In econometrics, the autoregressive conditional heteroskedasticity (ARCH) model is a statistical model for time series data that describes the variance of the current error term or innovation as 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. The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model. ARCH models are commonly employed in modeling financial time 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 predetermined (deterministic) given previous values.

Model specification

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 z_t and a time-dependent standard deviation \sigma_t characterizing the typical size of the terms so that The random variable z_t is a strong white noise process. The series \sigma_t^2 is modeled by An ARCH(q) model can be estimated using ordinary least squares. A method for testing whether the residuals \epsilon_t exhibit time-varying heteroskedasticity using the Lagrange multiplier test was proposed by Engle (1982). This procedure is as follows:

GARCH

If an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model. In that case, the GARCH (p, q) model (where p is the order of the GARCH terms ~\sigma^2 and q is the order of the ARCH terms ~\epsilon^2 ), 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:

NGARCH

NAGARCH

Nonlinear Asymmetric GARCH(1,1) (NAGARCH) is a model with the specification: For stock returns, parameter ~ \theta 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.

IGARCH

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 .

EGARCH

The exponential generalized autoregressive conditional heteroskedastic (EGARCH) model by Nelson & Cao (1991) is another form of the GARCH model. Formally, an EGARCH(p,q): where, is the conditional variance, \omega, \beta, \alpha, \theta and \lambda are coefficients. Z_{t} may be a standard normal variable or come from a generalized error distribution. The formulation for g(Z_{t}) allows the sign and the magnitude of Z_{t} to have separate effects on the volatility. This is particularly useful in an asset pricing context. Since may be negative, there are no sign restrictions for the parameters.

GARCH-M

The GARCH-in-mean (GARCH-M) model adds a heteroskedasticity term into the mean equation. It has the specification: The residual ~\epsilon_t is defined as:

QGARCH

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 ~\sigma_t is where z_t is i.i.d. and

GJR-GARCH

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 z_t is i.i.d., and where I_{t-1} = 0 if, and I_{t-1} = 1 if.

TGARCH model

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.

fGARCH

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.

COGARCH

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 z_t by the infinitesimal increments of a Lévy process, and the squared noise process z^2_t by the increments , where is the purely discontinuous part of the quadratic variation process of L. The result is the following system of stochastic differential equations: where the positive parameters \beta, \eta and \varphi are determined by \alpha_0, \alpha_1 and \beta_1. Now given some initial condition, the system above has a pathwise unique solution which is then called the continuous-time GARCH (COGARCH) model.

ZD-GARCH

Unlike GARCH model, the Zero-Drift GARCH (ZD-GARCH) model by Li, Zhang, Zhu and Ling (2018) lets the drift term ~\omega= 0 in the first order GARCH model. The ZD-GARCH model is to model, where z_t 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 ~\omega= 0, 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 ~\alpha_{1} and ~\beta_{1} can be estimated by the generalized QMLE method.

Spatial GARCH

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 ~s_i denotes the i-th spatial location and ~w_{iv} refers to the iv-th entry of a spatial weight matrix and w_{ii}=0 for. The spatial weight matrix defines which locations are considered to be adjacent.

Gaussian process-driven GARCH

In a different vein, the machine learning community has proposed the use of Gaussian process regression models to obtain a GARCH scheme. This results in a nonparametric modelling scheme, which allows for: (i) advanced robustness to overfitting, since the model marginalises over its parameters to perform inference, under a Bayesian inference rationale; and (ii) capturing highly-nonlinear dependencies without increasing model complexity.