Structural Breakpoints in Volatility in International Markets

and ψj,k(t) are scaled and translated versions of φ and ψ. As equation (3) indicates, the scale
or dilation factor is 2^j, whereas the translation or location parameter is 2^jk. As j gets larger,
so does the scale factor 2^j, and the functions φj,k(t) and ψj,k(t) get shorter and more spread
out. In other words, 2^j is a measure of the width of the functions φj,k(t) and ψj,k(t). Likewise,
as j increases, the translation step gets correspondingly larger in order to match the scale
parameter 2^j.

Applications of wavelet analysis commonly make use of a discrete wavelet
transform (DWT). The DWT calculates the coefficients of the approximation in (2) for a
discrete signal of final extent, f₁, f₂,.., f_n. That is, it maps the vector f=(f₁, f₂,...,f_n)' to a
vector ω of n wavelet coefficients that contains sJ,_k and dj,_k, j=1,2,..., J. The sJ,_k are called
the smooth coefficients and the dj,k are called the detail coefficients. Intuitively, the smooth
coefficients represent the underlying smooth behavior of the data at the coarse scale 2^J,
whereas the detail coefficients provide the coarse scale deviations from it.

When the length of the data n is divisible by 2^J, there are n/2 coefficients d1,k at the
finest scale 2¹=2. At the next finest scale, there are n/2² coefficients d2,k. Similarly, at the
coarsest scale, there are n/2^J dJ,k coefficients and n/2^J sJ,k coefficients. Altogether, there are

= n coefficients. The number of coefficients at a given scale is related to the

width of the wavelet function. For instance, at the finest scale, it takes n/2 terms for the
functions ψ1,k(t) to cover the interval 1≤t≤n.

The wavelet coefficients are ordered from coarse scales to fine scales in the vector
ω. If n is divisible by 2^J, ω will be given by

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