Simulation and estimation of log-GARCH models

Description

This package provides facilities for the simulation and estimation of univariate log-GARCH models, and for the multivariate CCC-log-GARCH(1,1) model, see Sucarrat, Gronneberg and Escribano (2013), Sucarrat and Escribano (2013), and Francq and Sucarrat (2013).

Let y[t] denote a financial return or the error of a regression at time t such that

y[t] = sigma[t]*z[t],

where sigma[t] > 0 is the conditional standard deviation or volatility at t, and where z[t] is an IID innovation with mean zero and unit variance. The log-volatility specifiction of the log-GARCH-X model is given by

ln sigma[t]^2 = intercept + Sum_i alpha_i * ln y[t-i]^2 + Sum_j beta_j *ln sigma[t-1]^2 + Sum_k lambda_k * x[t]_k,

where the conditioning x-variables can be contemporaneous and/or lagged. The lgarch package estimates this model via its ARMA-X representation, see Sucarrat, Gronneberg and Escribano (2013), and treats zeros on y as missing values, see Sucarrat and Escribano (2013).

Details

The main functions of the package are: lgarchSim, mlgarchSim, lgarch and mlgarch. The first two functions simulate from a univariate and a multivariate log-GARCH model, respectively, whereas the latter two estimate a univariate and a multivariate log-GARCH model, respectively.

The lgarch and mlgarch functions return an object (a list) of class 'lgarch' and 'mlgarch', respectively. In both cases a collection of methods can be applied to each of them: coef, fitted, logLik, print, residuals, summary and vcov. In addition, the function rss can be used to extract the Residual Sum of Squares of the estimated ARMA representation from an lgarch object.

The output produced by the lgarchSim and mlgarchSim functions, and by the fitted and residuals methods, are of the Z's ordered observations (zoo) class, see Zeileis and Grothendieck (2005), and Zeileis, Grothendieck and Ryan (2014). This means a range of time-series and plotting methods are available for these objects.

Author(s)

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

References

C. Francq and G. Sucarrat (2018), 'An Exponential Chi-Squared QMLE for Log-GARCH Models Via the ARMA Representation', Journal of Financial Econometrics 16, pp. 129-154 doi:10.1093/jjfinec/nbx032

G. Sucarrat and A. Escribano (2018), 'Estimation of Log-GARCH Models in the Presence of Zero Returns', European Journal of Finance 24, pp. 809-827, doi:10.1080/1351847X.2017.1336452

G. Sucarrat, S. Gronneberg and A. Escribano (2016), 'Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown', Computational Statistics and Data Analysis 100, pp. 582-594, doi:10.1016/j.csda.2015.12.005

Zeileis, A. and G. Grothendieck (2005), 'zoo: S3 Infrastructure for Regular and Irregular Time Series', Journal of Statistical Software 14, pp. 1-27

Zeileis, A., G. Grothendieck, J.A. Ryan and F. Andrews(2014), 'zoo: S3 Infrastructure for Regular and Irregular Time Series (Z's ordered observations)', R package version 1.7-11, https://CRAN.R-project.org/package=zoo/

See Also

lgarchSim, mlgarchSim, lgarch, mlgarch, coef.lgarch, coef.mlgarch, fitted.lgarch,
fitted.mlgarch, logLik.lgarch, logLik.mlgarch, print.lgarch, print.mlgarch,
residuals.lgarch, residuals.mlgarch, rss, summary.lgarch, summary.mlgarch, vcov.lgarch,
vcov.mlgarch and zoo

Examples

##simulate 500 observations w/default parameter values from
##a univariate log-garch(1,1):
set.seed(123)
y <- lgarchSim(500)

##estimate a log-garch(1,1):
mymod <- lgarch(y)

##print results:
print(mymod)

##extract coefficients:
coef(mymod)

##extract Gaussian log-likelihood (zeros excluded, if any) of the log-garch model:
logLik(mymod)

##extract Gaussian log-likelihood (zeros excluded, if any) of the arma representation:
logLik(mymod, arma=TRUE)

