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- 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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- The summability condition of the process t defined in (2) can be obtained by looking at t as a random walk process (analogously to Examples 3 and 5 in Berenguer-Rico and Gonzalo, 2014), where the innovation is zt = γtδt, where γt ∼ Bern(π) and δt ∼ N(0, σ2 δ ), so that E(zt) = E(γt) E(δt) = π 0 = 0, (19) Var(zt) = Var(γt) Var(δt) + Var(δt) E(γt)2 + Var(γt) E(δt)2 (20) = π(1 − π)σ2 δ + π2 σ2 δ + 0 = π σ2 δ . Therefore the partial sum T σδ √ π [Tr] ∑ t=1 zt d → W(r), (21) so that zt ∼ S(0) and zt ∼ I(0). Provided that t = ∑T t=1 zt, then T σδ √ π [Tr] ∑ t=1 t d → ∫ r W(r)dr, (22) so that t ∼ S(1) and t ∼ I(1), but with a slowly varying function equal to L(T) = 1 σδ √ π ,
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