ARCH and GARCH Processes

Explained: autoregressive conditional heteroskedasticity (ARCH) generalized ARCH (GARCH)

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):

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

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.

Related Internal Links
autoregressive moving-average process A type of stochastic process. autoregressive process A type of stochastic process. heteroskedasticity A condition where a stochastic process has non-constant second moments. moving-average process A type of stochastic process. random walk A discrete stochastic process whose increments form a white noise. stochastic volatility model A category of conditionally heteroskedastic stochastic processes. time series and stochastic processes An introductory article. volatility A metric of  variability in a stochastic process. volatility clustering A property of some stochastic processes that they experience periods of high and low variance. volatility skew A condition where implied volatilities vary by strike. white noise A simple form of stochastic process.
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Cited Papers
Alexander, Carol O. and A. M. Chibumba (1997). Multivariate orthogonal factor GARCH, working paper. Bollerslev, Tim (1986). Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307-328. Bollerslev, Tim (1990). Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model, Review of Economics and Statistics, 72, 498-505. Ding, Z. (1994). Time series analysis of speculative returns, PhD thesis, San Diego: University of California. Engle, Robert F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica, 50, 987-1008. Engle, Robert F. (2000). Dynamic conditional correlation—A simple class of multivariate GARCH models, working paper. Engle, Robert F. and K. F. Kroner (1995). Multivariate simultaneous generalized ARCH, Econometric Theory, 11, 122-150. Engle, Robert F. and Kevin Sheppard (2001). Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH, working paper. Klaassen, Franc (2000). Have exchange rates become more closely tied? Evidence from a new multivariate GARCH model, working paper.
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