On Black Monday, the return on the S&P 500 was −22.8 %. Ouch! This exercise attempts to answer the question, “what was the conditional probability of a return this small or smaller on Black Monday?” “Conditional” means given the information available the previous trading day. Run the following R code:
The S&P 500 returns are in the data set SP500 in the Ecdat package. The returns are the variable r500 (this is the only variable in this data set). Black Monday is the 1805th return in this data set. This code fits an AR(1)+GARCH(1,1) model to the last two years of data before Black Monday, assuming 253 trading days/year. The conditional distribution of the white noise is the t-distribution (called "std" in ugarchspec()). The code also plots the returns during these two years and on Black Monday. From the plot you can see that Black Monday was highly unusual. The parameter estimates are in coef(fit) and the sixth parameter is the degrees of freedom of the t-distribution. The ugarchforecast() function is used to predict one-step ahead, that is, to predict the return on Black Monday; the input variable n.ahead specifies how many days ahead to forecast, so n.ahead=5 would forecast the next five days. The object forecast will contain fitted(forecast), which is the conditional expected return on Black Monday, and sigma(forecast), which is the conditional standard deviation of the return on Black Monday.
(a) Use the information above to calculate the conditional probability of a return less than or equal to −0.228 on Black Monday.
(b) Compute and plot the standardized residuals. Also plot the ACF of the standardized residuals and their squares. Include all three plots with your work. Do the standardized residuals indicate that the AR(1)+GARCH(1,1) model fits adequately?
(c) Would an AR(1)+ARCH(1) model provide an adequate fit?
(d) Does an AR(1) model with a Gaussian conditional distribution provide an adequate fit? Use the arima() function to fit the AR(1) model. This function only allows a Gaussian conditional distribution.