# LS.whittle.loglik: Locally Stationary Whittle log-likelihood Function In LSTS: Locally Stationary Time Series

## 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

  1 2 3 4 5 6 7 8 9 10 11 12 13 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(θ) = \frac{1}{4π}\frac{1}{M} \int_{-π}^{π} \bigg\{log f_{θ}(u_j,λ) + \frac{I_N(u_j, λ)}{f_{θ}(u_j,λ)}\bigg\}\,dλ

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,…,M and λ the Fourier frequencies in the block (2\,π\,k/N, k = 1,…, N).

## References

For more information on theoretical foundations and estimation methods see \insertRefbrockwell2002introductionLSTS \insertRefpalma2010efficientLSTS

nlminb, LS.kalman

LSTS documentation built on July 29, 2021, 5:07 p.m.