Advanced Models

The Advanced models framework addresses the need to build a combo model. A combo model has two components: (1) conditional mean model (e.g. ARMAⁱ), and a conditional variance model (e.g. GARCHⁱ).

The Advanced (Combo) model is described as follow:

$x_t = \mu_t + a_t = f(x_{t-1}\,,...\,,x_{t-M},a_{t-1}\,,a_{t-2}\,,...\,,a_{t-N}\,,\sigma_{t}) + a_t$

$\sigma_t^2= V(\sigma_{t-1}^2\,,..\,,\sigma_{t-q}^2\,,a_{t-1}^2\,,..\,,a_{t-p}^2)$

Where:

$\left[\mu_t\right]$ is the conditional mean time series
$\left[a_t\right]$ is the innovation/shocks/residuals time series.
$a_t = \epsilon_t\times \sigma_t$
$\left[\epsilon\right] \sim i.i.d$
$f(.)$ is the conditional mean function (e.g. ARMA)
$V(.)$ is the conditional variance function (e.g. ARCH/GARCH)
$\left[\sigma_t^2\right]$ is the conditional variance time series

Remarks

In the combo model definition, $x_t$ is defined only in terms of its past observations. To bring exogenous factor, we can use a GLM model to capture the mean and expand the definition.
The standardized innovations/shocks (i.e. $\epsilon_t$ ) 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

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