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Version:

3.4

Date:

2025-03-26

Title:

Simulation, Estimation and Forecasting of Beta-Skew-t-EGARCH Models

Maintainer:

Genaro Sucarrat <genaro.sucarrat@bi.no>

Depends:

R (≥ 3.4.0), zoo

URL:

https://www.sucarrat.net/

Description:

Simulation, estimation and forecasting of first-order Beta-Skew-t-EGARCH models with leverage (one-component, two-component, skewed versions).

License:

GPL-2

NeedsCompilation:

yes

Packaged:

2025-03-26 10:32:32 UTC; sucarrat

Author:

Genaro Sucarrat [aut, cre]

Repository:

CRAN

Date/Publication:

2025-03-26 11:40:03 UTC

Simulation, estimation and forecasting of Beta-Skew-t-EGARCH models

Description

This package provides facilities for the simulation, estimation and forecasting of first order Beta-Skew-t-EGARCH models with leverage (one-component and two-component versions), see Harvey and Sucarrat (2014), and Sucarrat (2013).

Let y[t] denote a financial return at time t equal to

y[t] = sigma[t]*epsilon[t]

where sigma[t] > 0 is the scale or volatility (generally not equal to the conditional standard deviation), and where epsilon[t] is IID and t-distributed (possibly skewed) with df degrees of freedom. Then the first order log-volatility specifiction of the one-component Beta-Skew-t-EGARCH model can be parametrised as

sigma[t] = exp(lambda[t]),
lambda[t] = omega + lambdadagger,
lambdadagger[t] = phi1*lambdadagger[t-1] + kappa1*u[t-1] + kappastar*sign[-y]*(u[t-1]+1).

So the scale or volatility is given by sigma[t] = exp(lambda[t]). The omega is the unconditional or long-term log-volatility, phi1 is the GARCH parameter (|phi1| < 1 implies stability), kappa1 is the ARCH parameter, kappastar is the leverage or volatility-asymmetry parameter and u[t] is the conditional score or first derivative of the log-likelihood with respect to lambda. The score u[t] is zero-mean and IID, and (u[t]+1)/(df+1) is Beta distributed when there is no skew in the conditional density of epsilon[t]. The two-component specification is given by

sigma[t] = exp(lambda[t]),
lambda[t] = omega + lambda1dagger + lambda2dagger,
lambda1dagger[t] = phi1*lambdadagger[t-1] + kappa1*u[t-1],
lambda2dagger[t] = phi2*lambdadagger[t-1] + kappa2*u[t-1] + kappastar*sign[-y]*(u[t-1]+1).

The first component, lambda1dagger, is interpreted as the long-term component, whereas the second component, lambda2dagger, is interpreted as the short-term component.

Details

Package:	betategarch
Type:	Package
Version:	3.4
Date:	2025-03-26
License:	GPL-2
LazyLoad:	yes

The two main functions of the package are tegarchSim and tegarch. The first simulates a Beta-Skew-t-EGARCH models whereas the second estimates one. The second object returns an object (a list) of class 'tegarch', and a collection of methods can be applied to this class: coef.tegarch, fitted.tegarch, logLik.tegarch, predict.tegarch, print.tegarch, residuals.tegarch, summary.tegarch and vcov.tegarch. In addition, the output produced by the tegarchSim function and the fitted.tegarch and residuals.tegarch methods are of the Z's ordered observations (zoo) class, which means a range of time-series methods are available for these objects.

Author(s)

Genaro Sucarrat, https://www.sucarrat.net/

References

C. Fernandez and M. Steel (1998), 'On Bayesian Modeling of Fat Tails and Skewness', Journal of the American Statistical Association 93, pp. 359-371, doi:10.1080/01621459.1998.10474117

A. Harvey and G. Sucarrat (2014), 'EGARCH models with fat tails, skewness and leverage'. Computational Statistics and Data Analysis 76, pp. 320-338, doi:10.1016/j.csda.2013.09.022

G. Sucarrat (2013), 'betategarch: Simulation, Estimation and Forecasting of First-Order Beta-Skew-t-EGARCH models'. The R Journal (Volume 5/2), pp. 137-147, ,doi:10.32614/RJ-2013-034

Examples

##simulate 1000 observations from model with default parameter values:
set.seed(123)
y <- tegarchSim(1000)

##estimate and store as 'mymod':
mymod <- tegarch(y)

