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R/GEVStableGarch-stationarityAparch.R
In GEVStableGarch: ARMA-GARCH/APARCH Models with GEV and Stable Distributions
# Copyrights (C) 2014 Thiago do Rego Sousa <thiagoestatistico@gmail.com>
# This library is free software; you can redistribute it and/or
# modify it under the terms of the GNU Library General Public
# License as published by the Free Software Foundation; either
# version 2 of the License, or (at your option) any later version.
#
# This library is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Library General Public License for more details.
#
# You should have received a copy of the GNU Library General
# Public License along with this library; if not, write to the
# Free Foundation, Inc., 59 Temple Place, Suite 330, Boston,
# MA 02111-1307 USA
####################################################################################
# FUNCTION: DESCRIPTION:
#
# .stationarityAparch Compute the stationarity condition for the
# GARCH or APARCH model with several cond.
# distributions
#
# gsMomentAparch Evaluate the exact moments of the type
# E(z - gm|z|)^delta for several conditional
# distributions.
#
# .normMomentAparch Exact APARCH moments for the standard normal
#
# .stdMomentAparch Exact APARCH moments for the standard t-Student
#
# .skstdMomentAparch Exact APARCH moments for the standard skew
# t-Student defined by Fernandez and Steel (1998).
# Notice that this distribution is standardized
# with location zero and unit scale, but it is not
# standardized in the sense that it has a zero mean and
# unit variance. The exact formula for this expression
# would be very complicated to compute if we reescale
# the original distribution to (X - mu) / sigma and
# therefore, we choose to work with the raw distribution
# defined in Fernandez and Steel without this reparametrization
#
# .gedMomentAparch Exact APARCH moments for the standard GED distribution
# .gatMomentAparch Exact APARCH moments for the standard GAt distribution
#
# .stableS1MomentAparch Exact APARCH moments for the location zero and unit scale
# in S1 parametrization (see Nolan (1999)).
#
# .stableS1SymmetricMomentGarch Exact GARCH moments ( E |zt| for the symmetric stable (S1)
# distribution (S1 parametrization)
#
# .stableS1SymmetricMomentAparch Exact APARCH moments for the symmetric stable (S1) distribution
#
# .stableS1MomentPowerGarch Exact power-GARCH moments for the asymmetric stable (S1) distribution
#
# .trueAparchMomentsWurtz Function that evaluate moments with numerical integration.
# This function was used only to test the mathematical
# formulas implemented here
####################################################################################
.stationarityAparch <-
function (model = list(),
formula,
cond.dist = c("stableS1", "gev", "gat", "norm", "std", "sstd", "skstd", "ged"))
