ARCH and GARCH Processes

Autoregressive conditional heteroskedastic (ARCH) processes are a form of stochastic process that are widely used in finance and economics for modeling conditional heteroskedasticity and volatility clustering. First proposed by Engle (1982), ARCH processes are univariate conditionally heteroskedastic white noises. An ARCH(q) process X is defined by two interrelated formulas (see the notation conventions documentation):

	[1]
	[2]

where W is a standard normal Gaussian white noise. This means that the time t distribution of X, conditional on information available at time t–1, is normal, with constant mean 0 and a conditional variance that changes with time. Our notation indicates that is a variance at time t, but conditional on information available at time t–1. Formula [2] defines as a function of preceding values of X. Together, formulas [1] and [2] ensure that, if X takes on large positive or negative values at some point in time, its conditional variance will be elevated for subsequent points in time, thereby making it likely that X will also take on large positive or negative values at those times too. In this manner, an ARCH process models volatility clustering—periods of high or low volatility.

Bollerslev (1986) extended the model by allowing to also depend on its own past values. His generalized ARCH, or GARCH(p,q), process has form

	[3]
	[4]

See Hamilton (1994) for stationarity conditions. In applications, GARCH(1,1) processes are common. Exhibit 1 indicates a realization of the GARCH(1,1) process

	[5]
	[6]

GARCH(1,1) Process Exhibit 1

A realization of the GARCH(1,1) process defined by [5] and [6].

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.