##print estimates and standard errors:
print(mymod)

##graph of fitted volatility (conditional standard deviation):
plot(fitted(mymod))

##plot forecasts of volatility 1-step ahead up to 10-steps ahead:
plot(predict(mymod, n.ahead=10))

Extraction methods for 'tegarch' objects

Description

Extraction methods for objects of class 'tegarch' (i.e. the result of estimating a Beta-Skew-t-EGARCH model)

Usage

## S3 method for class 'tegarch'
coef(object, ...)
## S3 method for class 'tegarch'
fitted(object, verbose = FALSE, ...)
## S3 method for class 'tegarch'
logLik(object, ...)
## S3 method for class 'tegarch'
print(x, ...)
## S3 method for class 'tegarch'
residuals(object, standardised = TRUE, ...)
## S3 method for class 'tegarch'
summary(object, verbose = FALSE, ...)
## S3 method for class 'tegarch'
vcov(object, ...)

Arguments

object

an object of class 'tegarch'

x

an object of class 'tegarch'

verbose

logical. If FALSE (default) then only basic information is returned

standardised

logical. If TRUE (default) then the standardised residuals are returned. If FALSE then the scaled (by sigma) residuals are returned

...

additional arguments

Details

Empty

Value

coef:

A numeric vector containing the parameter estimates

fitted:

A zoo object. If verbose=FALSE (default), then the zoo object is a vector containing the fitted conditional standard deviations. If verbose = TRUE, then the zoo object is a matrix containing the return series y, fitted scale (sigma), fitted conditional standard deviation (stdev), fitted log-scale (lambda), dynamic component(s) (lambdadagger in the 1-component specification, lambda1dagger and lambda2dagger in the 2-compoment specification), the score (u), scaled residuals (epsilon) and standardised residuals (residstd)

logLik:

The value of the log-likelihood at the maximum

print:

Prints the most important parts of the estimation results

residuals:

A zoo object. If standardised = TRUE (default), then the zoo object is a vector with the standardised residuals. If standardised = FALSE, then the zoo vector contains the scaled residuals

summary:

A list. If verbose = FALSE, then only the most important entries are returned. If verbose = TRUE, then all entries apart from the 1st. (the y series) is returned

vcov:

The variance-covariance matrix of the estimated coefficents. The matrix is obtained by inverting the numerically estimated Hessian

Author(s)

Genaro Sucarrat, https://www.sucarrat.net/

References

C. Fernandez and M. Steel (1998), 'On Bayesian Modeling of Fat Tails and Skewness', Journal of the American Statistical Association 93, pp. 359-371, doi:10.1080/01621459.1998.10474117

A. Harvey and G. Sucarrat (2014), 'EGARCH models with fat tails, skewness and leverage'. Computational Statistics and Data Analysis 76, pp. 320-338, doi:10.1016/j.csda.2013.09.022

G. Sucarrat (2013), 'betategarch: Simulation, Estimation and Forecasting of First-Order Beta-Skew-t-EGARCH models'. The R Journal (Volume 5/2), pp. 137-147, ,doi:10.32614/RJ-2013-034

Examples


##simulate 1000 observations from model with default parameter values:
set.seed(123)
y <- tegarchSim(1000)

##estimate and store as 'mymodel':
mymod <- tegarch(y)

##print estimation result:
print(mymod)

##extract coefficients:
coef(mymod)

##extract log-likelihood:
logLik(mymod)

##plot fitted conditional standard deviations:
plot(fitted(mymod))

##plot all the fitted series:
plot(fitted(mymod, verbose=TRUE))

##histogram of standardised residuals:
hist(residuals(mymod))

The skewed t distribution

Description

Density, random number generation, mean, variance, skewness and kurtosis functions for the uncentred skewed t distribution. The skewing method is that of Fernandez and Steel (1998).

Usage

dST(y, df = 10, sd = 1, skew = 1, log = FALSE)
rST(n, df = 10, skew = 1)
STmean(df, skew = 1)
STvar(df, skew = 1)
STskewness(df, skew = 1)
STkurtosis(df, skew = 1)

Arguments

y

numeric vector of quantiles

n

integer, the number of observations

df

degrees of freedom, greater than 0 and less than Inf

sd

scale, greater than 0 and less than Inf

skew

skewness, greater than 0 and less than Inf. Symmetry obtains when skew = 1 (default).

log

logical. TRUE returns the natural log of the density value, FALSE (default) returns the density value.

