Abstract
Long memory, also called long range dependence (LRD), is commonly detected in the analysis of real-life time series data in many areas; for example, in finance, in econometrics, in hydrology, etc. Therefore the study of long-memory time series is of great value. The introduction of ARFIMA (fractionally autoregressive integrated moving average) process established a relationship between the fractional integration and long memory, and this model has found its power in long-term forecasting, hence it has become one of the most popular long-memory models in the statistical literature. Specifically, an ARFIMA(p,d,q) process X, is defined as follows: cD(B)(I - B)d X, = 8(B)c, , where cD(z)=l-~lz-•••-~pzP and 8(z)=1-B1z- .. •-Bqzq are polynomials of order p and q , respectively, with roots outside the unit circle; and c, is Gaussian white noise with a constant variance a2 . When c" X, is stationary and invertible. However, the a priori assumption on stationarity of real-life data is not reasonable. Therefore many statisticians have made their efforts to propose estimators applicable to the non-stationary case. Then questions arise that which estimator should be chosen for applications; and what we should pay attention to when using these estimators. Therefore we make a comprehensive finite sample comparison of semi-parametric Fourier and wavelet estimators under the non-stationary ARFIMA setting. ln light of this comparison study, we have that (i) without proper scale trimming the wavelet estimators are heavily biased and the y generally have an inferior performance to the Fourier ones; (ii) ail the estimators under investigation are robust to the presence of a linear time trend in levels of XI and the GARCH effects in variance of XI; (iii) the consistency of the estimators still holds in the presence of regime switches in levels of XI , however, it tangibly contaminates the estimation results. Moreover, the log-regression wavelet estimator works badly in this situation with small and medium sample sizes; and (iv) fully-extended local polynomial Whittle Fourier (fextLPWF) estimator is preferred for a practical utilization, and the fextLPWF estimator requires a wider bandwidth than the other Fourier estimators.