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, ...)
     predict (object, newdata, genuine = FALSE)
     coef (object)
     residuals (object)
     fitted (object)
     print (object, digits = max(3,.Options$digits-3))
     summary (object)
     plot (object, ask = interactive())
     print.summary (object, digits = max(3,.Options$digits-3),
                    signif.stars = .Options$show.signif.stars, ...)

_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?

  object: a fit from `garch'.

 newdata: a numeric vector or time series to compute GARCH predictons.
          Defaults to `eval(parse(text=object$series))'.

 genuine: a logical indicating whether a genuine prediction should be
          made, i.e., a prediction for which there is no target
          observation available.

digits, signif.stars: see `print.coefmat'.

     ask: Should the `plot' method work interactively? See
          `interactive'.

     ...: either additional arguments for `qr' when computing the
          asymptotic standard errors of `coef' (`garch'), or additional
          arguments for `print' (`print.summary.garch').

_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.

     `predict' returns +/- the conditional standard deviation
     predictions from a fitted GARCH model.

     `coef' returns the coefficient estimates.

     `residuals' returns the GARCH residuals, i.e., the time series
     used to fit the model divided by the computed conditional standard
     deviation predictions for this series. Under the assumption of
     conditional normality the residual series should be i.i.d.
     standard normal.  

     `fitted' returns +/- the conditional standard deviation
     predictions for the series which has been used to fit the model.

     `print', `summary', `plot', and `print.summary' provide the usual
     generic functions for fitted GARCH models. `summary' computes the
     asymptotic standard errors of the coefficient estimates from an
     outer-product approximation of the Hessian evaluated at the
     estimates, see Bollerslev (1986). It furthermore tests the
     residuals for normality and remaining ARCH effects, see
     `jarque.bera.test' and `Box.test'. `plot' graphically investigates
     normality and remaining ARCH effects for the residuals.

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

     For `garch' and its methods `print' and `plot' 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.


     For `predict' a bivariate time series (two-column matrix) of
     predictions. 

     For `coef', a numeric vector, for `residuals' and `fitted' a
     univariate (vector) and a bivariate time series (two-column
     matrix), respectively.

     For `summary' and `print.summary' a list of class
     `"summary.garch"'.

_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 Trans.  Math.
     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.

_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 = 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

