Package 'fGarch'

Modelling heterskedasticity in financial time series

Description

The Rmetrics fGarch package is a collection of functions to analyze and model heteroskedastic behavior in financial time series.

1 Introduction

GARCH, Generalized Autoregressive Conditional Heteroskedastic, models have become important in the analysis of time series data, particularly in financial applications when the goal is to analyze and forecast volatility.

For this purpose, the family of GARCH functions offers functions for simulating, estimating and forecasting various univariate GARCH-type time series models in the conditional variance and an ARMA specification in the conditional mean. The function garchFit is a numerical implementation of the maximum log-likelihood approach under different assumptions, Normal, Student-t, GED errors or their skewed versions. The parameter estimates are checked by several diagnostic analysis tools including graphical features and hypothesis tests. Functions to compute n-step ahead forecasts of both the conditional mean and variance are also available.

The number of GARCH models is immense, but the most influential models were the first. Beside the standard ARCH model introduced by Engle [1982] and the GARCH model introduced by Bollerslev [1986], the function garchFit also includes the more general class of asymmetric power ARCH models, named APARCH, introduced by Ding, Granger and Engle [1993]. The APARCH models include as special cases the TS-GARCH model of Taylor [1986] and Schwert [1989], the GJR-GARCH model of Glosten, Jaganathan, and Runkle [1993], the T-ARCH model of Zakoian [1993], the N-ARCH model of Higgins and Bera [1992], and the Log-ARCH model of Geweke [1986] and Pentula [1986].

There exist a collection of review articles by Bollerslev, Chou and Kroner [1992], Bera and Higgins [1993], Bollerslev, Engle and Nelson [1994], Engle [2001], Engle and Patton [2001], and Li, Ling and McAleer [2002] which give a good overview of the scope of the research.

2 Time series simulation

Functions to simulate artificial GARCH and APARCH time series processes.

garchSpec	specifies an univariate GARCH time series model
garchSim	simulates a GARCH/APARCH process

3 Parameter estimation

Functions to fit the parameters of GARCH and APARCH time series processes.

garchFit

fits the parameters of a GARCH process

Extractor Functions:

residuals	extracts residuals from a fitted "fGARCH" object
fitted	extracts fitted values from a fitted "fGARCH" object
volatility	extracts conditional volatility from a fitted "fGARCH" object
coef	extracts coefficients from a fitted "fGARCH" object
formula	extracts formula expression from a fitted "fGARCH" object

4 Forecasting

Functions to forcecast mean and variance of GARCH and APARCH processes.

predict

forecasts from an object of class "fGARCH"

5 Standardized distributions

This section contains functions to model standardized distributions.

Skew normal distribution:

[dpqr]norm	Normal distribution (base R)
[dpqr]snorm	Skew normal distribution
snormFit	fits parameters of Skew normal distribution

Skew generalized error distribution:

[dpqr]ged	Generalized error distribution
[dpqr]sged	Skew Generalized error distribution
gedFit	fits parameters of Generalized error distribution
sgedFit	fits parameters of Skew generalized error distribution

Skew standardized Student-t distribution:

[dpqr]std	Standardized Student-t distribution
[dpqr]sstd	Skew standardized Student-t distribution
stdFit	fits parameters of Standardized Student-t distribution
sstdFit	fits parameters of Skew standardized Student-t distribution

Absolute moments:

absMoments

computes absolute moments of these distribution

About Rmetrics

The fGarch Rmetrics package is written for educational support in teaching "Computational Finance and Financial Engineering" and licensed under the GPL.

Author(s)

Diethelm Wuertz [aut] (original code), Yohan Chalabi [aut], Tobias Setz [aut], Martin Maechler [ctb] (<https://orcid.org/0000-0002-8685-9910>), Chris Boudt [ctb] Pierre Chausse [ctb], Michal Miklovac [ctb], Georgi N. Boshnakov [cre, ctb]

Maintainer: Georgi N. Boshnakov <[email protected]>

---

Absolute moments of GARCH distributions

Description

Computes absolute moments of the standard normal, standardized GED, and standardized skew Student-t distributions.

Usage

absMoments(n, density = c("dnorm", "dged", "dstd"), ...)

Arguments

n	the order of the absolute moment, can be a vector to compute several absolute moments at once.
density	a character string naming a symmetric density function.
...	parameters passed to the density function.

Details

absMoments returns a numeric vector of length n with the values of the absolute moments, as specified by n, of the selected probability density function (pdf).

If density names one of the densities in the signature of absMoments, the moments are calculated from known formulas.

Otherwise, numerical integration is used and an attribute is attached to the results to report an estimate of the error. Note that the density is assumed symmetric wihtout a check.

Value

a numeric vector

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

ged, std

Examples

## absMoment -

absMoments(1, "dstd", nu = 6)
absMoments(1, "dstd", nu = 600)
absMoments(1, "dstd", nu = 60000)
absMoments(1, "dstd", nu = 600000)

absMoments(1, "dnorm")

## excess kurtosis of t_nu is  6/(nu - 4)
nu <- 6
absMoments(2*2, "dstd", nu = nu) / absMoments(2*1, "dstd", nu = nu)^2 - 3
6/(nu-4)

## 4th moment for t_4 is infinite
absMoments(4, "dstd", nu = 4)

absMoments(1, "dged", nu = 4)

---

GARCH coefficients methods

Description

Coefficients methods coef() for GARCH Models.

Methods

Methods for coef defined in package fGarch:

object = "fGARCH"

Extractor function for coefficients from a fitted GARCH model.

object = "fGARCHSPEC"

Extractor function for coefficients from a GARCH specification structure.

Note

coef is a generic function which extracts coefficients from objects returned by modeling functions.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

Examples

## garchSpec -
   # Use default parameters beside alpha:
   spec = garchSpec(model = list(alpha = c(0.05, 0.05)))
   spec
   coef(spec)

## garchSim -
   # Simulate an univariate "timeSeries" series from specification 'spec':
   x = garchSim(spec, n = 2000)
   x = x[,1]

## garchFit --
   fit = garchFit( ~ garch(1, 1), data = x, trace = FALSE)

## coef -
   coef(fit)

---

Class "fGARCH"

Description

The class 'fGARCH' represents a model of an heteroskedastic time series process.

Objects from the Class

Objects can be created by calls of the function garchFit. This object is a parameter estimate of an empirical GARCH process.

Slots

call:

Object of class "call": the call of the garch function.

formula:

Object of class "formula": a formula object specifying the mean and variance equations.

method:

Object of class "character": a string denoting the optimization method, by default "Max Log-Likelihood Estimation".

data:

Object of class "list": a list with one entry named x, containing the data of the time series to be estimated, the same as given by the input argument series.

fit:

Object of class "list": a list with the results from the parameter estimation. The entries of the list depend on the selected algorithm, see below.

residuals:

Object of class "numeric": a numeric vector with the (raw, unstandardized) residual values.

fitted:

Object of class "numeric": a numeric vector with the fitted values.

h.t:

Object of class "numeric":

a numeric vector with the conditional variances ($h_t = \sigma_t^\delta$).

sigma.t:

Object of class "numeric": a numeric vector with the conditional standard deviations.

title:

Object of class "character": a title string.

description:

Object of class "character": a string with a brief description.

Methods

plot

signature(x = "fGARCH", y = "missing"): plots an object of class "fGARCH".

show

signature(object = "fGARCH"): prints an object of class "fGARCH".

summary

signature(object = "fGARCH"): summarizes an object of class "fGARCH".

predict

signature(object = "fGARCH"): forecasts mean and volatility from an object of class "fGARCH".

fitted

signature(object = "fGARCH"): extracts fitted values from an object of class "fGARCH".

residuals

signature(object = "fGARCH"): extracts fresiduals from an object of class "fGARCH".

volatility

signature(object = "fGARCH"): extracts conditional volatility from an object of class "fGARCH".

coef

signature(object = "fGARCH"): extracts fitted coefficients from an object of class "fGARCH".

formula

signature(x = "fGARCH"): extracts formula expression from an object of class "fGARCH".

Author(s)

Diethelm Wuertz and Rmetrics Core Team

See Also

garchFit, garchSpec, garchFitControl

Examples

## simulate a time series, fit a GARCH(1,1) model, and show it:
x <- garchSim( garchSpec(), n = 500)
fit <- garchFit(~ garch(1, 1), data = x, trace = FALSE)
fit # == print(fit) and also == show(fit)

---

Time series datasets

Description

Datasets used in the examples, including DEM/GBP foreign exchange rates and data on SP500 index.

Format

dem2gbp is a data frame with one column "DEM2GBP" and 1974 rows (observations).

sp500dge is a data frame with one column "SP500DGE" and 17055 rows (observations).

Details

The data represent retuns. No further details have been recorded.

Further datasets are available in the packages that fGarch imports, see fBasicsData and TimeSeriesData.

