Package 'tvgarch'

Title: Time Varying GARCH Modelling
Description: Simulation, estimation and inference for univariate and multivariate TV(s)-GARCH(p,q,r)-X models, where s indicates the number and shape of the transition functions, p is the ARCH order, q is the GARCH order, r is the asymmetry order, and 'X' indicates that covariates can be included; see Campos-Martins and Sucarrat (2024) <doi:10.18637/jss.v108.i09>. In the multivariate case, variances are estimated equation by equation and dynamic conditional correlations are allowed. The TV long-term component of the variance as in the multiplicative TV-GARCH model of Amado and Terasvirta (2013) <doi:10.1016/j.jeconom.2013.03.006> introduces non-stationarity whereas the GARCH-X short-term component describes conditional heteroscedasticity. Maximisation by parts leads to consistent and asymptotically normal estimates.
Authors: Susana Campos-Martins [aut, cre], Genaro Sucarrat [ctb]
Maintainer: Susana Campos-Martins <[email protected]>
License: GPL (>= 2)
Version: 2.4.2
Built: 2025-03-09 03:25:15 UTC
Source: https://github.com/cran/tvgarch

Help Index

Time Varying GARCH Modelling

Description

Simulation, estimation and inference for univariate and multivariate TV(s)-GARCH(p,q,r)-X models, where s indicates the number and shape of the transition functions, p is the ARCH order, q is the GARCH order, r is the asymmetry order, and 'X' indicates that covariates can be included; see Campos-Martins and Sucarrat (2024) <doi:10.18637/jss.v108.i09>. The TV long-term component, as in the multiplicative TV-GARCH model of Amado and Terasvirta (2013) <doi:10.1016/j.jeconom.2013.03.006>, introduces non-stationarity whereas the GARCH-X short-term component describes conditional heteroscedasticity. Maximisation by parts leads to consistent and asymptotically normal estimates. In the multivariate case, conditional variances are estimated equation by equation and dynamic conditional correlations are allowed.

Details

Package: tvgarch
Type: Package
Version: 2.4.2
Date: 2024-04-03
License: GPL>=2

Author(s)

Susana Campos-Martins, https://sites.google.com/site/susanacamposmartins

Maintainer: Susana Campos-Martins
Contributor: Genaro Sucarrat

References

Cristina Amado and Timo Terasvirta (2013) Modelling volatility by variance decomposition, Journal of Econometrics 175, 142-153.

Susana Campos-Martins and Genaro Sucarrat (2024) Modeling Nonstationary Financial Volatility with the R Package tvgarch, Journal of Statistical Software 108, 1-38.

See Also

tvgarchTest, tvgarch, mtvgarch, tvgarchSim, mtvgarchSim

Examples

set.seed(123)

## Simulate from a TV(1)-GARCH(1,1) model (default):
ySim <- tvgarchSim(n = 1500)

## Test a GARCH(1,1) model against a TV(1)-GARCH(1,1) model:
yTest <- tvgarchTest(y = ySim)
yTest

## Estimate a TV(1)-GARCH(1,1) model (default):
yEst <- tvgarch(y = ySim)
yEst

Extraction functions for multivariate 'mtvgarch' objects

Description

Extraction functions for objects of class 'mtvgarch'.

Usage

## S3 method for class 'mtvgarch'
coef(object, spec = c("tvgarch", "garch", "tv", "cc"), ...)
  ## S3 method for class 'mtvgarch'
fitted(object, spec = c("tvgarch", "garch", "tv", "cc"), 
                             as.zoo = TRUE, ...)
  ## S3 method for class 'mtvgarch'
logLik(object, ...)
  ## S3 method for class 'mtvgarch'
nobs(object, ...)
  ## S3 method for class 'mtvgarch'
plot(x, spec = c("tvgarch", "garch", "tv"), ...)                           
  ## S3 method for class 'mtvgarch'
predict(object, n.ahead = 10, newxtv = NULL, 
                              newxreg = NULL, newindex = NULL, n.sim = 5000, 
                              as.zoo = TRUE, verbose = FALSE, ...)
  ## S3 method for class 'mtvgarch'
print(x, ...)
  ## S3 method for class 'mtvgarch'
quantile(x, probs = 0.025, type = 7, as.zoo = TRUE, ...)
  ## S3 method for class 'mtvgarch'
residuals(object, as.zoo = TRUE, ...)
  ## S3 method for class 'mtvgarch'
summary(object, ...)
  ## S3 method for class 'mtvgarch'
toLatex(object, digits = 4, ...)
  ## S3 method for class 'mtvgarch'
vcov(object, spec = c("tvgarch", "garch", "tv", "cc"), ...)

