Structural Breakpoints in Volatility in International Markets



Unlike Fourier series, wavelets are suitable building-block functions for signals
whose features change over time, and for non-smooth signals. A wavelet allows for
decomposing a signal into multi-resolution components: fine and coarse resolution
components.

There are father wavelets φ and mother wavelets ψ such that

φ(t)dt = 1           ψ(t)dt = 0                                                (1)

Father wavelets are good at representing the smooth and low-frequency parts of a
signal, whereas mother wavelets are good at representing the detailed and high-frequency
parts of a signal. The most commonly used wavelets are the orthogonal ones (i.e., haar,
daublets, symmelets, and coiflets). In particular, the orthogonal wavelet series
approximation to a continuous signal f(t) is given by

f(t)sJ,kφJ,k(t)+dJ,kψJ,k(t)+dJ-1,kψJ-1,k(t)+...+d1,kψ1,k(t)        (2)

kk   k     k

where J is the number of multi-resolution components or scales, and k ranges from 1 to the
number of coefficients in the corresponding component. The coefficients s
J,k, dJ,k,..., d1,k are
the wavelet transform coefficients, whereas the functions
φj,k(t) and ψj,k(t) are the
approximating wavelet functions. These functions are generated from
φ and ψ as follows

φj,k(t) = 2-j'2 φf t-2k )           ψik(t) = 2-j/2 ψ[^ t--jk 1                 (3)

The wavelet coefficients can be approximated by the following integrals
s
J,kφJ,k(t)f(t)dt           dj,k ψj,k(t)f(t)dt, j=1, 2,..., J                      (4)

These coefficients are a measure of the contribution of the corresponding wavelet
function to the total signal. On the other hand, the approximating wavelet functions
φj,k(t)



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