if we do not know anything about the past, the expected value of epsilon is still
zero.
This also makes sense.
This means that no matter what, whether we know the past epsilons or not, the expected value of the
next epsilon is always zero.
The next interesting property is that the covariance between any epsilon and any Epsilon at some other
point in time is zero.
So here are the first Epsilon is at time T and the other Epsilon is at time T minus S.
So as long as S is not zero, then these are two different epsilon's at two different points in time.
When this is the case, there are covariance is zero.
And as you recall, correlation is just covariance divided by some constant and thus the correlation
between any two epsilon's is also zero.
This makes sense since as you recall, when we look at the akef of log returns, they are in fact zero.
This model, therefore, seems to be accurate for modeling log returns, at least in this respect.
Now, one important fact to remember from your study of probability is this it's that absence of correlation
does not imply independence.
As you recall, if two random variables are independent, then they are uncorrelated, but the reverse
is not true.
So you might wonder, does this fact have any practical use?
And the answer is yes, because we're seeing an example at this moment.
The garden model is one example of where the epsilons are uncorrelated, but they are not independent.
And of course, we know that they are not independent because we have an equation relating past Epsilon's
to future epsilons.
OK, so as you recall, one way we can show that the epsilons are not independent is to plot the akef
of the squares.
When we do this, we see that there is, in fact, a relationship between Epsilon's at different times.
Now it's possible to derive the theoretical akef, but this is not necessary to appreciate the results.
The result is that for simple cases like Arche one, you'll see a geometric decay in the theoretical
HCF.
One way to make use of this fact is that if we see something similar in the sample akef, then we know
that the arch of the Gurche might be a good model.
OK, so considering these properties, does using Arch to model stock returns makes sense?
I think the answer is yes.
Let's consider the fact that the unconditional Menagh zero.
This makes sense because as we've seen, when we take the mean of stock returns, they are usually very
close to zero.
Of course, if they weren't, it would still be subtracted when we fit in a Arima or any other model
of the mean return.
How about the fact that the conditional means zero?
This also makes sense because as we've seen, even despite knowing past returns, the best guess for
the current return is still typically very close to zero.
OK, so how about the fact that the Akef shows no correlation with the legs?
This makes sense because when we look at the act of stock returns, this is precisely what we see.
In fact, this is why we can think of stock prices as following a random walk.
And how about the fact that the AKF does show correlation when we look at the squared returns?
Again, this makes sense.
We've seen volatility clustering where the magnitude of the return does, in fact depend on past return
magnitudes.
We haven't yet had a chance to show this in code, but we will very soon.
OK, so considering all these facts, I hope you can agree that Arche seems to be a pretty nice fit
for financial returns.