Arguments

object

an object of class 'mtvgarch'.
spec

specifies whether the function should extract specific results. If "tv", extracts results for the TV component and if "garch" extracts results for the GARCH-X component of TV-GARCH-X model. If "tvgarch", extracts results for TV-GARCH-X model. Only relevant for TV-GARCH-X models. Otherwise, extracts results for GARCH-X models. If "cc", extracts results concerning the conditional correlations. "cc" is not valid in plot.mtvgarch().
x

an object of class 'mtvgarch'.
as.zoo

logical. If TRUE, then the returned result is of class zoo.
n.ahead

integer that determines how many steps ahead predictions should be generated.
newxtv

NULL or vector with the out-of-sample transition variable. If NULL, out-of-sample component g equals intercept.g. Only relevant for TV-GARCH-X models.
newxreg

vector or matrix with the out-of-sample regressor values.
newindex

zoo-index for the out-of-sample predictions. If NULL (default), then 1:n.ahead is used.
n.sim

integer, the number of simulations.
verbose

logical. If TRUE, then the simulations - in addition to the predictions - are returned.
probs

vector of probabilities.
type

integer that determines the algorithm used to compute the quantile, see quantile.
digits

integer, the number of digits in the printed LaTeX code.
...

additional arguments.

Value

coef:

parameter estimates.
fitted:

fitted conditional variances and correlations.
logLik:

optimised log-likelihood (normal density) values.
nobs:

number of observations used in the estimation.
plot:

plots of the fitted conditional volatilities.
predict:

variance predictions. Column order differs when spillovers are allowed.
print:

print of the estimation results.
quantile:

fitted quantiles, i.e. the conditional standard deviation times the empirical quantile of the standardised innovations.
residuals:

volatility standardised residuals.
summary:

summary of estimation results.
vcov:

coefficient variance-covariance matrices.

Author(s)

Susana Campos-Martins

References

Cristina Amado and Timo Terasvirta (2013) Modelling volatility by variance decomposition, Journal of Econometrics 175, 142-153.

Christian Francq and Jean-Michel Zakoian (2016) Estimating multivariate volatility models equation by equation, J. R. Stat. Soc. Ser. B Stat. Methodol 78, 613-635.

Susana Campos-Martins and Genaro Sucarrat (2024) Modeling Nonstationary Financial Volatility with the R Package tvgarch, Journal of Statistical Software 108, 1-38.

See Also

mtvgarch, mtvgarchSim, tvgarch, garchx, zoo

Examples

set.seed(12345)

## Simulate from a bivariate CCC-TV(1)-GARCH(1,1) model (default):
mySim <- mtvgarchSim(n = 1500)

## Estimate a CCC-TV(1)-GARCH(1,1) model:
myEst <- mtvgarch(y = mySim)

## Print estimation results:
print(myEst)

## Extract and store conditional variances:
sigma2Est <- fitted(myEst)

## Plot:
plot(myEst)

## Generate predictions:
predict(myEst)

Extraction functions (S3 methods) for univarate 'tvgarch' objects

Description

Extraction functions (S3 methods) for objects of class 'tvgarch'.

Usage

## S3 method for class 'tvgarch'
coef(object, spec = c("tvgarch", "garch", "tv"), ...)
  ## S3 method for class 'tvgarch'
fitted(object, spec = c("tvgarch", "garch", "tv"), 
                            as.zoo = TRUE, ...)
  ## S3 method for class 'tvgarch'
logLik(object, ...)
  ## S3 method for class 'tvgarch'
nobs(object, ...)
  ## S3 method for class 'tvgarch'
plot(x, spec = c("tvgarch", "garch", "tv"), ...)                            
  ## S3 method for class 'tvgarch'
predict(object, n.ahead = 10, newxtv = NULL, 
                             newxreg = NULL, newindex = NULL, n.sim = 5000, 
                             as.zoo = TRUE, verbose = FALSE, ...)
  ## S3 method for class 'tvgarch'
print(x, ...)
  ## S3 method for class 'tvgarch'
quantile(x, probs = 0.025, names = TRUE, type = 7, 
                              as.zoo = TRUE, ...)
  ## S3 method for class 'tvgarch'
residuals(object, as.zoo = TRUE, ...)
  ## S3 method for class 'tvgarch'
summary(object, ...)
  ## S3 method for class 'tvgarch'
toLatex(object, digits = 4, ...)
  ## S3 method for class 'tvgarch'
vcov(object, spec = c("tvgarch", "garch", "tv"), ...)

Arguments

object

an object of class 'tvgarch'.
spec

specifies whether the function should extract specific results. If "tv", extracts results for the TV component and if "garch" extracts results for the GARCH-X component of TV-GARCH-X model. If "tvgarch", extracts results for TV-GARCH-X model. Only relevant for TV-GARCH-X models. Otherwise, extracts results for GARCH-X models.
x

an object of class 'tvgarch'.
as.zoo

logical. If TRUE, then the returned result is of class zoo.
n.ahead

integer that determines how many steps ahead predictions should be generated.
newxtv

NULL or vector with the out-of-sample transition variable. If NULL, out-of-sample component g equals intercept.g. Only relevant for TV-GARCH-X models.
newxreg

vector or matrix with the out-of-sample regressor values.
newindex

a zoo-index for the out-of-sample predictions. If NULL (default), then 1:n.ahead is used.
n.sim

integer, the number of simulations.
verbose

logical. If TRUE, then the simulations - in addition to the predictions - are returned.
probs

vector of probabilities.
names

logical, whether to return names or not.
type

integer that determines the algorithm used to compute the quantile, see quantile.
digits

integer, the number of digits in the printed LaTeX code.
...

additional arguments.

