# Multivariate

Analogously to the univariate case, an instance of MultivariateARCHModel contains a matrix of data (with observations in rows and assets in columns), and encapsulates information about the covariance specification (e.g., CCC or DCC), the mean specification, and the error distribution.

MultivariateARCHModels support many of the same methods as UnivariateARCHModels, with a few noteworthy differences: the prediction targets for predict are :covariances and :correlations for predicting $\Sigma_t$ and $R_t$, respectively, and the new functions covariances and correlations respectively return the in-sample estimates of $\Sigma_t$ and $R_t$.

## Covariance specifications

The dynamics of $\Sigma_t$ are modelled as subtypes of MultivariateVolatilitySpec.

### Conditional correlation models

The main challenge in multivariate ARCH modelling is the curse of dimensionality: allowing each of the $(d)(d+1)/2$ elements of $\Sigma_t$ to depend on the past returns of all $d$ other assets requires $O(d^4)$ parameters without imposing additional structure. Conditional correlation models approach this issue by decomposing $\Sigma_t$ as

$$$\Sigma_t=D_t R_t D_t,$$$

where $R_t$ is the conditional correlation matrix and $D_t$ is a diagonal matrix containing the volatilities of the individual assets, which are modelled as univariate ARCH processes.

#### DCC

The dynamic conditional correlation (DCC) model of Engle (2002) imposes a GARCH-type structure on the $R_t$. In particular, for a DCC(p, q) model (with covariance targeting),

$$$R_{ij, t} = \frac{Q_{ij,t}}{\sqrt{Q_{ii,t}Q_{jj,t}}},$$$

where

$$$Q_{t} \equiv\bar{Q}(1-\bar\alpha-\bar\beta)+\sum_{i=1}^{p} \beta_iQ_{t-i}+\sum_{i=1}^{q}\alpha_i\epsilon_{t-i}\epsilon_{t-i}^\mathrm{\scriptsize T},$$$

## Mean Specifications

The conditional mean of a MultivariateARCHModel is specified by a vector of MeanSpecs as described under Mean specifications.

## Multivariate Standardized Distributions

Multivariate standardized distributions subtype MultivariateStandardizedDistribution. Currently, only MultivariateStdNormal is available. Note that under mild assumptions, the Gaussian (quasi-)MLE consistently estimates the (multivariate) ARCH parameters even if Gaussianity is violated.