ARCH models are commonly employed in modeling financialtime series that exhibit time-varying volatility and volatility clustering, i.e. periods of swings interspersed with periods of relative calm. ARCH-type models are sometimes considered to be in the family of stochastic volatility models, although this is strictly incorrect since at time t the volatility is completely pre-determined (deterministic) given previous values.
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
The random variable is a strong white noise process. The series is modeled by
Obtain the squares of the error and regress them on a constant and q lagged values:
where q is the length of ARCH lags.
The null hypothesis is that, in the absence of ARCH components, we have for all . The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic T'R² follows distribution with q degrees of freedom, where is the number of equations in the model which fits the residuals vs the lags (i.e. ). If T'R² is greater than the Chi-square table value, we reject the null hypothesis and conclude there is an ARCH effect in the ARMA model. If T'R² is smaller than the Chi-square table value, we do not reject the null hypothesis.
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 ), following the notation of the original paper, 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 and GARCH errors.
Exponentially weighted moving average (EWMA) is an alternative model in a separate class of exponential smoothing models. As an alternative to GARCH modelling it has some attractive properties such as a greater weight upon more recent observations, but also drawbacks such as an arbitrary decay factor that introduces subjectivity into the estimation.
GARCH(p, q) model specification
The lag length p of a GARCH(p, q) process is established in three steps:
Estimate the best fitting AR(q) model
Compute and plot the autocorrelations of by
The asymptotic, that is for large samples, standard deviation of is . Individual values that are larger than this indicate GARCH errors. To estimate the total number of lags, use the Ljung-Box test until the value of these are less than, say, 10% significant. The Ljung-Box Q-statistic follows distribution with n degrees of freedom if the squared residuals are uncorrelated. It is recommended to consider up to T/4 values of n. The null hypothesis states that there are no ARCH or GARCH errors. Rejecting the null thus means that such errors exist in the conditional variance.
Nonlinear Asymmetric GARCH(1,1) (NAGARCH) is a model with the specification:
where and , which ensures the non-negativity and stationarity of the variance process.
For stock returns, parameter is usually estimated to be positive; in this case, it reflects a phenomenon commonly referred to as the "leverage effect", signifying that negative returns increase future volatility by a larger amount than positive returns of the same magnitude.
This model should not 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 imports a unit root in the GARCH process. The condition for this is
The exponential generalized autoregressive conditional heteroskedastic (EGARCH) model by Nelson & Cao (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):
Since may be negative, there are no sign restrictions for 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 asymmetric 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. The specification is one on conditional standard deviation instead of conditional variance:
where if , and if . Likewise, if , and if .
Hentschel's fGARCH model, 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.
In 2004, Claudia Klüppelberg, Alexander Lindner and Ross Maller proposed a continuous-time generalization of the discrete-time GARCH(1,1) process. The idea is to start with the GARCH(1,1) model equations
and then to replace the strong white noise process by the infinitesimal increments of a Lévy process, and the squared noise process by the increments , where
where the positive parameters , and are determined by , and . Now given some initial condition , the system above has a pathwise unique solution which is then called the continuous-time GARCH (COGARCH) model.
Unlike GARCH model, the Zero-Drift GARCH (ZD-GARCH) model by Li, Zhang, Zhu and Ling (2018)  lets the drift term in the first order GARCH model. The ZD-GARCH model is to model , where is i.i.d., and
The ZD-GARCH model does not require , and hence it nests the Exponentially weighted moving average (EWMA) model in "RiskMetrics". Since the drift term , the ZD-GARCH model is always non-stationary, and its statistical inference methods are quite different from those for the classical GARCH model. Based on the historical data, the parameters and can be estimated by the generalized QMLE method.
Spatial GARCH processes by Otto, Schmid and Garthoff (2018)  are considered as the spatial equivalent to the temporal generalized autoregressive conditional heteroscedasticity (GARCH) models. In contrast to the temporal ARCH model, in which the distribution is known given the full information set for the prior periods, the distribution is not straightforward in the spatial and spatiotemporal setting due to the interdependence between neighboring spatial locations. The spatial model is given by and
where denotes the -th spatial location and refers to the -th entry of a spatial weight matrix and for . The spatial weight matrix defines which locations are considered to be adjacent.
^Bollerslev, Tim; Russell, Jeffrey; Watson, Mark (May 2010). "Chapter 8: Glossary to ARCH (GARCH)"(PDF). Volatility and Time Series Econometrics: Essays in Honor of Robert Engle (1st ed.). Oxford: Oxford University Press. pp. 137–163. ISBN9780199549498. Retrieved 27 October 2017.
^Klüppelberg, C.; Lindner, A.; Maller, R. (2004). "A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour". Journal of Applied Probability. 41 (3): 601–622. doi:10.1239/jap/1091543413.