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Estimation of long memory in volatility using wavelets. (2015). Baruník, Jozef ; Barunik, Jozef ; Kraicova, Lucie .
In: FinMaP-Working Papers.
RePEc:zbw:fmpwps:33.

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  1. A FIEGARCH Transformation As discussed in Perez & Zaffaroni (2008) and Zaffaroni (2009), to estimate the FIEGARCH model using FWE, it is necessary to rewrite the model in a signal plus noise form, for which the spectral density can be derived. Let’s begin with the original FIEGARCH(1,d,2) model: t = zth 1/2 t (29) ln(ht) = ω + Φ(L)g(zt−1) (30) g(zt) = θzt + γ[|zt| − E(|zt|)] (31) Φ(L) = (1 − L)−d [1 + α[2](L)][β(L)]−1 (32) where for FIEGARCH(1, d, 2) α[2](L) = αL, β(L) = 1 − βL. Following Zaffaroni (2009), this can be rewritten as 2 t = z2 t ht (33) ln 2 t = ln z2 t + ln (ht) (34) ln(ht) = ω + Φ(L, )g(zt−1) (35) Φ(L)g(zt−1) = ∞ X s=0 Φsg(zt−s−1) (36) g(zt) = θzt + γ[|zt| − E(|zt|)], (37) which leads to ln 2 t = ln z2 t
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  2. π ), (45) where ψ(x) and Ψ(x) are digamma and trigamma functions respectively. Evaluated at Fourier frequencies, this spectral density occurs in both terms of the FWE objective function. For a generalization to zt following GED or Student-t distribution, see Perez & Zaffaroni (2008). B Spectral Density Estimation 0 0.1 0.2 0.3 0.4 0.5 (a) d=0.25 0.1 0.2 0.3 0.4 0.5 (b) d=45 0 0.1 0.2 0.3 0.4 0.5 4.5 5.5 6.5 7.5 (c) d=-0.25
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