Table 2 MSGARCH conditional distributions
Distribution | Equation | Label |
|---|---|---|
Normal | \(fN(\eta) = \frac {1}{\sqrt {2\pi }}\exp \left (-\frac {\eta ^{2}}{2}\right)\) | “norm” |
Student-t | \(fS(\eta ;\nu) = \frac {\Gamma \left (\frac {\eta + 1}{2} \right)}{\sqrt {\left (\nu -2\right)\pi }\Gamma \left (\frac {\nu }{2}\right)} \left (1 + \frac {\eta ^{2}}{\left (\nu -2\right)}\right)^{-\frac {\nu +1}{2}} \) | “std” |
GED | \(f_{GED}(\eta ;\nu) \equiv \frac {\nu \exp \left (-\frac {\left |\eta /\lambda \right |^{\nu }}{2} \right)}{\lambda 2^{\left (1+1/\nu \right)}\Gamma (1/\nu)}, \lambda \equiv \left (\frac {\Gamma (1/\nu)}{4^{1/\nu }\Gamma (3/\nu)} \right)^{1/2} \) | “ged” |
Skewed Normal | See Trottier and Ardia [44], equation 1 | “snorm” |
Skewed Student-t | See Trottier and Ardia [44], equation 1 | “sstd” |
Skewed GED | See Trottier and Ardia [44], equation 1 | “sged” |