"Frontmatter". In: Analysis of Financial Time Series

DURATION MODELS 201Series : xACF0.0 0.2 0.4 0.6 0.8 1.00 5 10 15 20 25 30LagSeries : yACF0.0 0.2 0.4 0.6 0.8 1.00 5 10 15 20 25 30LagFigure 5.10. The sample autocorrelation function of a simulated WACD(1, 1) series with 500observations: (a) the original series, and (b) the standardized residual series.The difference between the two models is evident. Finally, the sample ACF of thetwo simulated series are shown in Figure 5.10(a) and Figure 5.11(b), respectively.The serial dependence of the data is clearly seen.5.5.3 EstimationFor an ACD(r, s) model, let i o = max(r, s) and x t = (x 1 ,...,x t ) ′ . The likelihoodfunction of the durations x 1 ,...,x T is[ ]T∏f (x T | θ) = f (x i | F i−1 , θ) × f (x io | θ),i=i o +1where θ denotes the vector of model parameters, and T is the sample size. Themarginal probability density function f (x io | θ) of the previous equation is rathercomplicated for a general ACD model. Because its impact on the likelihood functionis diminishing as the sample size T increases, this marginal density is often ignored,resulting in the use of conditional likelihood method. For a WACD model, we usethe probability density function (pdf) of Eq. (5.55) and obtain the conditional log