time series volatility

GARCH

For a given time series

{% X_1,X_2,...,X_n %}

The errors are calculated as

{% \epsilon_i = X_i - \mu %}

Then, a GARCH model specifies

{% \sigma_t^2 = \omega + \sum_{i=1}^q \alpha_i\epsilon_{t-i}^2 + \sum_{j=1}^p \beta _j \sigma_{t-j}^2 %}