{
# Description:
# A function that evaluate stationarity conditions
# to guide parameter estimation in GARCH/APARCH models
# Make sure we have a garch or aparch with p > 0
# The stationarity condition for the APARCH is:
# sum ( E[ |z|-gm*z ] ^ delta * alpha + beta ) < 1
# and for the GARCH model is
# sum ( E[ |z| ] ^ delta * alpha + beta ) < 1
# where delta is usually equal to 2, except for the
# stable model where delta = 1.
# The expectation E[ |z| ] is equal to one for the following
# distributions: norm, std, sstd and ged.
# For the GAt, gev and stable conditional distribution we need to
# calculate this expectation, even for the GARCH model.
# Arguments:
# formula - an object returned by function .getFormula
# parm - a list with the model parameters as entries
# alpha - a vector of autoregressive coefficients
# of length p for the GARCH/APARCH specification,
# gm - a vector of leverage coefficients of
# length p for the APARCH specification,
# beta - a vector of moving average coefficients of
# length q for the GARCH/APARCH specification,
# delta - the exponent value used in the variance equation.
# skew - a numeric value listing the distributional
# skewness parameter.
# shape - a numeric value listing the distributional
# shape parameter.
# cond.dist - a character string naming the distribution
# function.
# Return Values:
# sum ( E[ |z|-gm*z ] ^ delta * alpha + beta ) - When it can be computed
# 1e99 - In case we the expectation E[ |z|-gm*z ] ^ delta could not be evaluated.
# FUNCTION:
# get parameters
alpha <- model$alpha
beta <- model$beta
delta <- model$delta
gm <- model$gm
skew <- model$skew
shape <- model$shape
# Conditional distribution
cond.dist = match.arg(cond.dist)
#print(t(model))
if(length(alpha) == 0 || length(alpha) != length(gm) || length(delta) != 1)
stop("Failed to verify conditions:
if(length(alpha) == 0 || length(delta) != 1)")
# We must have a model with a non zero garch/aparch order
if(formula$formula.order[3] == 0)
stop("Invalid model")
# garch model: sum ( E[ |z| ] ^ delta * alpha + beta ) < 1
if(formula$isAPARCH == FALSE)
{
if (any( c("norm", "std", "sstd", "ged") == cond.dist))
return(sum(alpha) + sum(beta))
if(cond.dist == "gev")
kappa = try(.gevMomentAparch(shape = shape, delta = 2, gm = 0), silent = TRUE)
if(cond.dist == "stableS1")
kappa = try(.stableS1MomentPowerGarch (shape = shape, skew = skew,
delta = 1), silent = TRUE)
if(cond.dist == "gat")
kappa = try(.gatMomentAparch(shape = shape, skew = skew,
delta = 2, gm = 0), silent = TRUE)
if(cond.dist == "skstd")
kappa = try(.skstdMomentAparch(shape = shape, skew = skew, delta = 2, gm = 0), silent = TRUE)
if( is.numeric(kappa))
return(kappa*sum(alpha) + sum(beta))
else
return(1e99)
} else {
# aparch model
kappa = rep(0,length(alpha))
for( i in 1:length(alpha))
{
if(cond.dist == "stableS1")
kappa[i] = try(.stableS1MomentAparch (shape = shape, skew = skew,
delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "gev")
kappa[i] = try(.gevMomentAparch(shape = shape, delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "gat")
kappa[i] = try(.gatMomentAparch(shape = shape, delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "norm")
kappa[i] = try(.normMomentAparch (delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "std")
kappa[i] = try(.stdMomentAparch(shape = shape, delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "skstd")
kappa[i] = try(.skstdMomentAparch(shape = shape, skew = skew, delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "ged")
kappa[i] = try(.gedMomentAparch(shape = shape, delta = delta, gm = gm[i]), silent = TRUE)
}
if( is.numeric(kappa) )
return(sum(kappa*alpha) + sum(beta))
else
return(1e99)
}
}
# ------------------------------------------------------------------------------
gsMomentAparch <- function(
cond.dist = c("stableS1", "gev", "gat", "norm", "std", "sstd", "skstd", "ged"),
shape = 1.5,
skew = 0,
delta = 1,
gm = 0)
{
# conditional distribution
cond.dist = match.arg(cond.dist)
if(cond.dist == "stableS1")
kappa = .stableS1MomentAparch (shape = shape, skew = skew,
delta = delta, gm = gm)
if(cond.dist == "gev")
kappa = .gevMomentAparch(shape = shape, delta = delta, gm = gm)
if(cond.dist == "gat")
kappa = .gatMomentAparch(shape = shape, delta = delta, skew = skew, gm = gm)
if(cond.dist == "norm")
kappa = .normMomentAparch (delta = delta, gm = gm)
if(cond.dist == "std")
kappa = .stdMomentAparch(shape = shape, delta = delta, gm = gm)
if(cond.dist == "sstd")
kappa = .sstdMomentAparch(shape = shape, skew = skew, delta = delta, gm = gm)
if(cond.dist == "skstd")
kappa = .skstdMomentAparch(shape = shape, skew = skew, delta = delta, gm = gm)
if(cond.dist == "ged")
kappa = .gedMomentAparch(shape = shape, delta = delta, gm = gm)
# Return
kappa
}
# ------------------------------------------------------------------------------
.normMomentAparch <- function(delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard normal distribution
# E[ (|z|-gm*z)^delta.
# Reference: A long memory property of stock market returns and a
# new model. Ding, Granger and Engle (1993), Appendix B.
# Error treatment of input parameters
if( ( abs ( gm ) >= 1 ) || ( delta <= 0 ) )
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following condition cannot be true
( abs ( gm ) >= 1 ) || ( delta <= 0 )")
result = 1 / sqrt ( 2 * pi ) * ( (1 + gm ) ^ delta + ( 1 - gm ) ^ delta ) *
2 ^ ( ( delta - 1 ) / 2 ) * gamma ( ( delta + 1 ) / 2 )
# Return
result
}
# ------------------------------------------------------------------------------
.stdMomentAparch <- function(shape = 3, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard t-Student distribution
# E[ (|z|-gm*z)^delta.
# Reference: Mittnik - (2000) - Conditional Density and Value-at-Risk
# Prediction of Asian Currency Exchange Rates
# Note that the formula provided by Mittnik is valid for the original t-Student
# distribution (which is nonStandardized).
# Since our objective is to calculate it for the standart t-Student distribution
# defined in Wurtz et al. (2006) - eq (13) we need to multiply the Mittnik
# formula by (sqrt((shape-2)/shape))^delta.
# The main drawback is that the standardized t-Student distribution
# is valid only for shape > 2 ( in which case the variance is finite)
# Error treatment of input parameters
if( (abs(gm) >= 1) || (delta <= 0) || (shape <= 2) || (delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
(abs(gm) >= 1) || (delta <= 0) || (shape <= 0)|| (delta >= shape)")
mittnikFormula = shape^(delta/2)*1/2/sqrt(pi)*( (1+gm)^delta+(1-gm)^delta )*
gamma((delta+1)/2)*(gamma(shape/2))^(-1)*gamma((shape-delta)/2)
factorToMultiply = (sqrt((shape-2)/shape))^delta
# Return
mittnikFormula*factorToMultiply
}
# ------------------------------------------------------------------------------
.skstdMomentAparch <- function(shape = 3, skew = 1, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard skew t-Student distribution
