Analysis of Financial Time Series, Third Edition

3.4 The ARCH Model

The first model that provides a systematic framework for volatility modeling is the ARCH model of Engle (1982). The basic idea of ARCH models is that (a) the shock at of an asset return is serially uncorrelated, but dependent, and (b) the dependence of at can be described by a simple quadratic function of its lagged values. Specifically, an ARCH(m) model assumes that

(3.5)

where {ϵt} is a sequence of independent and identically distributed (iid) random variables with mean zero and variance 1, α0 > 0, and αi ≥ 0 for i > 0. The coefficients αi must satisfy some regularity conditions to ensure that the unconditional variance of at is finite. In practice, ϵt is often assumed to follow the standard normal or a standardized Student-t or a generalized error distribution.

From the structure of the model, it is seen that large past squared shocks imply a large conditional variance for the innovation at. Consequently, at tends to assume a large value (in modulus). This means that, under the ARCH framework, large shocks tend to be followed by another large shock. Here I use the word tend because a large variance does not necessarily produce a large realization. It only says ...