 Hi everyone, it's MJ. And in this video, I want to give an overview of the generalized autoregressive conditional heteroscedastic time series model. And what we're going to be doing in this video is just talking about the very simple case where we have one and one. You can make these things a lot more complicated by increasing your parameters over here. But the logic kind of follows the math just gets a little bit messier. So what I mean by one by one, we have the following, the pink one is referring to the alpha here, and this x t minus one squared. This is the autoregressive part of the Gauss time series. And this other blue one over here, this parameter is saying, well, how much of the past variance we're looking at. So essentially what we have, we can show it over here. And our w symbol over here is our intercept. And all these values need to be greater than zero. But if we want our cock to be stable, we need the alpha plus the beta to be less than one. Now, essentially, what we're doing or what we're using this this model for is to measure volatility, specifically, this idea of volatility clustering. If you had to use a moving average to measure volatility, which is sometimes done, then volatility is going to be flat, you expect the volatility is going to be flat. However, what this model does, it allows for volatility clustering, which means sometimes, if there's a big movement, it's going to be followed by a big movement. If there's a small movement, it's going to be followed by a small movement. And there's going to be periods of this clustering. Now, it's important to know that this is the backwards or historical approach. And in another video, we're going to look at the forward or implied approach where you use the black skulls model. So this is a different technique. But anyway, coming back to our little model over here, you can see we're measuring volatility. We've got these two parameters. The alpha is kind of referring to the volatility's spikiness. You know, how quick are its reactions? And we're seeing now with the coronavirus, we're seeing a lot of spiky spikiness. And the whole, it just shows how crazy the volatility is. You can think of it as this crazy time series going up and around. The beta here is the volatility's persistence. You know, how long does it take to change? Now, like I said, these two values added together need to be less than one, and they both need to be greater than zero. And in normal market times, we normally see a 0.25 with alpha and a 0.7 with the beta. What I've actually got in the time series Udemy course is we do look at how we get to this GARC model and why it is the way it is. Like we actually derive it and we show why it's better than the moving average, specifically flat volatility versus volatility clustering. But what I want to do in this video is look more to the long term variants. So like I said, if W is the intercept, and if it's positive, what it's going to tell us is the volatility will be mean reverting, which means we're going to get a long term or a long run variance, which is going to be equal to this following formula over here. And like I said, in the time series course, we derive this formula. We talk about, you know, the exponential weightings of the auto regressive models and how we actually figure it out. What I want to do in this video is just look more at this long run variance and look at some of the maths here. So what we're going to be doing is very quickly looking at some of the maths. So we're saying, this is just our GARC formula over there. Let's maybe talk very quickly through this maths here. Essentially, what we're doing is we're replacing this W intercept with the one minus alpha minus beta, which is what we're doing over there. In the second line, we're just expanding that out. So we're getting V times one, and then we're subtracting it on our left hand side. And then we're going to see alpha V can be combined with this term over here. And that's how we're getting alpha XT minus one squared minus V plus beta sigma squared T minus one minus V. Then what we do is we do a little, we clean up the maths a little bit. And we accept this term that the expected value of our X squared T plus K minus one is equal to sigma squared N plus K minus one. And now if we take our expected values, we can kind of then join the alpha and the beta terms together. And then what we can do is we can then rearrange the maths back to get to the situation where the expected value of sigma squared T plus K is going to be equal to V plus alpha plus beta to the power of K times sigma squared T minus V. And this is the whole idea is since alpha plus beta is less than one, the second term is going to fall away as K gets larger. So as K gets larger, this area here is going to become zero, which means the expected value of sigma squared in the future is going to be equal to V. And this is the power of the Gauss model is once we fit this Gauss model to our time series, then we can use these three parameters to calculate what is the long run variant. And this long run variance is something that we can either use in our long term models, or if you want to have a more stochastic model, you could then even use the sigma squared where it's constantly fluctuating. But like I said, we talked more about the Gauss model in the Udemy time series course. What I just wanted to show you here is how you can use the Gauss model to figure out your variance, which like we said was very, very important when it comes to your models in market risk. So anyway, thanks guys so much for watching. And well, yeah, let me maybe show it to you here. This is kind of where we will now use our variance is in this ITER process model. And this is like I said, the backward looking approach using Gauss. Anyway, thanks so much for watching. In the next video, we're going to be looking more at the implied volatility aspect.