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

##extract and plot the fitted conditional standard deviation:
sdhat <- fitted(mymod)
plot(sdhat)

##extract and plot standardised residuals:
zhat <- residuals(mymod)
plot(zhat)

##extract and plot all the fitted series:
myhat <- fitted(mymod, verbose=TRUE)
plot(myhat)

##simulate from a two-dimensional ccc-log-garch(1,1) w/defaults:
set.seed(123)
yy <- mlgarchSim(2000)

##estimate a 2-dimensional ccc-log-garch(1,1):
myymod <- mlgarch(yy)

##print results:
print(myymod)

Extraction methods for 'lgarch' objects

Description

Extraction methods for objects of class 'lgarch' (i.e. the result of estimating a log-GARCH model)

Usage

## S3 method for class 'lgarch'
coef(object, arma = FALSE, ...)
## S3 method for class 'lgarch'
fitted(object, verbose = FALSE, ...)
## S3 method for class 'lgarch'
logLik(object, arma = FALSE, ...)
## S3 method for class 'lgarch'
print(x, arma = FALSE, ...)
## informal method for class 'lgarch'
rss(object, ...)
## S3 method for class 'lgarch'
residuals(object, arma = FALSE, ...)
## S3 method for class 'lgarch'
summary(object, ...)
## S3 method for class 'lgarch'
vcov(object, arma = FALSE, ...)

Arguments

Details

Note: The rss function is not a formal S3 method.

Value

Author(s)

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

See Also

lgarch

Examples

##simulate 500 observations w/default parameter values:
set.seed(123)
y <- lgarchSim(500)

##estimate a log-garch(1,1):
mymod <- lgarch(y)

##print results:
print(mymod)

##extract coefficients:
coef(mymod)

##extract Gaussian log-likelihood (zeros excluded) of the log-garch model:
logLik(mymod)

##extract the Residual Sum of Squares of the ARMA representation:
rss(mymod)

##extract log-likelihood (zeros excluded) of the ARMA representation:
logLik(mymod, arma=TRUE)

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

##extract and plot the fitted conditional standard deviation:
sdhat <- fitted(mymod)
plot(sdhat)

##extract and plot standardised residuals:
zhat <- residuals(mymod)
plot(zhat)

##extract and plot all the fitted series:
myhat <- fitted(mymod, verbose=TRUE)
plot(myhat)

Extraction methods for 'mlgarch' objects

Description

Extraction methods for objects of class 'mlgarch' (i.e. the result of estimating a multivariate CCC-log-GARCH model)

Usage

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

Arguments

Details

Empty

Value

Author(s)

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

See Also

mlgarch

Examples

##simulate from 2-dimensional model w/default parameter values:
set.seed(123)
y <- mlgarchSim(2000)

##estimate a 2-dimensional ccc-log-garch(1,1):
mymod <- mlgarch(y)

##print results:
print(mymod)

##extract ccc-log-garch coefficients:
coef(mymod)

##extract Gaussian log-likelihood (zeros excluded) of the ccc-log-garch model:
logLik(mymod)

##extract Gaussian log-likelihood (zeros excluded) of the varma representation:
logLik(mymod, varma=TRUE)

##extract variance-covariance matrix:
vcov(mymod)             #work in progress!
vcov(mymod, varma=TRUE)

##extract and plot the fitted conditional standard deviations:
sdhat <- fitted(mymod)
plot(sdhat)

##extract and plot standardised residuals:
zhat <- residuals(mymod)
plot(zhat)

Difference a vector or a matrix, with special treatment of zoo objects

Description

Similar to the diff function from the base package, but gdiff enables padding (e.g. NAs or 0s) of the lost entries. Contrary to the diff function in the base package, however, the default in gdiff is to pad (with NAs). The gdiff function is particularly suited for zoo objects, since their indexing is retained

Usage

gdiff(x, lag = 1, pad = TRUE, pad.value = NA)

Arguments

Value

A vector or matrix with the differenced values

Note

Empty

Author(s)

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

See Also

diff, glag, lag

Examples

##1st difference of a series:
x <- rnorm(5)
gdiff(x)

