If we are using the ARCH Model with high order, then it can be a good idea to use the GARCH Model. We can compute ARCH(1) und ARCH(10), and the second one would be the precise one, but the computation is x10 longer.

We can use some GARCH(m, s) model, that takes not only the last m errors, but also the last s volatilities. The precision of the GARCH(2, 1) is closer to ARCH(10)

The ARCH/GARCH model then can find serial correlation in the volatility and capture it by the parametrization of the variance equation.

Heteroscedastic return decomposition

$$ r_t = \mu_{t-1} + \epsilon_t; \quad \epsilon_t \equiv \sigma_{t-1} \, \times \, \eta_t; \; \eta_t \sim iid(0,1). $$

Variance Equation

$$ \sigma^2_{t-1} \overbrace{:=}^{GARCH(m,s)} \alpha_0 + \sum_{i=1}^m \alpha_i \, \times \, \epsilon^2_{t-i} \, + \, \sum_{j=1}^s \beta_j \, \times \, \sigma^2_{t-1-j}, \;with \; \\\\ \alpha_0 > 0, \alpha_i \geq 0, \forall i \in [1,...,m], \beta_i \geq 0, \forall j \in [1,...,s] and \sum_{i=1}^{max(m,s)} (\alpha_i + \beta_i) < 1 $$

We are using more information to calculate the volatility $\sigma_{t-1}$:

• $\alpha_0$ and the $\alpha_1$$...\alpha_m$ for GARCH(m, s) is the ARCH(m) part
• $\beta_1...\beta_s$ is the parameters of the information about previous volatilities

GARCH(m,s) Parametric Model

$$ \begin{align*}r^M_t &= \phi_0 + \phi_1 r^M_{t-1} + ... + \phi_M r^M_{t-p} + \theta_1 \epsilon_{t-1} + ...+\theta_q\epsilon_{t-q} \; + \; \epsilon_t; \; \\\\ & \text{with} \; \epsilon_t \sim N (0,\sigma^2_{t-1}) \; \text{and} \\\\\sigma^2_{t-1} &= \alpha_0 + \sum_{i=1}^m \alpha_i \epsilon^2_{t-i} \, + \, \sum_{j=1}^s \beta_j \sigma^2_{t-1-j}.\end{align*} $$

Joint log-likelihood

$$ \begin{align*}ln L(.) &= \sum_{t=1+max(p,q,m,s)}^T -\frac{1}{2} \ln (2\pi) - \frac{1}{2} \ln ( \sigma^2_{t-1} ) - \frac{1}{2} \, \frac{\epsilon^2_t}{ \sigma^2_{t-1} } \quad \text{with} \; \\\\\epsilon_t & \equiv r^M_t - \phi_0 - \phi_1 r^M_{t-1} - ... - \phi_p r^M_{t-p} - \theta_1 \epsilon_{t-1} - ... - \theta_q \epsilon_{t-q}, \; \text{and} \\ \\\sigma^2_{t-1} & \equiv \alpha_0 + \sum_{i=1}^m \alpha_i \, \times \, \epsilon^2_{t-i} \, + \, \sum_{j=1}^s \beta_j \, \times \, \sigma^2_{t-1-j} \quad \text{for} \, t>1 \\ \\& \text{and} \; \sigma^2_{0} \equiv \frac{\alpha_0}{ 1 - \sum_{i=1}^m \alpha_i - \sum_{j=1}^s \beta_j}.\end{align*} $$

E-GARCH (Exponential GARCH)

Negative return shocks tend to have bigger impact on volatility then the positive return shocks (Leverage effect)