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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.

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