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;[1] often the variance is related to the squares of the previous innovations.
[2] 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 (this is, when the time series exhibits heteroskedasticity).
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.
[3] To model a time series using an ARCH process, let
characterizing the typical size of the terms so that The random variable
exhibit time-varying heteroskedasticity using the Lagrange multiplier test was proposed by Engle (1982).
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.
The lag length p of a GARCH(p, q) process is established in three steps: Nonlinear Asymmetric GARCH(1,1) (NAGARCH) is a model with the specification:[6][7] 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.
[8] 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 exponential generalized autoregressive conditional heteroskedastic (EGARCH) model by Nelson & Cao (1991) is another form of the GARCH model.
may be a standard normal variable or come from a generalized error distribution.
The GARCH-in-mean (GARCH-M) model adds a heteroskedasticity term into the mean equation.
The Quadratic GARCH (QGARCH) model by Sentana (1995) is used to model asymmetric effects of positive and negative shocks.
Similar to QGARCH, the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model by Glosten, Jagannathan and Runkle (1993) also models asymmetry in the ARCH process.
Hentschel's fGARCH model,[12] 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
, where is the purely discontinuous part of the quadratic variation process of
The result is the following system of stochastic differential equations: where the positive parameters
which is then called the continuous-time GARCH (COGARCH) model.
[13] Unlike GARCH model, the Zero-Drift GARCH (ZD-GARCH) model by Li, Zhang, Zhu and Ling (2018) [14] lets the drift term
, and hence it nests the Exponentially weighted moving average (EWMA) model in "RiskMetrics".
Spatial GARCH processes by Otto, Schmid and Garthoff (2018) [15] 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 weight matrix defines which locations are considered to be adjacent.
In a different vein, the machine learning community has proposed the use of Gaussian process regression models to obtain a GARCH scheme.
[16] 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.