"Frontmatter". In: Analysis of Financial Time Series

DURATION MODELS 197(a) is the average durations of t i and, as expected, it exhibits a diurnal pattern. Part(b) is the average durations of t ∗ i(i.e., after the adjustment), and the diurnal patternis largely removed.5.5.1 The ACD ModelThe autoregressive conditional duration (ACD) model uses the idea of GARCH modelsto study the dynamic structure of the adjusted duration ti ∗ of Eq. (5.31). For easein notation, we define x i = ti ∗.Let ψ i = E(x i | F i−1 ) be the conditional expectation of the adjusted durationbetween the (i − 1)th and ith trades, where F i−1 is the information set available atthe (i − 1)th trade. In other words, ψ i is the expected adjusted duration given F i−1 .The basic ACD model is defined asx i = ψ i ɛ i , (5.33)where {ɛ i } is a sequence of independent and identically distributed non-negative randomvariables such that E(ɛ i ) = 1. In Engle and Russell (1998), ɛ i follows a standardexponential or a standardized Weibull distribution, and ψ i assumes the formψ i = ω +r∑γ j x i− j +j=1s∑ω j ψ i− j . (5.34)Such a model is referred to as an ACD(r, s) model. When the distribution of ɛ i isexponential, the resulting model is called an EACD(r, s) model. Similarly, if ɛ i followsa Weibull distribution, the model is a WACD(r, s) model. If necessary, readersare referred to Appendix A for a quick review of exponential and Weibull distributions.Similar to GARCH models, the process η i = x i − ψ i is a Martingale differencesequence [i.e., E(η i | F i−1 ) = 0], and the ACD(r, s) model can be written asj=1max(r,s) ∑x i = ω + (γ j + ω j )x i− j −j=1s∑ω j η i− j + η j , (5.35)which is in the form of an ARMA process with non-Gaussian innovations. It is understoodhere that γ j = 0for j > r and ω j = 0for j > s. Such a representation canbe used to obtain the basic conditions for weak stationarity of the ACD model. Forinstance, taking expectation on both sides of Eq. (5.35) and assuming weak stationarity,we haveE(x i ) =j=1ω1 − ∑ max(r,s)j=1(γ j + ω j ) .