Fat-Tails

To allow for a fat-tailed distribution, you can choose DISTRIB=T option which uses a multivariate Student-t distribution or DISTRIB=GED which uses a multivariate generalization of the GED. As with univariate models, you can choose a fixed “shape” (degrees of freedom for the t) with the SHAPE option, or, if you don’t use that option, the degrees of freedom will be estimated. This estimates an EWMA model with t distributed errors.

 

garch(p=1,q=1,mv=ewma,distrib=t) / xjpn xfra xsui

Asymmetry

The ASYMMETRIC option adds asymmetry effects to multivariate GARCH models. The precise form will depend upon the variance model you choose. For models where only the variances are directly governed by a GARCH model, each variance just has a term added as in the GJR model.

 

For the VECH models, define

\begin{equation} {\bf{v}}_{t - 1} = {\bf{u}}_{t - 1} \circ I_{u < 0} \left( {{\bf{u}}_{t - 1} } \right) \end{equation}

where \(\circ \) denotes the elementwise (Hadamard) product of the vectors. \(\bf{v}\) will be a copy of \(\bf{u}\) with positive elements zeroed out. With this, for the standard MV-GARCH (DVECH) model, the asymmetry terms in a GARCH(1,1) add the following to the formula for \(H_{ij,t} \):

\begin{equation} d_{ij} v_{i,t - 1} v_{j,t - 1} \end{equation}

for the BEKK, it’s

\begin{equation} {\bf{D'v}}_{t - 1} {\bf{v'}}_{t - 1} {\kern 1pt} {\bf{D}} \end{equation}

and for the full VECH, it’s

\begin{equation} {\bf{D}}vech\left( {{\bf{v}}_{t - 1} {\bf{v'}}_{t - 1} } \right) \end{equation}

 

Use the SIGNS option to change the behavior of I for each component. The default behavior is I=1 when \(u < 0\) and zero otherwise as noted above. This gives the behavior for the GJR model, where negative residuals increase variance given positive \(\bf{D}\) values. Use SIGNS to supply +1 values rather than -1 values for any components where you want the opposite behavior: I=1 for \(u > 0\) so positive residuals increase variance (for positive \(\bf{D}\)). This can differ among components, so signs=||-1.0,1.0|| has standard asymmetry behavior for the first variable, but reversed asymmetry for the second.

 

Note that, unlike univariate GARCH models, the choice for sign on the definition of \(\bf{v}\) matters because of the effect on the cross terms.

 

This estimates a CC-EGARCH with asymmetry.

 

garch(p=1,q=1,mv=cc,asymmetric,variances=exp) / xjpn xfra xsui

 

XREGRESSORS

As with univariate models, you add exogenous variance shifts by using the XREGRESSORS option and including a supplementary card listing the variables. (Again, don’t include the CONSTANT.) The same shift is applied to all variables. For the Restricted Covariance Models, XREG adds the shifts to each variance equation separately. For the DVECH and VECH, it adds for each X-regressor an extra “variance constant” term with the same layout as the model’s C term, but multiplied by the regressor.

 

Those are all fairly obvious. However, the proper way to handle an X-regressor with the BEKK models isn’t as clear. Employing the same idea as used for the DVECH and VECH runs into the problem that \({\bf{EE'}}\) is forcibly positive semi-definite. For instance, if that method were used and you add a Monday dummy, the model would force the variance to be higher (or at least no lower) for Monday than for any other day, even if the effect were, in fact, just the opposite. Instead, what RATS does by default is to replace the original variance constant with

\begin{equation} ({\bf{C}} + {\bf{E}}x_t )({\bf{C}} + {\bf{E}}x_t )' \end{equation}

where \(x_t\) is the X-regressor and \(\bf{E}\) (like \(\bf{C}\)) is a lower triangular matrix. This preserves the restriction that BEKK generates a positive definite matrix, and doesn’t prejudge the signs of the effects if you use a set of (mutually exclusive) dummies. You can use separate terms by including the option XBEKK=SEPARATE (the default is XBEKK=COMBINED) which replaces the constant term in the recursion with

\begin{equation} {\bf{CC'}} + {\bf{EE'}}x_t^2 \end{equation}

Copyright © 2024 Thomas A. Doan