[QuantEcon]MATLAB混编FORTRAN语言

$$
\Phi(L){(1 - L)^d}({y_t} - {\mu _t}) = \Theta (L){\varepsilon _t},
$$

$$
{\mu _t} = \mu  + \sum\limits_{i = 1}^{m - n} {{\delta _i}} {x_{i,t}} + \sum\limits_{i = m - n + 1}^m {\delta _i}{x_{i,t}}{\sigma _t} + \xi \sigma _t^k,
$$

$$
{z_t}(\theta ) = \frac{{{y_t} - \mu (\theta ,{x_t})}}{{\sigma (\theta ,{x_t})}},
$$

$$
\sigma _t^2 = \left( {\omega  + \sum\limits_{j = 1}^m {{\zeta _j}{v_{jt}}} } \right) + \sum\limits_{j = 1}^q {{\alpha _j}\varepsilon _{t - j}^2 + } \sum\limits_{j = 1}^p {{\beta _j}\sigma _{t - j}^2},
$$
with $\sigma_t^2$ denoting the conditional variance, $\omega$ the intercept
and $\varepsilon_t^2$ the residuals from the mean filtration process discussed
previously. The GARCH order is defined by $(q, p)$ (ARCH, GARCH), with possibly
\verb@m@ external regressors $v_j$ which are passed \emph{pre-lagged}.
If variance targeting is used, then $\omega$ is replaced by,

$$
{{\bar \sigma }^2}\left( {1 - \hat P} \right) - \sum\limits_{j = 1}^m {{\zeta _j}{{\bar v}_j}}
$$

where ${\bar \sigma}^2$ is the unconditional variance of $\varepsilon^2$ which
is consistently estimated by its sample counterpart at every iteration of the
solver following the mean equation filtration, and ${\bar v}_j$ represents the
sample mean of the $j^{th}$ external regressors in the variance equation
(assuming stationarity), and $\hat P$ is the persistence and defined below. If a
numeric value was provided to the \emph{variance.targeting} option in the specification
(instead of logical), this will be used instead of ${\bar \sigma }^2$ for the
calculation.\footnote{Note that this should represent a value related to the variance
in the plain vanilla GARCH model. In more general models such as the APARCH, this is
a value related to $\sigma^{\delta}$, which may not be obvious since $\delta$ is not
known prior to estimation, and therefore care should be taken in those cases.
Finally, if scaling is  used in the estimation (via the fit.control option), this value will
also be automatically scale adjusted by the routine.}

$$
\hat P = \sum\limits_{j = 1}^q {{\alpha _j}}  + \sum\limits_{j = 1}^p {{\beta _j}}.
$$

$$
{\log _e}\left( {\sigma _t^2} \right) = \left( {\omega  + \sum\limits_{j = 1}^m {{\zeta _j}{v_{jt}}} } \right) + \sum\limits_{j = 1}^q {\left( {{\alpha _j}{z_{t - j}} + {\gamma _j}\left( {\left| {{z_{t - j}}} \right| - E\left| {{z_{t - j}}} \right|} \right)} \right) + } \sum\limits_{j = 1}^p {{\beta _j}{{\log }_e}\left( {\sigma _{t - j}^2} \right)}
$$

$$ \hat P = \sum\limits_{j = 1}^q {{\alpha _j}}  + \sum\limits_{j = 1}^p {{\beta _j} + } \sum\limits_{j = 1}^q {{\gamma _j}\kappa }, $$

$$
\sigma _t^\delta  = \left( {\omega  + \sum\limits_{j = 1}^m {{\zeta _j}{v_{jt}}} } \right) + \sum\limits_{j = 1}^q {{\alpha _j}{{\left( {\left| {{\varepsilon _{t - j}}} \right| - {\gamma _j}{\varepsilon _{t - j}}} \right)}^\delta } + } \sum\limits_{j = 1}^p {{\beta _j}\sigma _{t - j}^\delta }
$$

$$
\sigma _t^\lambda  = \left( {\omega  + \sum\limits_{j = 1}^m {{\zeta _j}{v_{jt}}} } \right) + \sum\limits_{j = 1}^q {{\alpha _j}\sigma _{t - j}^\lambda {{\left( {\left| {{z_{t - j}} - {\eta _{2j}}} \right| - {\eta _{1j}}\left( {{z_{t - j}} - {\eta _{2j}}} \right)} \right)}^\delta } + } \sum\limits_{j = 1}^p {{\beta _j}\sigma _{t - j}^\lambda }
$$

which is a Box-Cox transformation for the conditional standard deviation whose
shape is determined by $\lambda$, and the parameter $\delta$ transforms the
absolute value function which it subject to rotations and shifts through the
$\eta_{1j}$ and $\eta_{2j}$ parameters respectively. Various submodels arise
from this model, and are passed to the \verb@ugarchspec@ 'variance.model' list
via the submodel option,