Value

coef:

parameter estimates.
fitted:

fitted conditional variance.
logLik:

optimised log-likelihood (normal density) value.
nobs:

the number of observations used in the estimation.
plot:

plot of the fitted conditional volatility.
predict:

variance predictions.
print:

print of the estimation results.
quantile:

fitted quantiles, i.e. the conditional standard deviation times the empirical quantile of the standardised innovations.
residuals:

volatility standardised residuals.
summary:

summary of estimation results.
vcov:

coefficient variance-covariance matrix.

Author(s)

Susana Campos-Martins

References

Cristina Amado and Timo Terasvirta (2013) Modelling volatility by variance decomposition, Journal of Econometrics 175, 142-153.

Cristina Amado and Timo Terasvirta (2014) Modelling changes in the unconditional variance of long stock return series, Journal of Empirical Finance 25, 15-35.

Susana Campos-Martins and Genaro Sucarrat (2024) Modeling Nonstationary Financial Volatility with the R Package tvgarch, Journal of Statistical Software 108, 1-38.

See Also

tvgarchTest, tvgarch, tvgarchSim, zoo

Examples

set.seed(123)

## Simulate from a TV(1)-GARCH(1,1) model (default):
ySim <- tvgarchSim(n = 1500)

## Estimate a TV(1)-GARCH(1,1) model:
yEst <- tvgarch(y = ySim)

## Print estimation results:
print(yEst)

## Extract and store conditional variances:
sigma2Est <- fitted(yEst)

## Plot:
plot(yEst)

## Generate predictions:
predict(yEst)

Extraction functions for univarate 'tvgarchTest' objects

Description

Extraction functions for objects of class 'tvgarchTest'. Results from the estimation of the model under the null hypothesis, i.e., a GARCH(1,1) model, can be extracted similar to an object of class 'tvgarch' with the exception of functions print.tvgarchTest() and summary.tvgarchTest().

Usage

## S3 method for class 'tvgarchTest'
coef(object, ...)
  ## S3 method for class 'tvgarchTest'
fitted(object, 
  as.zoo = TRUE, ...)
  ## S3 method for class 'tvgarchTest'
logLik(object, ...)
  ## S3 method for class 'tvgarchTest'
nobs(object, ...)
  ## S3 method for class 'tvgarchTest'
plot(x, ...)                           
  ## S3 method for class 'tvgarchTest'
predict(object, n.ahead = 10, newxreg = NULL, 
                                 newindex = NULL, n.sim = 5000,
                                 as.zoo = TRUE, verbose = FALSE, ...)
  ## S3 method for class 'tvgarchTest'
print(x, ...)
  ## S3 method for class 'tvgarchTest'
quantile(x, probs = 0.025, names = TRUE, type = 7, 
                                  as.zoo = TRUE, ...)
  ## S3 method for class 'tvgarchTest'
residuals(object, as.zoo = TRUE, ...)
  ## S3 method for class 'tvgarchTest'
summary(object, ...)
  ## S3 method for class 'tvgarchTest'
toLatex(object, digits = 4, ...)
  ## S3 method for class 'tvgarchTest'
vcov(object, ...)

Arguments

object

an object of class 'tvgarchTest'.
x

an object of class 'tvgarchTest'.
as.zoo

logical. If TRUE, then the returned result is of class zoo.
n.ahead

integer that determines how many steps ahead predictions should be generated.
newxreg

vector or matrix with the out-of-sample regressor values.
newindex

a zoo-index for the out-of-sample predictions. If NULL (default), then 1:n.ahead is used.
n.sim

integer, the number of simulations.
verbose

logical. If TRUE, then the simulations - in addition to the predictions - are returned.
probs

vector of probabilities.
names

logical, whether to return names or not.
type

integer that determines the algorithm used to compute the quantile, see quantile.
digits

integer, the number of digits in the printed LaTeX code.
...

additional arguments.

Value

coef:

parameter estimates.
fitted:

fitted conditional variance.
logLik:

optimised log-likelihood (normal density) value.
nobs:

the number of observations used in the estimation.
plot:

plot of the fitted conditional volatility.
predict:

variance predictions.
quantile:

fitted quantiles, i.e. the conditional standard deviation times the empirical quantile of the standardised innovations.
residuals:

volatility standardised residuals.
summary:

summary of test result.
vcov:

coefficient variance-covariance matrix.

