1. Wavelet Estimation of a Local Long Memory Parameter B. Whitcher EURANDOM, The Netherlands whitcher@eurandom.tue.nl M. J. Jensen University of Missouri - Columbia March 15, 2000 ASEG 2000, Perth, Western Australia

2. EURANDOMB. Whitcher2 Outline Motivation Locally Stationary Time Series Discrete Wavelet Transforms Local Wavelet Variance Vertical Ocean Shear Measurements

3. EURANDOMB. Whitcher3 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.

4. EURANDOMB. Whitcher4 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.

5. EURANDOMB. Whitcher5 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.

6. EURANDOMB. Whitcher6 Wavelet Basis Functions

7. EURANDOMB. Whitcher7 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

8. EURANDOMB. Whitcher8 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”.

9. EURANDOMB. Whitcher9 Vertical Ocean Shear

10. EURANDOMB. Whitcher10 Parameter Estimation

11. EURANDOMB. Whitcher11 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.