the form of an ARMA process with non-Gaussian innovations. It is understood here that γ j = 0 for j > r and ω j = 0 for 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(xi ) =
1−
ωmax(r,s)j=1
(γ j + ω j )
.
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HIGH - FREQUENCY DATA
Therefore, we assume ω > 0 and 1 > j (γ j + ω j ) because the expected duration ispositive. As another application of Eq. (5.35), we study properties of the EACD(1, 1)model.EACD(1, 1) ModelAn EACD(1, 1) model can be written asxi = ψi i ,
ψi = ω + γ1 xi−1 + ω1 ψi−1 ,
(5.36)
where i follows the standard exponential distribution. Using the moments of a standard exponential distribution in Appendix A, we have E( i ) = 1, Var( i ) = 1, andE( i2 ) = Var(xi ) + [E(xi )]2 = 2. Assuming that xi is weakly stationary (i.e., thefirst two moments of xi are time-invariant), we derive the variance of xi . First, takingexpectation of Eq. (5.36), we haveE(xi ) = E[E(ψi i | Fi−1 )] = E(ψi ),
E(ψi ) = ω + γ1 E(xi−1 ) + ω1 E(ψi−1 ).(5.37)
Under weak stationarity, E(ψi ) = E(ψi−1 ) so that Eq. (5.37) givesµx ≡ E(xi ) = E(ψi ) =
ω.1 − γ1 − ω1
(5.38)
Next, because E( i2 ) = 2, we have E(xi2 ) = E[E(ψi2 i2 | Fi−1 )] = 2E(ψi2 ).Taking square of ψi in Eq. (5.36) and expectation and using weak stationarity ofψi and xi , we have, after some algebra, thatE(ψi2 ) = µ2x ×
1 − (γ1 + ω1 )2.1 − 2γ12 − ω12 − 2γ1 ω1
(5.39)
Finally, using Var(xi ) = E(xi2 ) − [E(xi )]2 and E(xi2 ) = 2E(ψi2 ), we haveVar(xi ) = 2E(ψi2 ) − µ2x = µ2x ×
1 − ω12 − 2γ1 ω11 − ω12 − 2γ1 ω1 − 2γ12
,
where µx is defined in Eq. (5.38). This result shows that, to have time-invariantunconditional variance, the EACD(1, 1) model in Eq. (5.36) must satisfy 1 > 2γ12 +ω12 + 2γ1 ω1 . The variance of an WACD(1, 1) model can be obtained by using thesame techniques and the first two moments of a standardized Weibull distribution.ACD Models with a Generalized Gamma DistributionIn the statistical literature, intensity function is often expressed in terms of hazardfunction. As shown in Appendix B, the hazard function of an EACD model is constant over time and that of an WACD model is a monotonous function. These hazardfunctions are rather restrictive in application as the intensity function of stock trans-
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DURATION MODELS
actions might not be constant or monotone over time. To increase the flexibility of theassociated hazard function, Zhang, Russell, and Tsay (2001) employ a (standardized)generalized Gamma distribution for i . See Appendix A for some basic properties ofa generalized Gamma distribution. The resulting hazard function may assume various patterns, including U shape or inverted U shape. We refer to an ACD model withinnovations that follow a generalized Gamma distribution as a GACD(r, s) model.5.5.2 SimulationTo illustrate ACD processes, we generated 500 observations from the ACD(1, 1)modelxi = ψi i ,
ψi = 0.3 + 0.2xi−1 + 0.7ψi−1
(5.40)
using two different innovational distributions for i . In case 1, i is assumed to followa standardized Weibull distribution with parameter α = 1.5. In case 2, i follows a(standardized) generalized Gamma distribution with parameters κ = 1.5 and α =0.5.Figure 5.7(a) shows the time plot of the WACD(1, 1) series, whereas Figure 5.8(a)is the GACD(1, 1) series. Figure 5.9 plots the histograms of both simulated series.
0
2
4
dur6
8 10
(a) A simulated WACD(1,1) series
0
100
200
300
400
500
400
500
0
1
std-dur2
3
(b) Standardized series
0
100
200
300
Figure 5.7. A simulated WACD(1, 1) series in Eq. (5.40): (a) the original series, and (b) thestandardized series after estimation. There are 500 observations.