Author(s)

Susana Campos-Martins

References

Cristina Amado and Timo Terasvirta (2013) Modelling volatility by variance decomposition, Journal of Econometrics 175, 142-153.

Cristina Amado and Timo Terasvirta (2014) Modelling changes in the unconditional variance of long stock return series, Journal of Empirical Finance 25, 15-35.

Susana Campos-Martins and Genaro Sucarrat (2024) Modeling Nonstationary Financial Volatility with the R Package tvgarch, Journal of Statistical Software 108, 1-38.

See Also

tvgarchTest, tvgarchSim, tvgarch, zoo

Examples

set.seed(123)

## Simulate from a TV(1)-GARCH(1,1) model (default):
ySim <- tvgarchSim(n = 1500)

## Test a GARCH(1,1) model against a TV(1)-GARCH(1,1) model:
yTest <- tvgarchTest(y = ySim)

## Print test and estimation results:
print(yTest)

## Estimated number of locations
summary(yTest)

## Extract and plot estimation results for GARCH(1,1) used in the test:
sigma2Test <- fitted(yTest)
plot(yTest)

## Estimate a TV(s)-GARCH(1,1) model:
s <- summary(yTest)
yEst <- tvgarch(y = ySim, order.g  = s)

Compute all combinations of a hierarchy of models of n variables, and enumerate the combinations of the elements of a vector.

Description

combos produces a matrix of combinations of 1 to n variables in ascending order. combinations enumerates the possible combinations of a specified size from the elements of a vector.

Usage

combos(n)
  combinations(n, r, v = 1:n, set = TRUE, repeats.allowed = FALSE)

Arguments

n

an integer: the number of variables (combos) or the size of the source vector (combinations)
r

size of the target vectors
v

source vector. Defaults to 1:n
set

logical flag indicating whether duplicates should be removed from the source vector v. Defaults to TRUE.
repeats.allowed

logical flag indicating whether the constructed vectors may include duplicated values. Defaults to FALSE.

Details

combos lists hierarchy of all possible combinations of n variables in ascending order, starting with 1 variable, then all combinations of 2 variables, and so on until the one combination with all n variables. It is used by function tvgarch to constrain the size coefficients when s > 1 required to guarantee the variance is positive for all t.

When using combinations, the number of combinations increases rapidly with n and r! To use values of n above about 45, you will need to increase R's recursion limit. See the expression argument to the options command for details on how to do this. The source code is adapted from the function with the same name in the package gtools. There, it is stated that the code of the function is from an email by Brian D Ripley <[email protected]> to r-help dated Tue, 14 Dec 1999 11:14:04 +0000 (GMT) in response to Alex Ahgarin [email protected]. Original version was named "subsets" and was Written by Bill Venables

Value

combos:

a matrix with zeroes in empty elements and 1 in all full elements.
combinations:

a matrix where each row contains a vector of length r.

Author(s)

combos by Chris Walsh [email protected], with modifications by Susana Campos-Martins. Original versions of combinations by Bill Venables [email protected]. Extended to handle repeats.allowed by Gregory R. Warnes [email protected].

References

Venables, Bill. "Programmers Note", R-News, Vol 1/1, Jan. 2001. https://cran.r-project.org/doc/Rnews/

See Also

tvgarch

Examples

combos(3)

combinations(3,2,letters[1:3])
combinations(3,2,letters[1:3],repeats=TRUE)

Auxiliary functions

Description

Auxiliary functions used in the estimation of the multivariate TV(s)-GARCH(p,q,r)-X model. Not intended for the average user.

Usage

dccObj(par.dcc, z, sigma2, flag)

Arguments

par.dcc

numeric vector containing the ARCH- and GARCH-type coefficients in the dynamic conditional correlations.
z

matrix of standardized residuals.
sigma2

matrix of conditional variances.
flag

integer. If 0, returns a numeric vector with the values of the objective function; if 1 returns the the value of the objective function; if 2, returns the fitted variance components.

Value

The values of the objective function or fitted dynamic conditional correlations.

Author(s)

Susana Campos-Martins

References

Susana Campos-Martins and Genaro Sucarrat (2024) Modeling Nonstationary Financial Volatility with the R Package tvgarch, Journal of Statistical Software 108, 1-38.

Robert F. Engle (2002) Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models, Journal of Business and Economic Statistics 20, 339-350.

See Also

mtvgarch, fitted.mtvgarch, residuals.mtvgarch

Estimate a multivariate TV-GARCH-X model

Description

Equation by equation estimation of a multivariate multiplicative TV-GARCH-X model with dnamic conditional correlations. For each variance equation, the long-term or unconditional component (TV) and the short-term or conditional variance component (GARCH-X) are estimated separately using maximization by parts, where the iterative algorithm proceeds until convergence. Conditional on the variance estimates, the dynamic conditional correlations are estimated by maximum likelihood.