##1st difference with no padding:
gdiff(x, pad=FALSE)

##1st difference retaining the original zoo-index ordering:
gdiff(as.zoo(x))

##1st difference of a matrix:
y <- matrix(rnorm(8),4,2)
gdiff(y)

##2nd difference of the same matrix:
gdiff(y, lag=2)

Lag a vector or a matrix, with special treatment of zoo objects

Description

Similar to the lag function from the stats package, but glag enables padding (e.g. NAs or 0s) of the lost entries. Contrary to the lag function in the stats package, however, the default in glag is to pad (with NAs). The glag is particularly suited for zoo objects, since their indexing is retained

Usage

glag(x, k = 1, pad = TRUE, pad.value = NA)

Arguments

Value

A vector or matrix with the lagged values

Note

Empty

Author(s)

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

See Also

lag, gdiff, diff

Examples

##lag series with NA for the missing entries:
x <- rnorm(5)
glag(x)

##lag series with no padding:
x <- rnorm(5)
glag(x, pad=FALSE)

##lag series and retain the original zoo-index ordering:
x <- as.zoo(rnorm(5))
glag(x)

##lag two periods:
glag(x, k=2)

Estimate a log-GARCH model

Description

Fit a log-GARCH model by either (nonlinear) Least Squares (LS) or Quasi Maximum Likelihood (QML) via the ARMA representation. For QML either the Gaussian or centred exponential chi-squared distribution can be used as instrumental density, see Sucarrat, Gronneberg and Escribano (2013), and Francq and Sucarrat (2013). Zero-values on the dependent variable y are treated as missing values, as suggested in Sucarrat and Escribano (2013). Estimation is via the nlminb function, whereas a numerical estimate of the Hessian is obtained with optimHess for the computation of the variance-covariance matrix

Usage

lgarch(y, arch = 1, garch = 1, xreg = NULL, initial.values = NULL,
  lower = NULL, upper = NULL, nlminb.control = list(), vcov = TRUE,
  method=c("ls","ml","cex2"), mean.correction=FALSE,
  objective.penalty = NULL, solve.tol = .Machine$double.eps,
  c.code = TRUE)

Arguments

Value

A list of class 'lgarch'

Note

Empty

Author(s)

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

References

C. Francq and G. Sucarrat (2018), 'An Exponential Chi-Squared QMLE for Log-GARCH Models Via the ARMA Representation', Journal of Financial Econometrics 16, pp. 129-154 doi:10.1093/jjfinec/nbx032

G. Sucarrat and A. Escribano (2018), 'Estimation of Log-GARCH Models in the Presence of Zero Returns', European Journal of Finance 24, pp. 809-827, doi:10.1080/1351847X.2017.1336452

G. Sucarrat, S. Gronneberg and A. Escribano (2016), 'Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown', Computational Statistics and Data Analysis 100, pp. 582-594, doi:10.1016/j.csda.2015.12.005

See Also

lgarchSim, coef.lgarch, fitted.lgarch, logLik.lgarch, print.lgarch, residuals.lgarch and vcov.lgarch

Examples

##simulate 500 observations w/default parameter values:
set.seed(123)
y <- lgarchSim(500)

##estimate a log-garch(1,1) w/least squares:
mymod <- lgarch(y)

##estimate the same model, but w/cex2 method:
mymod2 <- lgarch(y, method="cex2")

##print results:
print(mymod); print(mymod2)

##extract coefficients:
coef(mymod)

##extract Gaussian log-likelihood (zeros excluded) of the log-garch model:
logLik(mymod)

##extract Gaussian log-likelihood (zeros excluded) of the arma representation:
logLik(mymod, arma=TRUE)

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

##extract and plot the fitted conditional standard deviation:
sdhat <- fitted(mymod)
plot(sdhat)

##extract and plot standardised residuals:
zhat <- residuals(mymod)
plot(zhat)

##extract and plot all the fitted series:
myhat <- fitted(mymod, verbose=TRUE)
plot(myhat)

Auxiliary functions

Description

lgarchObjective and lgarchRecursion1 are auxiliary functions called by lgarch. The functions are not intended for the average user.

