"Frontmatter". In: Analysis of Financial Time Series

DURATION MODELS 203reduces to that of a WACD(r, s) model in Eq. (5.41). This log likelihood functioncan be rewritten in many ways to simplify the estimation.Under some regularity conditions, the conditional maximum likelihood estimatesare asymptotically normal; see Engle and Russell (1998) and the references therein.In practice, simulation can be used to obtain finite-sample reference distributions forthe problem of interest once a duration model is specified.Example 5.3. (Simulated ACD(1,1) series continued) Consider the simulatedWACD(1,1) and GACD(1, 1) series of Eq. (5.40). We apply the conditional likelihoodmethod and obtain the results in Table 5.6. The estimates appear to be reasonable.Let ˆψ i be the 1-step ahead prediction of ψ i and ˆɛ i = x i / ˆψ i be the standardizedseries, which can be regarded as standardized residuals of the series. If the modelis adequately specified, {ˆɛ i } should behave as a sequence of independent and identicallydistributed random variables. Figure 5.7(b) and Figure 5.8(b) show the timeplot of ˆɛ i for both models. The sample ACF of ˆɛ i for both fitted models are shown inFigure 5.10(b) and Figure 5.11(b), respectively. It is evident that no significant serialcorrelations are found in the ˆɛ i series.Example 5.4. As an illustration of duration models, we consider the transactiondurations of IBM stock on five consecutive trading days from November 1 toNovember 7, 1990. Focusing on positive transaction durations, we have 3534 observations.In addition, the data have been adjusted by removing the deterministic componentin Eq. (5.32). That is, we employ 3534 positive adjusted durations as definedin Eq. (5.31).Figure 5.12(a) shows the time plot of the adjusted (positive) durations for the firstfive trading days of November 1990, and Figure 5.13(a) gives the sample ACF ofthe series. There exist some serial correlations in the adjusted durations. We fit aWACD(1, 1) model to the data and obtain the modelx i = ψ i ɛ i , ψ i = 0.169 + 0.064x i−1 + 0.885ψ i−1 , (5.43)Table 5.6. Estimation Results for Simulated ACD(1,1) Series with 500 Observations:(a) for WACD(1,1) Series and (b) for GACD(1,1) Series.(a) WACD(1,1) modelParameter ω γ 1 ω 1 αTrue 0.3 0.2 0.7 1.5Estimate 0.364 0.100 0.767 1.477Std Error (0.139) (0.025) (0.060) (0.052)(b) GACD(1,1) modelParameter ω γ 1 ω 1 α κTrue 0.3 0.2 0.7 0.5 1.5Estimate 0.401 0.343 0.561 0.436 2.077Std Error (0.117) (0.074) (0.065) (0.078) (0.653)