Analysis And Application Of Wavelet Theory In Non-Stationary Long-Memory Time Series

The assumption that real world processes exhibit a constant long memory struc-ture may not be reasonable. Time-varying long memory characteristics have been hypothesized or observed in the real world. To better capture the non-stationary behavior, we need to study the non-stationary models with the time-varying param-eters. In this thesis, we would like to provide the theory of the estimation on the time-varying long memory parameters based on the wavelet transformation.The ARFIMA(p, d, q) process which is frequently used in modelling long memory with long memory parameter d could be expressed as (1-B)dΦ(B)Xå'Œ=(?)(B)εt, where Φ(z)=1-φ1z-…-φpzp and (?)(z)=1+θ1z+…+θqzq are polynomials of order p and q, respectively, with roots outside the unit circle, B denotes the lag (backshift) operator and et is Gaussian white noise with the variance σ2. The study on this model could be divided into three cases:1. Fractional difference parameter d is constant with|d|<1/2, then the ARFIMA process is stationary;2. Fractional difference parameter d is constant with|d+≥1/2, then the ARFIMA process is non-stationary;3. Fractional difference parameter is a time-varying fuction d(t) with|d(t)|<1/2, then the ARFIMA process is non-stationary, which is also called locally stationary ARFIMA model.There exists a great amount of literature studying on the ARFIMA model under the first two cases, where the Fourier analysis is used as the main tool to estimate the fractional difference parameter in the frequency-domain. However the estimators based on the Fourier transformation are incapable of addressing any time-varying long memory behavior (case3).In this thesis, we propose a new wavelet-based estimator for the time-varying fractional difference parameter function d(t) of the locally stationary ARFIMA pro-cess, which is based on the log-log linear relationship between the local wavelet variances and wavelet scales. The consistency of the estimate is proved and the ro-bustness is verified via Monte Carlo simulations. The results suggest that this new method offers an interesting alternative competing framework in the description of the persistence dynamics of locally stationary models.We also improve the wavelet-based estimation method proposed by Cavanaugh et al.(2002) which is used to estimate the locally self-similar parameter H(t)=d(t)+1/2. As we know, the length of the data is required to be dyadic when we use the DWT. Therefore, we replace the DWT adopted in Cavanaugh et al.(2002) with the MODWT to construct a new method for the estimation of the time-varying self-similarity parameters. The new method relaxes the restriction on the dyadie sample size, and makes the estimation procedure easier thanks to the wavelet level decomposition of the MODWT. Besides, it has a better performance in the Monte Carlo simulations.Besides, we carry out a numbeg of Monte Carlo experiments to study the finite sample behavior of the locally stationary ARFIMA process. The accuracy of the wavelet-based estimation method is critically dependent on the number of chosen wavelet scales and that of partitioned sampling intervals used in the estimation. We provide a reference on the optimal choices of these two indices for different sample sizes through the grid search method and we also make a comparison among the estimation methods mentioned in the previous part.Furthermore, we make an investigation into the estimation method proposed by Whitcher and Jensen (2000) which is also based on the wavelet transformation in this thesis. We focus on the case that the long memory parameter function is discontinuous, where the estimated errors are significant near the discontinuous point. We give some suggestions on the choice of the wavelet scales in order to raise the estimated accuracy. In the end, an application on the vertical ocean shear measurements is made using the methods proposed in this thesis. The time-varying long memory charac-teristic in the the vertical ocean shear measurements is studied under two cases:1. the first differenced series is a locally stationary long memory process;2. the series is a locally self-similar process.