0
20
dur40
60
80
(a) A simulated GACD(1,1) series
0
100
200
300
400
500
400
500
0
std-dur5 10 15 20 25
(b) Standardized series
0
100
200
300
Figure 5.8. A simulated GACD(1, 1) series in Eq. (5.40): (a) the original series, and (b) thestandardized series after estimation. There are 500 observations.(b) GACD(1,1)
0
0
20
100
40
60
200
80
100
300
120
(a) WACD(1,1)
0
2
4
6z
8
10
0
20
40x
60
80
Figure 5.9. Histograms of simulated duration processes with 500 observations:(a) WACD(1, 1) model, and (b) GACD(1, 1) model200
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DURATION MODELS
ACF0.0 0.2 0.4 0.6 0.8 1.0
Series : x
0
5
10
15Lag
20
25
30
20
25
30
ACF0.0 0.2 0.4 0.6 0.8 1.0
Series : y
0
5
10
15Lag
Figure 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 xt = (x1 , . . . , xt ) . The likelihoodfunction of the durations x1 , . . . , x T isf (xT | θ) =
T
f (xi | Fi−1 , θ) × f (xio | θ),
i=i o +1
where θ denotes the vector of model parameters, and T is the sample size. Themarginal probability density function f (xio | 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
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HIGH - FREQUENCY DATA
0.0
ACF0.4
0.8
Series : x
0
5
10
15
20
25
15
20
25
Lag
0.0
ACF0.4
0.8
Series : y
0
5
10Lag
Figure 5.11. The sample autocorrelation function of a simulated GACD(1, 1) series with 500observations: (a) the original series, and (b) the standardized residual series.
likelihood function  α1(x | θ, xio ) =α ln  1 ++ lnαxii=i 0 +1 α   1 + α1 xixi ,+ α ln−ψiψiT
(5.41)
where ψi = ω + rj=1 γ j xi− j + sj=1 ω j ψi− j , θ = (ω, γ1 , . . . , γr , ω1 , . . . , ωs , α)and x = (xio +1 , . . . , x T ) . When α = 1, the (conditional) log likelihood functionreduces to that of an EACD(r, s) model.For a GACD(r, s) model, the conditional log likelihood function isαxi α(x | θ, xio ) =ln, (5.42)+(κα−1) ln(xi )−κα ln(λψi )−(κ)λψii=i +1T

o
where λ = (κ)/ (κ + α1 ) and the parameter vector θ now also includes κ. Asexpected, when κ = 1, λ = 1/ (1 + α1 ) and the log likelihood function in Eq. (5.42)
203
DURATION MODELS
reduces 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 likelihood method and obtain the results in Table 5.6. The estimates appear to be reasonable. Let ψ̂i be the 1-step ahead prediction of ψi and ˆi = xi /ψ̂i be the standardizedseries, which can be regarded as standardized residuals of the series. If the modelis adequately specified, {ˆi } should behave as a sequence of independent and identically distributed random variables. Figure 5.7(b) and Figure 5.8(b) show the timeplot of ˆi for both models. The sample ACF of ˆi 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 ˆi series.Example 5.4. As an illustration of duration models, we consider the transaction durations 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 component in 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 modelxi = ψi i ,
ψi = 0.169 + 0.064xi−1 + 0.885ψi−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
γ1
ω1
α
TrueEstimateStd Error
0.30.364(0.139)
0.20.100(0.025)
0.70.767(0.060)
1.51.477(0.052)
(b) GACD(1,1) modelParameter
ω
γ1
ω1
α
κ
TrueEstimateStd Error
0.30.401(0.117)
0.20.343(0.074)
0.70.561(0.065)
0.50.436(0.078)
1.52.077(0.653)
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HIGH - FREQUENCY DATA
0
10
adj-dur20 30
40
(a) Adjusted durations
1000
2000sequence
3000
0
1000
2000sequence
3000
norm-dur0 2 4 6 8 10 12 14
0
Figure 5.12. Time plots of durations for IBM stock traded in the first five trading days ofNovember 1990: (a) the adjusted series, and (b) the normalized innovations of an WACD(1, 1)model. There are 3534 nonzero durations.
