"Frontmatter". In: Analysis of Financial Time Series

92 CONDITIONAL HETEROSCEDASTIC MODELSisr t = 0.0222 + a t , σ 2t = 0.0121 + 0.3029a 2 t−1 , (3.11)where the standard errors of the parameters are 0.0019, 0.1443, and 0.0061, respectively.All the estimates are significant at the 5% level, but the t ratio of ˆα 1 is only2.10. The unconditional variance of a t is 0.0121/(1 − 0.3029) = 0.0174, which isclose to that obtained under normality. The Ljung–Box statistics of the standardizedshocks give Q(10) = 13.66 with p-value 0.19, confirming that the mean equationis adequate. However, the Ljung–Box statistics for the squared standardized shocksshow Q(10) = 23.83 with p value 0.008. The volatility equation is inadequate at the5% level. We refine the model by considering an ARCH(2) model and obtainr t = 0.0225 + a t . σ 2t = 0.0113 + 0.226a 2 t−1 + 0.108a2 t−2 , (3.12)where the standard errors of the parameters are 0.006, 0.002, 0.135, and 0.094,respectively. The coefficient of at−1 2 is marginally significant at the 10% level, butthat of at−2 2 is only slightly greater than its standard error. The Ljung–Box statisticsfor the squared standardized shocks give Q(10) = 8.82 with p value 0.55. Consequently,the fitted ARCH(2) model appears to be adequate.Comparing models (3.10), (3.11), and (3.12), we see that (a) using a heavy-taileddistribution for ɛ t reduces the ARCH effect, and (b) the difference among the threemodels is small for this particular instance. Finally, a more appropriate conditionalheteroscedastic model for this data set is a GARCH(1, 1) model, which is discussedin the next section.Example 3.2. Consider the percentage changes of the exchange rate betweenMark and Dollar in 10-minute intervals. The data are shown in Figure 3.2(a). Asshown in Figure 3.3(a), the series has no serial correlations. However, the samplePACF of the squared series at 2 shows some big spikes, especially at lags 1 and 3.There are some large PACF at higher lags, but the lower order lags tend to be moreimportant. Following the procedure discussed in the previous subsection, we specifyan ARCH(3) model for the series. Using the conditional Gaussian likelihood function,we obtain the fitted modelσ 2t = 0.22 × 10 −6 + 0.328a 2 t−1 + 0.073a2 t−2 + 0.103a2 t−3 ,where all the estimates are statistically significant at the 5% significant level, andthe standard errors of the parameters are 0.46 × 10 −8 , 0.0162, 0.0160, and 0.0147,respectively. Model checking, using the standardized shock ã t , indicates that themodel is adequate.Remark: The estimation of conditional heteroscedastic models of this chapteris carried out by the Regression Analysis of Time Series (RATS) package. There are