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Finite sample properties of the QMLE for the Log-ACD model: Application to Australian stocks
Journal of Econometrics 147 (2008) 163–185
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Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom
Finite sample properties of the QMLE for the Log-ACD model: Application to Australian stocks David Allen a,∗ , Felix Chan b , Michael McAleer c , Shelton Peiris d a
School of Accounting, Finance and Economics, Edith Cowan University, Australia
b
School of Economics and Finance, Curtin University of Technology, Australia
c
School of Economics and Commerce, University of Western Australia, Australia
d
Department of Mathematics and Statistics, University of Sydney, Australia
article
info
Article history: Available online 20 September 2008 Keywords: Conditional duration Asymmetry ACD Log-ACD Monte Carlo simulation
a b s t r a c t This paper concerns the properties of the Quasi Maximum Likelihood Estimator (QMLE) of the Logarithmic Autoregressive Conditional Duration (Log-ACD) model. Proofs of consistency and asymptotic normality of QMLE for the Log-ACD model with log-normal density are presented. This is an important issue as the Log-ACD is used widely for testing various market microstructure models and effects. Knowledge of the distribution of the QMLE is crucial for purposes of valid inference and diagnostic checking. The theoretical results developed in the paper are evaluated using Monte Carlo experiments. The experimental results also provide insights into the finite sample properties of the Log-ACD model under different distributional assumptions. Finally, this paper presents two extensions to the Log-ACD model to accommodate asymmetric effects. The usefulness of these novel models will be evaluated empirically using data from Australian stocks. © 2008 Elsevier B.V. All rights reserved.
1. Introduction An accurate description of the dynamics of duration between stock price changes has important implications and applications for the analysis of financial markets. Engle and Russell (1997) proposed the Autoregressive Conditional Duration (ACD) model, which assumes that the duration between price changes follows a process similar to that of Bollerslev’s (1986) Generalised Autoregressive Conditional Heteroscedasticity (GARCH) model. Both models are based on dynamic time series processes in the underlying variables. Due to the structural similarity between the GARCH and ACD models, Engle and Russell (1997) provided a proof of consistency and asymptotic normality for the QMLE of the ACD model following the approach of Lee and Hansen (1994), arguing that the theoretical results could be applied directly to the ACD model. Based on this result, researchers have subsequently proposed numerous extensions to the ACD model in a similar manner to the extensions of the GARCH model. These extensions include Bauwens and Giot’s (2000) Logarithmic ACD (Log-ACD) model, and Dufour and Engle’s (2000) Box-Cox ACD (BCACD) model and Exponential ACD (EXPACD) model, Zhang et al.’s (2001)
∗ Corresponding address: School of Accounting, Finance and Economics, Edith Cowan University, Faculty of Business and Law, 6027 Joondalup, WA, Australia. E-mail address: [email protected] (D. Allen). 0304-4076/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2008.09.020
Threshold ACD (TACD) model, and Hujer et al.’s (2003) Markov Switching ACD (MSACD) model. For a discussion of the structural and statistical properties of a variety of univariate and multivariate conditional volatility models, see McAleer (2005). This paper is concerned with the asymptotic and finite sample properties of the Quasi Maximum Likelihood Estimator (QMLE) for the Log-ACD model. The motivation of this paper is two-fold. First, Engle and Russell (1997) derived the asymptotic properties of the ACD model based on the results of Lee and Hansen (1994) for the GARCH model. However, such asymptotic properties are not yet available for the volatility counterpart of the Log-ACD model, namely Nelson’s (1991) Exponential GARCH (EGARCH) model. Therefore, the distribution of the QMLE for Log-ACD is