# Introduction

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`

.