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For this purpose, we use appropriate residuals obtained for the complete ICH
model.
First we consider the standardized residuals of the ICH model computed as
wi ¼ yi bE½YijF i 1; ZiŠ
bV½YijF i 1; ZiŠ
1 ;
2
where the estimated conditional mean is
bE½YijF i 1; ZiŠ
¼ X
j bPr½Yi¼jjFi1; ZiŠ:
j2Z
bPr½ Š represents the estimated counterpart of the conditional probability given
in Eq. (2.6) (augmented by the additional conditioning variable Z i). This probability
is calculated according to Eqs. (2.10), (2.13), (2.20), and (2.23). The estimated
conditional variance bV ½Š is obtained analogously. If the ICH model is
correctly specified, the standardized residuals evaluated at the true parameter
values should be uncorrelated in the first two moments with mean zero and unit
variance.
Figure 8 shows the autocorrelation function of the standardized residuals for
both stocks and Fig. 9 depicts the corresponding autocorrelation functions of the
squared standardized residuals. Comparing these autocorrelation functions with
those of the raw price changes (see Figs. 2 and 3), we observe that most of the serial
dependence in the first and second moments is captured by the ICH model.
Moreover, a comparison of the Ljung-Box Q-statistic for the residuals and squared
residuals with that for the price changes and squared price changes (see Table 5)
confirms that a large part of the dynamics can be explained by the ICH model.
JBX HAL
R. Liesenfeld et al.
Fig. 8 Autocorrelation functions of the residuals of the entire ICH models for Jack in the Box
Inc. (JBX) and Halliburton Company pffiffi
(HAL). The dashed lines mark off the approximate 99%
confidence interval 2:58 n