EXACT LOCAL WHITTLE ESTIMATION OF FRACTIONAL INTEGRATION WITH UNKNOWN MEAN AND TIME TREND
扩展了精确局部Whittle估计量，使其能处理未知均值和多项式时间趋势，并证明两步估计量在宽参数范围内具有渐近正态性，适用于经济数据分析。
Recently, Shimotsu and Phillips (2005, Annals of Statistics 33, 1890–1933) developed a new semiparametric estimator, the exact local Whittle (ELW) estimator, of the memory parameter ( d ) in fractionally integrated processes. The ELW estimator has been shown to be consistent, and it has the same $N(0,{\textstyle{1 \over 4}})$ asymptotic distribution for all values of d , if the optimization covers an interval of width less than 9/2 and the mean of the process is known. With the intent to provide a semiparametric estimator suitable for economic data, we extend the ELW estimator so that it accommodates an unknown mean and a polynomial time trend. We show that the two-step ELW estimator, which is based on a modified ELW objective function using a tapered local Whittle estimator in the first stage, has an $N(0,{\textstyle{1 \over 4}})$ asymptotic distribution for $d \in (- {\textstyle{1 \over 2}},2)$ (or $d \in (- {\textstyle{1 \over 2}},{\textstyle{7 \over 4}})$ when the data have a polynomial trend). Our simulation study illustrates that the two-step ELW estimator inherits the desirable properties of the ELW estimator.