See Also

data(package = "fBasics") and data(package = "timeSeries") for related datasets

Examples

data(dem2gbp)
head(dem2gbp)
tail(dem2gbp)
str(dem2gbp)
plot(dem2gbp[[1]])

data(sp500dge)
head(sp500dge)
tail(sp500dge)
str(sp500dge)
plot(sp500dge[[1]])

---

Class "fGARCHSPEC"

Description

Specification structure for an univariate GARCH time series model.

Objects from the Class

Objects can be created by calls of the function garchSpec. This object specifies the parameters of an empirical GARCH process.

Slots

call:

Object of class "call": the call of the garch function.

formula:

Object of class "formula": a list with two formula entries for the mean and variance equation.

model:

Object of class "list": a list with the model parameters.

presample:

Object of class "matrix": a numeric matrix with presample values.

distribution:

Object of class "character": a character string with the name of the conditional distribution.

rseed:

Object of class "numeric": an integer with the random number generator seed.

Methods

show

signature(object = "fGARCHSPEC"): prints an object of class 'fGARCHSPEC'.

Note

With Rmetrics Version 2.6.1 the class has been renamed from "garchSpec" to "fGARCHSPEC".

Author(s)

Diethelm Wuertz for the Rmetrics R-port

Examples

## garchSpec -
   spec = garchSpec()
   spec # print() or show() it

---

Extract GARCH model fitted values

Description

Extracts fitted values from a fitted GARCH object.

Details

The method for "fGARCH" objects extracts the @fitted value slot from an object of class "fGARCH" as returned by the function garchFit.

Methods

Methods for fitted defined in package fGarch:

object = "fGARCH"

Extractor function for fitted values.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

See Also

predict, residuals, garchFit, class fGARCH,

plot

Examples

## see examples for 'residuals()'

---

Extract GARCH model formula

Description

Extracts formula from a formula GARCH object.

Details

formula is a generic function which extracts the formula expression from objects returned by modeling functions.

The "fGARCH" method extracts the @formula expression slot from an object of class "fGARCH" as returned by the function garchFit.

The returned formula has always a left hand side. If the argument data was an univariate time series and no name was specified to the series, then the left hand side is assigned the name of the data.set. In the multivariate case the rectangular data object must always have column names, otherwise the fitting will be stopped with an error message

The class of the returned value depends on the input to the function garchFit who created the object. The returned value is always of the same class as the input object to the argument data in the function garchFit, i.e. if you fit a "timeSeries" object, you will get back from the function fitted also a "timeSeries" object, if you fit an object of class "zoo", you will get back again a "zoo" object. The same holds for a "numeric" vector, for a "data.frame", and for objects of class "ts", "mts".

In contrast, the slot itself returns independent of the class of the data input always a numeric vector, i.e. the function call rslot(object, "fitted") will return a numeric vector.

Methods

Methods for formula defined in package fGarch:

object = "fGARCH"

Extractor function for formula expression.

Note

(GNB) Contrary to the description of the returned value of the "fGARCH" method, it is always "numeric".

TODO: either implement the documented behaviour or fix the documentation.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

See Also

garchFit, class fGARCH

Examples

## garchFit -
   fit = garchFit(~garch(1, 1), data = garchSim(), trace = FALSE)
   
## formula - 
   formula(fit)

## A Bivariate series and mis-specified formula:
   x = garchSim(n = 500)
   y = garchSim(n = 500)
   z = cbind(x, y)
   colnames(z)
   class(z)
   ## Not run: 
   garchFit(z ~garch(1, 1), data = z, trace = FALSE)
   
## End(Not run)
   # Returns:
   # Error in .garchArgsParser(formula = formula, data = data, trace = FALSE) :  
   #   Formula and data units do not match.
   
## Doubled column names in data set - formula can't fit:
   colnames(z) <- c("x", "x")
   z[1:6,]
   ## Not run: 
   garchFit(x ~garch(1, 1), data = z, trace = FALSE)
   
## End(Not run)
   # Again the error will be noticed:
   # Error in garchFit(x ~ garch(1, 1), data = z) : 
   #   Column names of data are not unique.

## Missing column names in data set - formula can't fit:
   z.mat <- as.matrix(z)
   colnames(z.mat) <- NULL
   z.mat[1:6,]
   ## Not run: 
   garchFit(x ~ garch(1, 1), data = z.mat, trace = FALSE)
   
## End(Not run)
   # Again the error will be noticed:
   # Error in .garchArgsParser(formula = formula, data = data, trace = FALSE) : 
   #   Formula and data units do not match

---

Class 'fUGARCHSPEC'

Description

Class 'fUGARCHSPEC'.

Objects from the Class

Objects can be created by calls of the form new("fUGARCHSPEC", ...).

Slots

model:

Object of class "list" ~~

distribution:

Object of class "list" ~~

optimization:

Object of class "list" ~~

documentation:

Object of class "list" ~~

Methods

No methods defined with class "fUGARCHSPEC" in the signature.

Note

(GNB) This class seems to be meant for internal use by the package.

See Also

class "fGARCH"

Examples

showClass("fUGARCHSPEC")

---

Univariate or multivariate GARCH time series fitting

Description

Estimates the parameters of a univariate ARMA-GARCH/APARCH process, or — experimentally — of a multivariate GO-GARCH process model. The latter uses an algorithm based on fastICA(), inspired from Bernhard Pfaff's package gogarch.

Usage

garchFit(formula = ~ garch(1, 1), data,
	init.rec = c("mci", "uev"),
	delta = 2, skew = 1, shape = 4,
	cond.dist = c("norm", "snorm", "ged", "sged",
                      "std", "sstd", "snig", "QMLE"),
	include.mean = TRUE, include.delta = NULL, include.skew = NULL,
        include.shape = NULL,
        leverage = NULL, trace = TRUE,
	
	algorithm = c("nlminb", "lbfgsb", "nlminb+nm", "lbfgsb+nm"),
	hessian = c("ropt", "rcd"),
        control = list(),
        title = NULL, description = NULL, ...)

garchKappa(cond.dist = c("norm", "ged", "std", "snorm", "sged", "sstd", "snig"),
           gamma = 0, delta = 2, skew = NA, shape = NA)

.gogarchFit(formula = ~garch(1, 1), data, init.rec = c("mci", "uev"),
            delta = 2, skew = 1, shape = 4,
            cond.dist = c("norm", "snorm", "ged", "sged",
                          "std", "sstd", "snig", "QMLE"),
            include.mean = TRUE, include.delta = NULL, include.skew = NULL,
            include.shape = NULL,
            leverage = NULL, trace = TRUE,
            algorithm = c("nlminb", "lbfgsb", "nlminb+nm", "lbfgsb+nm"),
            hessian = c("ropt", "rcd"),
            control = list(),
            title = NULL, description = NULL, ...)

Arguments

algorithm	a string parameter that determines the algorithm used for maximum likelihood estimation.
cond.dist	a character string naming the desired conditional distribution. Valid values are "dnorm", "dged", "dstd", "dsnorm", "dsged", "dsstd" and "QMLE". The default value is the normal distribution. See Details for more information.
control	control parameters, the same as used for the functions from nlminb, and 'bfgs' and 'Nelder-Mead' from optim.
data	an optional timeSeries or data frame object containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which armaFit is called. If data is an univariate series, then the series is converted into a numeric vector and the name of the response in the formula will be neglected.
delta	a numeric value, the exponent delta of the variance recursion. By default, this value will be fixed, otherwise the exponent will be estimated together with the other model parameters if include.delta=FALSE.
description	optional character string with a brief description.
formula	formula object describing the mean and variance equation of the ARMA-GARCH/APARCH model. A pure GARCH(1,1) model is selected e.g., for formula = ~garch(1,1). To specify an ARMA(2,1)-APARCH(1,1) process, use ~ arma(2,1) + aparch(1,1).
gamma	APARCH leverage parameter entering into the formula for calculating the expectation value.
hessian	a string denoting how the Hessian matrix should be evaluated, either hessian ="rcd", or "ropt". The default, "rcd" is a central difference approximation implemented in R and "ropt" uses the internal R function optimhess.
include.delta	a logical determining if the parameter for the recursion equation delta will be estimated or not. If false, the shape parameter will be kept fixed during the process of parameter optimization.
include.mean	this flag determines if the parameter for the mean will be estimated or not. If include.mean=TRUE this will be the case, otherwise the parameter will be kept fixed durcing the process of parameter optimization.
include.shape	a logical flag which determines if the parameter for the shape of the conditional distribution will be estimated or not. If include.shape=FALSE then the shape parameter will be kept fixed during the process of parameter optimization.
include.skew	a logical flag which determines if the parameter for the skewness of the conditional distribution will be estimated or not. If include.skew=FALSE then the skewness parameter will be kept fixed during the process of parameter optimization.
init.rec	a character string indicating the method how to initialize the mean and varaince recursion relation.
leverage	a logical flag for APARCH models. Should the model be leveraged? By default leverage=TRUE.
shape	a numeric value, the shape parameter of the conditional distribution.
skew	a numeric value, the skewness parameter of the conditional distribution.
title	a character string which allows for a project title.
trace	a logical flag. Should the optimization process of fitting the model parameters be printed? By default trace=TRUE.
...	additional arguments to be passed.

Details

"QMLE" stands for Quasi-Maximum Likelihood Estimation, which assumes normal distribution and uses robust standard errors for inference. Bollerslev and Wooldridge (1992) proved that if the mean and the volatility equations are correctly specified, the QML estimates are consistent and asymptotically normally distributed. However, the estimates are not efficient and “the efficiency loss can be marked under asymmetric ... distributions” (Bollerslev and Wooldridge (1992), p. 166). The robust variance-covariance matrix of the estimates equals the (Eicker-White) sandwich estimator, i.e.

$V = H^{-1} G^{\prime} G H^{-1},$

where $V$ denotes the variance-covariance matrix, $H$ stands for the Hessian and $G$ represents the matrix of contributions to the gradient, the elements of which are defined as

$G_{t,i} = \frac{\partial l_{t}}{\partial \zeta_{i}},$

where $t_{t}$ is the log likelihood of the t-th observation and $\zeta_{i}$ is the i-th estimated parameter. See sections 10.3 and 10.4 in Davidson and MacKinnon (2004) for a more detailed description of the robust variance-covariance matrix.

Value

for garchFit, an S4 object of class "fGARCH". Slot @fit contains the results from the optimization.

for .gogarchFit(): Similar definition for GO-GARCH modeling. Here, data must be multivariate. Still “preliminary”, mostly undocumented, and untested(!). At least mentioned here...

Author(s)

Diethelm Wuertz for the Rmetrics R-port,
R Core Team for the 'optim' R-port,
Douglas Bates and Deepayan Sarkar for the 'nlminb' R-port,
Bell-Labs for the underlying PORT Library,
Ladislav Luksan for the underlying Fortran SQP Routine,
Zhu, Byrd, Lu-Chen and Nocedal for the underlying L-BFGS-B Routine.

Martin Maechler for cleaning up; mentioning .gogarchFit().

References

ATT (1984); PORT Library Documentation, http://netlib.bell-labs.com/netlib/port/.

Bera A.K., Higgins M.L. (1993); ARCH Models: Properties, Estimation and Testing, J. Economic Surveys 7, 305–362.

Bollerslev T. (1986); Generalized Autoregressive Conditional Heteroscedasticity, Journal of Econometrics 31, 307–327.

Bollerslev T., Wooldridge J.M. (1992); Quasi-Maximum Likelihood Estimation and Inference in Dynamic Models with Time-Varying Covariance, Econometric Reviews 11, 143–172.

Byrd R.H., Lu P., Nocedal J., Zhu C. (1995); A Limited Memory Algorithm for Bound Constrained Optimization, SIAM Journal of Scientific Computing 16, 1190–1208.

Davidson R., MacKinnon J.G. (2004); Econometric Theory and Methods, Oxford University Press, New York.

Engle R.F. (1982); Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica 50, 987–1008.

Nash J.C. (1990); Compact Numerical Methods for Computers, Linear Algebra and Function Minimisation, Adam Hilger.

Nelder J.A., Mead R. (1965); A Simplex Algorithm for Function Minimization, Computer Journal 7, 308–313.

Nocedal J., Wright S.J. (1999); Numerical Optimization, Springer, New York.

See Also

garchSpec, garchFitControl, class "fGARCH"

Examples

## UNIVARIATE TIME SERIES INPUT:
   # In the univariate case the lhs formula has not to be specified ...

   # A numeric Vector from default GARCH(1,1) - fix the seed:
   N = 200
   x.vec = as.vector(garchSim(garchSpec(rseed = 1985), n = N)[,1])
   garchFit(~ garch(1,1), data = x.vec, trace = FALSE)

   # An univariate timeSeries object with dummy dates:
   stopifnot(require("timeSeries"))
   x.timeSeries = dummyDailySeries(matrix(x.vec), units = "GARCH11")
   garchFit(~ garch(1,1), data = x.timeSeries, trace = FALSE)

## Not run: 
   # An univariate zoo object:
   require("zoo")
   x.zoo = zoo(as.vector(x.vec), order.by = as.Date(rownames(x.timeSeries)))
   garchFit(~ garch(1,1), data = x.zoo, trace = FALSE)

## End(Not run)

   # An univariate "ts" object:
   x.ts = as.ts(x.vec)
   garchFit(~ garch(1,1), data = x.ts, trace = FALSE)

## MULTIVARIATE TIME SERIES INPUT:
   # For multivariate data inputs the lhs formula must be specified ...

   # A numeric matrix binded with dummy random normal variates:
   X.mat = cbind(GARCH11 = x.vec, R = rnorm(N))
   garchFit(GARCH11 ~ garch(1,1), data = X.mat)

   # A multivariate timeSeries object with dummy dates:
   X.timeSeries = dummyDailySeries(X.mat, units = c("GARCH11", "R"))
   garchFit(GARCH11 ~ garch(1,1), data = X.timeSeries)

## Not run: 
   # A multivariate zoo object:
   X.zoo = zoo(X.mat, order.by = as.Date(rownames(x.timeSeries)))
   garchFit(GARCH11 ~ garch(1,1), data = X.zoo)

## End(Not run)

   # A multivariate "mts" object:
   X.mts = as.ts(X.mat)
   garchFit(GARCH11 ~ garch(1,1), data = X.mts)

## MODELING THE PERCENTUAL SPI/SBI SPREAD FROM LPP BENCHMARK:

   stopifnot(require("timeSeries"))
   X.timeSeries = as.timeSeries(data(LPP2005REC))
   X.mat = as.matrix(X.timeSeries)
   ## Not run: X.zoo = zoo(X.mat, order.by = as.Date(rownames(X.mat)))
   X.mts = ts(X.mat)
   garchFit(100*(SPI - SBI) ~ garch(1,1), data = X.timeSeries)
   # The remaining are not yet supported ...
   # garchFit(100*(SPI - SBI) ~ garch(1,1), data = X.mat)
   # garchFit(100*(SPI - SBI) ~ garch(1,1), data = X.zoo)
   # garchFit(100*(SPI - SBI) ~ garch(1,1), data = X.mts)

## MODELING HIGH/LOW RETURN SPREADS FROM MSFT PRICE SERIES:

   X.timeSeries = MSFT
   garchFit(Open ~ garch(1,1), data = returns(X.timeSeries))
   garchFit(100*(High-Low) ~ garch(1,1), data = returns(X.timeSeries))

## GO-GARCH Modelling  (not yet!!) % FIXME

  ## data(DowJones30, package="fEcofin") # no longer exists
  ## X = returns(as.timeSeries(DowJones30)); head(X)
  ## N = 5; ans = .gogarchFit(data = X[, 1:N], trace = FALSE); ans
  ## [email protected]

---

Control GARCH fitting algorithms

Description

Control parameters for the GARCH fitting algorithms.

Usage

garchFitControl(
    llh = c("filter", "internal", "testing"),
    nlminb.eval.max = 2000,
    nlminb.iter.max = 1500,
    nlminb.abs.tol = 1.0e-20,
    nlminb.rel.tol = 1.0e-14,
    nlminb.x.tol = 1.0e-14,
    nlminb.step.min = 2.2e-14,
    nlminb.scale = 1,
    nlminb.fscale = FALSE,
    nlminb.xscale = FALSE,
    sqp.mit = 200,
    sqp.mfv = 500,
    sqp.met = 2,
    sqp.mec = 2,
    sqp.mer = 1,
    sqp.mes = 4,
    sqp.xmax = 1.0e3,
    sqp.tolx = 1.0e-16,
    sqp.tolc = 1.0e-6,
    sqp.tolg = 1.0e-6,
    sqp.told = 1.0e-6,
    sqp.tols = 1.0e-4,
    sqp.rpf = 1.0e-4,
    lbfgsb.REPORT = 10,
    lbfgsb.lmm = 20,
    lbfgsb.pgtol = 1e-14,
    lbfgsb.factr = 1,
    lbfgsb.fnscale = FALSE,
    lbfgsb.parscale = FALSE,
    nm.ndeps = 1e-14,
    nm.maxit = 10000,
    nm.abstol = 1e-14,
    nm.reltol = 1e-14,
    nm.alpha = 1.0,
    nm.beta = 0.5,
    nm.gamma = 2.0,
    nm.fnscale = FALSE,
    nm.parscale = FALSE)

Arguments

llh	llh = c("filter", "internal", "testing")[1], defaults to "filter".
nlminb.eval.max	maximum number of evaluations of the objective function, defaults to 200.
nlminb.iter.max	maximum number of iterations, defaults to 150.
nlminb.abs.tol	absolute tolerance, defaults to 1e-20.
nlminb.rel.tol	relative tolerance, defaults to 1e-10.
nlminb.x.tol	X tolerance, defaults to 1.5e-8.
nlminb.fscale	defaults to FALSE.
nlminb.xscale	defaulkts to FALSE.
nlminb.step.min	minimum step size, defaults to 2.2e-14.
nlminb.scale	defaults to 1.
sqp.mit	maximum number of iterations, defaults to 200.
sqp.mfv	maximum number of function evaluations, defaults to 500.
sqp.met	specifies scaling strategy: sqp.met=1 - no scaling, sqp.met=2 - preliminary scaling in 1st iteration (default), sqp.met=3 - controlled scaling, sqp.met=4 - interval scaling, sqp.met=5 - permanent scaling in all iterations.
sqp.mec	correction for negative curvature: sqp.mec=1 - no correction, sqp.mec=2 - Powell correction (default).
sqp.mer	restarts after unsuccessful variable metric updates: sqp.mer=0 - no restarts, sqp.mer=1 - standard restart.
sqp.mes	interpolation method selection in a line search: sqp.mes=1 - bisection, sqp.mes=2 - two point quadratic interpolation, sqp.mes=3 - three point quadratic interpolation, sqp.mes=4 - three point cubic interpolation (default).
sqp.xmax	maximum stepsize, defaults to 1.0e+3.
sqp.tolx	tolerance for the change of the coordinate vector, defaults to 1.0e-16.
sqp.tolc	tolerance for the constraint violation, defaults to 1.0e-6.
sqp.tolg	tolerance for the Lagrangian function gradient, defaults to 1.0e-6.
sqp.told	defaults to 1.0e-6.
sqp.tols	defaults to 1.0e-4.
sqp.rpf	value of the penalty coefficient, default to1.0D-4. The default velue may be relatively small. Therefore, larger value, say one, can sometimes be more suitable.
lbfgsb.REPORT	the frequency of reports for the "BFGS" and "L-BFGS-B" methods if control$trace is positive. Defaults to every 10 iterations.
lbfgsb.lmm	an integer giving the number of BFGS updates retained in the "L-BFGS-B" method, It defaults to 5.
lbfgsb.factr	controls the convergence of the "L-BFGS-B" method. Convergence occurs when the reduction in the objective is within this factor of the machine tolerance. Default is 1e7, that is a tolerance of about 1.0e-8.
lbfgsb.pgtol	helps control the convergence of the "L-BFGS-B" method. It is a tolerance on the projected gradient in the current search direction. This defaults to zero, when the check is suppressed.
lbfgsb.fnscale	defaults to FALSE.
lbfgsb.parscale	defaults to FALSE.
nm.ndeps	a vector of step sizes for the finite-difference approximation to the gradient, on par/parscale scale. Defaults to 1e-3.
nm.maxit	the maximum number of iterations. Defaults to 100 for the derivative-based methods, and 500 for "Nelder-Mead". For "SANN" maxit gives the total number of function evaluations. There is no other stopping criterion. Defaults to 10000.
nm.abstol	the absolute convergence tolerance. Only useful for non-negative functions, as a tolerance for reaching zero.
nm.reltol	relative convergence tolerance. The algorithm stops if it is unable to reduce the value by a factor of reltol * (abs(val) + reltol) at a step. Defaults to sqrt(.Machine$double.eps), typically about 1e-8.
nm.alpha, nm.beta, nm.gamma	scaling parameters for the "Nelder-Mead" method. alpha is the reflection factor (default 1.0), beta the contraction factor (0.5), and gamma the expansion factor (2.0).
nm.fnscale	an overall scaling to be applied to the value of fn and gr during optimization. If negative, turns the problem into a maximization problem. Optimization is performed on fn(par) / nm.fnscale.
nm.parscale	a vector of scaling values for the parameters. Optimization is performed on par/parscale and these should be comparable in the sense that a unit change in any element produces about a unit change in the scaled value.

Value

a list

Author(s)

Diethelm Wuertz for the Rmetrics R-port,
R Core Team for the 'optim' R-port,
Douglas Bates and Deepayan Sarkar for the 'nlminb' R-port,
Bell-Labs for the underlying PORT Library,
Ladislav Luksan for the underlying Fortran SQP Routine,
Zhu, Byrd, Lu-Chen and Nocedal for the underlying L-BFGS-B Routine.

See Also

garchFit

Examples

##

---

Simulate univariate GARCH/APARCH time series

Description

Simulates univariate GARCH/APARCH time series.

Usage

garchSim(spec = garchSpec(), n = 100, n.start = 100, extended = FALSE)

Arguments

spec	a specification object of class "fGARCHSPEC" as returned by garchSpec. See also below for further details.
n	length of the output series, an integer value, by default n=100.
n.start	length of ‘burn-in’ period, by default 100.
extended	logical parameter specifying what to return. If FALSE, return the univariate GARCH/APARCH time series. If TRUE, return a multivariate time series containing also the volatility and conditional innovations time series.

Details

garchSim simulates an univariate GARCH or APARCH time series process as specified by argument spec. The default model specifies Bollerslev's GARCH(1,1) model with normally distributed innovations.

spec is an object of class "fGARCHSPEC" as returned by the function garchSpec. It comes with a slot @model which is a list of just the numeric parameter entries. These are recognized and extracted for use by the function garchSim.

One can estimate the parameters of a GARCH process from empirical data using the function garchFit and then simulate statistically equivalent GARCH processes with the same set of model parameters using the function garchSim.

Value

the simulated time series as an objects of class "timeSeries" with attribute "spec" containing the specification of the model.

If extended is TRUE, then the time series is multivariate and contains also the volatility, sigma, and the conditional innovations, eps.

Note

An undocumented feature (so, it should not be relied on) is that the returned time series is timed so that the last observation is the day before the date when the function is executed. This probably should be controlled by an additional argument in garchSim.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

See Also

garchSpec, garchFit

Examples

## garchSpec -
   spec = garchSpec()
   spec

## garchSim -
   # Simulate a "timeSeries" object:
   x = garchSim(spec, n = 50)
   class(x)
   print(x) 
   
## More simulations ...

   # Default GARCH(1,1) - uses default parameter settings
   spec = garchSpec(model = list())
   garchSim(spec, n = 10)
   
   # ARCH(2) - use default omega and specify alpha, set beta=0!
   spec = garchSpec(model = list(alpha = c(0.2, 0.4), beta = 0))
   garchSim(spec, n = 10)
   
   # AR(1)-ARCH(2) - use default mu, omega
   spec = garchSpec(model = list(ar = 0.5, alpha = c(0.3, 0.4), beta = 0))
   garchSim(spec, n = 10)
   
   # AR([1,5])-GARCH(1,1) - use default garch values and subset ar[.]
   spec = garchSpec(model = list(mu = 0.001, ar = c(0.5,0,0,0,0.1)))
   garchSim(spec, n = 10)
   
   # ARMA(1,2)-GARCH(1,1) - use default garch values
   spec = garchSpec(model = list(ar = 0.5, ma = c(0.3, -0.3)))  
   garchSim(spec, n = 10)
   
   # GARCH(1,1) - use default omega and specify alpha/beta
   spec = garchSpec(model = list(alpha = 0.2, beta = 0.7))
   garchSim(spec, n = 10)
   
   # GARCH(1,1) - specify omega/alpha/beta
   spec = garchSpec(model = list(omega = 1e-6, alpha = 0.1, beta = 0.8))
   garchSim(spec, n = 10)
   
   # GARCH(1,2) - use default omega and specify alpha[1]/beta[2]
   spec = garchSpec(model = list(alpha = 0.1, beta = c(0.4, 0.4)))
   garchSim(spec, n = 10)
   
   # GARCH(2,1) - use default omega and specify alpha[2]/beta[1]
   spec = garchSpec(model = list(alpha = c(0.12, 0.04), beta = 0.08))
   garchSim(spec, n = 10)
   
   # snorm-ARCH(1) - use defaults with skew Normal
   spec = garchSpec(model = list(beta = 0, skew = 0.8), cond.dist = "snorm")
   garchSim(spec, n = 10)
   
   # sged-GARCH(1,1) - using defaults with skew GED
   model = garchSpec(model = list(skew = 0.93, shape = 3), cond.dist = "sged")
   garchSim(model, n = 10)
   
   # Taylor Schwert GARCH(1,1) - this belongs to the family of APARCH Models
   spec = garchSpec(model = list(delta = 1))
   garchSim(spec, n = 10)
   
   # AR(1)-t-APARCH(2, 1) - a little bit more complex specification ...
   spec = 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")
   garchSim(spec, n = 10)

   garchSim(spec, n = 10, extended = TRUE)

---

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

cond.dist	a character string naming the desired conditional distribution. Valid values are "norm", "ged", "std", "snorm", "sged", "sstd". The default value is "norm", the standard normal distribution.
model	a list of GARCH model parameters, see section ‘Details’. The default model=list() specifies Bollerslev's GARCH(1,1) model with normal conditional distributed innovations.
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.
rseed	single integer argument, the seed for the intitialization of the random number generator for the innovations. If rseed=NULL, the default, then the state of the random number generator is not touched by this function.

Details

The function garchSpec specifies a GARCH or APARCH time series process which we can use for simulating artificial GARCH and/or APARCH models. This is very useful for testing the GARCH parameter estimation results, since your model parameters are known and well specified.

Argument model is a list of model parameters. For the GARCH part of the model they are:

omega

the constant coefficient of the variance equation, by default 1e-6;

alpha

the value or vector of autoregressive coefficients, by default 0.1, specifying a model of order 1;

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:

delta

a positive number, the power of sigma in the volatility equation, it is 2 for GARCH models;

gamma

the leverage parameters, a vector of length alpha, containing numbers in the interval $(0,1)$.

The values for the linear part (conditional mean) are:

mu

the mean value, by default NULL;

ar

the autoregressive ARMA coefficients, by default NULL;

ma

the moving average ARMA coefficients, by default NULL.

The parameters for the conditional distributions are:

skew

the skewness parameter (also named "xi"), by default 0.9, effective only for the "dsnorm", the "dsged", and the "dsstd" skewed conditional distributions;

shape

the shape parameter (also named "nu"), by default 2 for the "dged" and "dsged", and by default 4 for the "dstd" and "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:

        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. ‘Formula’ describes the formula expression specifying the generating process, ‘Model’ lists the associated model parameters, ‘Distribution’ the type of the conditional distribution function in use, ‘Distributional Parameters’ lists the distributional parameter (if any), and the ‘Presample’ shows the presample input matrix.

If we have specified presample = NULL in the argument list, then the presample is generated automatically by default as norm-AR()-GARCH() process.

Value

an object of class "fGARCHSPEC"

Author(s)

Diethelm Wuertz for the Rmetrics R-port

See Also

garchSim, garchFit

Examples

## garchSpec -

   # 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")

---

Standardized generalized error distribution

Description

Functions to compute density, distribution function, quantile function and to generate random variates for the standardized generalized error distribution.

Usage

dged(x, mean = 0, sd = 1, nu = 2, log = FALSE)
pged(q, mean = 0, sd = 1, nu = 2)
qged(p, mean = 0, sd = 1, nu = 2)
rged(n, mean = 0, sd = 1, nu = 2)

Arguments

x, q	a numeric vector of quantiles.
p	a numeric vector of probabilities.
n	number of observations to simulate.
mean	location parameter.
sd	scale parameter.
nu	shape parameter.
log	logical; if TRUE, densities are given as log densities.

Details

The standardized GED is defined so that for a given sd it has the same variance, sd^2, for all values of the shape parameter, see the reference by Wuertz et al below.

dged computes the density, pged the distribution function, qged the quantile function, and rged generates random deviates from the standardized-t distribution with the specified parameters.

Value

numeric vector

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347–370.

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

Wuertz D., Chalabi Y. and Luksan L. (????); Parameter estimation of ARMA models with GARCH/APARCH errors: An R and SPlus software implementation, Preprint, 41 pages, https://github.com/GeoBosh/fGarchDoc/blob/master/WurtzEtAlGarch.pdf

See Also

gedFit, absMoments, sged (skew GED),

gedSlider for visualization

Examples

## sged -
   par(mfrow = c(2, 2))
   set.seed(1953)
   r = rsged(n = 1000)
   plot(r, type = "l", main = "sged", col = "steelblue")
   
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
   box()
   x = seq(min(r), max(r), length = 201)
   lines(x, dsged(x), lwd = 2)
   
   # Plot df and compare with true df:
   plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue",
     ylab = "Probability")
   lines(x, psged(x), lwd = 2)
   
   # Compute quantiles:
   round(qsged(psged(q = seq(-1, 5, by = 1))), digits = 6)

---

Generalized error distribution parameter estimation

Description

Function to fit the parameters of the generalized error distribution.

Usage

gedFit(x, ...)

Arguments

x	a numeric vector of quantiles.
...	parameters parsed to the optimization function nlm.

Value

gedFit returns a list with the following components:

par	The best set of parameters found.
objective	The value of objective corresponding to par.
convergence	An integer code, 0 indicates successful convergence.
message	A character string giving any additional information returned by the optimizer, or NULL. For details, see PORT documentation.
iterations	Number of iterations performed.
evaluations	Number of objective function and gradient function evaluations.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347–370.

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

ged, sgedFit

Examples

## rged -
   set.seed(1953)
   r = rged(n = 1000)
       
## gedFit -
   gedFit(r)

---

Generalized error distribution slider

Description

Displays interactively the dependence of the GED distribution on its parameters.

Usage

gedSlider(type = c("dist", "rand"))

Arguments

type	a character string denoting which interactive plot should be displayed. Either a distribution plot type = "dist", the default value, or a random variates plot, type = "rand".

Value

a Tcl object

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347–370.

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

ged, gedFit

Examples

## Not run:   
## gedSlider -
   require(tcltk)
   gedSlider("dist")
   gedSlider("rand")

## End(Not run)

---

GARCH plot methods

Description

Plot methods for GARCH modelling.

Usage

## S4 method for signature 'fGARCH,missing'
plot(x, which = "ask", ...)

Arguments

x	an object of class "fGARCH".
which	a character string or a vector of positive integers specifying which plot(s) should be displayed, see section ‘Details’.
...	optional arguments to be passed.

Details

The plot method for "fGARCH" objects offers a selection of diagnostic, exploratory, and presentation plots from a menu. Argument which can be used to request specific plots. This is particularly useful in scripts.

If which is of length larger than one, all requested plots are produced. For this to be useful, the graphics window should be split beforehand in subwindows, e.g., using par(mfrow = ...), par(mfcol = ...), or layout() (see section ‘Examples’). If this is not done, then only the last plot will be visible.

The following graphs are available:

1	Time SeriesPlot
2	Conditional Standard Deviation Plot
3	Series Plot with 2 Conditional SD Superimposed
4	Autocorrelation function Plot of Observations
5	Autocorrelation function Plot of Squared Observations
6	Cross Correlation Plot
7	Residuals Plot
8	Conditional Standard Deviations Plot
9	Standardized Residuals Plot
10	ACF Plot of Standardized Residuals
11	ACF Plot of Squared Standardized Residuals
12	Cross Correlation Plot between $r^2$ and r
13	Quantile-Quantile Plot of Standardized Residuals
14	Series with -VaR Superimposed
15	Series with -ES Superimposed
16	Series with -VaR & -ES Superimposed

Author(s)

Diethelm Wuertz for the Rmetrics R-port;

VaR and ES graphs were added by Georgi N. Boshnakov in v4033.92

See Also

fGARCH method for tsdiag,

garchFit, class fGARCH,

predict, fitted, residuals VaR ES

plot

Examples

## simulate a Garch(1,1) time series
x <- garchSim(n = 200)
head(x) 

## fit GARCH(1,1) model
fit <- garchFit(formula = ~ garch(1, 1), data = x, trace = FALSE)

## Not run: 
## choose plots interactively
plot(fit)

## End(Not run)
   
## Batch Plot:
plot(fit, which = 3)

## a 2 by 2 matrix of plots
op <- par(mfrow = c(2,2))          # prepare 2x2 window
plot(fit, which = c(10, 11, 3, 16)) # plot
par(op)                            # restore the previous layout

---

GARCH prediction function

Description

Predicts a time series from a fitted GARCH object.

Usage

## S4 method for signature 'fGARCH'
predict(object, n.ahead = 10, trace = FALSE, mse = c("cond","uncond"),
        plot=FALSE, nx=NULL, crit_val=NULL, conf=NULL, ..., p_loss = NULL)

Arguments

n.ahead	an integer value, denoting the number of steps to be forecasted, by default 10.
object	an object of class "fGARCH" as returned by the function garchFit.
trace	a logical flag. Should the prediction process be traced? By default trace=FALSE.
mse	If set to "cond", meanError is defined as the conditional mean errors $\sqrt{E_t[x_{t+h}-E_t(x_{t+h})]^2}$. If set to "uncond", it is defined as $\sqrt{E[x_{t+h}-E_t(x_{t+h})]^2}$.
plot	If set to TRUE, the confidence intervals are computed and plotted
nx	The number of observations to be plotted along with the predictions. The default is round(n*0.25), where n is the sample size.
crit_val	The critical values for the confidence intervals when plot is set to TRUE. The intervals are defined as $\hat{x}_{t+h}$ + crit_val[2] * meanError and $\hat{x}_{t+h}$ + crit_val[1] * meanError if two critical values are provided and $\hat{x}_{t+h} \pm$ crit_val * meanError if only one is given. If you do not provide critical values, they will be computed automatically.
conf	The confidence level for the confidence intervals if crit_val is not provided. By default it is set to 0.95. The critical values are then computed using the conditional distribution that was chosen to create the object with garchFit using the same shape and skew parameters. If the conditionnal distribution was set to "QMLE", the critical values are computed using the empirical distribution of the standardized residuals.
...	additional arguments to be passed.
p_loss	if not null, compute predictions for VaR and ES for loss level p_loss (typically, 0.05 or 0.01).

Details

The predictions are returned as a data frame with columns "meanForecast", "meanError", and "standardDeviation". Row h contains the predictions for horizon h (so, n.ahead rows in total).

If plot = TRUE, the data frame contain also the prediction limits for each horizon in columns lowerInterval and upperInterval.

If p_loss is not NULL, predictions of Value-at-Risk (VaR) and Expected Shortfall (ES) are returned in columns VaR and ES. The data frame has attribute "p_loss" containing p_loss. Typical values for p_loss are 0.01 and 0.05.

These are somewhat experimental and the arguments and the returned values may change.

Value

a data frame containing n.ahead rows and 3 to 7 columns, see section ‘Details’

Author(s)

Diethelm Wuertz for the Rmetrics R-port

See Also

predict in base R

fitted, residuals,

plot, garchFit, class fGARCH,

Examples

## Parameter Estimation of Default GARCH(1,1) Model
set.seed(123)
fit = garchFit(~ garch(1, 1), data = garchSim(), trace = FALSE)
fit

## predict
predict(fit, n.ahead = 10)
predict(fit, n.ahead = 10, mse="uncond")

## predict with plotting: critical values = +/- 2
predict(fit, n.ahead = 10, plot=TRUE, crit_val = 2)

## include also VaR and ES at 5%
predict(fit, n.ahead = 10, plot=TRUE, crit_val = 2, p_loss = 0.05)

## predict with plotting: automatic critical values 
## for different conditional distributions
set.seed(321)
fit2 = garchFit(~ garch(1, 1), data = garchSim(), trace=FALSE, cond.dist="sged")

## 95% confidence level
predict(fit2, n.ahead=20, plot=TRUE) 

set.seed(444)
fit3 = garchFit(~ garch(1, 1), data = garchSim(), trace=FALSE, cond.dist="QMLE")

## 90% confidence level and nx=100
predict(fit3, n.ahead=20, plot=TRUE, conf=.9, nx=100)

---

Extract GARCH model residuals

Description

Extracts residuals from a fitted GARCH object.

Usage

## S4 method for signature 'fGARCH'
residuals(object, standardize = FALSE)

Arguments

object	an object of class "fGARCH" as returned by garchFit.
standardize	a logical, indicating if the residuals should be standardized.

Details

The "fGARCH" method extracts the @residuals slot from an object of class "fGARCH" as returned by the function garchFit and optionally standardizes them, using conditional standard deviations.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

See Also

fitted, predict, garchFit, class fGARCH,

Examples

stopifnot(require("timeSeries"))
## Swiss Pension fund Index
data(LPP2005REC, package = "timeSeries")
x <- as.timeSeries(LPP2005REC)

## Fit LPP40 Bechmark:
fit <- garchFit(LPP40 ~ garch(1, 1), data = 100*x, trace = FALSE)
fit

fitted <- fitted(fit)
head(fitted)
class(fitted)

res <- residuals(fit)
head(res)
class(res)

---

Skew generalized error distribution

Description

Functions to compute density, distribution function, quantile function and to generate random variates for the skew generalized error distribution.

Usage

dsged(x, mean = 0, sd = 1, nu = 2, xi = 1.5, log = FALSE)
psged(q, mean = 0, sd = 1, nu = 2, xi = 1.5)
qsged(p, mean = 0, sd = 1, nu = 2, xi = 1.5)
rsged(n, mean = 0, sd = 1, nu = 2, xi = 1.5)

Arguments

mean, sd, nu, xi	location parameter mean, scale parameter sd, shape parameter nu, skewness parameter xi.
n	the number of observations.
p	a numeric vector of probabilities.
x, q	a numeric vector of quantiles.
log	a logical; if TRUE, densities are given as log densities.

Details

The distribution is standardized as discussed in the reference by Wuertz et al below.

Value

d* returns the density, p* returns the distribution function, q* returns the quantile function, and r* generates random deviates,
all values are numeric vectors.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347–370.

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

Wuertz D., Chalabi Y. and Luksan L. (????); Parameter estimation of ARMA models with GARCH/APARCH errors: An R and SPlus software implementation, Preprint, 41 pages, https://github.com/GeoBosh/fGarchDoc/blob/master/WurtzEtAlGarch.pdf

See Also

sgedFit (fit), sgedSlider (visualize),

ged (symmetric GED)

Examples

## sged -
   par(mfrow = c(2, 2))
   set.seed(1953)
   r = rsged(n = 1000)
   plot(r, type = "l", main = "sged", col = "steelblue")
   
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
   box()
   x = seq(min(r), max(r), length = 201)
   lines(x, dsged(x), lwd = 2)
   
   # Plot df and compare with true df:
   plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue",
     ylab = "Probability")
   lines(x, psged(x), lwd = 2)
   
   # Compute quantiles:
   round(qsged(psged(q = seq(-1, 5, by = 1))), digits = 6)

---

Skew generalized error distribution parameter estimation

Description

Function to fit the parameters of the skew generalized error distribution.

Usage

sgedFit(x, ...)

Arguments

x	a numeric vector of quantiles.
...	parameters parsed to the optimization function nlm.

Value

sgedFit returns a list with the following components:

par	The best set of parameters found.
objective	The value of objective corresponding to par.
convergence	An integer code. 0 indicates successful convergence.
message	A character string giving any additional information returned by the optimizer, or NULL. For details, see PORT documentation.
iterations	Number of iterations performed.
evaluations	Number of objective function and gradient function evaluations.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347–370.

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

sged, sgedSlider

Examples

## rsged -
   set.seed(1953)
   r = rsged(n = 1000)
       
## sgedFit -
   sgedFit(r)

---

Skew GED distribution slider

Description

Displays interactively the dependence of the skew GED distribution on its parameters.

Usage

sgedSlider(type = c("dist", "rand"))

Arguments

type	a character string denoting which interactive plot should be displayed. Either a distribution plot type="dist", the default value, or a random variates plot, type="rand".

Value

a Tcl object

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347–370.

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

sged, sgedFit

Examples

## Not run:   
## sgedSlider -
   require(tcltk)
   sgedSlider("dist")
   sgedSlider("rand")

## End(Not run)

---

Skew normal distribution

Description

Functions to compute density, distribution function, quantile function and to generate random variates for the skew normal distribution.

The distribution is standardized as discussed in the reference by Wuertz et al below.

Usage

dsnorm(x, mean = 0, sd = 1, xi = 1.5, log = FALSE)
psnorm(q, mean = 0, sd = 1, xi = 1.5)
qsnorm(p, mean = 0, sd = 1, xi = 1.5)
rsnorm(n, mean = 0, sd = 1, xi = 1.5)

Arguments

x, q	a numeric vector of quantiles.
p	a numeric vector of probabilities.
n	the number of observations.
mean	location parameter.
sd	scale parameter.
xi	skewness parameter.
log	a logical; if TRUE, densities are given as log densities.

Details

dsnorm computed the density, psnorm the distribution function, qsnorm the quantile function, and rsnorm generates random deviates.

Value

numeric vector

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

Wuertz D., Chalabi Y. and Luksan L. (????); Parameter estimation of ARMA models with GARCH/APARCH errors: An R and SPlus software implementation, Preprint, 41 pages, https://github.com/GeoBosh/fGarchDoc/blob/master/WurtzEtAlGarch.pdf

See Also

snormFit (fit), snormSlider (visualize),

sstd (skew Student-t), sged (skew GED)

Examples

## snorm -
   # Ranbdom Numbers:
   par(mfrow = c(2, 2))
   set.seed(1953)
   r = rsnorm(n = 1000)
   plot(r, type = "l", main = "snorm", col = "steelblue")
   
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
   box()
   x = seq(min(r), max(r), length = 201)
   lines(x, dsnorm(x), lwd = 2)
   
   # Plot df and compare with true df:
   plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue",
     ylab = "Probability")
   lines(x, psnorm(x), lwd = 2)
   
   # Compute quantiles:
   round(qsnorm(psnorm(q = seq(-1, 5, by = 1))), digits = 6)

---

Skew normal distribution parameter estimation

Description

Fits the parameters of the skew normal distribution.

Usage

snormFit(x, ...)

Arguments

x	a numeric vector of quantiles.
...	parameters passed to the optimization function nlm.

Value

snormFit returns a list with the following components:

par	The best set of parameters found.
objective	The value of objective corresponding to par.
convergence	An integer code. 0 indicates successful convergence.
message	A character string giving any additional information returned by the optimizer, or NULL. For details, see PORT documentation.
iterations	Number of iterations performed.
evaluations	Number of objective function and gradient function evaluations.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

snormFit (fit), snormSlider (visualize), absMoments

Examples

## rsnorm -
   set.seed(1953)
   r = rsnorm(n = 1000)
       
## snormFit -
   snormFit(r)

---

Skew normal distribution slider

Description

Displays interactively the dependence of the skew Normal distribution on its parameters.

Usage

snormSlider(type = c("dist", "rand"))

Arguments

type	a character string denoting which interactive plot should be displayed. Either a distribution plot type="dist", the default value, or a random variates plot, type="rand".

Value

a Tcl object

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

snormFit (fit), snorm,

Examples

## Not run:   
## snormSlider -
   require(tcltk)
   snormSlider("dist")
   snormSlider("rand")

## End(Not run)

---

Skew Student-t distribution

Description

Functions to compute density, distribution function, quantile function and to generate random variates for the skew Student-t distribution.

Usage

dsstd(x, mean = 0, sd = 1, nu = 5, xi = 1.5, log = FALSE)
psstd(q, mean = 0, sd = 1, nu = 5, xi = 1.5)
qsstd(p, mean = 0, sd = 1, nu = 5, xi = 1.5)
rsstd(n, mean = 0, sd = 1, nu = 5, xi = 1.5)

Arguments

x, q	a numeric vector of quantiles.
p	a numeric vector of probabilities.
n	number of observations to simulate.
mean	location parameter.
sd	scale parameter.
nu	shape parameter (degrees of freedom).
xi	skewness parameter.
log	logical; if TRUE, densities are given as log densities.

Details

The distribution is standardized as discussed in the reference by Wuertz et al below.

dsstd computes the density, psstd the distribution function, qsstd the quantile function, and rsstd generates random deviates.

Value

numeric vector

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

Wuertz D., Chalabi Y. and Luksan L. (????); Parameter estimation of ARMA models with GARCH/APARCH errors: An R and SPlus software implementation, Preprint, 41 pages, https://github.com/GeoBosh/fGarchDoc/blob/master/WurtzEtAlGarch.pdf

See Also

sstdFit (fit), sstdSlider (visualize)

Examples

## sstd -
   par(mfrow = c(2, 2))
   set.seed(1953)
   r = rsstd(n = 1000)
   plot(r, type = "l", main = "sstd", col = "steelblue")
   
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
   box()
   x = seq(min(r), max(r), length = 201)
   lines(x, dsstd(x), lwd = 2)
   
   # Plot df and compare with true df:
   plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue",
     ylab = "Probability")
   lines(x, psstd(x), lwd = 2)
   
   # Compute quantiles:
   round(qsstd(psstd(q = seq(-1, 5, by = 1))), digits = 6)

---

Skew Student-t distribution parameter estimation

Description

Fits the parameters of the skew Student-t distribution.

Usage

sstdFit(x, ...)

Arguments

x	a numeric vector of quantiles.
...	parameters passed to the optimization function nlm.

Value

sstdFit returns a list with the following components:

par	The best set of parameters found.
objective	The value of objective corresponding to par.
convergence	An integer code. 0 indicates successful convergence.
message	A character string giving any additional information returned by the optimizer, or NULL. For details, see PORT documentation.
iterations	Number of iterations performed.
evaluations	Number of objective function and gradient function evaluations.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

sstd, stdFit

Examples

## sstd -
   set.seed(1953)
   r = rsstd(n = 1000)

       
## sstdFit -
   sstdFit(r)

---

Skew Student-t distribution slider

Description

Displays interactively the dependence of the skew Student-t distribution on its parameters.

Usage

sstdSlider(type = c("dist", "rand"))

Arguments

type	a character string denoting which interactive plot should be displayed. Either a distribution plot type="dist", the default value, or a random variates plot, type="rand".

Value

a Tcl object

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

sstd, sstdFit

Examples

## Not run:   
## sstdSlider -
   require(tcltk)
   sstdSlider("dist")
   sstdSlider("rand")

## End(Not run)

---

Diagnostic plots and statistics for fitted GARCH models

Description

Produce diagnostics for fitted GARCH/APARCH models. The method offers several tests, plots of autocorrelations and partial autocorrelations of the standardised conditional residuals, ability to control which graphs are produced (including interactively), as well as their layout.

Usage

## S3 method for class 'fGARCH'
tsdiag(object, gof.lag = NULL, ask = FALSE, ..., plot = c(4L, 5L, 7L), 
       layout = NULL)

Arguments

object	an object from class "fGARCH", as returned by garchFit.
gof.lag	maximal lag for portmanteau tests.
ask	if TRUE present a menu of available plots, see Details.
...	not used.
plot	if TRUE all available plots; a vector of positive integers specifies a subset of the available plots.
layout	a list with arguments for graphics::layout for the plots. The default plots the autocorrelations of the standardised conditional residuals and their squares, as well as a QQ-plot for the fitted conditional distribution.

Details

Compute and graph diagnostics for fitted ARMA-GARCH/APARCH models.

plot can be TRUE to ask for all plots or a vector of positive integers specifying which plots to consider. Currently the following options are available:

1	Residuals
2	Conditional SDs
3	Standardized Residuals
4	ACF of Standardized Residuals
5	ACF of Squared Standardized Residuals
6	Cross Correlation between r^2 and r
7	QQ-Plot of Standardized Residuals

The default produces plots of autocorrelations and partial autocorrelations of the standardised conditional residuals, as well as a QQ-plot for the fitted conditional distribution. If plot is TRUE, you probably need also ask = TRUE.

If argument plot is of length two the graphics window is split into 2 equal subwindows. Argument layout can still be used to change this. If argument plot is of length one the graphics window is not split at all.

In interactive sessions, if the number of requested graphs (as specified by argument plot) is larger than the number of graphs specified by the layout (by default 3), the function makes the first graph and then presents a menu of the requested plots.

Argument layout can be used to change the layout of the plot, for example to put two graphs per plot, see the examples. Currently it should be a list of arguments for layout, see ?layout. Don't call layout youself, as that will change the graphics device prematurely.

The computed results are returned (invisibly). This is another difference from stats::tsdiag which doesn't return them.

Value

(experimental, may change) a list with components:

residuals	standardised conditional residuals,
gof	goodness-of-fit tests, pretending parameters are known,
gof_composite	goodness-of-fit tests taking into account that the parameters are estimated.

Only components that are actually computed are included, the rest are NULL or absent.

Author(s)

Georgi N. boshnakov

See Also

fGARCH method for plot,

tsdiag

Examples

set.seed(20230612)
x <- garchSim(n = 200)
fit <- garchFit(formula = ~ garch(1, 1), data = x, trace = FALSE)
fit_test <- tsdiag(fit)
fit_test

## 2x2 matrix with acf of r, r^2 on diag, cor(r,r^2) below it, and qq-plot
tsdiag(fit, plot = c(4, 6, 7, 5), layout = list(matrix(1:4, nrow = 2)))

---

Standardized Student-t distribution

Description

Functions to compute density, distribution function, quantile function and to generate random variates for the standardized Student-t distribution.

Usage

dstd(x, mean = 0, sd = 1, nu = 5, log = FALSE)
pstd(q, mean = 0, sd = 1, nu = 5)
qstd(p, mean = 0, sd = 1, nu = 5)
rstd(n, mean = 0, sd = 1, nu = 5)

Arguments

x, q	a numeric vector of quantiles.
p	a numeric vector of probabilities.
n	number of observations to simulate.
mean	location parameter.
sd	scale parameter.
nu	shape parameter (degrees of freedom).
log	logical; if TRUE, densities are given as log densities.

Details

The standardized Student-t distribution is defined so that for a given sd it has the same variance, sd^2, for all degrees of freedom. For comparison, the variance of the usual Student-t distribution is nu/(nu-2), where nu is the degrees of freedom. The usual Student-t distribution is obtained by setting sd = sqrt(nu/(nu - 2)).

Argument nu must be greater than 2. Although there is a default value for nu, it is rather arbitrary and relying on it is strongly discouraged.

dstd computes the density, pstd the distribution function, qstd the quantile function, and rstd generates random deviates from the standardized-t distribution with the specified parameters.

Value

numeric vector

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

Wuertz D., Chalabi Y. and Luksan L. (2006); Parameter estimation of ARMA models with GARCH/APARCH errors: An R and SPlus software implementation, Preprint, 41 pages, https://github.com/GeoBosh/fGarchDoc/blob/master/WurtzEtAlGarch.pdf

See Also

stdFit (fit). stdSlider (visualize),

absMoments

Examples

## std -

pstd(1, sd = sqrt(5/(5-2)), nu = 5) == pt(1, df = 5) # TRUE

   par(mfrow = c(2, 2))
   set.seed(1953)
   r = rstd(n = 1000)
   plot(r, type = "l", main = "sstd", col = "steelblue")
   
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
   box()
   x = seq(min(r), max(r), length = 201)
   lines(x, dstd(x), lwd = 2)
   
   # Plot df and compare with true df:
   plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue",
     ylab = "Probability")
   lines(x, pstd(x), lwd = 2)
   
   # Compute quantiles:
   round(qstd(pstd(q = seq(-1, 5, by = 1))), digits = 6)

---

Student-t distribution parameter estimation

Description

Fits the parameters of the standardized Student-t distribution.

Usage

stdFit(x, ...)

Arguments

x	a numeric vector of quantiles.
...	parameters parsed to the optimization function nlm.

Value

stdFit returns a list with the following components:

par	The best set of parameters found.
objective	The value of objective corresponding to par.
convergence	An integer code. 0 indicates successful convergence.
message	A character string giving any additional information returned by the optimizer, or NULL. For details, see PORT documentation.
iterations	Number of iterations performed.
evaluations	Number of objective function and gradient function evaluations.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

See Also

std, stdSlider

Examples

## std -
   set.seed(1953)
   r = rstd(n = 1000)

       
## stdFit -
   stdFit(r)

---

Student-t distribution slider

Description

Displays interactively the dependence of the Student-t distribution on its parameters.

Usage

stdSlider(type = c("dist", "rand"))

Arguments

type	a character string denoting which interactive plot should be displayed. Either a distribution plot type="dist", the default value, or a random variates plot, type="rand".

Value

a Tcl object

Author(s)

Diethelm Wuertz for the Rmetrics R-port

See Also

std, stdFit,

Examples

## Not run:   
## stdSlider -
   require(tcltk)
   stdSlider("dist")
   stdSlider("rand")

## End(Not run)

---

GARCH summary methods

Description

Summary methods for GARCH modelling.

Methods

Methods for summary defined in package fGarch:

object = "fGARCH"

Summary function for objects of class "fGARCH".

How to read a diagnostic summary report?

The first five sections return the title, the call, the mean and variance formula, the conditional distribution and the type of standard errors:

 
        Title:
         GARCH Modelling 
        
        Call:
         garchFit(~ garch(1, 1), data = garchSim(), trace = FALSE) 
        
        Mean and Variance Equation:
         ~arch(0)
        
        Conditional Distribution:
         norm 
        
        Std. Errors:
         based on Hessian
        

The next three sections return the estimated coefficients and an error analysis including standard errors, t values, and probabilities, as well as the log Likelihood values from optimization:

        Coefficient(s):
                  mu         omega        alpha1         beta1  
        -5.79788e-05   7.93017e-06   1.59456e-01   2.30772e-01  
        
        Error Analysis:
                 Estimate  Std. Error  t value Pr(>|t|)
        mu     -5.798e-05   2.582e-04   -0.225    0.822
        omega   7.930e-06   5.309e-06    1.494    0.135
        alpha1  1.595e-01   1.026e-01    1.554    0.120
        beta1   2.308e-01   4.203e-01    0.549    0.583
        
        Log Likelihood:
         -843.3991    normalized:  -Inf 
        

The next section provides results on standardized residuals tests, including statistic and p values, and on information criterion statistic including AIC, BIC, SIC, and HQIC:

 
        Standardized Residuals Tests:
                                        Statistic p-Value    
         Jarque-Bera Test   R    Chi^2  0.4172129 0.8117146  
         Shapiro-Wilk Test  R    W      0.9957817 0.8566985  
         Ljung-Box Test     R    Q(10)  13.05581  0.2205680  
         Ljung-Box Test     R    Q(15)  14.40879  0.4947788  
         Ljung-Box Test     R    Q(20)  38.15456  0.008478302
         Ljung-Box Test     R^2  Q(10)  7.619134  0.6659837  
         Ljung-Box Test     R^2  Q(15)  13.89721  0.5333388  
         Ljung-Box Test     R^2  Q(20)  15.61716  0.7400728  
         LM Arch Test       R    TR^2   7.049963  0.8542942  
         
        Information Criterion Statistics:
                 AIC      BIC      SIC     HQIC 
            8.473991 8.539957 8.473212 8.500687  
        

Author(s)

Diethelm Wuertz for the Rmetrics R-port

Examples

## garchSim -
   x = garchSim(n = 200)

## garchFit - 
   fit = garchFit(formula = x ~ garch(1, 1), data = x, trace = FALSE)
   summary(fit)

---

Compute Value-at-Risk (VaR) and expected shortfall (ES)

Description

Compute Value-at-Risk (VaR) and Expected Shortfall (ES) for a fitted GARCH-APARCH model.

Usage

## S3 method for class 'fGARCH'
VaR(dist, p_loss = 0.05, ..., tol)

## S3 method for class 'fGARCH'
ES(dist, p_loss = 0.05, ...)

Arguments

dist	an object from class "fGARCH", obtained from garchFit().
p_loss	level, default is 0.05.
...	not used.
tol	tollerance

Details

We provide methods for the generic functions cvar::VaR and cvar::ES.

Note

We use the traditional definition of VaR as the negated lower quantile. For example, if $X$ are returns on an asset, VAR${}_\alpha$ = $-q_\alpha$, where $q_\alpha$ is the lower $\alpha$ quantile of $X$. Equivalently, VAR${}_\alpha$ is equal to the lower $1-\alpha$ quantile of $-X$ (the loss series). For details see the vignette in package cvar availalble at https://cran.r-project.org/package=cvar/vignettes/Guide_cvar.pdf (or by calling vignette("Guide_cvar", package = "cvar")).

If you wish to overlay the VaR or ES over returns, just negate the VaR/ES, see the examples.

See Also

VaR and ES in package cvar

Examples

## simulate a time series of returns
x <- garchSim( garchSpec(), n = 500)
class(x)
## fit a GARCH model
fit <- garchFit(~ garch(1, 1), data = x, trace = FALSE)

head(VaR(fit))
head(ES(fit))

## use plot method for fitted GARCH models
plot(fit, which = 14) # VaR
plot(fit, which = 15) # ES
plot(fit, which = 16) # VaR & ES
## plot(fit) # choose the plot interactively

## diy plots

## overlay VaR and ES over returns
## here x is from class 'timeSeries', so we convert VaR/ES to timeSeries
## don't forget to negate the result of VaR()/ES(),
plot(x)
lines(timeSeries(-VaR(fit)), col = "red")
lines(timeSeries(-ES(fit)), col = "blue")

## alternatively, plot losses (rather than returns) and don't negate VaR()/ES()
plot(-x)
lines(timeSeries(VaR(fit)), col = "red")
lines(timeSeries(ES(fit)), col = "blue")

---

Extract GARCH model volatility

Description

Extracts volatility from a fitted GARCH object.

Usage

## S3 method for class 'fGARCH'
volatility(object, type = c("sigma", "h"), ...)

Arguments

object	an object of class "fGARCH" as returned by garchFit().
type	a character string denoting if the conditional standard deviations "sigma" or the variances "h" should be returned.
...	currently not used.

Details

volatility is an S3 generic function for computation of volatility, see volatility for the default method.

The method for "fGARCH" objects, described here, extracts the volatility from slot @sigma.t or @h.t of an "fGARCH" object usually obtained from the function garchFit().

The class of the returned value depends on the input to the function garchFit who created the object. The returned value is always of the same class as the input object to the argument data in the function garchFit, i.e. if you fit a "timeSeries" object, you will get back from the function fitted also a "timeSeries" object, if you fit an object of class "zoo", you will get back again a "zoo" object. The same holds for a "numeric" vector, for a "data.frame", and for objects of class "ts", "mts".

In contrast, the slot itself always contains a numeric vector, independently of the class of the input data input, i.e. the function call slot(object, "fitted") will return a numeric vector.

Methods

Methods for volatility defined in package fGarch:

object = "fGARCH"

Extractor function for volatility or standard deviation from an object of class "fGARCH".

Note

(GNB) Contrary to the description of the returned value of the "fGARCH" method, it is always "numeric".

TODO: either implement the documented behaviour or fix the documentation.

Author(s)

Diethelm Wuertz for the Rmetrics R-port

See Also

garchFit, class fGARCH

Examples

## Swiss Pension fund Index -
   stopifnot(require("timeSeries")) # need package 'timeSeries'
   x = as.timeSeries(data(LPP2005REC, package = "timeSeries"))

## garchFit
   fit = garchFit(LPP40 ~ garch(1, 1), data = 100*x, trace = FALSE)
   fit

## volatility -
   # Standard Deviation:
   vola = volatility(fit, type = "sigma")
   head(vola)
   class(vola)
   # Variance:
   vola = volatility(fit, type = "h")
   head(vola)
   class(vola)

## slot -
   vola = slot(fit, "sigma.t")
   head(vola)
   class(vola)
   vola = slot(fit, "h.t")
   head(vola)
   class(vola)

---