Usage

lgarchObjective(pars, aux)
lgarchRecursion1(pars, aux)

Arguments

Details

To understand the structure and content of pars and aux, see the source code of the lgarch function

Value

lgarchObjectivek() returns the value of the objective function (either the log-likelihood or the residual sum of squares) used in estimating the ARMA representation. lgarchRecursion1 returns the residuals of the ARMA representation associated with the ARMA parameters pars

Author(s)

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

References

C. Francq and G. Sucarrat (2018), 'An Exponential Chi-Squared QMLE for Log-GARCH Models Via the ARMA Representation', Journal of Financial Econometrics 16, pp. 129-154 doi:10.1093/jjfinec/nbx032

G. Sucarrat, S. Gronneberg and A. Escribano (2016), 'Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown', Computational Statistics and Data Analysis 100, pp. 582-594, doi:10.1016/j.csda.2015.12.005

See Also

lgarch

Simulate from a univariate log-GARCH model

Description

Simulate the y series (typically a financial return or the error in a regression) from a log-GARCH model. Optionally, the conditional standard deviation, the standardised error (z) and their logarithmic transformations are also returned.

Usage

lgarchSim(n, constant = 0, arch = 0.05, garch = 0.9, xreg = NULL,
  backcast.values = list(lnsigma2 = NULL, lnz2 = NULL, xreg = NULL),
  check.stability = TRUE, innovations = NULL, verbose = FALSE,
  c.code=TRUE)

Arguments

Details

Empty

Value

A zoo vector of length n if verbose = FALSE (default), or a zoo matrix with n rows if verbose = TRUE.

Author(s)

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

References

G. Sucarrat, S. Gronneberg and A. Escribano (2016), 'Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown', Computational Statistics and Data Analysis 100, pp. 582-594, doi:10.1016/j.csda.2015.12.005

See Also

mlgarchSim, lgarch, mlgarch and zoo

Examples

##simulate 500 observations w/default parameter values:
set.seed(123)
y <- lgarchSim(500)

##simulate the same series, but with more output:
set.seed(123)
y <- lgarchSim(500, verbose=TRUE)
head(y)

##plot the simulated values:
plot(y)

##simulate w/conditioning variable:
x <- rnorm(500)
y <- lgarchSim(500, xreg=0.05*x)

##simulate from a log-GARCH with a simple form of leverage:
z <- rnorm(500)
zneg <- as.numeric(z < 0)
zneglagged <- glag(zneg, pad=TRUE, pad.value=0)
y <- lgarchSim(500, xreg=0.05*zneglagged, innovations=z)

##simulate from a log-GARCH w/standardised t-innovations:
set.seed(123)
n <- 500
df <- 5
z <- rt(n, df=df)/sqrt(df/(df-2))
y <- lgarchSim(n, innovations=z)

Estimate a multivariate CCC-log-GARCH(1,1) model

Description

Fit a multivariate Constant Conditional Correlation (CCC) log-GARCH(1,1) model with multivariate Gaussian Quasi Maximum Likelihood (QML) via the VARMA representation, see Sucarrat, Gronneberg and Escribano (2013). Zero-values on y are treated as missing values, as suggested in Sucarrat and Escribano (2013). Estimation is via the nlminb function, whereas a numerical estimate of the Hessian is obtained with optimHess for the computation of the variance-covariance matrix

Usage

mlgarch(y, arch = 1, garch = 1, xreg = NULL, initial.values = NULL,
  lower = NULL, upper = NULL, nlminb.control = list(), vcov = TRUE,
  objective.penalty = NULL, solve.tol = .Machine$double.eps, c.code = TRUE)

Arguments

Value

A list of class 'mlgarch'

Note

Empty

Author(s)

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

References

G. Sucarrat and A. Escribano (2018), 'Estimation of Log-GARCH Models in the Presence of Zero Returns', European Journal of Finance 24, pp. 809-827, doi:10.1080/1351847X.2017.1336452

