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\name{garchSpec}
\alias{garchSpec}
\concept{GARCH model}
\concept{APARCH model}
\concept{asymmetric power ARCH model}
\concept{ARMA-GARCH model}
\concept{ARMA-APARCH model}
\title{Univariate GARCH/APARCH time series specification}
\description{
Specifies an univariate ARMA-GARCH or ARMA-APARCH time series model.
}
\usage{
garchSpec(model = list(), presample = NULL,
cond.dist = c("norm", "ged", "std", "snorm", "sged", "sstd"),
rseed = NULL)
}
\arguments{
\item{model}{
a list of GARCH model parameters, see section \sQuote{Details}.
The default \code{model=list()} specifies Bollerslev's
GARCH(1,1) model with normal conditional distributed innovations.
}
\item{presample}{
a numeric three column matrix with start values for the series,
for the innovations, and for the conditional variances. For an
ARMA(m,n)-GARCH(p,q) process the number of rows must be at least
max(m,n,p,q)+1, longer presamples are truncated. Note, all presamples
are initialized by a normal-GARCH(p,q) process.
}
\item{cond.dist}{
a character string naming the desired conditional distribution.
Valid values are \code{"norm"}, \code{"ged"}, \code{"std"},
\code{"snorm"}, \code{"sged"}, \code{"sstd"}. The default value
is \code{"norm"}, the standard normal distribution.
}
\item{rseed}{
single integer argument, the seed for the intitialization of
the random number generator for the innovations. If
\code{rseed=NULL}, the default, then the state of the random
number generator is not touched by this function.
}
}
\details{
\code{garchSpec} specifies a GARCH or APARCH time series model which
can be used for simulating artificial GARCH and/or APARCH time
series. This is very useful for testing the GARCH parameter estimation
results, since the model parameters are known and well specified.
Argument \code{model} is a list of model parameters. For the GARCH
part of the model they are:
\describe{
\item{\code{omega}}{the constant coefficient of the variance
equation, by default \code{1e-6};}
\item{\code{alpha}}{the value or vector of autoregressive
coefficients, by default 0.1, specifying a model of order 1;}
\item{\code{beta}}{the value or vector of variance coefficients, by
default 0.8, specifying a model of order 1.}
}
If the model is APARCH, then the following additional parameters are
available:
\describe{
\item{delta}{a positive number, the power of sigma in the volatility
equation, it is 2 for GARCH models;}
\item{gamma}{the leverage parameters, a vector of length
\code{alpha}, containing numbers in the interval \eqn{(0,1)}.}
}
The values for the linear part (conditional mean) are:
\describe{
\item{\code{mu}}{the mean value, by default NULL;}
\item{\code{ar}}{the autoregressive ARMA coefficients, by default NULL;}
\item{\code{ma}}{the moving average ARMA coefficients, by default NULL.}
}
The parameters for the conditional distributions are:
\describe{
\item{\code{skew}}{the skewness parameter (also named "xi"), by
default 0.9, effective only for the \code{"dsnorm"}, the
\code{"dsged"}, and the \code{"dsstd"} skewed conditional
distributions;}
\item{\code{shape}}{the shape parameter (also named "nu"), by
default 2 for the \code{"dged"} and \code{"dsged"}, and by default
4 for the \code{"dstd"} and \code{"dsstd"} conditional
distributions.}
}
For example, specifying a subset AR(5[1,5])-GARCH(2,1) model with a
standardized Student-t distribution with four degrees of freedom will
return the following printed output:
\preformatted{
garchSpec(model = list(ar = c(0.5,0,0,0,0.1), alpha =
c(0.1, 0.1), beta = 0.75, shape = 4), cond.dist = "std")
Formula:
~ ar(5) + garch(2, 1)
Model:
ar: 0.5 0 0 0 0.1
omega: 1e-06
alpha: 0.1 0.1
beta: 0.75
Distribution:
std
Distributional Parameter:
nu = 4
Presample:
time z h y
0 0 -0.3262334 2e-05 0
-1 -1 1.3297993 2e-05 0
-2 -2 1.2724293 2e-05 0
-3 -3 0.4146414 2e-05 0
-4 -4 -1.5399500 2e-05 0
}
Its interpretation is as follows. \sQuote{Formula} describes the
formula expression specifying the generating process, \sQuote{Model}
lists the associated model parameters, \sQuote{Distribution} the type
of the conditional distribution function in use,
\sQuote{Distributional Parameters} lists the distributional parameter
(if any), and the \sQuote{Presample} shows the presample input matrix.
If we have specified \code{presample = NULL} in the argument list,
then the presample is generated automatically by default as
norm-AR()-GARCH() process.
}
\value{
an object of class \code{"\linkS4class{fGARCHSPEC}"}
}
\author{
Diethelm Wuertz for the Rmetrics \R-port
}
\seealso{
\code{\link{garchSim}},
\code{\link{garchFit}}
}
\examples{
# Normal Conditional Distribution:
spec = garchSpec()
spec
# Skewed Normal Conditional Distribution:
spec = garchSpec(model = list(skew = 0.8), cond.dist = "snorm")
spec
# Skewed GED Conditional Distribution:
spec = garchSpec(model = list(skew = 0.9, shape = 4.8), cond.dist = "sged")
spec
## More specifications ...
# Default GARCH(1,1) - uses default parameter settings
garchSpec(model = list())
# ARCH(2) - use default omega and specify alpha, set beta=0!
garchSpec(model = list(alpha = c(0.2, 0.4), beta = 0))
# AR(1)-ARCH(2) - use default mu, omega
garchSpec(model = list(ar = 0.5, alpha = c(0.3, 0.4), beta = 0))
# AR([1,5])-GARCH(1,1) - use default garch values and subset ar[.]
garchSpec(model = list(mu = 0.001, ar = c(0.5,0,0,0,0.1)))
# ARMA(1,2)-GARCH(1,1) - use default garch values
garchSpec(model = list(ar = 0.5, ma = c(0.3, -0.3)))
# GARCH(1,1) - use default omega and specify alpha/beta
garchSpec(model = list(alpha = 0.2, beta = 0.7))
# GARCH(1,1) - specify omega/alpha/beta
garchSpec(model = list(omega = 1e-6, alpha = 0.1, beta = 0.8))
# GARCH(1,2) - use default omega and specify alpha[1]/beta[2]
garchSpec(model = list(alpha = 0.1, beta = c(0.4, 0.4)))
# GARCH(2,1) - use default omega and specify alpha[2]/beta[1]
garchSpec(model = list(alpha = c(0.12, 0.04), beta = 0.08))
# snorm-ARCH(1) - use defaults with skew Normal
garchSpec(model = list(beta = 0, skew = 0.8), cond.dist = "snorm")
# sged-GARCH(1,1) - using defaults with skew GED
garchSpec(model = list(skew = 0.93, shape = 3), cond.dist = "sged")
# Taylor Schwert GARCH(1,1) - this belongs to the family of APARCH Models
garchSpec(model = list(delta = 1))
# AR(1)-t-APARCH(2, 1) - a little bit more complex specification ...
garchSpec(model = list(mu = 1.0e-4, ar = 0.5, omega = 1.0e-6,
alpha = c(0.10, 0.05), gamma = c(0, 0), beta = 0.8, delta = 1.8,
shape = 4, skew = 0.85), cond.dist = "sstd")
}
\keyword{models}
\keyword{ts}
|
