ARCH, GARCH, and ARIMA

 

Auto-Regressive Conditional Heteroscedasticity (ARCH)

ARCH was developed by an economist Robert F. Engle III having won the 2003 Nobel Memorial Prize in Economic Sciences for its achievement.

Dependent variable: the variance error terms of the first regression:

The explanatory variable X can be the lagged dependent variables and/or the other variables: 

Then, it find the coefficient γ of the lagged squared error terms with reference to the log-likelihood: 

 

Generalised Auto-Regressive Conditional Heteroscedasticity (GARCH)

GARCH assumes the variance of the error term symmetrically varies depending the average size of the error terms in pervious time steps. It adds the lagged variance on the explanatory variable of the second regression with reference to the log-likelihood for finding the coefficients γ and δ:

 

Auto-Regressive Integrated Moving-Average (ARIMA)

Another popular auto-regressive method (AR) + Integrated differences (I) + moving-averaging (MA). 

Ref.: https://people.duke.edu/~rnau/411arim.htm 

Integrated (I): Using the differentiating a non-stationary time series to attempt to transform it to a stationary series. 

Moving-Average (MR) model: Using the past forecasted error terms instead of the lags of the dependent variable. 

 Limitation of these AR methods

Ref. Modeling daily realized futures volatility with singular spectrum analysis    Dimitrios D. Thomakosa, Tao Wanga, Luc T. Willeb; ∗ Received 29 November 2001 https://doi.org/10.1016/S0378-4371(02)00845-2

Limited as the forecasting model: Whereas it provides the magnitude of the fluctuations e.g., volatility, their distribution is assumed to be symmetric. The error of the estimation expands for the future estimate

The coefficients of the lagged variable contain non-periodic noise terms, which disrupt the estimation of the important periodicities represented by these coefficients.

SSA and Prony-like methods can extract the cyclical physical components that contain important information about the periodicity and its magnitude by distinguishing them from the noise components.

SSA performs better than various auto-regressive methods and Hodrick-Prescott filter (Noise-filtering method) introduced in "Singular spectrum analysis for real-time financial cycles
measurement   Maximilien Coussin 1 https://doi.org/10.1016/j.jimonfin.2021.102532"