Fit a Generalized Autoregressive Conditional Heteroscedastic GARCH(p,
q) time series model to the data by computing the maximum-likelihood
estimates of the conditionally normal model.
a two dimensional integer vector giving the orders of the
model to fit. order[2] corresponds to the ARCH part and
order[1] to the GARCH part.
series
name for the series. Defaults to
deparse(substitute(x)).
control
a list of control parameters as set up by garch.control.
maxiter
gives the maximum number of log-likelihood function
evaluations maxiter and the maximum number of iterations
2*maxiter the optimizer is allowed to compute.
trace
logical. Trace optimizer output?
start
If given this numeric vector is used as the initial estimate
of the GARCH coefficients. Default initialization is to set the
GARCH parameters to slightly positive values and to initialize the
intercept such that the unconditional variance of the initial GARCH
is equal to the variance of x.
grad
character indicating whether analytical gradients or a numerical
approximation is used for the optimization.
abstol
absolute function convergence tolerance.
reltol
relative function convergence tolerance.
xtol
coefficient-convergence tolerance.
falsetol
false convergence tolerance.
...
additional arguments for qr when computing
the asymptotic covariance matrix.
Details
garch uses a Quasi-Newton optimizer to find the maximum
likelihood estimates of the conditionally normal model. The first
max(p, q) values are assumed to be fixed. The optimizer uses a hessian
approximation computed from the BFGS update. Only a Cholesky factor
of the Hessian approximation is stored. For more details see Dennis
et al. (1981), Dennis and Mei (1979), Dennis and More (1977), and
Goldfarb (1976). The gradient is either computed analytically or
using a numerical approximation.
Value
A list of class "garch" with the following elements:
order
the order of the fitted model.
coef
estimated GARCH coefficients for the fitted model.
n.likeli
the negative log-likelihood function evaluated at the
coefficient estimates (apart from some constant).
n.used
the number of observations of x.
residuals
the series of residuals.
fitted.values
the bivariate series of conditional standard
deviation predictions for x.
series
the name of the series x.
frequency
the frequency of the series x.
call
the call of the garch function.
vcov
outer product of gradient estimate of the asymptotic-theory
covariance matrix for the coefficient estimates.
Author(s)
A. Trapletti, the whole GARCH part;
D. M. Gay, the FORTRAN optimizer
References
A. K. Bera and M. L. Higgins (1993):
ARCH Models: Properties, Estimation and Testing.
J. Economic Surveys7 305–362.
T. Bollerslev (1986):
Generalized Autoregressive Conditional Heteroscedasticity.
Journal of Econometrics31, 307–327.
R. F. Engle (1982):
Autoregressive Conditional Heteroscedasticity with Estimates of the
Variance of United Kingdom Inflation.
Econometrica50, 987–1008.
J. E. Dennis, D. M. Gay, and R. E. Welsch (1981):
Algorithm 573 — An Adaptive Nonlinear Least-Squares
Algorithm.
ACM Transactions on Mathematical Software7, 369–383.
J. E. Dennis and H. H. W. Mei (1979):
Two New Unconstrained Optimization Algorithms which use Function and
Gradient Values.
J. Optim. Theory Applic.28, 453–482.
J. E. Dennis and J. J. More (1977):
Quasi-Newton Methods, Motivation and Theory.
SIAM Rev.19, 46–89.
D. Goldfarb (1976):
Factorized Variable Metric Methods for Unconstrained Optimization.
Math. Comput.30, 796–811.
See Also
summary.garch for summarizing GARCH model fits;
garch-methods for further methods.
Examples
n <- 1100
a <- c(0.1, 0.5, 0.2) # ARCH(2) coefficients
e <- rnorm(n)
x <- double(n)
x[1:2] <- rnorm(2, sd = sqrt(a[1]/(1.0-a[2]-a[3])))
for(i in 3:n) # Generate ARCH(2) process
{
x[i] <- e[i]*sqrt(a[1]+a[2]*x[i-1]^2+a[3]*x[i-2]^2)
}
x <- ts(x[101:1100])
x.arch <- garch(x, order = c(0,2)) # Fit ARCH(2)
summary(x.arch) # Diagnostic tests
plot(x.arch)
data(EuStockMarkets)
dax <- diff(log(EuStockMarkets))[,"DAX"]
dax.garch <- garch(dax) # Fit a GARCH(1,1) to DAX returns
summary(dax.garch) # ARCH effects are filtered. However,
plot(dax.garch) # conditional normality seems to be violated