where { i } is a sequence of independent and identically distributed random variatesthat follow the standardized Weibull distribution with parameter α̂ = 0.879(0.012),where 0.012 is the estimated standard error. Standard errors of the estimates inEq. (5.43) are 0.039, 0.010, and 0.018, respectively. All t ratios of the estimatesare greater than 4.2, indicating that the estimates are significant at the 1% level.Figure 5.12(b) shows the time plot of ˆi = xi /ψ̂i , and Figure 5.13(b) provides thesample ACF of ˆi . The Ljung–Box statistics show Q(10) = 4.96 and Q(20) = 10.75for the ˆi series. Clearly, the standardized innovations have no significant serial correlations. In fact, the sample autocorrelations of the squared series {ˆi2 } are also smallwith Q(10) = 6.20 and Q(20) = 11.16, further confirming lack of serial dependencein the normalized innovations. In addition, the mean and standard deviation of a standardized Weibull distribution with α = 0.879 are 1.00 and 1.14, respectively. Thesenumbers are close to the sample mean and standard deviation of {ˆi }, which are 1.01and 1.22, respectively. The fitted model seems adequate.In model (5.43), the estimated coefficients show γ̂1 + ω̂1 ≈ 0.949, indicatingcertain persistence in the adjusted durations. The expected adjusted duration is0.169/(1 − 0.064 − 0.885) = 3.31 seconds, which is close to the sample mean 3.29of the adjusted durations. The estimated α of the standardized Weibull distribution
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DURATION MODELS
ACF0.0 0.2 0.4 0.6 0.8 1.0
Series : x
0
10
20
30
20
30
Lag
ACF0.0 0.2 0.4 0.6 0.8 1.0
Series : epsi
0
10Lag
Figure 5.13. The sample autocorrelation function of adjusted durations for IBM stock tradedin the first five trading days of November 1990: (a) the adjusted series, and (b) the normalizedinnovations for a WACD(1, 1) model.
is 0.879, which is less than but close to 1. Thus, the conditional hazard function ismonotonously decreasing at a slow rate.If a generalized Gamma distribution function is used for the innovations, then thefitted GACD(1, 1) model isxi = ψi i ,
ψi = 0.141 + 0.063xi−1 + 0.897ψi−1 ,
(5.44)
where { i } follows a standardized, generalized Gamma distribution in Eq. (5.56)with parameters κ = 4.248(1.046) and α = 0.395(0.053), where the number inparentheses 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 arestatistically significant at the 1% level. Again, the normalized innovational process{ˆi } and its squared series have no significant serial correlation, where ˆi = xi /ψ̂ibased on model (5.44). Specifically, for the ˆi process, we have Q(10) = 4.95 andQ(20) = 10.28. For the ˆi2 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 thatof the WACD(1, 1) model in Eq. (5.43). Similarly, the persistence parameter γ̂1 + ω̂1of model (5.44) is also slightly higher at 0.96.
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HIGH - FREQUENCY DATA
Remark: Estimation of EACD models can be carried out by using programs forARCH models with some minor modification; see Engle and Russell (1998). In thisbook, we use either the RATS program or some Fortran programs developed by theauthor to estimate the duration models. Limited experience indicates that it is harderto estimate a GACD model than an EACD or a WACD model. RATS programs usedto estimate WACD and GACD models are given in Appendix C.