# E[ (|z|-gm*z)^delta.
# Reference: fGarch: Wurtz and the package Skewt uses the one defined
# in Fernandez, C. and Steel, M. F. J. (1998). On Bayesian modeling of fat
# tails and skewness, J. Am. Statist. Assoc. 93, 359-371.
# Another reference: http://www.timberlake.co.uk/slaurent/G@RCH/Book63.html
# Note: This is not the APARCH moments for the 'dsstd' function from fGarch package.
# This is the skew t-Student defined by Fernandez and Steel without the
# reparametrization introduced by Wurtz et al. (2006).
# but theThe point here is that the SPLUS FinMetrics
# also uses this distribution without reparametrization.
# Indeed, Mittnik et al. (2000) also uses his t3-distribution (GAt) without such
# reparametrization. These distributions are in fact standardized, but the
# point is that the assymetry parameter plays a crucial role in the calculation
# of the APARCH moments.
if( (abs(gm) >= 1) || (delta <= 0) || (shape <= 2) ||
(delta >= shape) || (skew <= 0))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
(abs(gm) >= 1) || (delta <= 0) || (shape <= 2) ||
(delta >= shape) || (skew <= 0)")
a = skew^(-1-delta)*(1+gm)^delta + skew^(1+delta)*(1-gm)^delta
b = gamma((delta+1)/2)*gamma((shape-delta)/2)*(shape-2)^((1+delta)/2)
c = (skew + 1/skew)*sqrt((shape-2)*pi)*gamma(shape/2)
# Return
a*b/c
}
# ------------------------------------------------------------------------------
.gatMomentAparch <- function(shape = c(3,1), skew = 1, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard GAt distribution
# E[ (|z|-gm*z)^delta.
# Reference: Mittnik - (2000) - Conditional Density and Value-at-Risk
# Prediction of Asian Currency Exchange Rates.
if( (abs(gm) >= 1) || (delta <= 0) || (shape[1] <= 0) || (shape[2] <= 0) ||
(delta >= (shape[1])*(shape[2]) ) || (skew <= 0))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
(abs(gm) >= 1) || (delta <= 0) || (shape[1] <= 0) || (shape[2] <= 0)
(delta >= shape[1]*shape[2]) || (skew <= 0)")
nu = shape[1]
d = shape[2]
theta = skew
a = ( 1 + gm ) ^ delta * theta ^ ( - delta - 1 ) + ( 1 - gm ) ^ delta * theta ^ ( delta + 1 )
b = nu ^ ( delta / d ) * beta ( ( delta + 1 ) / d , nu - delta / d )
c = ( 1 / theta + theta ) * beta( 1 / d , nu )
# Return
a * b / c
}
# ------------------------------------------------------------------------------
.gedMomentAparch <- function(shape = 3, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard skew t-Student distribution
# E[ (|z|-gm*z)^delta.
# Reference: http://www.timberlake.co.uk/slaurent/G@RCH/Book63.html
if( (abs(gm) >= 1) || (delta <= 0))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
")
lambda = sqrt(gamma(1/shape)*2^(-2/shape)/gamma(3/shape))
((1+gm)^delta + (1-gm)^delta)*2^((delta-shape)/shape)*
gamma((delta+1)/shape)*lambda^delta/gamma(1/shape)
}
# ------------------------------------------------------------------------------
.gevMomentAparch <- function(shape = 0.3, delta = 1.2, gm = 0)
{
# FIND A BETTER WAY TO CALCULATE IT INSTEAD OF USING THE INTEGRATION FUNCTION
# Description:
# Returns the following Expectation for a standard (location zero and scale 1)
# GEV distribution
# E[ (|z|-gm*z)^delta.
# Reference: GEVStableGarch papper
if( ( abs( gm ) >= 1 ) || ( delta <= 0 ))
# if( ( abs( gm ) >= 1 ) || ( delta <= 0 ) || ( shape <= -0.5 ) )
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
( abs( gm ) >= 1 ) || ( delta <= 0 )")
xi = shape
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * dgev(x, xi = xi)
}
# Compute Moment by Integration
I = integrate(e, lower = -Inf, upper = +Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
# ------------------------------------------------------------------------------
.sstdMomentAparch <- function(shape = 4, skew = 1, delta = 1.2, gm = 0)
{
# FIND A BETTER WAY TO CALCULATE IT INSTEAD OF USING THE INTEGRATION FUNCTION
# Description:
# Returns the following Expectation for the sstd distribution implemented
# inside package fGarch
# sstd distribution
# E[ (|z|-gm*z)^delta.
# Reference: fGarch package
if( ( abs( gm ) >= 1 ) || ( delta <= 0 ) || ( shape <= 2 ) || ( skew <= 0 ))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
( abs( gm ) >= 1 ) || ( delta <= 0 ) || ( shape <= 2 ) || ( skew <= 0 )")
nu = shape
xi = skew
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * dsstd(x, nu = nu, xi = xi)
}
# Compute Moment by Integration
I = integrate(e, lower = -Inf, upper = Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
# ------------------------------------------------------------------------------
.stableS1MomentAparch <- function(shape = 1.5, skew = 0.5, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard stable distribution in S1 parametrization
# E[ (|z|-gm*z)^delta.
# This formula uses S(alpha,skew,1,0;pm) where pm = 1.
# Reference: The GEVStableGarch papper on JSS.
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta <= 1) || (delta >= shape)
Note: This function do not accept the normal case, i.e, alpha = 2")
# Calculate the expression
sigma.til <- (1 + (skew*tan(shape*pi/2))^2)^(1/2/shape)
k.shape <- shape - 2 # since skew > 1
skew.til <- 2/pi/(shape-2)*atan(skew*tan((shape-2)*pi/2))
g1 <- gamma((1/2 + skew.til*k.shape/2/shape)*(-delta))
g2 <- gamma(1/2 - skew.til*k.shape/2/shape + (1/2 + skew.til*k.shape/2/shape)*(delta+1))
g3 <- gamma((1/2 - skew.til*k.shape/2/shape)*(-delta))
g4 <- gamma(1/2 + skew.til*k.shape/2/shape + (1/2 - skew.til*k.shape/2/shape)*(delta+1))
kappa <- 1/shape/sigma.til*sigma.til^(delta+1)*gamma(delta+1)*gamma(-delta/shape)*
(1/g1/g2*(1-gm)^delta + 1/g3/g4*(1+gm)^delta)
# Return
kappa
}
# ------------------------------------------------------------------------------
.stableS0MomentAparch <- function(shape = 1.5, skew = 0.5, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for standard stable distribution in S0 parametrization
