R: Locally Stationary Whittle log-likelihood Function

LS.whittle.loglik {LSTS}

R Documentation

Locally Stationary Whittle log-likelihood Function

Description

This function computes Whittle estimator for LS-ARMA and LS-ARFIMA models, in data with mean zero. If mean is not zero, then it is subtracted to data.

Usage

LS.whittle.loglik(
  x,
  series,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE
)

Arguments

`x`	(type: numeric) parameter vector.
`series`	(type: numeric) univariate time series.
`order`	(type: numeric) vector corresponding to `ARMA` model entered.
`ar.order`	(type: numeric) AR polimonial order.
`ma.order`	(type: numeric) MA polimonial order.
`sd.order`	(type: numeric) polinomial order noise scale factor.
`d.order`	(type: numeric) `d` polinomial order, where `d` is the `ARFIMA` parameter.
`include.d`	(type: numeric) logical argument for `ARFIMA` models. If `include.d=FALSE` then the model is an ARMA process.
`N`	(type: numeric) value corresponding to the length of the window to compute periodogram. If `N=NULL` then the function will use `N = \textrm{trunc}(n^{0.8})`, see Dahlhaus (1998) where `n` is the length of the `y` vector.
`S`	(type: numeric) value corresponding to the lag with which will go taking the blocks or windows.
`include.taper`	(type: logical) logical argument that by default is `TRUE`. See `periodogram`.

Details

The estimation of the time-varying parameters can be carried out by means of the Whittle log-likelihood function proposed by Dahlhaus (1997),

L_n(\theta) = \frac{1}{4\pi}\frac{1}{M} \int_{-\pi}^{\pi} \bigg\{log f_{\theta}(u_j,\lambda) + \frac{I_N(u_j, \lambda)}{f_{\theta}(u_j,\lambda)}\bigg\}\,d\lambda

where M is the number of blocks, N the length of the series per block, n =S(M-1)+N, S is the shift from block to block, u_j =t_j/n, t_j =S(j-1)+N/2, j =1,\ldots,M and \lambda the Fourier frequencies in the block (2\,\pi\,k/N, k = 1,\ldots, N).

References