In econometrics, AutoRegressive Conditional Heteroskedasticity (ARCH) models are used to characterize and model observed time series. They are used whenever there is reason to believe that, at any point in a series, the terms will have a characteristic size, or variance. In particular ARCH models assume the variance of the current error term or innovation to be a function of the actual sizes of the previous time periods' error terms: often the variance is related to the squares of the previous innovations.
Such models are often called ARCH models (Engle, 1982), although a variety of other acronyms are applied to particular structures of model which have a similar basis. ARCH models are employed commonly in modeling financial time series that exhibit time-varying volatility clustering, i.e. periods of swings followed by periods of relative calm.
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Suppose one wishes to model a time series using an ARCH process. Let denote the error terms (return residuals, with respect to a mean process) i.e. the series terms. These are split into a stochastic piece and a time-dependent standard deviation characterizing the typical size of the terms so that
where is a random variable drawn from a Gaussian distribution centered at 0 with standard deviation equal to 1. (i.e. ) and where the series are modeled by
and where and .
An ARCH(q) model can be estimated using ordinary least squares. A methodology to test for the lag length of ARCH errors using the Lagrange multiplier test was proposed by Engle (1982). This procedure is as follows:
If an autoregressive moving average model (ARMA model) is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH, Bollerslev(1986)) model.
In that case, the GARCH(p, q) model (where p is the order of the GARCH terms and q is the order of the ARCH terms ) is given by
Generally, when testing for heteroskedasticity in econometric models, the best test is the White test. However, when dealing with time series data, this means to test for ARCH errors (as described above) and GARCH errors (below).
Prior to GARCH there was EWMA which has now been superseded by GARCH, although some people utilise both.
The lag length p of a GARCH(p, q) process is established in three steps:
Nonlinear GARCH (NGARCH) also known as Nonlinear Asymmetric GARCH(1,1) (NAGARCH) was introduced by Engle and Ng in 1993.
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For stock returns, parameter is usually estimated to be positive; in this case, it reflects the leverage effect, signifying that negative returns increase future volatility by a larger amount than positive returns of the same magnitude.[1][2]
This model shouldn't be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.
Integrated Generalized Autoregressive Conditional Heteroskedasticity IGARCH is a restricted version of the GARCH model, where the persistent parameters sum up to one, and therefore there is a unit root in the GARCH process. The condition for this is
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The exponential general autoregressive conditional heteroskedastic (EGARCH) model by Nelson (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):
where , is the conditional variance, , , , and are coefficients, and may be a standard normal variable or come from a generalized error distribution. The formulation for allows the sign and the magnitude of to have separate effects on the volatility. This is particularly useful in an asset pricing context.[3]
Since may be negative there are no (fewer) restrictions on the parameters.
The GARCH-in-mean (GARCH-M) model adds a heteroskedasticity term into the mean equation. It has the specification:
The residual is defined as
The Quadratic GARCH (QGARCH) model by Sentana (1995) is used to model symmetric effects of positive and negative shocks.
In the example of a GARCH(1,1) model, the residual process is
where is i.i.d. and
Similar to QGARCH, The Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model by Glosten, Jagannathan and Runkle (1993) also models asymmetry in the ARCH process. The suggestion is to model where is i.i.d., and
where if , and if .
The Threshold GARCH (TGARCH) model by Zakoian (1994) is similar to GJR GARCH, and the specification is one on conditional standard deviation instead of conditional variance:
where if , and if . Likewise, if , and if .
Hentschel's fGARCH model,[4] also known as Family GARCH, is an omnibus model that nests a variety of other popular symmetric and asymmetric GARCH models including APARCH, GJR, AVGARCH, NGARCH, etc.
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