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
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
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
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).