still unknown, which is particularly important as the ACD model is often used for testing hypotheses about the market microstructure. Thus, the distribution of QMLE is crucial for purposes of valid inference and diagnostic checking. Second, in the GARCH literature the standardised residuals are often assumed to be normally distributed. The QMLE based on the normal density has been proven to be consistent and asymptotically normal under fairly general conditions by Ling and McAleer (2003). However, the assumption of normality cannot be applied to ACD (or Log-ACD) models as the standardised residuals of these models are required to be positive. The Weibull, exponential, generalised gamma and log-normal are four of the most widely used probability density functions (pdf). A natural question is the nature of the statistical
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 1 Summary statistics for eight companies on the Australian stock exchange. Statistics
ANZ
CBA
BHP
WPL
CML
WOW
TLS
QAN
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Observations
80.1535 36 18485 1 232.1636 51.39532 3739.022 16384
57.55475 27 18316 1 168.9323 78.25914 8343.062 16384
100.3229 32 18560 1 312.1668 33.61695 1721.953 14854
133.8996 52 18191 1 367.0297 29.96407 1363.638 11130
287.579 109 18229 1 604.7731 11.50926 261.3184 5104
136.1386 53 18220 1 407.2284 26.17681 990.3336 11105
461.3966 123 18302 1 973.3416 6.607736 78.29639 3172
463.057 152 15798 1 915.73 6.082173 64.82925 3122
Table 2a True Distribution: Lognormal; Sample Size: 500; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal 0.214 0.214 0.347 0.060 0.035 0.025 3.179
Weibull 0.205 0.204 0.353 0.035 0.039 0.104 3.127
G. Gamma 0.181 0.191 0.393 −0.028 0.067 −0.788 3.688
Exponential 0.198 0.196 0.379 0.011 0.047 0.176 3.221
Normal 0.214 0.214 0.347 0.060 0.035 0.025 3.179
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
0.369 0.408 3.739 −4.598 1.027 −0.196 3.374 36.600 0.000
0.066 0.121 3.412 −5.903 1.107 −0.290 3.443 66.504 0.000
−7.656 −0.084
−0.142 −0.079
0.185 0.232 3.517 −4.578 1.060 −0.208 3.278 31.216 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.646 0.654 0.868 0.084 0.071 −0.868 5.620
0.669 0.676 0.882 0.154 0.076 −0.738 4.617
0.482 0.675 0.940 −1.026 0.544 −2.241 6.223
0.681 0.692 0.961 −0.076 0.099 −1.991 13.953
0.646 0.654 0.868 0.084 0.071 −0.868 5.620
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.686 −0.730
−0.296 −0.374
−134.088 −0.215
−1.482 −0.089
−0.478 −0.530
4.133
5.433
−3.784
−3.815
0.956 0.323 3.764 124.676 0.000
1.083 0.384 3.582 115.953 0.000
properties of QMLE for the (Log-)ACD model based on a variety of alternative probability distributions. This paper will present the proofs of consistency and asymptotic normality of QMLE for the Log-ACD model with log-normal density. The theoretical results developed in the paper are evaluated using Monte Carlo experiments. The experimental results also provide insights into the finite sample properties of the Log-ACD model under different distributional assumptions. Moreover, the relationship between the Log-ACD and the Autoregressive–Moving Average (ARMA) model will also be discussed in details. As the structural and statistical properties of the ARMA process are well established, the properties for the ARMA model will also apply to the Log-ACD model. Moreover, the statistical properties of the ARMA model with exogenous variables (ARMAX) will also apply to the Log-ACD model with exogenous variables. This is particularly important as ACD models with exogenous variables are often used for purposes of testing hypotheses about mar-
3.242
−1.620E+04 299.533
−52.959 2857.967 1.018E+09 0.000
14.251
−1.131E+05 3.316E+03 −29.428 906.741 1.023E+08 0.000
6.725
−15.993 1.253
−1.079 13.049 1.317E+04 0.000
9.021
−3.973E+03 72.677
−54.546 2.981E+03 1.108E+09 0.000
4.223
−3.691 1.018 0.279 3.517 72.233 0.000