# E[ (|z|-gm*z)^delta.
# This formula uses S(alpha,skew,1,0;pm) where pm = 0.
# Reference: The GEVStableGarch papper on JSS.
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta <= 1) || (delta >= shape)
Note: This function do not accept the normal case, i.e, alpha = 2")
# Compute Moment by Integration
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * stable::dstable.quick(x = x, alpha = shape,
beta = skew, param = 0)
}
I = integrate(e, lower = -Inf, upper = +Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
.stableS2MomentAparch <- function(shape = 1.5, skew = 0.5, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for standard stable distribution in S2 parametrization
# E[ (|z|-gm*z)^delta.
# This formula uses S(alpha,skew,1,0;pm) where pm = 2.
# Reference: The GEVStableGarch papper on JSS.
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta <= 1) || (delta >= shape)
Note: This function do not accept the normal case, i.e, alpha = 2")
# Compute Moment by Integration
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * stable::dstable.quick(x = x, alpha = shape,
beta = skew, param = 2)
}
I = integrate(e, lower = -Inf, upper = +Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
.gevMomentAparch <- function(shape = 0.3, delta = 1.2, gm = 0)
{
# FIND A BETTER WAY TO CALCULATE IT INSTEAD OF USING THE INTEGRATION FUNCTION
# Description:
# Returns the following Expectation for a standard (location zero and scale 1)
# GEV distribution
# E[ (|z|-gm*z)^delta.
# Reference: GEVStableGarch papper
if( ( abs( gm ) >= 1 ) || ( delta <= 0 ))
# if( ( abs( gm ) >= 1 ) || ( delta <= 0 ) || ( shape <= -0.5 ) )
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
( abs( gm ) >= 1 ) || ( delta <= 0 )")
xi = shape
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * dgev(x, xi = xi)
}
# Compute Moment by Integration
I = integrate(e, lower = -Inf, upper = +Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
# ------------------------------------------------------------------------------
# ------------------------------------------------------------------------------
.stableS1SymmetricMomentGarch <- function(shape = 1.5)
{
# See Mittinik et al. (1995) for the definition of
# the stable garch model with conditional symmetric stable distribution
# This formula uses S(alpha,skew,1,0;pm) where pm = 1.
if( (shape <= 1) || (shape > 2) )
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 0) || (shape > 2)")
# Return
gamma(1 - 1/shape)*2/pi
}
# ------------------------------------------------------------------------------
.stableS1SymmetricMomentAparch <- function(shape = 1.5, delta = 1.2, gm = 0)
{
# See Diongue - 2008 (An investigation of the stable-Paretian Asymmetric Power GARCH model)
# This formula uses S(alpha,0,1,0;pm) where pm = 1.
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || (delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || (delta >= shape))")
# k(delta): At this point we have delta > 1
if(delta == 1)
k = pi/2
else
k = gamma(1-delta)*cos(pi*delta/2)
result = k^(-1)*gamma(1-delta/shape)*1/2*((1+gm)^delta+(1-gm)^delta)
# Return
result
}
# ------------------------------------------------------------------------------
.stableS1MomentPowerGarch <- function(shape = 1.5, skew = 0.5, delta = 1.2)
{
# See Mittnik et al. (2002) - Stationarity of stable power-GARCH processes
# This formula uses S(alpha,skew,1,0;pm) where pm = 1 (tests reveald)
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || ( abs(skew) > 1) ||
(delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || ( abs(skew) > 1) ||
(delta >= shape)")
# k(delta): At this point we have delta > 1
if(delta == 1)
k = pi/2
else
k = gamma(1-delta)*cos(pi*delta/2)
# tal(shape,skew)
tau = skew*tan(shape*pi/2)
# lambda(shape,skew,delta): eq (10) Mittnik et al. (2002)
lambda = k^(-1)*gamma(1-delta/shape)*(1 + tau^2)^(delta/2/shape)*cos(delta/shape*atan(tau))
lambda
}
# ------------------------------------------------------------------------------
# ------------------------------------------------------------------------------
.trueAparchMomentsWurtz <-
function(fun = "norm", gm = 0, delta = 1, lower = -Inf, upper = Inf, ...)
{
# This function was originally from package fGarch. We
# changed it to return only the aparch moment expression.
# Description:
# Computes APARCH moments using Integration
# Arguments:
# fun - name of density functions of APARCH innovations
# alpha, gm - numeric value or vector of APARCH coefficients,
# must be of same length
# beta - numeric value or vector of APARCH coefficients
# delta - numeric value of APARCH exponent
# Note:
# fun is one of: norm, snorn, std, sstd, ged, sged, snig
# FUNCTION:
# Match Density Function:
fun = match.fun(fun)
# Persisgtence Function: E(|z|-gm z)^delta
e = function(x, gm, delta, ...) {
(abs(x)-gm*x)^delta * fun(x, ...)
}
# Compute Moment by Integration
I = integrate(e, lower = lower, upper = upper, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta, ...)
