Autoregressive conditional heteroskedasticity

An ARCH model ( autoregressive conditional heteroscedasticity ) is a stochastic model for time series analysis, with the aid of particular mathematical financial time series can be described by a non-constant volatility. It starts from the assumption that the conditional variance of the random model error depends on the realized random error of the previous period, so that large and small errors tend to occur in groups. ARCH models have been developed by Robert F. Engle in the 1980s. In 2003 he won the Nobel Prize in Economics was awarded for this achievement.

Definition

A time series is ARCH ( P) time-series, if it is defined recursively by

Being with real, non-negative parameters, and the process is a so-called white noise, that is, and for all.

Properties

For ARCH models, the following statements apply:

• The conditional on the past and expected values ​​conditional variances are:

• An ARCH ( p) time series is exactly then (weakly ) stationary if all roots of the characteristic polynomial

• A stationary ARCH ( p) - time series has stationary expected value and their autocorrelation disappears: for. The formula applies to their stationary variance

• Is a stationary ARCH ( p) - time series, applies to the, then the squared process is an AR time series.

Generalizations

The idea of ​​the ARCH model was further developed in various ways and is now quite naturally to the more advanced methods of econometrics.

A generalization of the GARCH models ( generalized autoregressive conditional heteroscedasticity ), which were developed in 1986 by Tim Bollerslev. Here, the conditional variance depends not only on the history of the time series, but also from their own past.