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Multi-jumps. (2014). Renò, Roberto ; Caporin, Massimiliano ; Kolokolov, Aleksey.
In: Marco Fanno Working Papers.
RePEc:pad:wpaper:0185.

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  1. ! 0 (the stochastic bandwidth can be dealt as in Theorem 2.3 of Corsi et al., 2010), write: SRV(X(i) ) TRVN (X(i) ) = n X j=1 ⇣ jX(i) ⌘2 K jX(i) H (i) j ,n ! I{| j X(i) |H (i) j ,n} ! = X j X(i) Hj ,n 1 ⇣ jX(i) ⌘2 K jX(i) H (i) j ,n ! ! + X j X(i) Hj ,n >1 ⇣ jX(i) ⌘2 K jX(i) H (i) j ,n ! (mean value theorem) = X j X(i) Hj ,n 1 ⇣ jX(i) ⌘2 K0 ⇠j H (i) j ,n ! jX(i) H (i) j ,n + n X j=1 ⇣ jX(i) ⌘2 K jX(i) H (i) j ,n ! I{| j X(i) |>H (i) j ,n} ,
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  2. A = 0, (30) where the last implication follows from the independence of ⌘i j from ⌘k j when i 6= k. The rest follows as in Podolskij and Ziggel (2010). ⌅ B Quadratic variation measures and implementation B.1 Kernel and bandwidth selection Numerical experiments show that the test is more stable when the kernel is smooth, and that the kernel shape is not crucial. In simulations and empirical work we use a (normalized) Gaussian kernel K(x) = e x2 /2 . The bandwidth process is expressed as a function of the local variance, as follows: Ht,n = hn b (i) t r
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  42. T (Mancini and Gobbi, 2012, straightforward generalization of Theorem 4.2). On ⌦MJ,N T e An,1 tends to P X (1) t ... X (N) t 6=0 ⇣ X (i) t ⌘2 . Indeed, e An,1 di↵ers from the realized variance of the process X(i)0 by a finite number of asymptotically vanishing terms, where X(i)0 is defined by the sum of continuous part of X(i) and the process of multi-jumps: X(i)0 = X(i) X tT X (i) t I{ X (i) t 6=0 T X (1) t ... X (N) t =0} . (25) The proof for An,1 then follows by the continuous mapping theorem as for Theorem 3.1. ⌅ Proof of Theorem 3.3 Given Theorems 3.1 and 3.2, it is sufficient to show that a vector of statistics S0 (X(1) ), ..., S0 (X(N) ), where S0 (X(i) ) = Pn j=1 jX (i) c 1 ⌘i j p
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