5.6 NONLINEAR DURATION MODELSNonlinear features are also commonly found in high-frequency data. As an illustration, we apply some nonlinearity tests discussed in Chapter 4 to the normalized innovations ˆi of the WACD(1, 1) model for the IBM transaction durations inExample 5.4; see Eq. (5.43). Based on an AR(4) model, the test results are given inpart (a) of Table 5.7. As expected from the model diagnostics of Example 5.4, theOri-F test indicates no quadratic nonlinearity in the normalized innovations. However, the TAR-F test statistics suggest strong nonlinearity.Based on the test results in Table 5.7, we entertain a threshold duration modelwith two regimes for the IBM intraday durations. The threshold variable is xt−1 (i.e.,lag-1 adjusted duration). The estimated threshold value is 3.79. The fitted thresholdWACD(1, 1) model is xi = ψi i , where
ψi =
0.020 + 0.257xi−1 + 0.847ψi−1 ,
i ∼ w(0.901)
if xi−1 ≤ 3.79
1.808 + 0.027xi−1 + 0.501ψi−1 ,
i ∼ w(0.845)
if xi−1 > 3.79,(5.45)
Table 5.7. Nonlinearity Tests for IBM Transaction Durations from November 1 toNovember 7, 1990. Only Intraday Durations Are Used. The Number in the Parentheses of Tar-F Tests Denotes Time Delay.(a) Normalized innovations of a WACD(1,1) modelType
Ori-F
Tar-F(1)
Tar-F(2)
Tar-F(3)
Tar-F(4)
Testp value
0.3430.969
3.2880.006
3.1420.008
3.1280.008
0.2970.915
(b) Normalized innovations of a threshold WACD(1,1) modelType
Ori-F
Tar-F(1)
Tar-F(2)
Tar-F(3)
Tar-F(4)
Testp value
0.1630.998
0.7460.589
1.8990.091
1.7520.119
0.2700.929
THE PCD MODEL
207
where w(α) denotes a standardized Weibull distribution with parameter α. The number of observations in the two regimes are 2503 and 1030, respectively. In Eq. (5.45),the standard errors of the parameters for the first regime are 0.043, 0.041, 0.024,and 0.014, whereas those for the second regime are 0.526, 0.020, 0.147, and 0.020,respectively.Consider the normalized innovations ˆi = xi /ψ̂i of the threshold WACD(1, 1)model in Eq. (5.45). We obtain Q(12) = 9.8 and Q(24) = 23.9 for ˆi and Q(12) =8.0 and Q(24) = 16.7 for ˆi2 . Thus, there are no significant serial correlations in theˆi and ˆi2 series. Furthermore, applying the same nonlinearity tests as before to thisnewly normalized innovational series ˆi , we detect no nonlinearity; see part (b) ofTable 5.7. Consequently, the two-regime threshold WACD(1, 1) model in Eq. (5.45)is adequate.If we classify the two regimes as heavy and thin trading periods, then the thresholdmodel suggests that the trading dynamics measured by intraday transaction durationsare different between heavy and thin trading periods for IBM stock even after theadjustment of diurnal pattern. This is not surprising as market activities are oftendriven by arrivals of news and other information.The estimated threshold WACD(1, 1) model in Eq. (5.45) contains some insignificant parameters. We refine the model and obtain the result:0.225xi−1 + 0.867ψi−1 ,i ∼ w(0.902) if xi−1 ≤ 3.79ψi =1.618 + 0.614ψi−1 ,if xi−1 > 3.79.i ∼ w(0.846)All of the estimates of the refined model are highly significant. The Ljung–Boxstatistics of the standardized innovations ˆi = xi /ψ̂i show Q(10) = 5.91(0.82)and Q(20) = 16.04(0.71) and those of ˆi2 give Q(10) = 5.35(0.87) and Q(20) =15.20(0.76), where the number in parentheses is the p value. Therefore, the refinedmodel is adequate. The RATS program used to estimate the prior model is given inAppendix C.5.7 BIVARIATE MODELS FOR PRICE CHANGE AND DURATIONIn this section, we introduce a model that considers jointly the process of pricechange and the associated duration. As mentioned before, many intraday transactionsof a stock result in no price change. Those transactions are highly relevant to tradingintensity, but they do not contain direct information on price movement. Therefore,to simplify the complexity involved in modeling price change, we focus on transactions that result in a price change and consider a price change and duration (PCD)model to describe the multivariate dynamics of price change and the associated timeduration.We continue to use the same notation as before, but the definition is changed totransactions with a price change. Let ti be the calendar time of the ith price changeof an asset. As before, ti is measured in seconds from midnight of a trading day. LetPti be the transaction price when the ith price change occurred and ti = ti −ti−1 be
