garch                package:tseries                R Documentation

_F_i_t _G_A_R_C_H _M_o_d_e_l_s _t_o _T_i_m_e _S_e_r_i_e_s

_D_e_s_c_r_i_p_t_i_o_n:

     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.

_U_s_a_g_e:

     garch(x, order = c(1, 1), coef = NULL, itmax = 200, eps = NULL,
           grad = c("analytical","numerical"), series = NULL,
           trace = TRUE, ...)

_A_r_g_u_m_e_n_t_s:

       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 the initial GARCH is equal to the variance of 'x'.

   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 ('1e2*eps'). Default value is the
          machine epsilon, see 'Machine'.

    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'.

_D_e_t_a_i_l_s:

     '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.

_V_a_l_u_e:

     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.

_A_u_t_h_o_r(_s):

     A. Trapletti, the whole GARCH part; D. M. Gay, the FORTRAN
     optimizer

_R_e_f_e_r_e_n_c_e_s:

     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.

_S_e_e _A_l_s_o:

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

_E_x_a_m_p_l_e_s:

     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