# Return Value:
I
}
################################################################################
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# Copyrights (C) 2014 Thiago do Rego Sousa <thiagoestatistico@gmail.com>
# This library is free software; you can redistribute it and/or
# modify it under the terms of the GNU Library General Public
# License as published by the Free Software Foundation; either
# version 2 of the License, or (at your option) any later version.
#
# This library is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Library General Public License for more details.
#
# You should have received a copy of the GNU Library General
# Public License along with this library; if not, write to the
# Free Foundation, Inc., 59 Temple Place, Suite 330, Boston,
# MA 02111-1307 USA
####################################################################################
# FUNCTION: DESCRIPTION:
#
# .stationarityAparch Compute the stationarity condition for the
# GARCH or APARCH model with several cond.
# distributions
#
# gsMomentAparch Evaluate the exact moments of the type
# E(z - gm|z|)^delta for several conditional
# distributions.
#
# .normMomentAparch Exact APARCH moments for the standard normal
#
# .stdMomentAparch Exact APARCH moments for the standard t-Student
#
# .skstdMomentAparch Exact APARCH moments for the standard skew
# t-Student defined by Fernandez and Steel (1998).
# Notice that this distribution is standardized
# with location zero and unit scale, but it is not
# standardized in the sense that it has a zero mean and
# unit variance. The exact formula for this expression
# would be very complicated to compute if we reescale
# the original distribution to (X - mu) / sigma and
# therefore, we choose to work with the raw distribution
# defined in Fernandez and Steel without this reparametrization
#
# .gedMomentAparch Exact APARCH moments for the standard GED distribution
# .gatMomentAparch Exact APARCH moments for the standard GAt distribution
#
# .stableS1MomentAparch Exact APARCH moments for the location zero and unit scale
# in S1 parametrization (see Nolan (1999)).
#
# .stableS1SymmetricMomentGarch Exact GARCH moments ( E |zt| for the symmetric stable (S1)
# distribution (S1 parametrization)
#
# .stableS1SymmetricMomentAparch Exact APARCH moments for the symmetric stable (S1) distribution
#
# .stableS1MomentPowerGarch Exact power-GARCH moments for the asymmetric stable (S1) distribution
#
# .trueAparchMomentsWurtz Function that evaluate moments with numerical integration.
# This function was used only to test the mathematical
# formulas implemented here
####################################################################################
.stationarityAparch <-
function (model = list(),
formula,
cond.dist = c("stableS1", "gev", "gat", "norm", "std", "sstd", "skstd", "ged"))
{
# Description:
# A function that evaluate stationarity conditions
# to guide parameter estimation in GARCH/APARCH models
# Make sure we have a garch or aparch with p > 0
# The stationarity condition for the APARCH is:
# sum ( E[ |z|-gm*z ] ^ delta * alpha + beta ) < 1
# and for the GARCH model is
# sum ( E[ |z| ] ^ delta * alpha + beta ) < 1
# where delta is usually equal to 2, except for the
# stable model where delta = 1.
# The expectation E[ |z| ] is equal to one for the following
# distributions: norm, std, sstd and ged.
# For the GAt, gev and stable conditional distribution we need to
# calculate this expectation, even for the GARCH model.
# Arguments:
# formula - an object returned by function .getFormula
# parm - a list with the model parameters as entries
# alpha - a vector of autoregressive coefficients
# of length p for the GARCH/APARCH specification,
# gm - a vector of leverage coefficients of
# length p for the APARCH specification,
# beta - a vector of moving average coefficients of
# length q for the GARCH/APARCH specification,
# delta - the exponent value used in the variance equation.
# skew - a numeric value listing the distributional
# skewness parameter.
# shape - a numeric value listing the distributional
# shape parameter.
# cond.dist - a character string naming the distribution
# function.
# Return Values:
# sum ( E[ |z|-gm*z ] ^ delta * alpha + beta ) - When it can be computed
# 1e99 - In case we the expectation E[ |z|-gm*z ] ^ delta could not be evaluated.
# FUNCTION:
# get parameters
alpha <- model$alpha
beta <- model$beta
delta <- model$delta
gm <- model$gm
skew <- model$skew
shape <- model$shape
# Conditional distribution
cond.dist = match.arg(cond.dist)
#print(t(model))
if(length(alpha) == 0 || length(alpha) != length(gm) || length(delta) != 1)
stop("Failed to verify conditions:
if(length(alpha) == 0 || length(delta) != 1)")
# We must have a model with a non zero garch/aparch order
if(formula$formula.order[3] == 0)
stop("Invalid model")
# garch model: sum ( E[ |z| ] ^ delta * alpha + beta ) < 1
if(formula$isAPARCH == FALSE)
{
if (any( c("norm", "std", "sstd", "ged") == cond.dist))
return(sum(alpha) + sum(beta))
if(cond.dist == "gev")
kappa = try(.gevMomentAparch(shape = shape, delta = 2, gm = 0), silent = TRUE)
if(cond.dist == "stableS1")
kappa = try(.stableS1MomentPowerGarch (shape = shape, skew = skew,
delta = 1), silent = TRUE)
if(cond.dist == "gat")
kappa = try(.gatMomentAparch(shape = shape, skew = skew,
delta = 2, gm = 0), silent = TRUE)
if(cond.dist == "skstd")
kappa = try(.skstdMomentAparch(shape = shape, skew = skew, delta = 2, gm = 0), silent = TRUE)
if( is.numeric(kappa))
return(kappa*sum(alpha) + sum(beta))
else
return(1e99)
} else {
# aparch model
kappa = rep(0,length(alpha))
for( i in 1:length(alpha))
