# garch

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##### Fit GARCH Models to Time Series

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.

Keywords
ts
##### Usage
garch(x, order = c(1, 1), coef = NULL, itmax = 200, eps = NULL,
grad = c("analytical","numerical"), series = NULL,
trace = TRUE, ...)
##### Arguments
x
a numeric vector or time series.
order
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.
coef
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
itmax
gives the maximum number of log-likelihood function evaluations itmax and the maximum number of iterations 2*itmax the optimizer is allowed to compute.
eps
defines the absolute (max(1e-20, eps^2)) and relative function convergence tolerance (max(1e-10, eps^(2/3))), the coefficient-convergence tolerance (sqrt(eps)), and the false convergence tolerance (
indicates if the analytical gradient or a numerical approximation is used for the optimization.
series
name for the series. Defaults to deparse(substitute(x)).
trace
trace optimizer output?
...
additional arguments for qr when computing the asymptotic standard errors of coef.
##### 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:
• orderthe order of the fitted model.
• coefestimated GARCH coefficients for the fitted model.
• n.likelithe negative log-likelihood function evaluated at the coefficient estimates (apart from some constant).
• n.usedthe number of observations of x.
• residualsthe series of residuals.
• fitted.valuesthe bivariate series of conditional standard deviation predictions for x.
• seriesthe name of the series x.
• frequencythe frequency of the series x.
• callthe call of the garch function.
• asy.se.coefthe asymptotic-theory standard errors of the coefficient estimates.

##### References

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 Least-Squares 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): 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.

summary.garch for summarizing GARCH model fits; garch-methods for further methods.

• garch
##### 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
Documentation reproduced from package tseries, version 0.10-1, License: GPL (see file COPYING)

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