G. Sucarrat, S. Gronneberg and A. Escribano (2016), 'Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown', Computational Statistics and Data Analysis 100, pp. 582-594, doi:10.1016/j.csda.2015.12.005

See Also

lgarchSim, coef.lgarch, fitted.lgarch, logLik.lgarch, print.lgarch, residuals.lgarch and vcov.lgarch

Examples

##simulate from a 2-dimensional ccc-log-garch(1,1) w/defaults:
set.seed(123)
y <- mlgarchSim(1000)

##estimate a 2-dimensional ccc-log-garch(1,1):
mymod <- mlgarch(y)

##print results:
print(mymod)

##extract ccc-log-garch coefficients:
coef(mymod)

##extract Gaussian log-likelihood (zeros excluded) of the ccc-log-garch model:
logLik(mymod)

##extract Gaussian log-likelihood (zeros excluded) of the varma representation:
logLik(mymod, varma=TRUE)

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

##extract and plot the fitted conditional standard deviations:
sdhat <- fitted(mymod)
plot(sdhat)

##extract and plot standardised residuals:
zhat <- residuals(mymod)
plot(zhat)

Auxiliary functions

Description

mlgarchObjective and mlgarchRecursion1 are auxiliary functions called by mlgarch. The functions are not intended for the average user.

Usage

mlgarchObjective(pars, aux)
mlgarchRecursion1(pars, aux)

Arguments

Details

To understand the structure and content of pars and aux, see the source code of the mlgarch function

Value

mlgarchObjective returns the log-likelihood of the VARMA representation, whereas mlgarchRecursion1 returns the residuals of the VARMA representation associated with the VARMA parameters pars

Author(s)

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

References

G. Sucarrat, S. Gronneberg and A. Escribano (2016), 'Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown', Computational Statistics and Data Analysis 100, pp. 582-594, doi:10.1016/j.csda.2015.12.005

See Also

mlgarch and lgarchObjective

Simulate from a multivariate log-GARCH(1,1) model

Description

Simulate the y series (typically a collection of financial returns or regression errors) from a log-GARCH model. Optionally, the conditional standard deviation and the standardised error, together with their logarithmic transformations, are also returned.

Usage

mlgarchSim(n, constant = c(0,0), arch = diag(c(0.1, 0.05)),
  garch = diag(c(0.7, 0.8)), xreg = NULL,
  backcast.values = list(lnsigma2 = NULL, lnz2 = NULL, xreg = NULL),
  innovations = NULL, innovations.vcov = diag(rep(1,
  length(constant))), check.stability = TRUE, verbose = FALSE)

Arguments

Details

Empty

Value

A zoo matrix with n rows.

Author(s)

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

References

G. Sucarrat, S. Gronneberg and A. Escribano (2016), 'Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown', Computational Statistics and Data Analysis 100, pp. 582-594, doi:10.1016/j.csda.2015.12.005

See Also

lgarchSim, mlgarch and zoo

Examples

##simulate from a multivariate ccc-log-garch(1,1) w/defaults:
set.seed(123)
y <- mlgarchSim(500)

##simulate the same series, but with more output:
set.seed(123)
y <- mlgarchSim(500, verbose=TRUE)
head(y)

##plot the simulated values:
plot(y)

Random number generation from the multivariate normal distribution

Description

This function is a speed-optimised version of the rmnorm function from the mnormt package of Adelchi Azzalini (2013).

Usage

rmnorm(n, mean = NULL, vcov = 1)

Arguments

Details

Empty

Value

A matrix of n rows

Author(s)

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

References

Adelchi Azzalini (2013): 'mnormt: The multivariate normal and t distributions', R package version 1.4-7, https://CRAN.R-project.org/package=mnormt

Examples

##generate from univariate standardised normal:
z1 <- rmnorm(100)

##generate from bivariate, independent standardised normal:
z2 <- rmnorm(100, vcov=diag(c(1,1)))

##generate from bivariate, dependent standardised normal:
z3 <- rmnorm(100, vcov=cbind(c(1,0.3),c(0.3,1)))