{
if(cond.dist == "stableS1")
kappa[i] = try(.stableS1MomentAparch (shape = shape, skew = skew,
delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "gev")
kappa[i] = try(.gevMomentAparch(shape = shape, delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "gat")
kappa[i] = try(.gatMomentAparch(shape = shape, delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "norm")
kappa[i] = try(.normMomentAparch (delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "std")
kappa[i] = try(.stdMomentAparch(shape = shape, delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "skstd")
kappa[i] = try(.skstdMomentAparch(shape = shape, skew = skew, delta = delta, gm = gm[i]), silent = TRUE)
if(cond.dist == "ged")
kappa[i] = try(.gedMomentAparch(shape = shape, delta = delta, gm = gm[i]), silent = TRUE)
}
if( is.numeric(kappa) )
return(sum(kappa*alpha) + sum(beta))
else
return(1e99)
}
}
# ------------------------------------------------------------------------------
gsMomentAparch <- function(
cond.dist = c("stableS1", "gev", "gat", "norm", "std", "sstd", "skstd", "ged"),
shape = 1.5,
skew = 0,
delta = 1,
gm = 0)
{
# conditional distribution
cond.dist = match.arg(cond.dist)
if(cond.dist == "stableS1")
kappa = .stableS1MomentAparch (shape = shape, skew = skew,
delta = delta, gm = gm)
if(cond.dist == "gev")
kappa = .gevMomentAparch(shape = shape, delta = delta, gm = gm)
if(cond.dist == "gat")
kappa = .gatMomentAparch(shape = shape, delta = delta, skew = skew, gm = gm)
if(cond.dist == "norm")
kappa = .normMomentAparch (delta = delta, gm = gm)
if(cond.dist == "std")
kappa = .stdMomentAparch(shape = shape, delta = delta, gm = gm)
if(cond.dist == "sstd")
kappa = .sstdMomentAparch(shape = shape, skew = skew, delta = delta, gm = gm)
if(cond.dist == "skstd")
kappa = .skstdMomentAparch(shape = shape, skew = skew, delta = delta, gm = gm)
if(cond.dist == "ged")
kappa = .gedMomentAparch(shape = shape, delta = delta, gm = gm)
# Return
kappa
}
# ------------------------------------------------------------------------------
.normMomentAparch <- function(delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard normal distribution
# E[ (|z|-gm*z)^delta.
# Reference: A long memory property of stock market returns and a
# new model. Ding, Granger and Engle (1993), Appendix B.
# Error treatment of input parameters
if( ( abs ( gm ) >= 1 ) || ( delta <= 0 ) )
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following condition cannot be true
( abs ( gm ) >= 1 ) || ( delta <= 0 )")
result = 1 / sqrt ( 2 * pi ) * ( (1 + gm ) ^ delta + ( 1 - gm ) ^ delta ) *
2 ^ ( ( delta - 1 ) / 2 ) * gamma ( ( delta + 1 ) / 2 )
# Return
result
}
# ------------------------------------------------------------------------------
.stdMomentAparch <- function(shape = 3, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard t-Student distribution
# E[ (|z|-gm*z)^delta.
# Reference: Mittnik - (2000) - Conditional Density and Value-at-Risk
# Prediction of Asian Currency Exchange Rates
# Note that the formula provided by Mittnik is valid for the original t-Student
# distribution (which is nonStandardized).
# Since our objective is to calculate it for the standart t-Student distribution
# defined in Wurtz et al. (2006) - eq (13) we need to multiply the Mittnik
# formula by (sqrt((shape-2)/shape))^delta.
# The main drawback is that the standardized t-Student distribution
# is valid only for shape > 2 ( in which case the variance is finite)
# Error treatment of input parameters
if( (abs(gm) >= 1) || (delta <= 0) || (shape <= 2) || (delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
(abs(gm) >= 1) || (delta <= 0) || (shape <= 0)|| (delta >= shape)")
mittnikFormula = shape^(delta/2)*1/2/sqrt(pi)*( (1+gm)^delta+(1-gm)^delta )*
gamma((delta+1)/2)*(gamma(shape/2))^(-1)*gamma((shape-delta)/2)
factorToMultiply = (sqrt((shape-2)/shape))^delta
# Return
mittnikFormula*factorToMultiply
}
# ------------------------------------------------------------------------------
.skstdMomentAparch <- function(shape = 3, skew = 1, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard skew t-Student distribution
# E[ (|z|-gm*z)^delta.
# Reference: fGarch: Wurtz and the package Skewt uses the one defined
# in Fernandez, C. and Steel, M. F. J. (1998). On Bayesian modeling of fat
# tails and skewness, J. Am. Statist. Assoc. 93, 359-371.
# Another reference: http://www.timberlake.co.uk/slaurent/G@RCH/Book63.html
# Note: This is not the APARCH moments for the 'dsstd' function from fGarch package.
# This is the skew t-Student defined by Fernandez and Steel without the
# reparametrization introduced by Wurtz et al. (2006).
# but theThe point here is that the SPLUS FinMetrics
# also uses this distribution without reparametrization.
# Indeed, Mittnik et al. (2000) also uses his t3-distribution (GAt) without such
# reparametrization. These distributions are in fact standardized, but the
# point is that the assymetry parameter plays a crucial role in the calculation
# of the APARCH moments.
if( (abs(gm) >= 1) || (delta <= 0) || (shape <= 2) ||
(delta >= shape) || (skew <= 0))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
(abs(gm) >= 1) || (delta <= 0) || (shape <= 2) ||
(delta >= shape) || (skew <= 0)")
a = skew^(-1-delta)*(1+gm)^delta + skew^(1+delta)*(1-gm)^delta
b = gamma((delta+1)/2)*gamma((shape-delta)/2)*(shape-2)^((1+delta)/2)
c = (skew + 1/skew)*sqrt((shape-2)*pi)*gamma(shape/2)
# Return
a*b/c
}
# ------------------------------------------------------------------------------
.gatMomentAparch <- function(shape = c(3,1), skew = 1, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard GAt distribution
