Definition of ARCH

Definition: ARCH stands for Autoregressive Conditional Heteroskedasticity. It is a technique used in finance to model asset price volatility over time. It is observed in much time series data on asset prices that there are periods when variance is high and periods where variance is low. The ARCH econometric model for this (introduced by Engle (1982)) is that the variance of the series itself is an AR (autoregressive) time series, often a linear one.

Formally, per Bollerslev et al 1992 and Engle (1982):

An ARCH model is a discrete time stochastic process {e_t} of the form:

e_t = z_ts_t

where the z_t's are iid over time, E(z_t)=0, var(z_t)=1, and s_t is positive and time-varying. Usually s_t is further modeled to be an autoregressive process.

According to Andersen and Bollerslev 1995/6/7, "ARCH models are usually estimated by maximum likelihood techniques." They almost always give a leptokurtic distrbution of asset returns even if one assumes that each period's returns are normal, because the variance is not the same each period. Even ARCH models, however, do not usually generate enough kurtosis in equity returns to match U.S. stock data.(Econterms)

Terms related to ARCH:

• Leptokurtic
• AR

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Books on ARCH:
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Journal Articles on ARCH:

• Engle, R.F.. 1982. "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation."
  Econometrica. 50: 987-1008.