Wavelet Estimation of a Local Long Memory Parameter

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Wavelet Estimation of a Local Long Memory
Parameter

• B. Whitcher
• EURANDOM, The Netherlands
• whitcher_at_eurandom.tue.nl
• M. J. Jensen
• University of Missouri - Columbia

March 15, 2000 ASEG 2000, Perth, Western Australia
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Outline

• Motivation
• Locally Stationary Time Series
• Discrete Wavelet Transforms
• Local Wavelet Variance
• Vertical Ocean Shear Measurements

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Motivation

• Long-range dependence is everywhere.
• Want to generalise current time series models
• fractional ARIMA
• Popular method of estimation is ordinary
  least-squares (OLS).
• Propose local version of the OLS estimator based
  on wavelet coefficients.
• Compare it to an adapted global estimator.

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Locally Stationary Long-Memory Model

• Define to be a stochastic process given by
• Time-varying generalisation of Box Jenkins
  model.
• Long-memory parameter
• Spectrum for has the property
• Log-linear relation between spectrum and
  frequency.

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Discrete Wavelet Transform

• Project observations onto wavelet functions.
• Common wavelets are the Haar and Daubechies.
• Decompose process on a scale-by-scale basis.
• Multiresolution analysis.
• Appealing for the physical sciences.
• Also captures features locally in time.
• Allows us to estimate time-varying structure.

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Wavelet Basis Functions
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Comparison of Transforms

• DWT
• Orthonormal transform
• Filter and downsample
• Decorrelates LMPs
• Poor time resolution
• Inferior statistical properties
• Not used here

• Maximal Overlap DWT
• NOT orthogonal
• Filter, no downsample
• Correlated coefficients
• Better time resolution
• Better statistical properties
• Used to construct local wavelet variance

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Local Wavelet Variance

• Intuitive definition of the wavelet variance
• Local wavelet variance is estimated
  by
• is the width of the central portion.
• is the offset of the central portion.

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Vertical Ocean Shear
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Parameter Estimation
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Conclusions

• Methodology
• Introduced new time series model.
• Developed wavelet-based estimation procedure.
• Results
• Quantified time-varying persistence in vertical
  ocean shear measurements.
• Outperformed global estimator on partitioned data
• Future Research
• Quantify variability of estimator.
• Weighted least squares or Maximum Likelihood.