garchx {garchx}

Estimate a GARCH-X model

Description

Quasi Maximum Likelihood (ML) estimation of a GARCH(q,p,r)-X model, where q is the GARCH order, p is the ARCH order, r is the asymmetry (or leverage) order and 'X' indicates that covariates can be included. Note that the underlying estimation theory assumes the covariates are stochastic. The estimation procedure will, in general, provide consistent estimates when the standardised innovations are not normal or independent (or both), see Francq and Thieu (2018).

Usage

garchx(y, order = c(1,1), arch = NULL, garch = NULL, asym = NULL,
  xreg = NULL, vcov.type = c("ordinary", "robust"),
  initial.values = NULL, backcast.values = NULL, lower = 0,
  upper = +Inf, control = list(), hessian.control = list(),
  solve.tol = .Machine$double.eps, estimate = TRUE, c.code = TRUE,
  penalty.value = NULL, sigma2.min = .Machine$double.eps,
  objective.fun = 1, turbo = FALSE)

Arguments

y

numeric vector, time-series or zoo object. Missing values in the beginning and at the end of the series is allowed, as they are removed with the na.trim command

order

integer vector of length 1, 2 or 3, for example c(1,1,1). The first entry controls the GARCH order, the second the ARCH order and the third the ASYM (asymmetry/leverage) order

arch

NULL or numeric vector containing the ARCH-terms to include. Note: If not NULL, then the value of the ARCH argument overrides the value of the first entry in the order argument

garch

NULL or numeric vector containing the GARCH-terms to include. Note: If not NULL, then the value of the GARCH argument overrides the value of the second entry in the order argument

asym

NULL or numeric vector containing the ASYM-terms (asymmetry/leverage terms) to include. Note: If not NULL, then the value of the ASYM argument overrides the value of the third entry in the order argument

xreg

numeric vector, time-series or zoo object. Missing values in the beginning and at the end of the series is allowed, as they are removed with the na.trim command

vcov.type

character, either "ordinary" or "robust", see vcov.garchx

initial.values

NULL or a numeric vector with the initial parameter values passed on to the optimisation routine, nlminb. If NULL, the default, then the values are chosen automatically

backcast.values

NULL or a non-negative numeric. The backcast value is used to initiate the forward recursion of the conditional variance. If NULL (default), then the value is chosen automatically (currently the average of y squared is used). If backcast.values is a non-negative numeric, then the initial recursion values are all set to this value

lower

numeric vector, either of length 1 or the number of parameters to be estimated, see nlminb

upper

numeric vector, either of length 1 or the number of parameters to be estimated, see nlminb

control

a list passed on to the control argument of nlminb

hessian.control

a list passed on to the control argument of optimHess

solve.tol

numeric value passed on to the tol argument of solve, which is called whenever the coefficient variance-coariance matrix is computed. The value controls the toleranse for detecting linear dependence between columns when inverting a matrix

estimate

logical, if TRUE then estimation is carried out. If FALSE, then the initial.values are used

c.code

logical, if TRUE then compiled C code is used in the forward recursion

penalty.value

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. The penalty.value is the value returned by the objective function garchxObjective in the presence of NA or +/-Inf values

sigma2.min

numeric with default .Machine$double.eps. To avoid taking taking the log of a very small value when computing the log-likelihood, sigma2.min is used as the lower bound of the fitted conditional variances, see the code of garchxObjective

objective.fun

numeric, either 1 or 0

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.garchx, fitted.garchx and residuals.garchx, respectively

Value

A list of class 'garchx'

Author(s)

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

References

Christian Francq and Le Quien Thieu (2018): 'QML inference for volatility models with covariates', Econometric Theory, doi:10.1017/S0266466617000512
Christian Francq and Jean-Michel Zakoian (2019): 'GARCH Models', 2nd Edition, Wiley

See Also

garchxSim, nlminb, optimHess, coef.garchx

Examples

##simulate from a garch(1,1):
set.seed(123)
y <- garchxSim(1000)

##estimate garch(1,1) model:
mymod <- garchx(y)

##print estimation results:
print(mymod)

##extract coefficients:
coef(mymod)

##extract and store conditional variances:
sigma2hat <- fitted(mymod)

##extract log-likelihood:
logLik(mymod)

##extract and store standardised residuals:
etahat <- residuals(mymod)

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

##generate predictions:
predict(mymod)