Usage

mtvgarch(y, order.g = c(1, 1), order.h = NULL, order.x = NULL, 
initial.values = list(), xtv = NULL, xreg = NULL, opt = 2, upper.speed = NULL,
tvgarch = FALSE, dcc = FALSE, turbo = TRUE, trace = FALSE)

Arguments

y

numeric matrix, time series or zoo object.
order.g

integer matrix with each row indicating the order.g for each series; number of locations in each transition function of the TV components.
order.h

integer matrix with each row indicating the order.h for each series; the first column controls the GARCH order, the second the ARCH order and the third the asymmetry order of the GARCH-X components. If NULL, the default, all series are assumed to follow a GARCH(1,1,0).
order.x

NULL or binary matrix indicating which xreg variables should be included as covariates in the GARCH-X components. If provided and xreg is NULL, then the selected volatility spillovers are included as covariates.
initial.values

a list containing the initial parameter values passed on to the optimisation routines (constrOptim for the TV component and nlminb for the GARCH-X component). If list(), the default, then the values are chosen automatically. TV component: intercept.g - NULL or numeric vector, size - NULL or numeric matrix containing the size initial coefficients, speed - NULL or numeric matrix containing the speed initial coefficients, location - NULL or numeric matrix containing the location initial coefficients. GARCH-X component: intercept.h - numeric vector, arch - NULL or numeric matrix containing the ARCH initial coefficients, garch - NULL or numeric matrix containing the GARCH-type initial coefficients, asym - NULL or numeric matrix containing the asymmetry-type initial coefficients, par.xreg - NULL or numeric matrix containing the X-type initial coefficients, and R - initial correlation coefficients.
xtv

NULL or numeric vector, time series or zoo object to include as the transition variable in the TV component. If NULL, calendar time, scaled between 0 and 1, is used as the transition variable.
xreg

numeric vector, time series or zoo object to include as covariates in the GARCH-X component.
opt

integer indicating whether the speed parameter in the TV component should be scaled. If 0, no scaling; if 1, speed/sd(xtv); if 2, exp(speed).
upper.speed

NULL or numeric scalar that sets the upper bound for speed in each transition function. If NULL, the default, the upper bound is 10000 for all transition functions. If numeric scalar, upper.speed is used for all transition functions.
tvgarch

logical. If TRUE, the full parameter set is estimated in one final step as well as the standard errors. If FALSE (default), estimates from last iteration are reported instead.
dcc

logical. If TRUE, dynamic conditional correlations are estimated. If FALSE (default), then the conditional correlations are constant.
turbo

logical. If FALSE (default), then the coefficient variance-covariance is computed during estimation, and the fitted values and residuals are attached to the returned object. If TRUE, then these operations are skipped, and hence estimation is faster. Note, however, that if turbo is set to TRUE, then the coefficient-covariance, fitted values and residuals can still be extracted subsequent to estimation with vcov.mtvgarch(), fitted.mtvgarch() and residuals.mtvgarch(), respectively.
trace

logical. If TRUE all output is printed.

Value

An object of class 'mtvgarch'.

Author(s)

Susana Campos-Martins

References

Cristina Amado and Timo Terasvirta (2013) Modelling volatility by variance decomposition, Journal of Econometrics 175, 142-153.

Susana Campos-Martins and Genaro Sucarrat (2024) Modeling Nonstationary Financial Volatility with the R Package tvgarch, Journal of Statistical Software 108, 1-38.

Christian Francq and Jean-Michel Zakoian (2016) Estimating multivariate volatility models equation by equation, J. R. Stat. Soc. Ser. B Stat. Methodol 78, 613-635.

Robert F. Engle (2002) Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models, Journal of Business and Economic Statistics 20, 339-350.

See Also

tvgarch, garchx, nlminb, constrOptim

Examples

set.seed(12345)

## Simulate from a bivariate CCC-TV(1)-GARCH(1,1) model (default):
mySim <- mtvgarchSim(n = 1000)

## Estimate a CCC-TV(1)-GARCH(1,1) model (default):
myEst <- mtvgarch(y = mySim)

## Print estimation results:
print(myEst)

## Extract coefficients:
coef(myEst)

## Plot conditional volatilities:
plot(myEst)

## Generate predictions:
predict(myEst)

Simulate from a multivariate TV-GARCH-X model

Description

Simulate from a multivariate multiplicative TV(s)-GARCH(p,q,r)-X model.

