Locally Stationary Whittle log-likelihood Function

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.

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

For more information on theoretical foundations and estimation methods see Brockwell PJ, Davis RA, Calder MV (2002). Introduction to time series and forecasting, volume 2. Springer. Palma W, Olea R, others (2010). “An efficient estimator for locally stationary Gaussian long-memory processes.” The Annals of Statistics, 38(5), 2958--2997.

See also

nlminb, LS.kalman