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% MULTIWAVE Matlab Toolbox
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%                                           Achard & Gannaz (2014)
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This package computes an estimation of the long memory parameters and the long-run covariance matrix using a multivariate model (Lobato, 1999; Shimotsu 2007). Two semi-parametric methods are implemented: a Fourier based approach (Shimotsu 2007) and a wavelet based approach (Achard and Gannaz 2014).


Authors
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The package is provided by S. Achard and I. Gannaz. For any question or remark, please contact Sophie.Achard@gipsa-lab.grenoble-inp.fr or Irene.Gannaz@insa-lyon.fr.

The functions computenj.m, DWTexact.m, makescalingfunction.m and scalingfilter.m were implemented by G. Fay, E. Moulines, F. Roueff and M.S. Taqqu.

The functions mfw_eval.m and mfw_cov.m were based on Shimotsu's codes, available at http://shimotsu.web.fc2.com/Site/Matlab_Codes.html

S. Achard and I. Gannaz are grateful to G. Fay, E. Moulines, F. Roueff, M.S. Taqqu and Shimotsu for kindly providing their codes.



References
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S. Achard, I. Gannaz (2014) Multivariate wavelet Whittle estimation in long-range dependence.
arXiv, http://arxiv.org/abs/1412.0391

G. Fay, E. Moulines, F. Roueff, M. S. Taqqu (2009) Estimators of long-memory: Fourier versus
wavelets. Journal of Econometrics, vol. 151, N. 2, pages 159-177.

K. Shimotsu (2007) Gaussian semiparametric estimation of multivariate fractionally integrated pro-
cesses Journal of Econometrics Vol. 137, N. 2, pages 277-310.



Description of the package
--------------------------

The file example.m contains an example of a simulation of a bivariate ARFIMA process. It provides the estimation of the long-range dependence parameters jointly with the long-run covariance matrix using Fourier-based and wavelet-based Whittle estimation procedures. This example so illustrates the application of the main functions of the package.

We provide here a very brief summary of the functions provided in this package. Please see at the beginning of the files for a more detailed description of the functions utilities and for the input and output parameters.


	- Simulation

- fracdiff.m applies a vectorial fractional differencing procedure.
- varma.m computes a realisation of a multivariate ARMA process.
- varfima.m computes a realisation of a multivariate ARFIMA process and evaluates the long-run covariance matrix and the long-memory parameters associated.


	- Wavelet transform

- scalingfilter.m defines the wavelet filter (only Daubechies' wavelets are available).
- computenj.m computes the number of wavelet coefficients for each individual scale.
- DWTexact.m provides the wavelet transform of the data.
- psi_hat_exact.m gives the Fourier transform of the wavelet function.
- K_eval.m evaluates the values of the integrals K_{l,m}(d) = int( u^(d_l+d_m) |\psi_hat(u)|^2 du ).


	- Fourier-based estimation

- mfw.m computes the multivariate Fourier Whittle estimators of the long-range dependence parameters and the long-run covariance matrix.
- mfw_cov.m computes the multivariate Fourier-based Whittle estimator for the long-run covariance matrix for a given value of the long-range dependence d.
- mfw_eval.m returns the value of the multivariate Fourier Whittle criterion with respect to d at a given value of d.


	- Wavelet-based estimation directly on the data

- mww.m computes the multivariate Wavelet Whittle estimators of the long-range dependence parameters and the long-run covariance matrix.
- mww_cov.m computes the multivariate Wavelet-based Whittle estimator for the long-run covariance matrix for a given value of the long-range dependence d.
- mww_eval.m returns the value of the multivariate Wavelet Whittle criterion with respect to d at a given value of d.


	- Wavelet-based estimation applied on the wavelet transform of the data

- mww_wav.m computes the multivariate Wavelet Whittle estimators of the long-range dependence parameters and the lon-run covariance matrix, given the wavelet transform of the data.
- mww_wav_cov.m computes the multivariate Wavelet-based Whittle estimator for the long-run covariance matrix for a given value of the long-range dependence d, given the wavelet transform of the data.
- mww_wav_eval.m returns the value of the multivariate Wavelet Whittle criterion with respect to d at a given value of d, given the wavelet transform of the data.