# E[ (|z|-gm*z)^delta.
# Reference: Mittnik - (2000) - Conditional Density and Value-at-Risk
# Prediction of Asian Currency Exchange Rates.
if( (abs(gm) >= 1) || (delta <= 0) || (shape[1] <= 0) || (shape[2] <= 0) ||
(delta >= (shape[1])*(shape[2]) ) || (skew <= 0))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
(abs(gm) >= 1) || (delta <= 0) || (shape[1] <= 0) || (shape[2] <= 0)
(delta >= shape[1]*shape[2]) || (skew <= 0)")
nu = shape[1]
d = shape[2]
theta = skew
a = ( 1 + gm ) ^ delta * theta ^ ( - delta - 1 ) + ( 1 - gm ) ^ delta * theta ^ ( delta + 1 )
b = nu ^ ( delta / d ) * beta ( ( delta + 1 ) / d , nu - delta / d )
c = ( 1 / theta + theta ) * beta( 1 / d , nu )
# Return
a * b / c
}
# ------------------------------------------------------------------------------
.gedMomentAparch <- function(shape = 3, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard skew t-Student distribution
# E[ (|z|-gm*z)^delta.
# Reference: http://www.timberlake.co.uk/slaurent/G@RCH/Book63.html
if( (abs(gm) >= 1) || (delta <= 0))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
")
lambda = sqrt(gamma(1/shape)*2^(-2/shape)/gamma(3/shape))
((1+gm)^delta + (1-gm)^delta)*2^((delta-shape)/shape)*
gamma((delta+1)/shape)*lambda^delta/gamma(1/shape)
}
# ------------------------------------------------------------------------------
.gevMomentAparch <- function(shape = 0.3, delta = 1.2, gm = 0)
{
# FIND A BETTER WAY TO CALCULATE IT INSTEAD OF USING THE INTEGRATION FUNCTION
# Description:
# Returns the following Expectation for a standard (location zero and scale 1)
# GEV distribution
# E[ (|z|-gm*z)^delta.
# Reference: GEVStableGarch papper
if( ( abs( gm ) >= 1 ) || ( delta <= 0 ))
# if( ( abs( gm ) >= 1 ) || ( delta <= 0 ) || ( shape <= -0.5 ) )
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
( abs( gm ) >= 1 ) || ( delta <= 0 )")
xi = shape
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * dgev(x, xi = xi)
}
# Compute Moment by Integration
I = integrate(e, lower = -Inf, upper = +Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
# ------------------------------------------------------------------------------
.sstdMomentAparch <- function(shape = 4, skew = 1, delta = 1.2, gm = 0)
{
# FIND A BETTER WAY TO CALCULATE IT INSTEAD OF USING THE INTEGRATION FUNCTION
# Description:
# Returns the following Expectation for the sstd distribution implemented
# inside package fGarch
# sstd distribution
# E[ (|z|-gm*z)^delta.
# Reference: fGarch package
if( ( abs( gm ) >= 1 ) || ( delta <= 0 ) || ( shape <= 2 ) || ( skew <= 0 ))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
( abs( gm ) >= 1 ) || ( delta <= 0 ) || ( shape <= 2 ) || ( skew <= 0 )")
nu = shape
xi = skew
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * dsstd(x, nu = nu, xi = xi)
}
# Compute Moment by Integration
I = integrate(e, lower = -Inf, upper = Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
# ------------------------------------------------------------------------------
.stableS1MomentAparch <- function(shape = 1.5, skew = 0.5, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for a standard stable distribution in S1 parametrization
# E[ (|z|-gm*z)^delta.
# This formula uses S(alpha,skew,1,0;pm) where pm = 1.
# Reference: The GEVStableGarch papper on JSS.
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta <= 1) || (delta >= shape)
Note: This function do not accept the normal case, i.e, alpha = 2")
# Calculate the expression
sigma.til <- (1 + (skew*tan(shape*pi/2))^2)^(1/2/shape)
k.shape <- shape - 2 # since skew > 1
skew.til <- 2/pi/(shape-2)*atan(skew*tan((shape-2)*pi/2))
g1 <- gamma((1/2 + skew.til*k.shape/2/shape)*(-delta))
g2 <- gamma(1/2 - skew.til*k.shape/2/shape + (1/2 + skew.til*k.shape/2/shape)*(delta+1))
g3 <- gamma((1/2 - skew.til*k.shape/2/shape)*(-delta))
g4 <- gamma(1/2 + skew.til*k.shape/2/shape + (1/2 - skew.til*k.shape/2/shape)*(delta+1))
kappa <- 1/shape/sigma.til*sigma.til^(delta+1)*gamma(delta+1)*gamma(-delta/shape)*
(1/g1/g2*(1-gm)^delta + 1/g3/g4*(1+gm)^delta)
# Return
kappa
}
# ------------------------------------------------------------------------------
.stableS0MomentAparch <- function(shape = 1.5, skew = 0.5, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for standard stable distribution in S0 parametrization
