garch  R Documentation 
Fit a Generalized Autoregressive Conditional Heteroscedastic GARCH(p, q) time series model to the data by computing the maximumlikelihood estimates of the conditionally normal model.
garch(x, order = c(1, 1), series = NULL, control = garch.control(...), ...)
garch.control(maxiter = 200, trace = TRUE, start = NULL,
grad = c("analytical","numerical"), abstol = max(1e20, .Machine$double.eps^2),
reltol = max(1e10, .Machine$double.eps^(2/3)), xtol = sqrt(.Machine$double.eps),
falsetol = 1e2 * .Machine$double.eps, ...)
x 
a numeric vector or time series. 
order 
a two dimensional integer vector giving the orders of the
model to fit. 
series 
name for the series. Defaults to

control 
a list of control parameters as set up by 
maxiter 
gives the maximum number of loglikelihood function
evaluations 
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 
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 
coefficientconvergence tolerance. 
falsetol 
false convergence tolerance. 
... 
additional arguments for 
garch
uses a QuasiNewton 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.
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 loglikelihood function evaluated at the coefficient estimates (apart from some constant). 
n.used 
the number of observations of 
residuals 
the series of residuals. 
fitted.values 
the bivariate series of conditional standard
deviation predictions for 
series 
the name of the series 
frequency 
the frequency of the series 
call 
the call of the 
vcov 
outer product of gradient estimate of the asymptotictheory covariance matrix for the coefficient estimates. 
A. Trapletti, the whole GARCH part; D. M. Gay, the FORTRAN optimizer
A. K. Bera and M. L. Higgins (1993): ARCH Models: Properties, Estimation and Testing. J. Economic Surveys 7 305–362.
T. Bollerslev (1986): Generalized Autoregressive Conditional Heteroscedasticity. Journal of Econometrics 31, 307–327.
R. F. Engle (1982): Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 50, 987–1008.
J. E. Dennis, D. M. Gay, and R. E. Welsch (1981): Algorithm 573 — An Adaptive Nonlinear LeastSquares Algorithm. ACM Transactions on Mathematical Software 7, 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): QuasiNewton Methods, Motivation and Theory. SIAM Rev. 19, 46–89.
D. Goldfarb (1976): Factorized Variable Metric Methods for Unconstrained Optimization. Math. Comput. 30, 796–811.
summary.garch
for summarizing GARCH model fits;
garchmethods
for further methods.
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]/(1.0a[2]a[3])))
for(i in 3:n) # Generate ARCH(2) process
{
x[i] < e[i]*sqrt(a[1]+a[2]*x[i1]^2+a[3]*x[i2]^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
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