Figure 5.13. The sample autocorrelation function of adjusted durations for IBM stock traded in the first five trading days of November 1990: (a) the adjusted series, and (b) the normalized innovations for a WACD(1, 1) model.

is 0.879, which is less than but close to 1. Thus, the conditional hazard function is monotonously decreasing at a slow rate.

If a generalized Gamma distribution function is used for the innovations, then the fitted GACD(1, 1) model is xi = f €i, fi = 0.141 + 0.063xi-1 + 0.897f _i, (5.44)

where {ei} follows a standardized, generalized Gamma distribution in Eq. (5.56) with parameters k = 4.248(1.046) and a = 0.395(0.053), where the number in parentheses denotes estimated standard error. Standard errors of the three parameters in Eq. (5.44) are 0.041, 0.010, and 0.019, respectively. All of the estimates are statistically significant at the 1% level. Again, the normalized innovational process {ei} and its squared series have no significant serial correlation, where ei = xi /fi based on model (5.44). Specifically, for the ei process, we have Q (10) = 4.95 and Q(20) = 10.28. For the e? series, we have Q(10) = 6.36 and Q(20) = 10.89.

The expected duration of model (5.44) is 3.52, which is slightly greater than that of the WACD(1, 1) model in Eq. (5.43). Similarly, the persistence parameter y\ +a>i of model (5.44) is also slightly higher at 0.96.

Remark: Estimation of EACD models can be carried out by using programs for ARCH models with some minor modification; see Engle and Russell (1998). In this book, we use either the RATS program or some Fortran programs developed by the author to estimate the duration models. Limited experience indicates that it is harder to estimate a GACD model than an EACD or a WACD model. RATS programs used to estimate WACD and GACD models are given in Appendix C.

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