Usage

mtvgarchSim(n, m = 2, order.g = c(1,1), order.h = c(1,1,0, 1,1,0), 
order.x = NULL, intercept.g = c(1.2,1), size = c(3,5), speed = c(10,25), 
location = c(0.5,0.8), intercept.h = c(0.2,0.3), arch = c(0.10,0.05), 
garch = c(0.80,0.90), asym = NULL, xtv = NULL, xreg = NULL, par.xreg = NULL,  
R = c(1,0.6,0.6,1), dcc = FALSE, par.dcc = NULL, opt = 0, as.zoo = TRUE, 
verbose = FALSE, innovations = NULL)

Arguments

n

integer.
m

integer indicating the dimenson of the multivariate series.
order.g

integer matrix with each row indicating the number of locations in each transition function of the TV components; m rows and max.s columns.
order.h

integer matrix with each row indicating the order.h for each series; the first column controls the GARCH order, the second the ARCH order and the third the asymmetry order of the GARCH-X components.
order.x

NULL or binary matrix indicating which xreg variables should be included as covariates in the GARCH-X components. If provided and xreg is NULL, volatility spillovers for the selected series are included as covariates.
intercept.g

NULL or numeric vector.
size

NULL or numeric matrix containing the size coefficients. Only relevant for TV-GARCH models.
speed

NULL or numeric matrix containing the speed coefficients. Only relevant for TV-GARCH models.
location

NULL or numeric matrix containing the location coefficients; m rows and max.c columns. Only relevant for TV-GARCH models.
intercept.h

numeric matrix.
arch

NULL or numeric matrix containing the ARCH coefficients.
garch

NULL or numeric matrix containing the GARCH-type coefficients.
asym

NULL or numeric matrix containing the asymmetry-type coefficients.
xtv

NULL or numeric vector, time series or zoo object to include as the transition variable in the TV component. If NULL, calendar time, scaled between 0 and 1, is used as the transition variable. Only relevant for TV-GARCH models.
xreg

numeric vector, matrix, time series or zoo object to include as covariates in the GARCH-X component.
par.xreg

NULL or numeric matrix containing the covariates initial coefficients.
R

matrix of (constant) conditional correlations.
dcc

logical. If TRUE, dynamic conditional correlations are estimated. If FALSE (default), then the conditional correlations are constant.
par.dcc

numeric vector containing the ARCH- and GARCH-type coefficients in the dynamic conditional correlations.
opt

integer indicating whether the speed parameter in the TV component should be scaled. If 0, no scaling; if 1, speed/sd(xtv); if 2, exp(speed). Only relevant for TV-GARCH models.
as.zoo

logical. If TRUE, then the returned result is of class zoo.
verbose

logical, if TRUE, the conditional variance and the innovations are also returned.
innovations

NULL or numeric matrix with the innovations. If NULL, then standard normal innovations are generated with rnorm.

Value

An object of class 'zoo' (if as.zoo = TRUE), otherwise a matrix or a list (if verbose = TRUE), with the simulated values.

Author(s)

Susana Campos-Martins

See Also

mtvgarch, tvgarch, garchx, zoo

Examples

set.seed(12345)

## Simulate from a bivariate CCC-TV(1)-GARCH(1,1) model (default):
mySim1 <- mtvgarchSim(n = 1500)

## Simulate from a bivariate CCC-TV(1)-GARCH(1,1)-X model
## (with volatility spillovers)
mySim2 <- mtvgarchSim(n = 1500, order.x = c(0,1,1,0), par.xreg = 
c(0.03,0.04))

Estimate a TV-GARCH-X model

Description

Quasi Maximum Likelihood (ML) estimation of a univariate multiplicative TV(s)-GARCH(p,q,r)-X model, where s indicates the number and the shape of the transition functions, r is the asymmetry order, p is the ARCH order, q is the GARCH order, and 'X' indicates that covariates can be included. Any transition variable, deterministic or stochastic, can be used to drive the transitions between the variance states. The TV long-term component introduces non-stationarity in the variance process, where the GARCH-X short-term component describes conditional heteroscedasticity. Maximization by parts leads to consistent and asymptotically normal estimates.

Usage

tvgarch(y, order.g = 1, order.h = c(1,1,0), xtv = NULL, xreg = NULL, 
        initial.values = list(), opt = 2, upper.speed = NULL, tvgarch = FALSE,
        turbo = FALSE, trace = FALSE)

Arguments

y

numeric vector, time series or zoo object.
order.g

integer vector of length s indicating the number of locations in each transition function of the TV component. Indicates whether a stationary GARCH or a nonstationary GARCH, i.e., TV-GARCH, shall be estimated.
order.h

integer vector of the form c(p,q,r). The first entry controls the GARCH order, the second the ARCH order and the third the asymmetry order of the GARCH-X component.
initial.values

a list containing the initial parameter values passed on to the optimisation routines (constrOptim for the TV component and nlminb for the GARCH-X component). If list(), the default, then the values are chosen automatically. TV component: intercept.g - NULL or numeric, size - NULL or numeric vector containing the size initial coefficients, speed - NULL or numeric vector containing the speed initial coefficients, location - NULL or numeric vector containing the location initial coefficients. GARCH-X or GARCH-X component of TV-GARCH-X: intercept.h - numeric, arch - NULL or numeric vector containing the ARCH initial coefficients, garch - NULL or numeric vector containing the GARCH-type initial coefficients, asym - NULL or numeric vector containing the assymetry-type initial coefficients, and par.xreg - NULL or numeric vector containing the X-type initial coefficients.
xtv

