This is the fourth part of the series on volatility modelling. For other parts of the series follow the tag volatility.

In this exercise set we will explore GARCH-M and E-GARCH models. We will also use these models to generate rolling window forecasts, bootstrap forecasts and perform simulations.

Answers to the exercises are available here.

Exercise 1
Load the rugarch and the FinTS packages. Next, load the m.ibmspln dataset from the FinTS package. This dataset contains monthly excess returns of the S&P500 index and IBM stock from Jan-1926 to Dec-1999 (Ruey Tsay (2005) Analysis of Financial Time Series, 2nd ed. ,Wiley, chapter 3).
Also, load the forecast package which we will use for autocorrelation graphs.

Exercise 2
Estimate a GARCH(1,1)-M model for the S&P500 excess returns series. Determine if the effect of volatility on asset returns is significant.

Exercise 3
Excess IBM stock returns are defined as a regular zoo variable. Convert this to a time series variable with correct dates.

Exercise 4
Plot the absolute and squared excess IBM stock returns along with its ACF and PACF graphs and determine the appropriate model configuration.

Exercise 5
The exponential GARCH model incorporates asymmetric effects for positive and negative asset returns. Estimate an AR(1)-EGARCH(1,1) model for the IBM series.

Exercise 6
Using the results from exercise-5, get rolling window forecasts starting from the 800th observation and refit the model after every three observations.

Exercise 7
Estimate an AR(1)-GARCH(1,1) model for the IBM series and get a bootstrap forecast for the next 50 periods with 500 replications.

Exercise 8
Plot the forecasted returns and sigma with bootstrap error bands.

Exercise 9
We can use Monte-Carlo simulation to get a distribution of the parameter estimates. Using the fitted model from exercise-7, run the simulation for 500 periods for a horizon of 2000 periods.

Exercise 10
Plot the density functions of the parameter estimates.