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

In this exercise set we will use the `dmbp`

dataset from part-1 and extend our analysis to GARCH (Generalized Autoregressive Conditional Heteroscedasticity) models.

Answers to the exercises are available here.

**Exercise 1**

Load the `rugarch`

package and the `dmbp`

dataset (Bollerslev, T. and Ghysels, E. 1996, Periodic Autoregressive Conditional Heteroscedasticity, Journal

of Business and Economic Statistics, 14, 139–151). This dataset has daily logarithmic nominal returns for Deutsche-mark / Pound. There is also a dummy variable to indicate non-trading days.

Define the daily return as a time series variable.

**Exercise 2**

We will first simulate and analyze a GARCH process. Use the `ugarchspec`

function to define an GARCH(1,1) process. The return has a simple mean specification with mean=0. The variance follows as AR-1 process with constant=0.2, AR-1 coefficient = 0.2 and MA-1 coefficient = 0.6.

**Exercise 3**

Simulate the defined GARCH process for 500 time periods. Exercises 4 to 6 use this simulated data.

**Exercise 4**

Plot the returns vs time and note the apparent unpredictability. Plot the path of conditional sigma vs time and note that there is some persistence over time.

**Exercise 5**

Plot the ACF of returns and squared returns. Note that there is no auto-correlation between returns but squared returns have significant first degree autocorrelation.

**Exercise 6**

Test for ARCH effects using the Ljung Box test for the simulated data.

**Exercise 7**

We will now use the currency returns data. Use the `ugarchfit`

function to estimate a GARCH(1,1) model for the data. The return has a simple mean specification with mean=0.

**Exercise 8**

Plot the fit diagnostics graphs.

**Exercise 9**

Fit an AR(1)-GARCH(1,1) model to the data.

**Exercise 10**

Plot the fit diagnostics graphs for the new model.

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