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Refined inference on long memory in realized volatility. (English) Zbl 1359.91033
Summary: There is an emerging consensus in empirical finance that realized volatility series typically display long range dependence with a memory parameter $$(d)$$ around 0.4 [T. G. Andersen et al., Stochastic volatility. Selected readings. Oxford: Oxford University Press. Adv. Texts Econom., 451–479 (2005; Zbl 1126.91354); M. Martens et al., Modeling and forecasting S&P 500 volatility: Long memory, structural breaks and nonlinearity. Tinbergen Institute Discussion Paper 2004-067/4 (2004)]. The present article provides some illustrative analysis of how long memory may arise from the accumulative process underlying realized volatility. The article also uses results ot the authors [Econom. Theory, 20, No. 3, 464–484 (2004; Zbl 1061.62022); Econom. J. 8, No. 3, 367–379 (2005; Zbl 1083.62091)] to refine statistical inference about $$d$$ by higher order theory. Standard asymptotic theory has an $$O(n^{-1/2})$$ error rate for error rejection probabilities, and the theory used here refines the approximation to an error rate of $$o(n^{-1/2})$$. The new formula is independent of unknown parameters, is simple to calculate and user-friendly. The method is applied to test whether the reported long memory parameter estimates of Andersen et al. (loc. cit.) and Martens et al. (loc. cit.) differ significantly from the lower boundary $$(d = 0.5)$$ of nonstationary long memory, and generally confirms earlier findings.

##### MSC:
 91G70 Statistical methods; risk measures 62G20 Asymptotic properties of nonparametric inference 62P05 Applications of statistics to actuarial sciences and financial mathematics
##### Keywords:
Edgeworth expansion; long memory; realized volatility
longmemo
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