Structural Breakpoints in Volatility in International Markets

15

Figure 4 illustrates the turbulence experienced by North America over June-
September 1998. The left-hand side panel depicts recomposed crystal D4—the component
that would explain the violation of variance homogeneity over 1997-1998—, whereas the
right-hand side panel shows the return series. The 95-percent out-of-sample value at risk
was computed by fitting a GARCH (1,1) model to each series over January 1997-May
1998. As the right-hand shows, the accumulative actual loss on a $10,000 portfolio invested
on the North America index exceeds the 95-percent value at risk for about the first forty
observations (June and mid-July).

[Figure 4]

Regarding interest rates, breakpoints are still detected by the wavelet variance test
after controlling for conditional heteroskedasticity and serial correlation (Table 5). In
particular, there is evidence, at the 5 percent significance level, of variance shifts at scale 1
of the 30-day and 60-day interest rates over February 2001-April 2002. However, variance
homogeneity is never rejected at any scale for the 90-day and 180-interest rates over the
same period. That is to say, switching from an inflation-linked to a nominal monetary
policy interest rate would have affected only the volatility of short nominal interest rates,
according to this method.

[Table 5]

An alternative way to test for structural stability is by using a Wald test of the sort in
Hamilton (1994, chapter 14). In particular, if n is the sample size, and n0 is a known
breakpoint, a Wald test of the null hypothesis that θ1=θ2, is given by
л         Λ                   ^                       л               л         Λ

n(θ₁,∏0 -θ2,n-no)'(∏^-1Vι,no + (1 -π)^-1V_λn-n_o)^-1(θ₁,n_o -θ₂,_n-n_o)~^→X²(q) (14)

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