# E[ (|z|-gm*z)^delta.
# This formula uses S(alpha,skew,1,0;pm) where pm = 0.
# Reference: The GEVStableGarch papper on JSS.
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta <= 1) || (delta >= shape)
Note: This function do not accept the normal case, i.e, alpha = 2")
# Compute Moment by Integration
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * stable::dstable.quick(x = x, alpha = shape,
beta = skew, param = 0)
}
I = integrate(e, lower = -Inf, upper = +Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
.stableS2MomentAparch <- function(shape = 1.5, skew = 0.5, delta = 1.2, gm = 0)
{
# Description:
# Returns the following Expectation for standard stable distribution in S2 parametrization
# E[ (|z|-gm*z)^delta.
# This formula uses S(alpha,skew,1,0;pm) where pm = 2.
# Reference: The GEVStableGarch papper on JSS.
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || ( abs(skew) > 1) || (abs(gm) >= 1) ||
(delta <= 1) || (delta >= shape)
Note: This function do not accept the normal case, i.e, alpha = 2")
# Compute Moment by Integration
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * stable::dstable.quick(x = x, alpha = shape,
beta = skew, param = 2)
}
I = integrate(e, lower = -Inf, upper = +Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
.gevMomentAparch <- function(shape = 0.3, delta = 1.2, gm = 0)
{
# FIND A BETTER WAY TO CALCULATE IT INSTEAD OF USING THE INTEGRATION FUNCTION
# Description:
# Returns the following Expectation for a standard (location zero and scale 1)
# GEV distribution
# E[ (|z|-gm*z)^delta.
# Reference: GEVStableGarch papper
if( ( abs( gm ) >= 1 ) || ( delta <= 0 ))
# if( ( abs( gm ) >= 1 ) || ( delta <= 0 ) || ( shape <= -0.5 ) )
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true
( abs( gm ) >= 1 ) || ( delta <= 0 )")
xi = shape
e = function(x, gm, delta) {
(abs(x)-gm*x)^delta * dgev(x, xi = xi)
}
# Compute Moment by Integration
I = integrate(e, lower = -Inf, upper = +Inf, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta)
# Return
as.numeric(I[1])
}
# ------------------------------------------------------------------------------
# ------------------------------------------------------------------------------
.stableS1SymmetricMomentGarch <- function(shape = 1.5)
{
# See Mittinik et al. (1995) for the definition of
# the stable garch model with conditional symmetric stable distribution
# This formula uses S(alpha,skew,1,0;pm) where pm = 1.
if( (shape <= 1) || (shape > 2) )
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 0) || (shape > 2)")
# Return
gamma(1 - 1/shape)*2/pi
}
# ------------------------------------------------------------------------------
.stableS1SymmetricMomentAparch <- function(shape = 1.5, delta = 1.2, gm = 0)
{
# See Diongue - 2008 (An investigation of the stable-Paretian Asymmetric Power GARCH model)
# This formula uses S(alpha,0,1,0;pm) where pm = 1.
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || (delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || (delta >= shape))")
# k(delta): At this point we have delta > 1
if(delta == 1)
k = pi/2
else
k = gamma(1-delta)*cos(pi*delta/2)
result = k^(-1)*gamma(1-delta/shape)*1/2*((1+gm)^delta+(1-gm)^delta)
# Return
result
}
# ------------------------------------------------------------------------------
.stableS1MomentPowerGarch <- function(shape = 1.5, skew = 0.5, delta = 1.2)
{
# See Mittnik et al. (2002) - Stationarity of stable power-GARCH processes
# This formula uses S(alpha,skew,1,0;pm) where pm = 1 (tests reveald)
# Error treatment of input parameters
if( (shape <= 1) || (shape > 2) || ( abs(skew) > 1) ||
(delta >= shape))
stop("Invalid parameters to calculate the expression E[ (|z|-gm*z)^delta ].
The following conditions cannot be true.
(shape <= 1) || (shape > 2) || ( abs(skew) > 1) ||
(delta >= shape)")
# k(delta): At this point we have delta > 1
if(delta == 1)
k = pi/2
else
k = gamma(1-delta)*cos(pi*delta/2)
# tal(shape,skew)
tau = skew*tan(shape*pi/2)
# lambda(shape,skew,delta): eq (10) Mittnik et al. (2002)
lambda = k^(-1)*gamma(1-delta/shape)*(1 + tau^2)^(delta/2/shape)*cos(delta/shape*atan(tau))
lambda
}
# ------------------------------------------------------------------------------
# ------------------------------------------------------------------------------
.trueAparchMomentsWurtz <-
function(fun = "norm", gm = 0, delta = 1, lower = -Inf, upper = Inf, ...)
{
# This function was originally from package fGarch. We
# changed it to return only the aparch moment expression.
# Description:
# Computes APARCH moments using Integration
# Arguments:
# fun - name of density functions of APARCH innovations
# alpha, gm - numeric value or vector of APARCH coefficients,
# must be of same length
# beta - numeric value or vector of APARCH coefficients
# delta - numeric value of APARCH exponent
# Note:
# fun is one of: norm, snorn, std, sstd, ged, sged, snig
# FUNCTION:
# Match Density Function:
fun = match.fun(fun)
# Persisgtence Function: E(|z|-gm z)^delta
e = function(x, gm, delta, ...) {
(abs(x)-gm*x)^delta * fun(x, ...)
}
# Compute Moment by Integration
I = integrate(e, lower = lower, upper = upper, subdivisions = 1000,
rel.tol = .Machine$double.eps^0.25,
gm = gm, delta = delta, ...)
# Return Value:
I
}
################################################################################
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