standardized residuals, ηt, are resampled with replacement to form the resampled standardized residual series {ηt*: t=18,...,T}. The bootstrapped log conditional variance series is equal to
{ln(ht*)=α0+g(ηt-1*)+β1 ln(ht-1*):t=19,...,T }.
We set h18* equal to the unconditional variance. Under the assumption that the ηt are iid N(0,1) the unconditional variance of εt is equal to
| E(ht)={exp(α0) E[exp | (g(ηt-1))]} |
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{εt*=ηt* ht*: t=19,...,T} and the bootstrapped return series is equal to
{rt*=α+φ16 rt-16*+εt*: t=19,...,T}. For t=2,...,18 we set rt*=rt. The bootstrapped price series is equal to {Pt*=exp(rt*) Pt-1*: t=2,...,T} and P1*=P1. Table 2.10 contains the estimation results for the exponential GARCH model.
Structural break in trend
Figure 2.4 reveals that the LIFFE cocoa futures price series contains an upward trend in the period January 5, 1983 until February 4, 1985, when the price peaks, and a downward trend in the period February 5, 1985 until December 31, 1987. Therefore, we split the first subperiod in two periods, which separately contain the upward and downward trend. By doing this we allow for a structural change in the return process. The final bootstrap procedure we consider will simulate comparison series that will have the same change in trends. For the first period we estimate and bootstrap the autoregressive model (2.11). We don't find signs of volatility clustering for this period. However on the second period we find significant volatility clustering and therefore we estimate and bootstrap the following GARCH model:
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