When we estimated the GARCH model with normal errors, we run some diagnostic tests. We would do the same diagnostic test here. Under this second specification of the GARCH model, the fitted values of Epsilon_t, which we call Zts, should have the following statistical property. It's mean should be 0. Its standard deviation should be 1. It's skewness should also be 0, and now it's kurtosis should be 3 plus 6 divided by the quantity Nu minus 4, where Nu is the estimated parameter. In this case, Nu was estimated to be 6.13. So that gives us the kurtosis of 5.08. The actual values of these four statistics are in black on the slide. The mean is close to 0. The standard deviation is close to 1. However, as in the GARCH model with normal errors, the Zts here are also left-skewed, and they have a kurtosis larger than three. Now here we do not need to run the Jarque-Berra test for normality since we already know that the t distribution is not normal. Here is the second set of diagnostic tests. The graph on the left has the autocorrelation coefficients of the fitted Epsilons. It shows no evidence of serial correlation. The graph on the right has the autocorrelation coefficients of the absolute values of the fitted Epsilons. It also shows no evidence of volatility clustering. Let's quickly review what the two sets of diagnostic tests tell us. The GARCH 1-1 model with t distribution can explain volatility clustering in the data and also most of the kurtosis. The one aspect of the data that we have not explained is the left skewness. Now it is possible to further extend the GARCH model to model the left skewness in our data. For example, we can add another parameter to the t distribution that allows it to become a skewed distribution. In fact, there are many, many extensions of the GARCH model to pick up different statistical features of the data. But the math gets very, very ugly so we will not go any further. As you will see later, the GARCH 1-1 model with t errors turns out to be a reasonably good model for our data. Let's take a few minutes to explore some of the output from the estimated GARCH model. Remember in the safe one variable, there is a second column labeled S, which is the standard deviation of the log return each day. Here is the graph of these fitted volatilities. Now in drawing this graph, I have multiplied the fitted volatility, S, by the square root of 252. This annualizes this volatility in the same way we annualize interest rates. As you can see, the fitted volatility changes over time. It jumped to higher than 75 percent in October of 1987, and it did so again in 2008 during the great recession. So the graph clearly tells us that volatility is changing over time. That means value at risk and expected shortfall are also changing over time. When the fitted volatility is high, the value at risk and expected shortfall are also going to be high. When the fitted volatility is low, the value at risk and expected shortfall are also going to be low. Now look closely at the end of the sample period. The fitted volatility is quite low, certainly lower than the average of the sample. What does that mean? I hope you're thinking what I'm thinking. The value at risk and expected shortfall at the end of 2017 are going to be low relative to the overall sample. Remember this point. We will come back to it very soon. Now as a reality check, we can compare the fitted volatility from the GARCH model with the implied volatility of options on the S&P 500 Index. This series is called VIX or volatility index. Let me explain what this comparison is opposed to tell us. This is an index based on options on the S&P 500 Index. The VIX is the forecast of volatility over the next 30 days. In this graph, we plot the log of x on the x-axis and the log of fitted volatility on the y-axis. If they are forecasting the same thing, all the points should be on the 45-degree line, which is the red line in the graph. Well, there are many points that are not under red line, so the VIX and the fitted volatility from the GARCH model are not exactly the same thing. But the points do lie along the red line. This means that the VIX and the fitted volatility are strongly positively correlated with each other. When the VIX is high, the fitted volatility is also high. When the VIX is low, the fitted volatility is also low. There is another interesting aspect of this graph. Most of the points are below the 45-degree red line. This says that the VIX is generally larger than the fitted volatility. This tells us that options on the S&P tends to over-predict future volatility relative to the GARCH model. That will be true, however, if option bias are paying an additional premium beyond the predicted volatility. To tell you the truth, I'm delighted to see the high degree of correlation between the VIX and the fitted volatility. It gives me quite a bit of assurance that the GARCH model is a sensible model for volatility.