Details

Empty

Value

dST:

a numeric value, either the density value or the natural log of the density value

rST:

a numeric vector with n random numbers

STmean:

The mean of an uncentred skewed t variable

STvar:

The variance of an uncentred skewed t variable

STskewness:

3rd. moment of a standardised skewed t variable

STkurtosis:

4th. moment of a standardised skewed t variable

Note

Empty

Author(s)

Genaro Sucarrat, https://www.sucarrat.net/

References

C. Fernandez and M. Steel (1998), 'On Bayesian Modeling of Fat Tails and Skewness', Journal of the American Statistical Association 93, pp. 359-371, doi:10.1080/01621459.1998.10474117

Examples


##generate 1000 random numbers from the skewed t:
set.seed(123)
eps <- rST(500, df=5) #symmetric t
eps <- rST(500, df=5, skew=0.8) #skewed to the left
eps <- rST(500, df=5, skew=2) #skewed to the right

##compare empirical mean with analytical:
mean(eps)
STmean(5, skew=2)

##compare empirical variance with analytical:
var(eps)
STvar(5, skew=2)

Daily Apple stock returns

Description

The dataset contains two variables, day and nasdaqret. Day is the date of the return and nasdaqret is the daily (closing value) log-return in percent of the Apple stock over the period 10 September 1985 - 10 May 2011 (a total of 6835 observations).

Usage

data(nasdaq)

Format

A data frame with 3215 observations:

day: a factor
nasdaqret: a numeric vector

Details

The data is studied in more detail in Harvey and Sucarrat (2014), and in Sucarrat (2013).

Source

The source of the original raw data is http://yahoo.finance.com/.

References

A. Harvey and G. Sucarrat (2014), 'EGARCH models with fat tails, skewness and leverage'. Computational Statistics and Data Analysis 76, pp. 320-338, doi:10.1016/j.csda.2013.09.022

G. Sucarrat (2013), 'betategarch: Simulation, Estimation and Forecasting of First-Order Beta-Skew-t-EGARCH models'. The R Journal (Volume 5/2), pp. 137-147, ,doi:10.32614/RJ-2013-034

Examples


data(nasdaq) #load data into workspace
mymod <- tegarch(nasdaq[,"nasdaqret"]) #estimate volatility model of Apple returns
print(mymod)

Generate volatility forecasts n-steps ahead

Description

Generates volatility forecasts from a model fitted by tegarch (i.e. a Beta-Skew-t-EGARCH model)

Usage

## S3 method for class 'tegarch'
predict(object, n.ahead = 1, initial.values = NULL, n.sim = 10000,
  verbose = FALSE, ...)

Arguments

object

an object of class 'tegarch'.

n.ahead

the number of steps ahead for which prediction is required.

initial.values

a vector containing the initial values of lambda and lambdadagger (lambda1dagger and lambda2dagger for 2-component models). If NULL (default) then the fitted values associated with the last return-observation are used

n.sim

number of simulated skew t variates.

verbose

logical. If FALSE (default) then only the conditional standard deviations are returned. If TRUE then also the scale is returned.

...

additional arguments

Details

The forecast formulas of exponential ARCH models are much more complicated than those of ordinary or non-exponential ARCH models. This is particularly the case when the conditional density is skewed. The forecast formula of the conditional scale of the Beta-Skew-t-EGARCH model is not available in closed form. Accordingly, some terms (expectations involving the skewed t) are estimated numerically by means of simulation.

Value

A zoo object. If verbose = FALSE, then the zoo object is a vector with the forecasted conditional standard deviations. If verbose = TRUE, then the zoo object is a matrix with forecasts of both the conditional scale and the conditional standard deviation

Author(s)

Genaro Sucarrat, http://www.sucarrat.net/

References

C. Fernandez and M. Steel (1998), 'On Bayesian Modeling of Fat Tails and Skewness', Journal of the American Statistical Association 93, pp. 359-371, doi:10.1080/01621459.1998.10474117

A. Harvey and G. Sucarrat (2014), 'EGARCH models with fat tails, skewness and leverage'. Computational Statistics and Data Analysis 76, pp. 320-338, doi:10.1016/j.csda.2013.09.022

