GARCH

A GARCH process is similar to an EWMA (exponentially weighted moving average) used for predicting volatility (rather variance). In most cases the process used is GARCH(1,1). GARCH(p,q) means that we calculate the variance from p observations of u² and the most recent predictions of the variance σ².

$\sigma^2_t = \gamma V_L + \alpha u^2_{t-1} + \beta \sigma^2_{t-1}$

where $\gamma + \alpha + \beta = 1$

V_L is the long-run average variance

the coefficients α, β and γ assign weights to the predictors.

Often the formula is written as

$\sigma^2_t = \omega + \alpha u^2_{t-1} + \beta \sigma^2_{t-1}$

since $\gamma = 1 - \alpha - \beta$ we must ensure that $\alpha + \beta < 1$ , otherwise ω is negative.

β can be interpreted as the decay rate of the contribution of previous values of the variance. The model is mean reverting.

GARCH(p,q)

(from the book Introductory Econometrics for Finance by Chris Brooks)

$\sigma^2_t = \alpha_0 + \sum_{i=1}^{q} \alpha_iu^2_{t-i} + \sum_{j=1}^{p} \beta_j \sigma^2_{t-j}$

Estimation of GARCH models using maximum likelihood

Given the parameters (α, β, ω) of the GARCH model we estimate how well it produced the observed data.