4.9 GARCH Processes
Engle (1982) proposes autoregressive conditional heteroskedastic (ARCH) processes. These are univariate conditionally heteroskedastic white noises. An ARCH(q) process W has conditional distribution
[4.72]
[4.73]
where, as explained in Section 4.6.1, the “t – 1″ over the tilde in [4.72] indicates that the distribution is conditional on information available at time t – 1.
Bollerslev (1986) extends the model by allowing t | t–1σ2 to also depend on its own past values. His generalized ARCH, or GARCH(p,q), process has form
[4.74]
[4.75]
See Hamilton (1994) for stationarity conditions. In applications, GARCH(1,1) processes are common. Exhibit 4.17 indicates a realization of the GARCH(1,1) process
[4.76]
[4.77]
GARCH processes are often estimated by maximum likelihood.
There have been many attempts to generalize GARCH models to multiple dimensions. Attempts include:
- the vech and BEKK models of Engle and Kroner (1995),
- the CCC-GARCH of Bollerslev (1990),
- the orthogonal GARCH of Ding (1994), Alexander and Chibumba (1997), and Klaassen (2000), and
- the DCC-GARCH of Engle (2000), and Engle and Sheppard (2001).
With some of these approaches, the number of parameters that must be specified becomes unmanageable as dimensionality n increases. With some, estimation requires considerable user intervention or entails other challenges. Some require assumptions that are difficult to reconcile with phenomena to be modeled. This is an area of ongoing research. To illustrate general techniques, we present two of the above models.