GARCH Toolbox

The Default Model

The GARCH Toolbox default model is the simple (yet common) conditional mean model with GARCH(1,1) Gaussian innovations, based on Eq. (2-8) and Eq. (2-9).

(2-12)

(2-13)

In the conditional mean model, Eq. (2-12), the returns, y_t, consist of a simple constant, plus an uncorrelated, white noise disturbance, _t. This model is often sufficient to describe the conditional mean in a financial return series. Most financial return series do not require the comprehensiveness that an ARMAX model provides.

In the conditional variance model, Eq. (2-13), the variance forecast, _t², consists of a constant plus a weighted average of last period's forecast, _t-1², and last period's squared disturbance, _t-1². Although financial return series, as defined in Eq. (2-10) and Eq. (2-11), typically exhibit little correlation, the squared returns often indicate significant correlation and persistence. This implies correlation in the variance process, and is an indication that the data is a candidate for GARCH modeling.

Although simplistic, the default model shown in Eq. (2-12) and Eq. (2-13) has several benefits:

• It represents a parsimonious model that requires you to estimate only four parameters (C, , G₁, and A₁). According to Box and Jenkins [7], the fewer parameters to estimate, the less that can go wrong. Elaborate models often fail to offer real benefits when forecasting (see Hamilton [12], page 109).
• The simple GARCH(1,1) model captures most of the variability in most return series. Small lags for P and Q are common in empirical applications. Typically, GARCH(1,1), GARCH(2,1), or GARCH(1,2) models are adequate for modeling volatilities even over long sample periods (see Bollerslev, Chou, and Kroner [5], pages 10 and 22).

Conventions and Clarifications

Analysis and Estimation Example Using the Default Model