ket microstructures. Therefore, these properties are crucial for ensuring valid inferences and diagnostic checking. The second part of the paper will propose two alternative methods for accommodating asymmetric effects. The Log-ACD model assumes that the duration between price movements is affected only by the previous duration but not by the direction of the price change. However, since the market frequently has a different attitude to price rises as compared with price falls, it is important to examine how the direction of the price movement affects the future duration. Although many alternative asymmetric ACD models have been proposed in the literature, including Zhang et al.’s (2001) Threshold ACD (TACD) model and Hujer et al.’s (2003) Markov Switching ACD (MSACD) model, these models often lack structural and/or statistical properties and can be difficult to estimate. Moreover, using these models for purposes of testing hypotheses about market microstructure is not always straightforward. However, the
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
165
Table 2b True Distribution: Lognormal; Sample Size: 1000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.207 0.207 0.302 0.123 0.025 0.108 3.173
Weibull
G. Gamma
0.202 0.203 0.305 0.119 0.028 0.082 2.981
Exponential
0.190 0.196 0.351 −0.022 0.051 −1.694 7.362
0.199 0.198 0.317 0.100 0.033 0.127 3.046
Normal 0.207 0.207 0.302 0.123 0.025 0.108 3.173
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
0.275 0.302 3.816 −4.003 1.023 −0.130 3.241 15.799 0.000
0.035 0.101 3.297 −4.052 1.081 −0.241 3.149 31.736 0.000
−9.491 −0.101
−0.152 −0.065
0.128 0.158 3.609 −4.363 1.056 −0.150 3.196 16.067 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.675 0.677 0.819 0.496 0.045 −0.311 3.312
0.686 0.689 0.828 0.465 0.050 −0.397 3.453
0.601 0.692 0.879 −1.010 0.383 −3.792 15.789
0.693 0.697 0.868 0.370 0.059 −0.559 4.044
0.675 0.677 0.819 0.496 0.045 −0.311 3.312
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.493 −0.539
−0.181 −0.258
−85.761 −0.120
0.013 −0.026 5.251 −4.328E+00 1.162 0.317 3.735E+00 117.927 0.000
−0.324 −0.383
4.437
5.454
−3.571
−3.381
0.989 0.324 3.755 123.900 0.000
3.785
3.489
−2.484E+04 453.635
2.215
−54.683 2.993E+03 1.120E+09
−33.697 1.564E+03 3.050E+08
0.000
0.000
5.834
−1.862E+05 3.471E+03 −51.878 2.765E+03 9.550E+08
1.073 0.403 3.633 131.321 0.000
0.000
methods proposed in this paper are simple and straightforward to implement in practice. Moreover, the structural and statistical properties of the Threshold Autoregressive (TAR) model and ARMAX model can be applied directly to the two new asymmetric models that are proposed in this paper. The empirical performance of the models will be examined using tick-by-tick data for eight Australian shares that are traded on the Australia Stock Exchange (ASX). The paper is organised as follows. Section 2 discusses the LogACD model and the distributions that are most frequently used for obtaining the QMLE. A novel method of estimation is also proposed. The statistical properties of QMLE for the Log-ACD model with log-normal density will be analysed in details. Section 3 provides the Monte Carlo experiments and numerical results for the finite sample properties. Section 4 presents two new models for accommodating asymmetric effects. The data are discussed in Section 5. Empirical examples and estimates are given in Section 6. Finally, Section 7 contains some concluding comments.
4.825
−3.478 1.042 0.324 3.635 102.968 0.000
where xi is the duration and εi is the independently and identically distributed (iid) innovation. The connection between the structures of the ACD and GARCH models is obvious. √ Considering yi = xi (which always holds as duration is always positive), then yi essentially follows a GARCH (p,q) process. As xi is positive for all i, it is natural to assume that ψi and εi are both positive. Mathematically, εi can follow any distribution F (x) with probability P (x < 0) = 0. A sufficient condition to ensure the positivity of ψi is ω > 0, αi > 0 ∀i = 1, . . . , p and βi ≥ 0 ∀i = 1, . . . , q. This condition is identical to that of the GARCH model for ensuring that the conditional variance is positive. However, this sufficient condition may be too restrictive in some cases. Bauwens and Giot (2000) resolved this issue by proposing the Logarithmic ACD (Log-ACD) model as follows: xi = exp(ϕi )εi
ϕi = ω +
p X
2. Model specifications and theoretical results
αj log xi−j +
q X
j =1
βj ϕi−j .