G. Sucarrat (2013), 'betategarch: Simulation, Estimation and Forecasting of First-Order Beta-Skew-t-EGARCH models'. The R Journal (Volume 5/2), pp. 137-147, ,doi:10.32614/RJ-2013-034

Examples


##simulate series with 500 observations:
set.seed(123)
y <- tegarchSim(500, omega=0.01, phi1=0.9, kappa1=0.1, kappastar=0.05, df=10, skew=0.8)

##estimate a 1st. order Beta-t-EGARCH model and store the output in mymod:
mymod <- tegarch(y)

##plot forecasts of volatility 1-step ahead up to 10-steps ahead:
plot(predict(mymod, n.ahead=10))

Estimate first order Beta-Skew-t-EGARCH models

Description

Fits a first order Beta-Skew-t-EGARCH model to a univariate time-series by exact Maximum Likelihood (ML) estimation. Estimation is via the nlminb function

Usage

tegarch(y, asym = TRUE, skew = TRUE, components = 1, initial.values = NULL,
  lower = NULL, upper = NULL, hessian = TRUE, lambda.initial = NULL,
  c.code = TRUE, logl.penalty = NULL, aux = NULL, ...)

Arguments

y

numeric vector, typically a financial return series.

asym

logical. TRUE (default) includes leverage or volatility asymmetry in the log-scale specification

skew

logical. TRUE (default) enables and estimates the skewness in conditional density (epsilon). The skewness method is that of Fernandez and Steel (1998)

components

Numeric value, either 1 (default) or 2. The former estimates a 1-component model, the latter a 2-component model

initial.values

NULL (default) or a vector with the initial values. If NULL, then the values are automatically chosen according to model (with or without skewness, 1 or 2 components, etc.)

lower

NULL (default) or a vector with the lower bounds of the parameter space. If NULL, then the values are automatically chosen

upper

NULL (default) or a vector with the upper bounds of the parameter space. If NULL, then the values are automatically chosen

hessian

logical. If TRUE (default) then the Hessian is computed numerically via the optimHess function. Setting hessian=FALSE speeds up estimation, which might be particularly useful in simulation. However, it also slows down the extraction of the variance-covariance matrix by means of the vcov method.

lambda.initial

NULL (default) or a vector with the initial value(s) of the recursion for lambda and lambdadagger. If NULL then the values are chosen automatically

c.code

logical. TRUE (default) is faster since it makes use of compiled C-code

logl.penalty

NULL (default) or a numeric value. If NULL then the log-likelihood value associated with the initial values is used. Sometimes estimation can result in NA and/or +/-Inf values, which are fatal for simulations. The value logl.penalty is the value returned by the log-likelihood function in the presence of NA or +/-Inf values

aux

NULL (default) or a list, se code. Useful for simulations (speeds them up)

...

further arguments passed to the nlminb function

Value

Returns a list of class 'tegarch' with the following elements:

y

the series used for estimation.

date

date and time of estimation.

initial.values

initial values used in estimation.

lower

lower bounds used in estimation.

upper

upper bounds used in estimation.

lambda.initial

initial values of lambda provided by the user, if any.

model

type of model estimated.

hessian

the numerically estimated Hessian.

sic

the value of the Schwarz (1978) information criterion.

par

parameter estimates.

objective

value of the log-likelihood at the maximum.

convergence

an integer code. 0 indicates successful convergence, see the documentation of nlminb.

iterations

number of iterations, see the documentation of nlminb.

evaluations

number of evaluations of the objective and gradient functions, see the documentation of nlminb.

message

a character string giving any additional information returned by the optimizer, or NULL. For details, see PORT documentation and the nlminb documentation.

NOTE

an additional message returned if one tries to estimate a 2-component model without leverage.

Note

Empty

Author(s)

Genaro Sucarrat, http://www.sucarrat.net/

References

C. Fernandez and M. Steel (1998), 'On Bayesian Modeling of Fat Tails and Skewness', Journal of the American Statistical Association 93, pp. 359-371, doi:10.1080/01621459.1998.10474117

A. Harvey and G. Sucarrat (2014), 'EGARCH models with fat tails, skewness and leverage'. Computational Statistics and Data Analysis 76, pp. 320-338, doi:10.1016/j.csda.2013.09.022

