Adaptive Semiparametric Estimation of the Memory Parameter

✍ Scribed by Liudas Giraitis; Peter M Robinson; Alexander Samarov

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 239 KB
• Volume: 72
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1999.1865

✦ Synopsis

In Giraitis, Robinson, and Samarov (1997), we have shown that the optimal rate for memory parameter estimators in semiparametric long memory models with degree of ``local smoothness'' ; is n &r( ;) , r( ;)=;Â(2;+1), and that a logperiodogram regression estimator (a modified Geweke and Porter-Hudak (1983) estimator) with maximum frequency m=m( ;) Ä n 2r( ;) is rate optimal. The question which we address in this paper is what is the best obtainable rate when ; is unknown, so that estimators cannot depend on ;. We obtain a lower bound for the asymptotic quadratic risk of any such adaptive estimator, which turns out to be larger than the optimal nonadaptive rate n &r( ;) by a logarithmic factor. We then consider a modified log-periodogram regression estimator based on tapered data and with a data-dependent maximum frequency m=m( ; ), which depends on an adaptively chosen estimator ; of ;, and show, using methods proposed by Lepskii (1990) in another context, that this estimator attains the lower bound up to a logarithmic factor. On one hand, this means that this estimator has nearly optimal rate among all adaptive (free from ;) estimators, and, on the other hand, it shows