Structural Breakpoints in Volatility in International Markets

We work with four stock indices and four interest rates series. The stock indices are
taken from Morgan Stanley: Emerging Asia (China, India, Indonesia, Korea, Malaysia,
Pakistan, Philippines, Taiwan, and Thailand), Europe (Austria, Belgium, Denmark,
Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Netherlands, Norway,
Portugal, Spain, Sweden, and the United Kingdom), Latin America (Argentina, Brazil,
Chile, Colombia, Mexico, Peru, and Venezuela), and North America (Canada and the
United States). All indices are free-float adjusted by market capitalization, and are
expressed in U.S. dollars. Index values are measured at closing time. The interest rates
series correspond with interest rates paid on deposits for 30-day and 60 day maturities
(nominal), and 90-day and 180-day maturities (inflation indexed). The sample period is
1997-2002, and the data are measured on a daily frequency. Descriptive statistics are given
in Table 1.

[Table 1]

The energy concentration function for a vector x=(x₁, x₂, ..., x_n)' is defined by
K
∑x(²i)

Ex(K) = ʒ---                                              (8)

∑xi²
i=1

where x(i) is the ith-largest absolute value in x. That is, the energy in a given crystal is
calculated as the sum of squares of all of its elements over the sum of squares of all
observations in the original time series. One appealing characteristic of the DWT is that it is
an energy preserving transform. This means that the energy in all the DWT coefficients
equals the energy in the original time series.

For our data, the coefficients at the two finest scales 2¹ and 2² (i.e., d1 and d2)
concentrate in all cases over 60 percent of the energy. For instance, the daily return on

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