Session 9 Additional exercises

9.1 Exercises

Exercise 1

Describe the behaviour of the ACF and PACF for ARMA models. How does the behaviour of the ACF/PACF help identifying an ARMA model?

Exercise 2

Show that the following AR(2) model is stationary:

\[ y_t = 0.6 y_{t-1} - 0.08 y_{t-2} + \varepsilon_t \]

Exercise 3

Show that the kurtosis of an ARCH(1,1) process is

\[ \mathbb{E}[\sigma_t^4] = \sigma^4(1+\alpha^2) - \alpha \mu_4 \sigma^4 \frac{1-\alpha^2}{1-\alpha^2 \mu_4} \]

Exercise 4

Consider the process \[ y_t = \delta + \sum_{s=1}^t \varepsilon_s + x_t \]

where \(\varepsilon_t\) is i.i.d. and \(x_t\) is a covariance-stationary process with \(\mathbb{E}[x_t] = 0\). Show that \(y_t\) is an \(I(1)\) process.

Exercise 5

Let \(x_t\) be a covariance-stationary process. Show that the first difference \(\Delta x_t\) is also covariance stationary.

Exercise 6

Derive the optimal \(h\)-step ahead forecast for the conditional mean using an AR(1) model.

Exercise 7

Derive the MA(\(\infty\)) representation of an AR(1) model.