NULL or numeric vector, time series or zoo object to include as the transition variable in the TV component. If NULL, calendar time, scaled between 0 and 1, is used as the transition variable. Not relevant for stationary GARCH.
xreg

numeric vector, time series or zoo object to include as covariates in the GARCH-X component.
opt

integer indicating whether the speed parameter in the TV component should be scaled. If 0, no scaling; if 1, speed/sd(xtv); if 2, exp(speed). Only relevant for TV-GARCH models.
upper.speed

NULL or numeric scalar that sets the upper bound for speed in each transition function. If NULL, the default, the upper bound is 10000 for all transition functions. If numeric scalar, upper.speed is used for all transition functions.
tvgarch

logical. If TRUE, the full parameter set is estimated in one final step as well as the standard errors. If FALSE (default), estimates from last iteration are reported instead.
turbo

logical. If FALSE (default), then the coefficient variance-covariance is computed during estimation, and the fitted values and residuals are attached to the returned object. If TRUE, then these operations are skipped, and hence estimation is faster. Note, however, that if turbo is set to TRUE, then the coefficient-covariance, fitted values and residuals can still be extracted subsequent to estimation with vcov.tvgarch(), fitted.tvgarch() and residuals.tvgarch(), respectively.
trace

logical. If TRUE all output is printed when estimating a TV-GARCH.

Value

An object of class 'tvgarch'.

Author(s)

Susana Campos-Martins

References

Cristina Amado and Timo Terasvirta (2013) Modelling volatility by variance decomposition, Journal of Econometrics 175, 142-153.

Cristina Amado and Timo Terasvirta (2014) Modelling changes in the unconditional variance of long stock return series, Journal of Empirical Finance 25, 15-35.

Susana Campos-Martins and Genaro Sucarrat (2024) Modeling Nonstationary Financial Volatility with the R Package tvgarch, Journal of Statistical Software 108, 1-38.

See Also

garchx, tvgarchSim, nlminb, constrOptim

Examples

set.seed(123)

## Simulate from a TV(1)-GARCH(1,1) model (default):
ySim <- tvgarchSim(n = 1500)

## Estimate a TV(1)-GARCH(1,1) model:
yEst <- tvgarch(y = ySim)

## Print estimation results:
print(yEst)

## Extract coefficients:
coef(yEst)

## Plot conditional volatilities:
plot(yEst)

## Extract log-likelihood:
logLik(yEst)

## Extract and store standardised residuals:
etaEst <- residuals(yEst)

## Generate predictions:
predict(yEst)

Auxiliary functions

Description

Auxiliary functions used in the estimation of the univariate and multivariate TV(s)-GARCH(p,q,r)-X model. Not intended for the average user.

Usage

tv(speed, location, xtv = NULL, n = NULL, opt = 0, 
  order.g = NULL, as.zoo = TRUE, verbose = FALSE)
  tvObj(par.g, fixed.par.g, xtv, opt, order.g, fixed.h, y, iter0, flag)
  garchObj(par.h, xreg, order.h, fixed.g, y, flag)
  tvgarchObj(par, fixed.par.g, y, order.g, xtv, opt, iter.fit.h, flag)

Arguments

speed

NULL or numeric vector with the values of the speed coefficients.
location

NULL or numeric vector with the values of the location coefficients.
xtv

NULL or numeric vector, time series or zoo object to include as the transition variable in the TV component. If NULL, a continuous variable bounded between 0 and 1 for n observations is constructed and used as the transition variable.
n

integer indicating the number of observations of the continuous transition variable bounded between 0 and 1.
opt

integer, indicates whether the speed parameter in the TV component should be scaled. If 0, no scaling; if 1, speed/sd(xtv); if 2, exp(speed). For function tv(), the default is 0.
order.g

a scalar in tv() and an integer vector of length s in tvObj() indicating the number of locations in each transition function of the TV component. For function tv(), it defaults to NULL.
as.zoo

logical. If TRUE, then the returned result is of class zoo.
verbose

logical, if TRUE, the values of not only the logistic transition function but also the transition variable are returned.
par.g

numeric vector with the values of the parameters in the TV component. If iter0=TRUE, par.g takes the form c(intercept.g, size, speed, location); if iter0=FALSE, then par.g=c(speed,size, location) and the values of the fixed parameters are provided using fixed.par.g).

fixed.par.g

NULL or numeric vector with the values of the parameters fixed in the TV component, i.e., intercept.g.
par

NULL or numeric vector with the values of the parameters in the TV-GARCH-X model.
fixed.h

numeric vector, time series or zoo containing the values of GARCH-X component).
y

numeric vector, time series or zoo object.
iter0

logical. If FALSE, some parameters in the TV component are fixed during the iterative estimation.
flag

integer. If 0, returns a numeric vector with the values of the objective function; if 1 returns the the value of the objective function; if 2, returns the fitted variance components.
par.h

numeric vector with the values of the parameters in the GARCH-X component.
order.h

integer vector of the form c(p,q,r). The first entry controls the GARCH order, the second the ARCH order and the third the asymmetry order of the GARCH-X component.
xreg

numeric vector, times eries or zoo object to include as covariates in the GARCH-X component.
fixed.g

numeric vector, time series or zoo containing the values of TV component).
iter.fit.h

a list of class 'garchx'.

