tseries (version 0.9-4)

garch: Fit GARCH Models to Time Series

Description

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.

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 (

grad

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.

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.
• asy.se.coef
  the asymptotic-theory standard errors of the coefficient estimates.

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.

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.

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