D. Nelson (1991): 'Conditional Heteroskedasticity in Asset Returns: A New Approach', Econometrica 59, pp. 347-370.

G. Schwarz (1978), 'Estimating the Dimension of a Model', The Annals of Statistics 6, pp. 461-464.

G. Sucarrat (2013), 'betategarch: Simulation, Estimation and Forecasting of First-Order Beta-Skew-t-EGARCH models'. The R Journal (Volume 5/2), pp. 137-147, ,doi:10.32614/RJ-2013-034

Examples

##simulate series with 500 observations:
set.seed(123)
y <- tegarchSim(500, omega=0.01, phi1=0.9, kappa1=0.1, kappastar=0.05,
  df=10, skew=0.8)

##estimate a 1st. order Beta-t-EGARCH model and store the output in mymod:
mymod <- tegarch(y)

##print estimates and standard errors:
print(mymod)

##graph of fitted volatility (conditional standard deviation):
plot(fitted(mymod))

##graph of fitted volatility and more:
plot(fitted(mymod, verbose=TRUE))

##plot forecasts of volatility 1-step ahead up to 20-steps ahead:
plot(predict(mymod, n.ahead=20))

##full variance-covariance matrix:
vcov(mymod)

Auxiliary functions

Description

tegarchLogl, tegarchLogl2, tegarchRecursion and tegarchRecursion2 are auxiliary functions called by tegarch, and which are not intended to be used for the average user. Henceforth they are thusonly scarcely documented, but most should either be self-explanatory (for the non-average user!) or more or less documented in relation with the tegarch and tegarchSim functions.

Usage

##the '2' relates to the 2-component specification:
tegarchLogl(y, pars, lower = -Inf, upper = Inf, lambda.initial = NULL,
  logl.penalty = -1e+100, c.code = TRUE, aux = NULL)
tegarchLogl2(y, pars, lower = -Inf, upper = Inf, lambda.initial = NULL,
  logl.penalty = -1e+101, c.code = TRUE, aux = NULL)
tegarchRecursion(y, omega = 0.1, phi1 = 0.4, kappa1 = 0.2, kappastar = 0.1,
  df = 10, skew = 0.6, lambda.initial = NULL, c.code = TRUE, verbose = FALSE,
  aux = NULL)
tegarchRecursion2(y, omega = 0.1, phi1 = 0.4, phi2 = 0.2, kappa1 = 0.05,
  kappa2 = 0.1, kappastar = 0.02, df = 10, skew = 0.6, lambda.initial = NULL,
  c.code = TRUE, verbose = FALSE, aux = NULL)

Arguments

y

numeric vector, typically a financial return series

omega

numeric

phi1

numeric, must be less than 1 in absolute value

phi2

numeric, must be less than 1 in absolute value

kappa1

numeric

kappa2

numeric

kappastar

numeric

df

numeric, the value of df (degrees of freedom)

skew

numeric (positive), the value of skew (skewness parameter)

verbose

logical. If FALSE (default) then only lambda is returned. If TRUE then a matrix with y and the fitted values of, amongst other, sigma, the log-scale (lambda), the conditional standard deviation (stdev), u, epsilon and the standardised residuals (residstd) are returned

pars

numeric vector, the parameter values

lower

numeric vector, the lower bounds used during estimation

upper

numeric vector, the upper bounds used during estimation

lambda.initial

NULL (default) or initial value(s) of the recursion for lambda. If NULL, then the values are chosen automatically

logl.penalty

numeric value

c.code

logical. TRUE (default) is faster since it makes use of compiled C-code

aux

NULL (default) or a list, se tegarch code

Details

tegarchLogl and tegarchLogl2 return the value of the log-likelihood for a 1-component and 2-component model, respectively.

Value

tegarchLogl:

The log-likelihood value (i.e. a numeric) of a 1-component specification

tegarchLogl2:

The log-likelihood value (i.e. a numeric) of a 2-component specification

tegarchRecursion:

A numeric vector containing the lambda values if verbose=FALSE (default). If verbose=TRUE then a matrix then a matrix with y and the fitted values of sigma, the log-scale (lambda), the conditional standard deviation (stdev), u, epsilon and the standardised residuals (residstd) are returned

tegarchRecursion2:

A numeric vector containing the lambda values if verbose=FALSE (default). If verbose=TRUE, then a matrix then a matrix with y and the fitted values of sigma, the log-scale (lambda), the conditional standard deviation (stdev), u, epsilon and the standardised residuals (residstd) are returned

Author(s)