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HIGH - FREQUENCY DATA
the time duration between price changes. In addition, let Ni be the number of tradesin the time interval (ti−1 , ti ) that result in no price change. This new variable is usedto represent trading intensity during a period of no price change. Finally, let Di bethe direction of the ith price change with Di = 1 when price goes up and Di = −1when the price comes down, and let Si be the size of the ith price change measuredin ticks. Under the new definitions, the price of a stock evolves over time byPti = Pti−1 + Di Si ,
(5.46)
and the transactions data consist of { ti , Ni , Di , Si } for the ith price change. ThePCD model is concerned with the joint analysis of ( ti , Ni , Di , Si ).Remark: Focusing on transactions associated with a price change can reducethe sample size dramatically. For example, consider the intraday data of IBM stockfrom November 1, 1990 to January 31, 1991. There were 60,265 intraday trades, butonly 19,022 of them resulted in a price change. In addition, there is no diurnal patternin time durations between price changes.To illustrate the relationship among the price movements of all transactions andthose of transactions associated with a price change, we consider the intraday tradings of IBM stock on November 21, 1990. There were 726 transactions on that dayduring the normal trading hours, but only 195 trades resulted in a price change. Figure 5.14 shows the time plot of the price series for both cases. As expected, the priceseries are the same.The PCD model decomposes the joint distribution of ( ti , Ni , Di , Si ) given Fi−1asf ( ti , Ni , Di , Si | Fi−1 )= f (Si | Di , Ni , ti , Fi−1 ) f (Di | Ni , ti , Fi−1 ) f (Ni |
ti , Fi−1 ) f ( ti | Fi−1 ).(5.47)
This partition enables us to specify suitable econometric models for the conditionaldistributions and, hence, to simplify the modeling task. There are many ways tospecify models for the conditional distributions. A proper specification might dependon the asset under study. Here we employ the specifications used by McCulloch andTsay (2000), who use generalized linear models for the discrete-valued variables anda time series model for the continuous variable ln( ti ).For the time duration between price changes, we use the modelln( ti ) = β0 + β1 ln( ti−1 ) + β2 Si−1 + σ i ,
(5.48)
where σ is a positive number and { i } is a sequence of iid N (0, 1) random variables.This is a multiple linear regression model with lagged variables. Other explanatoryvariables can be added if necessary. The log transformation is used to ensure thepositiveness of time duration.
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THE PCD MODEL
price112.5 113.0 113.5 114.0
(a) All transactions•••••••••••••••• •••• ••••••••••••• ••••••••••••• ••••••••••••••• ••••• •••••••••• •• •••• ••• •••• ••••••••••••••• •••••• •••••• •••••••••• ••••••••••••••••••• •••• •••••• •••• •• •••••• •••• ••••••••••••••••• ••••••••••• ••••••••••• •• • ••••••••• ••• • ••••••• ••••••••••••• • ••• • ••••••••• •• ••••• • ••••••••••••••• •••••••• •3500040000450005000055000seconds
price112.5 113.0 113.5 114.0
(b) Transactions with a price change•• •••• •• ••• •••••••• ••• ••• • • •••••• • ••••• • •• ••• •••••••••••• ••• •• ••• •••••• •••• •••••••• ••••••• ••••• ••• •• • •• •• • ••• • ••••• • ••••• • •••• • • • • ••• • •••••• •• • •• •• • •• • ••• • • •• • • ••• • •• •• •3500040000450005000055000seconds
Figure 5.14. Time plots of the intraday transaction prices of IBM stock on November 21,1990: (a) all transactions, and (b) transactions that resulted in a price change.
The conditional model for Ni is further partitioned into two parts because empirical data suggest a concentration of Ni at 0. The first part of the model for Ni is thelogit modelp(Ni = 0 |
ti , Fi−1 ) = logit[α0 + α1 ln( ti )],
(5.49)
where logit(x) = exp(x)/[1 + exp(x)], whereas the second part of the model isNi | (Ni > 0, ti , Fi−1 ) ∼ 1 + g(λi ),
λi =
exp[γ0 + γ