Value

The values of the objective function or fitted variance components.

Author(s)

Susana Campos-Martins

See Also

tvgarch, fitted.tvgarch, residuals.tvgarch

Simulate from a univariate TV-GARCH-X model

Description

Simulate from a univariate multiplicative TV(s)-GARCH(p,q,r)-X model.

Usage

tvgarchSim(n, order.g = 1, order.h = c(1,1,0),
           intercept.g = 1.2, size = 5, speed = 25, location = 0.5, xtv = NULL, 
           intercept.h = 0.2, arch = 0.1, garch = 0.8, asym = NULL, xreg = NULL,
           opt = 0, as.zoo = TRUE, verbose = FALSE, innovations = NULL)

Arguments

n

integer.
order.g

integer vector of length s indicating the number of locations in each transition function of the TV component.
order.h

integer vector of the form c(p,q,r). The first entry controls the GARCH order, the second the ARCH order and the third the asymmetry order of the GARCH-X component.
intercept.g

NULL or numeric with the value of the intercept in the TV component.
size

NULL or numeric vector with the values of the size coefficients.
speed

NULL or numeric vector with the values of the speed coefficients.
location

NULL or numeric vector with the values of the location coefficients.
xtv

NULL or numeric vector, time series or zoo object to include as the transition variable in the TV component. If NULL, calendar time, scaled between 0 and 1, is used as the transition variable.
opt

integer indicating whether the speed parameter in the TV component should be scaled. If 0, no scaling; if 1, speed/sd(xtv); if 2, exp(speed).
intercept.h

numeric with the value of the intercept in the GARCH-X component.
arch

NULL or numeric vector with the values of the ARCH-coefficients.
garch

NULL or numeric vector with the values of the GARCH-coefficients.
asym

NULL or numeric vector with the values of the asymmetry-coefficients.
xreg

NULL or numeric vector with the values of the X-term.
as.zoo

logical. If TRUE, then the returned result is of class zoo.
verbose

logical, if TRUE, the conditional variance and innovations are also returned.
innovations

NULL or numeric vector with the innovations. If NULL, then standard normal innovations are generated with rnorm.

Value

An object of class 'zoo' (if as.zoo = TRUE), otherwise a vector or a matrix (if verbose = TRUE), with the simulated values.

Author(s)

Susana Campos-Martins

See Also

tvgarch, garchx, zoo

Examples

set.seed(123)

## Simulate from a TV(1)-GARCH(1,1) model (default):
ySim1 <- tvgarchSim(n = 1500)

## Simulate from a TV(2)-GARCH(1,1) model:
ySim2 <- tvgarchSim(n = 1500, order.g = c(1,2), size = c(0.5,-0.4), 
                    speed = c(1.5,2), location = c(0.2, 0.5,0.8))

## Simulate from a GARCH(1,1) model:
ySim3 <- tvgarchSim(n = 1500, order.g = NULL)

## Simulate from a TV(1)-GARCH(1,1,1)-X model:
ySim4 <- tvgarchSim(n = 1500, order.h = c(1,1,1), asym = 0.025, xreg = ySim3^2)

Test of a multiplicative time-varying GARCH model

Description

Compute the non-robust and robust Lagrange-Multiplier (LM-)type test statistics for examining the null hypothesis of constant long-term variance, GARCH(1,1), against the alternative of a smoothly changing long-term component, TV-GARCH(1,1).

Usage

tvgarchTest(y, xtv = NULL, alpha = 0.05)

Arguments

y

numeric vector, time series or zoo object.
xtv

NULL or numeric vector, time series or zoo object to include as the transition variable in the TV component. If NULL, calendar time scaled between 0 and 1 is used as the transition variable.
alpha

the significance level.

Value

An object of class 'tvgarchTest'.

Author(s)

Susana Campos-Martins

References

Cristina Amado and Timo Terasvirta (2017) Specification and testing of multiplicative time varying GARCH models with applications, Econometric Reviews 36:4, 421-446.

See Also

tvgarch, garchx, tvgarchSim

Examples

set.seed(12345)

## Simulate from a TV(1)-GARCH(1,1) model (default):
ySim <- tvgarchSim(n = 1500)

## Test of a TV(1)-GARCH(1,1) model:
yTest <- tvgarchTest(y = ySim)
orderG1 <- summary(yTest)

## Estimate a TV(1)-GARCH(1,1) model:
yEst <- tvgarch(y = ySim, order.g = orderG1)