Genaro Sucarrat, http://www.sucarrat.net/

References

C. Fernandez and M. Steel (1998), 'On Bayesian Modeling of Fat Tails and Skewness', Journal of the American Statistical Association 93, pp. 359-371, doi:10.1080/01621459.1998.10474117

A. Harvey and G. Sucarrat (2014), 'EGARCH models with fat tails, skewness and leverage'. Computational Statistics and Data Analysis 76, pp. 320-338, doi:10.1016/j.csda.2013.09.022

G. Sucarrat (2013), 'betategarch: Simulation, Estimation and Forecasting of First-Order Beta-Skew-t-EGARCH models'. The R Journal (Volume 5/2), pp. 137-147, ,doi:10.32614/RJ-2013-034

Simulate from a first order Beta-Skew-t-EGARCH model

Description

Simulate the y series (typically interpreted as a financial return or the error in a regression) from a first order Beta-Skew-t-EGARCH model. Optionally, the conditional scale (sigma), log-scale (lambda), conditional standard deviation (stdev), dynamic components (lambdadagger in the 1-component specification, lambda1dagger and lambda2dagger in the 2-component specification), score (u) and centred innovations (epsilon) are also returned.

Usage

tegarchSim(n, omega = 0, phi1 = 0.95, phi2 = 0, kappa1 = 0.01, kappa2 = 0,
  kappastar = 0, df = 10, skew = 1, lambda.initial = NULL, verbose = FALSE)

Arguments

n

integer, length of y (i.e. no of observations)

omega

numeric, the value of omega

phi1

numeric, the value of phi1

phi2

numeric, the value of phi2

kappa1

numeric, the value of kappa1

kappa2

numeric, the value of kappa2

kappastar

numeric, the value of kappastar

df

numeric, the value of df (degrees of freedom)

skew

numeric, the value of skew (skewness parameter

lambda.initial

NULL (default) or initial value(s) of the recursion for lambda or log-volatility. If NULL then the values are chosen automatically

verbose

logical, TRUE or FALSE (default). If TRUE then a matrix with n rows containing y, sigma, lambda, lambdadagger, u and epsilon is returned. If FALSE then only y is returned

Details

Empty

Value

A zoo vector of length n or a zoo matrix with n rows, depending on the value of verbose.

Author(s)

Genaro Sucarrat, http://www.sucarrat.net/

References

C. Fernandez and M. Steel (1998), 'On Bayesian Modeling of Fat Tails and Skewness', Journal of the American Statistical Association 93, pp. 359-371, doi:10.1080/01621459.1998.10474117

A. Harvey and G. Sucarrat (2014), 'EGARCH models with fat tails, skewness and leverage'. Computational Statistics and Data Analysis 76, pp. 320-338, doi:10.1016/j.csda.2013.09.022

G. Sucarrat (2013), 'betategarch: Simulation, Estimation and Forecasting of First-Order Beta-Skew-t-EGARCH models'. The R Journal (Volume 5/2), pp. 137-147, ,doi:10.32614/RJ-2013-034

Examples


##1-component specification: simulate series with 500 observations:
set.seed(123)
y <- tegarchSim(500, omega=0.01, phi1=0.9, kappa1=0.1, kappastar=0.05,
  df=10, skew=0.8)

##simulate the same series, but with more output (volatility, log-volatility or
##lambda, lambdadagger, u and epsilon)
set.seed(123)
y <- tegarchSim(500, omega=0.01, phi1=0.9, kappa1=0.1, kappastar=0.05, df=10, skew=0.8,
  verbose=TRUE)
  
##plot the simulated values:
plot(y)

##2-component specification: simulate series with 500 observations:
set.seed(123)
y <- tegarchSim(500, omega=0.01, phi1=0.95, phi2=0.9, kappa1=0.01, kappa2=0.05,
  kappastar=0.03, df=10, skew=0.8)

Simulation, estimation and forecasting of Beta-Skew-t-EGARCH models

Description

Details

Author(s)

References

Examples

Extraction methods for 'tegarch' objects

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

The skewed t distribution

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Daily Apple stock returns

Description

Usage

Format

Details

Source

References

Examples

Generate volatility forecasts n-steps ahead

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Estimate first order Beta-Skew-t-EGARCH models

Description

Usage

Arguments

Value

Note

Author(s)

References

See Also

Examples

Auxiliary functions

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Simulate from a first order Beta-Skew-t-EGARCH model

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples