So in this lecture, we will be studying Arch.
So why do we call this arch?
Well, Arch stands for Auto Regressive Conditional Heteros Cadastre City.
You can see how this might be a bit difficult to say.
Thus, we simply call it arch.
OK, so let's try to make sense of this name.
Firstly, we've seen that stock returns are heteros get stick.
This means that their variance changes over time.
This is clear because sometimes it's high, but sometimes it's low.
Compare this to completely random idy noise, which would be used to generate a random walk.
We would call this homo sadistic because the variance is the same over time.
Hetero means different.
Homo means same.
OK, so that's why it's hetero sadistic.
What about conditional?
Well, we can see that not only does the variance of the return a change over time, it seems to be
dependent on nearby values.
That is, when we see large returns, they tend to be surrounded by other large returns.
When we see small returns, they tend to be surrounded by other small returns.
So it would make sense to build a model where the current variance is conditioned on previous values.
Furthermore, since it's conditioned on the previous values of the same time series, it's also auto
regressive.
OK, so that explains why we call this auto regressive conditional heteros cadastre.
Basically, it means we're going to build a model for the variance of a time series which can change
over time and its value depends on previous values.
OK, so before we even begin to look at Gargash, let's recall how we typically model a time series.
Most models we build have this form.
We say that Y of T is equal to some function of past values of Y and some model parameters, plus some
noise epsilon of T.
OK, so normally Epsilon of T is assumed to be something like Gaussian white noise.
We typically assume that it's a zero and its variance is constant and that we cannot predict it because
of this.
Our attention typically turns to the first part of this equation, which is what I've called F.
Basically our job is to find theta, which determines that, well, in this section things are a bit
different.
Now we are not concerned with F, but rather we are concerned with Epsilon.
So all throughout this course we've been trying to build a model for F.
Now we are going to build a model for Epsilon.
So another way to think of this is that a time series can be modeled by two separate models are and
Gargash Arima is used to model the mean of the Time series.
This part is assumed to have zero variance, on the other hand, is used to model the variance of the
TIME series.
This part is assumed to have zero mean.
When you put these together, you get both the mean and the variance model by two separate components.
So another thing to consider is why should we build an auto regressive model to begin with?
We'd like to have a bit more evidence to motivate this decision.
So, as you recall, one tool that we use to determine the order of an auto regressive model is the
Akef or Sieff.
Now, recall what happens when you take the akef of stock returns.
We can think of a stock log returns as epsilon of T, since we've seen that in A1 process is often the
best Arima model for a stock price time series.
So if we fit in, I want to the log price then the epsilons or just the log returns.
OK, so what does the AKF look like?
Well, we've seen that there are zero statistically significant lags.
The same thing happens when we look at the PSA and also recall this is the same thing we would see with
completely random Gaussian white noise.
And as such, we've seen that stock returns are very close to a random walk.
But something interesting happens when we plot the Akef for the square of the log returns, when we
do this, we find that there are, in fact, statistically significant lags.
This suggests that the square of the log returns are predictable using pass values.
Now, at this point, you'll just have to take my word that this is true, but we will be demonstrating
this in code.
Otherwise, if you're eager to observe this yourself and as an exercise, please try it on your own.