Consider a sample of daily asset returns $\{r_t\}_{t\in\{1,\ldots,T\}}$. All models covered in this package share the same basic structure, in that they decompose the return into a conditional mean and a mean-zero innovation. In the univariate case,

\[r_t=\mu_t+a_t,\quad \mu_t\equiv\mathbb{E}[r_t\mid\mathcal{F}_{t-1}],\quad \sigma_t^2\equiv\mathbb{E}[a_t^2\mid\mathcal{F}_{t-1}],\]

$z_t\equiv a_t/\sigma_t$ is identically and independently distributed according to some law with mean zero and unit variance, and $\\{\mathcal{F}_t\\}$ is the natural filtration of $\\{r_t\\}$ (i.e., it encodes information about past returns). In the multivariate case, $r_t\in\mathbb{R}^d$, and the general model structure is

\[r_t=\mu_t+a_t,\quad \mu_t\equiv\mathbb{E}[r_t\mid\mathcal{F}_{t-1}],\quad \Sigma_t\equiv\mathbb{E}[a_ta_t^\mathrm{\scriptsize T}]\mid\mathcal{F}_{t-1}].\]

ARCH models specify the conditional volatility $\sigma_t$ (or in the multivariate case, the conditional covariance matrix $\Sigma_t$) in terms of past returns, conditional (co)variances, and potentially other variables.

This package represents an ARCH model as an instance of either UnivariateARCHModel or MultivariateARCHModel. These are subtypes ARCHModel and implement the interface of StatisticalModel from StatsBase.