\name{MultiGASFit}
\alias{MultiGASFit}
\title{
	Estimate multivariate GAS models
}
\description{
	Estimate multivariate GAS models by Maximum Likelihood.
}
\usage{
MultiGASFit(GASSpec, data, fn.optimizer = fn.optim, Compute.SE = TRUE)
}
\arguments{
\item{GASSpec}{
An object of the class \link{mGASSpec} created using the function \link{MultiGASSpec}}
%
\item{data}{ \code{matrix} (or something coercible to that using \code{as.matrix()}) of
dimension TxN containing the multivariate time series of observations.
It can also be an object of the class \code{ts}, \code{xts} or \code{zoo}.}
%
\item{fn.optimizer}{\code{function}. This is a generic optimization function that can be provided by the user.
By default \code{fn.optimizer = fn.optim} where \link{fn.optim} is
a wrapper to the \link{optim} function. See Details for user defined optimization routines.}
%
\item{Compute.SE}{\code{logical}. Should asymptotic Standard Errors be computed? By default \code{Compute.SE = TRUE}}
}
\details{
Maximum Likelihood estimation of GAS models is an on-going research topic.
General results are reported by Blasques et al. (2014b), Blasques et al. (2014a) and Harvey
(2013), while results for specific models have been derived by Blasques et al. (2014c) and
Andres (2014).\cr
%

Starting values for the optimizer are chosen in the following way: (i) estimate the static
version of the model (i.e., with A = 0 and B = 0) and set the initial value of the
intercept parameter accordingly, and (ii) perform a grid search for the
coefficients contained in A and B. Further technical details are presented in Section 3.2 of Ardia et. al. (2016a).\cr
%

The user is free to employ his/her own optimization routine via the \code{fn.optimizer} argument. \code{fn.optimizer}
accepts a \code{function} object. The user provided optimizer has to satisfy strict requirements. The arguments of the
\code{fn.optimizer} are : i) \code{par0} a vector of starting values, ii) \code{data} the data provided, iii) \code{GASSpec}
an object of the class \link{uGASSpec}, and iv) \code{FUN} the likelihood function. The output of \code{fn.optimizer} has
to be an object of the class \code{list} with four named elements: i) \code{pars}: a \code{numeric} vector
where the estimated parameters are stored, ii) \code{value}: a \code{numeric} containing the value of the negative log likelihood
evaluated at its minimum, iii) \code{hessian}, a \code{numeric} matrix containing the Hessian matrix evaluated at
the minimum of the negative log likelihood, this is used for inferential purposes, and iv) \code{convergence} a \code{numeric} variable  reporting information about the convergence of the optimization. This quantity is printed by the
\code{show()} and \code{summary()} methods. \code{convergence = 0} has to indicates successful completion.

The user is allowed to not include the last two elements of the output of the \code{fn.optimizer} function, that is, the values
\code{hessian = NULL} and \code{convergence = NULL} are admissible. In the case of \code{hessian = NULL}, the Hessian matrix is
evaluated numerically using the \link{hessian} function in the \code{numDeriv} package of Gilbert and Varadhan (2016). If the provided hessian is not positive definite, a try with the hessian evaluation used by the BFGS quasi-Newton implementation in the function \link{optim} is made.

By default, the \code{optim} optimizer with \code{method = "BFGS"} is employed.

}
\value{
An object of the class \link{mGASFit}.
}
\references{
Ardia D, Boudt K and Catania L (2016a).
"Generalized Autoregressive Score Models in R: The GAS Package."
\url{http://ssrn.com/abstract=2825380}.\cr

Blasques F, Koopman SJ, Lucas A (2014a).
"Maximum Likelihood Estimation for Correctly Specified Generalized Autoregressive Score Models: Feedback Effects, Contraction Conditions and Asymptotic Properties." techreport TI 14-074/III, Tinbergen Institute.
\url{http://www.tinbergen.nl/discussionpaper/?paper=2332}.

Blasques F, Koopman SJ, Lucas A (2014b).
"Maximum Likelihood Estimation for Generalized Autoregressive Score Models."
techreport TI 2014-029/III, Tinbergen Institute.
\url{http://www.tinbergen.nl/discussionpaper/?paper=2286}.\cr

Blasques F, Koopman SJ, Lucas A, Schaumburg J (2014c).
"Spillover Dynamics for Systemic Risk Measurement using Spatial Financial Time Series Models."
techreport TI 2014-103/III, Tinbergen Institute.
\url{http://www.tinbergen.nl/discussionpaper/?paper=2369}.\cr

Creal D, Koopman SJ, Lucas A (2013).
"Generalized Autoregressive Score Models with Applications."
Journal of Applied Econometrics, 28(5), 777-795.
\doi{10.1002/jae.1279}.\cr

Ghalanos A, Theussl S (2016).
"Rsolnp: General Non-Linear Optimization using Augmented Lagrange Multiplier Method."
\url{https://cran.r-project.org/package=Rsolnp}.\cr

Gilbert P, Varadhan R (2016).
numDeriv: Accurate Numerical Derivatives. R package 2016.8-1,
\url{https://CRAN.R-project.org/package=numDeriv}.

Harvey AC (2013).
Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series.
Cambridge University Press.\cr

Ye Y (1988).
Interior Algorithms for Linear, Quadratic, and Linearly Constrained Convex Programming.
Ph.D. thesis, Stanford University.
}
\author{Leopoldo Catania}
%
\examples{
\dontrun{
# Specify an GAS model with multivariate Student-t
# conditional distribution and time-varying scales and correlations

library("GAS")

data("StockIndices")

GASSpec = MultiGASSpec(Dist = "mvt", ScalingType = "Identity",
                       GASPar = list(scale = TRUE, correlation = TRUE))

Fit = MultiGASFit(GASSpec, StockIndices)

Fit
}
}