

garch(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 Het-
     eroscedastic 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 (obj, newdata, genuine = FALSE)
     coef (obj)
     residuals (obj)
     fitted (obj)
     print (obj, digits = max(3,.Options$digits-3))
     summary (obj)
     plot (obj)
     print.summary (obj, digits = max(3,.Options$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 ini-
          tial 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 func-
          tion 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 rel-
          ative function convergence tolerance
          (`max(1e-10,eps^(2/3))'), the coefficient-conver-
          gence tolerance (`sqrt(eps)'), and the false con-
          vergence tolerance (`1e2*eps'). Default value is
          the machine epsilon, see `Machine'.

    grad: indicates if the analytical gradient or a numeri-
          cal approximation is used for the optimization.

  series: name for the series. Defaults to `deparse(substi-
          tute(x))'.

   trace: trace optimizer output?

     ...: additional arguments for `qr' when computing the
          asymptotic standard errors of `coef'.

     obj: a fit from `garch'.

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

     ...: additional arguments for `print'.

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

     `garch' uses a Quasi-Newton optimizer to find the maxi-
     mum-likelihood estimates of the conditionally normal
     model. The first max(p,q) values are assumed to be
     fixed. The optimizer uses a hessian approximation com-
     puted 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 devia-
     tion 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 approx-
     imation of the Hessian evaluated at the estimates, see
     Bollerslev (1986). It furthermore tests the residuals
     for normality and remaining ARCH effects, see `jar-
     que.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 con-
          stant).

  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 for-
     tran optimizer

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

     A. K. Bera and M. L. Higgins (1993): ARCH Models: Prop-
     erties, Estimation and Testing. J. Economic Surveys 7
     305-362.

     T. Bollerslev (1986): Generalized Autoregressive Condi-
     tional Heteroscedasticity. Journal of Econometrics 31,
     307-327.

     R. F. Engle (1982): Autoregressive Conditional Het-
     eroscedasticity with Estimates of the Variance of
     United Kingdom Inflation. Econometrica 50, 987-1008.

     J. E. Dennis, D. M. Gay, and R. E. Welsch (1981): Algo-
     rithm 573 - An Adaptive Nonlinear Least-Squares Algo-
     rithm. ACM Trans.  Math. Software 7, 369-383.

     J. E. Dennis and H. H. W. Mei (1979): Two New Uncon-
     strained 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 Meth-
     ods, 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

