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Does the ARFIMA really shift?. (2017). Santucci de Magistris, Paolo ; Grassi, Stefano ; Delle Monache, Davide.
In: CREATES Research Papers.
RePEc:aah:create:2017-16.

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  1. , R = (1, ζ1, . . . , ζm)′ , Q = σ2 ξ . (30) The ML estimator, ψ̂ = arg max ℓ(YT ; ψ), based on the truncated representation, as shown to be consistent, asymptotically Gaussian and efficient for m = Tβ with β ≥ 1/2; see Chan and Palma (1998). Here we adopt the truncated AR(m) representation with m = √ T.
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  2. A Appendix A.1 Summability order of the level shift component t In order to study the order of integration of the non-linear process t defined in (2), we use the concept of summability as introduced by Berenguer-Rico and Gonzalo (2014). Definition 1. A stochastic process {t : t ∈ N} is said to be summable of order β, or S(β), if there exists a slowly varying function L(T) and a deterministic sequence mt, such that ST = T1/2+β L(T) T ∑ t=1 (t − mt) = Op(1) where β is the minimum real number that makes ST bounded in probability. Under mild regularity conditions, it holds that if a process t is I(β) with d ≥ 0, then it is S(β).
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