State-space estimation of

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IV. Results

Our data series is gathered from the International Financial Statistics from January, 1980 through August, 1995 (187 observations). The exchange rate used in the construction of the series representing the deviations from uncovered interest parity (i.e., xt in 6) was the end-of-the-month quotations of the yen/Deutschemark exchange rate. The interest rates used were monthly observations of the daily averages of the overnight money market interest rates. The generated xt is plotted in Figure 3. The correlogram and descriptive statistics for the series are provided in Figures 4 and 5, respectively, which show that the series is easily consistent with white noise generated data.

The series was reversed and each span of 20 observations was investigated as described above for evidence of a bubble. Of the 168 spans considered along the 187 observation series, one area of the series shows several sequential spans where bubble activity seems likely. Specifically, the possibility is that the last period before a bubble burst is either January, February, or April of 1990. However, the largest improvements in both the bubble and the trimmed bubble specifications over the non-bubble specification is for a bubble that would have burst between the April and May of 1990 observations. The improvement in the maximized log-likelihood for both the bubble and trimmed bubble over the non-bubble specification are no less than 10.15. In the Monte Carlo experiments not one of the 100,000 simulated white noise spans had this great of difference in maximized log-likelihoods. If there had been just one, then the implied probability of finding this large of difference when considering the 168 spans would be only 0.168%. Therefore, the possibility that this portion of the observed series is not white noise is highly likely. Figure 3 includes a plot of the estimated bubble which is characterized by the estimates of  (=0.876) and h (=1.445). As one would expect, the Q statistic on the residual (i.e., et in 9) improves (i.e., falls) notably for all lags over the Q statistic for the observed series.

Since the estimate of  is found by essentially interpreting the reversed data, the estimated bubble actually grew at a rate of [(1/ - 1)  100]% per period. Therefore, the estimate of =0.876 implies that the bubble was growing at 14.2% per month which is clearly very large. This indicates that those rational investors who were speculating in the exchange or interest rate markets were requiring/receiving a large premium to compensate themselves for bearing the risk that the bubble might burst.

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