Structural Breakpoints in Volatility in International Markets

^{z s}J '

^d J

^d J-1

where

s_J = (s_J,1,s_J,2,...,s_J,n/2J)'

dJ = (dJ,1,dJ,2,...,d_J,n/2J)'
dJ-1 =   (dJ-1,1,dJ-1,2,...,d_J-1,n/2J-1)'

d1 =        (d1,1,d1,2,...,d_1,n/2 )'

Each of the sets of coefficients s_j, d∣.d₁ is called a crystal.

Expression (2) can be rewritten as

f(t) ≈ S_j(t)+D_j(t)+D_j-1(t)+...+D1(t)                                           (6)

where

SJ(t) = ∑sJ,kφJ,k(t)                                                            (7a)

k

DJ(t)=∑dj,kψJ,k(t)                                                       (7b)

k

are denominated the smooth signal and the detail signals, respectively.

The terms in expression (6) represent a decomposition of the signal into orthogonal
signal components S∣(t), D∣(t), D∣ ₁(t), ...,D₁(t) at different scales. These terms are
components of the signal at different resolutions. That is why the approximation in (6) is
called a multi-resolution decomposition (MRD).

III   Data and Estimation Results

3.1   Description of the Data

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