Journal of Statistical Theory and Practice

, Volume 1, Issue 2, pp 149–166 | Cite as

Weighted Averages and Local Polynomial Estimation for Fractional Linear ARCH Processes

  • Jan BeranEmail author
  • Yuanhua Feng


We consider local polynomial regression estimation for time series defined by Yi = μ(ti)+Ui where Ui is a stationary zero mean process with Wold decomposition Ui = C(B)Zi. The innovations Zi are assumed to be generated by a LARCH process, and may thus exhibit long-range correlations in volatility. The linear dependence structure, defined by the filter C(B) includes short memory, long memory and antipersistence. A central limit theorem for weighted averages, with triangular arrays of weights, and a limit theorem for local polynomial estimates of μ(ν)(i) are derived. The asymptotic distribution of μ(ν)(ti) turns out to be unaffected by long-range dependence in volatility. The question of optimal regression weights is also addressed.

AMS Subject Classification

62G08 and 62M10 


long memory LARCH process volatility central limit theorem location estimation local polynomial estimation 


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Copyright information

© Grace Scientific Publishing 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of KonstanzKonstanzGermany
  2. 2.of Mathematical and Computer SciencesHeriot-Watt UniversityEdinburghUK

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