Okay. Let's recall what we have learned about the ordering of our data. We saw that the log returns of the Wilshire 5,000 index does not have serial correlation. That means, it is difficult to predict the mean of future returns. However, we also saw that the log returns of the Wilshire 5,000 index have strong volatility clustering. In other words, volatility which means the same thing as the standard deviation of returns is changing over time in a predictable manner. What we need is a statistical model of volatility, which allows us to predict the future volatility or standard deviation of returns. I have already mentioned that Robert Engle proposed the ARCH model in 1982, which won him then 2003 Nobel Prize in Economics. Remember what ARCH stands for. It stands for, Autoregressive Conditional Heteroscedasticity. Engle student, Tim Bolleraslev extended the ARCH model to the generalized ARCH model, or GARCH model In 1986. The GARCH model has many extensions done by many other researchers. We will use the simplest GARCH model, called GARCH (1,1). As you will see, this simple GARCH model does a very good job of modeling the volatility clustering in our data. Now, here's the model that is called GARCH (1,1)-normal model. Let me take some time to explain what this model is. There are three equations in this GARCH model. Let me write down all three equations, and I'll explain the notation and how they work. The first equation on this slide is called the mean equation. The second equation is the variance equation. The third equation is the distribution equation. Now, let me tell you what the notation is and what the equations mean. R sub t is the return series that has time-varying volatility. The mean equation says that r sub t is the sum of two parts: The first part, a sub zero is the expected return of this return series. The second part is the unexpected return, which is the product of two variables. The square root of h sub t multiply two epsilon sub t. H sub t is the variance of r sub t. It changes over time, according to the variance equation. In each period t, the variance h sub t is the sum of three terms. The first term, alpha sub zero is a constant number. This is the smallest possible variance you will see. The second term is beta sub one multiplied to the variance of period t minus one. The third term in the variance equation is also a product of a constant term, alpha sub one and the random variable epsilon, but from the last period. In that distribution equation, epsilon is a standard normal variable. Now, let me highlight the variance equation in red, and spend a few minutes to explain how this equation can give us time-varying volatility, which is predictable. First, I want to explain what makes variance change over time. Can you guess it? A culprit, if you want to call it that, is the square of epsilon t minus one in the variance equation. Remember, epsilon is a random variable which is different every period. As epsilon changes over time, it is causing the variance to change over time as well. Next, let me explain why the variance equation gives us volatility clustering. To get volatility clustering, I need beta sub one and alpha sub one to be positive numbers. What happens when we get a large epsilon in period t minus one. By the way, it doesn't matter whether epsilon t minus one is large and positive or large and negative, we just need the square of epsilon t minus one to be large. If alpha sub one is positive, then a large epsilon t minus one, last period will make the variance, this period h sub t large. Now, let's see what happens next period, time t plus one. Remember, today's variance, h sub t is large already. When beta sub one is positive, that is going to make the variance next period h sub t plus one large as well. This gives us a volatility clustering. I will give you a numerical example in a few minutes. Let me show you that the GARCH (1,1) model ness the constant variance model as a special case. Remember, the constant variance model is what we have been assuming implicitly this far. So, assume beta sub one and alpha sub one are both zero in the variance equation. In that case, the GARCH model now becomes the following: the mean equation is the same as before, that doesn't change. The variance equation now has only one term. Distribution equation is also the same as before. I claim that this is exactly the normal model of log returns, which is the first statistical model of return that we worked with. Can you see why this is the case? So, let me recall what the normal model of log return was, and here it is. The return is a constant term mu, which is the mean return plus the product of sigma, which is the standard deviation of return times epsilon, a random variable with the standard normal distribution. Is it more clear now? Well, mu is just a sub zero, and sigma is just the square root of alpha sub zero. So, the two models are identical. It is nice that the GARCH (1,1) model ness the constant variance model as a special case. So, if the volatility of our data does not change over time, we will find that out when we estimate the GARCH (1,1) model. Now as I have promised a couple of slides earlier, let me pick a simple example to show you why the GARCH (1,1) gives us volatility clustering. For this demonstration, I will set beta sub one to be one-half, and alpha sub one also to be one-half. The GARCH model now becomes the following. Again, the mean equation is the same as before. The variance equation still has three terms and I have replaced beta sub one with one-half and alpha sub one with one-half. The distribution equation is also the same as before. Now, suppose that in period one, epsilon one is large and negative, say, minus two. The square of epsilon one then equals four. The variance equation tells us that the variance in period two is alpha sub zero plus one-half the variance in period one plus two. Whatever the variance in period one is, the variance in period two will be unusually large because of the third term, which is two. A large epsilon in one period gives rise to a large variance in the following period. We can see what happens to the variance in period three. Using the variance equation again, we see that the variance in period three which is h sub three is alpha sub zero plus one-half the variance in period two plus some other terms which are all positive. If the variance in period two is unusually large, then the variance in period three is also going to be unusually large. This is what gives rise to volatility clustering. Now, let's estimate the GARCH model for our data.