fracdiff              package:fracdiff              R Documentation

_M_L _E_s_t_i_m_a_t_e_s _f_o_r _F_r_a_c_t_i_o_n_a_l_l_y-_D_i_f_f_e_r_e_n_c_e_d _A_R_I_M_A (_p,_d,_q) _m_o_d_e_l_s

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

     Calculates the maximum likelihood estimators of the parameters of
     a fractionally-differenced ARIMA (p,d,q) model, together (if
     possible) with their estimated covariance and correlation matrices
     and standard errors, as well as the value of the maximized
     likelihood.  The likelihood is approximated using the fast and
     accurate method of Haslett and Raftery (1989).

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

     fracdiff(x, nar = 0, nma = 0,
              ar = rep(NA, max(nar, 1)), ma = rep(NA, max(nma, 1)),
              dtol = NULL, drange = c(0, 0.5), h, M = 100)

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

       x: time series (numeric vector) for the ARIMA model

     nar: number of autoregressive parameters p.

     nma: number of moving average parameters q.

      ar: initial autoregressive parameters

      ma: initial moving average parameters

    dtol: interval of uncertainty for d.  If 'dtol' is negative or
          NULL, the fourth root of machine precision will be used.
          'dtol' will be altered if necessary by the program.

  drange: interval over which the likelihood function is to be
          maximized as a function of d.

       h: size of finite difference interval for numerical derivatives.
          By default (or if negative), 'h = min(0.1, eps.5 * (1+
          abs(cllf)))', where 'clff := log. max.likelihood' (as
          returned) and 'eps.5 := sqrt(.Machine$double.neg.eps)'
          (typically 1.05e-8).

          This only influences the cov, cor, and std.error
          computations; see also 'fracdiff.var'. 

       M: number of terms in the likelihood approximation (see Haslett
          and Raftery 1989).

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

     a list containing the following elements : 

log.likelihood: logarithm of the maximum likelihood

       d: optimal fractional-differencing parameter

      ar: vector of optimal autoregressive parameters

      ma: vector of optimal moving average parameters

covariance.dpq: covariance matrix of the parameter estimates (order :
          d, ar, ma).

stderror.dpq: standard errors of the parameter estimates  c(d, ar, ma).

correlation.dpq: correlation matrix of the parameter estimates (order :
          d, ar, ma).

       h: size of interval used for numerical derivatives.

    dtol: interval of uncertainty for d.

       M: as input.

hessian.dpq: ......

_M_e_t_h_o_d:

     The optimization is carried out in two levels:
      an outer univariate unimodal optimization in d over the interval
     'drange' (typically [0,.5]), using Brent's 'fmin' algorithm), and
      an inner nonlinear least-squares optimization in the AR and MA
     parameters to minimize white noise variance (uses the MINPACK
     subroutine 'lm'DER). written by Chris Fraley (March 1991).

_N_o_t_e:

     Ordinarily, 'nar' and 'nma' should not be too large (say < 10) to
     avoid degeneracy in the model.  The function 'fracdiff.sim' is
     available for generating test problems.

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

     J. Haslett and A. E. Raftery (1989) Space-time Modelling with
     Long-memory Dependence: Assessing Ireland's Wind Power Resource
     (with Discussion); _Applied Statistics_ *38*, 1-50.

     R. Brent (1973) _Algorithms for Minimization without Derivatives_,
     Prentice-Hall

     J. J. More, B. S. Garbow, and K. E. Hillstrom (1980) _Users Guide
     for MINPACK-1_, Technical Report ANL-80-74, Applied Mathematics
     Division, Argonne National Laboratory.

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

     'fracdiff.sim'

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

     ts.test <- fracdiff.sim( 5000, ar = .2, ma = -.4, d = .3)
     fracdiff( ts.test$series, nar = length(ts.test$ar), nma = length(ts.test$ma))

