The Advanced (Combo) model is described as follow:
- is the conditional mean time series
- is the innovation/shocks/residuals time series.
- is the conditional mean function (e.g. ARMA)
- is the conditional variance function (e.g. ARCH/GARCH)
- is the conditional variance time series
- In the combo model definition, is defined only in terms of its past observations. To bring exogenous factor, we can use a GLMi model to capture the mean and expand the definition.
- The standardized innovations/shocks (i.e. ) can be modeled to follow either a Gaussian or a leptokurtic distribution (e.g. student's t, GEDi, etc.).
- The number of free parameters in our combo model is the sum of the two components model parameters minus two(2).
For instance, an ARMA-GARCH combo model will have the following: (1) p+q+1 free parameters from the ARMA(p,q) , and (2) M+N+1 free parameters from the GARCH(M,N) components