Analysis of Financial Time Series, Third Edition

3.5 The GARCH Model

Although the ARCH model is simple, it often requires many parameters to adequately describe the volatility process of an asset return. For instance, consider the monthly excess returns of S&P 500 index of Example 3.3. An ARCH(9) model is needed for the volatility process. Some alternative model must be sought. Bollerslev (1986) proposes a useful extension known as the generalized ARCH (GARCH) model. For a log return series rt, let at = rt − μt be the innovation at time t. Then at follows a GARCH(m, s) model if

(3.14)

where again {ϵt} is a sequence of iid random variables with mean 0 and variance 1.0, α0 > 0, αi ≥ 0, βj ≥ 0, and . Here it is understood that αi = 0 for i > m and βj = 0 for j > s. The latter constraint on αi + βi implies that the unconditional variance of at is finite, whereas its conditional variance evolves over time. As before, ϵt is often assumed to follow a standard normal or standardized Student-t distribution or generalized error distribution. Equation (3.14) reduces to a pure ARCH(m) model if s = 0. The αi and βj are referred to as ARCH and GARCH parameters, respectively.

To understand properties of GARCH models, it is informative to use the ...