(2)
j=1
Note that E (εi ) = ν > 0, so that
2.1. ACD model Engle and Russell (1994) proposed the Autoregressive Conditional Duration (ACD) model as follows: x i = ψ i εi ,
εi ∼ iid p q X X ψi = ω + αj xi−j + βj ψi−j , j=1
−103.182
j=1
(1)
E (xi | Ii−1 ) = ν exp(ϕi ), where Ii is the information set available up to the ith price change. Let exp(φi ) = ν exp(ϕi ), with
φi = $ +
p X j =1
αj log xi−j +
q X j =1
βj φi−j ,
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 2c True Distribution: Lognormal; Sample Size: 3000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.203 0.203 0.260 0.158 0.014 0.058 2.973
Weibull
G. Gamma
0.201 0.201 0.262 0.145 0.016 0.070 3.073
Exponential
0.189 0.198 0.279 −0.005 0.048 −2.971 11.900
Normal
0.200 0.199 0.269 0.132 0.019 0.085 3.197
0.203 0.203 0.260 0.158 0.014 0.058 2.973
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
0.189 0.178 3.546 −3.234 1.010 −0.014 2.953 0.381 0.826
0.029 0.063 3.288 −4.130 1.057 −0.139 3.131 11.858 0.003
−96.143 −0.058
−0.062 −0.014
0.092 0.094 3.338 −3.551 1.008 −0.078 2.974 3.135 0.209
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.691 0.691 0.770 0.587 0.025 −0.150 3.134
0.695 0.696 0.792 0.567 0.027 −0.160 3.238
0.603 0.696 0.807 −1.004 0.390 −3.834 15.818
0.697 0.698 0.811 0.550 0.032 −0.212 3.365
0.691 0.691 0.770 0.587 0.025 −0.150 3.134
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.331 −0.367
−0.129 −0.171
−114.526 −0.080
−0.025 −0.018
−0.207 −0.256
3.565
4.636
−3.992
−4.136
1.004 0.108 3.043 6.067 0.048
3.780
φi = $ +
j =1
βj φi−j ,
1.051 0.274 3.381 55.757 0.000
3.954
−3.795
1.297
1.004 0.198 3.161 22.945 0.000
−6.329 178.028 3.849E+06 0.000
1. Lognormal distribution:
(3)
1
√
xs 2π
exp −
1
log(x) − m
2
2 !
s
.
(6)
2. Weibull distribution:
j =1
ε
T X
6.207
−34.982
0.000
where ηi = νi . Eq. (3) is more convenient for purposes of obtaining the QMLE as E (ηi ) = 1 and hence avoids potential identification problems. The parameters in model (3) can be estimated by the Maximum Likelihood method. Let l(θ ) be the log-likelihood function with parameter vector λ = (θ 0 , s0 )0 so that l(λ) =
6.687 1.785E+03 0.000
4.425
f1 (x) =
αj log xi−j +
1.133
−0.415
−1.073E+05 2.461E+03 −35.410 1.395E+03 2.430E+08
xi = exp(φi )ηi q X
−11.657
0.000
where $ = ω + log ν . Then E (xi | Ii−1 ) = exp(φi ) and Eq. (2) can be rewritten as p X
3.875
−2.562E+05 4.691E+03 −54.278 2.962E+03 1.100E+09
f2 (x) =
g
x−m
s
g −1
s
exp −
x−m s
g
.
(7)
3. Generalised Gamma distribution: f3 (x) =
g
0 (k)s
x kg −1 s
x g
exp −
s
.
(8)
4. Exponential distribution: log f (xi , λ),
(4)
f4 (x) = s exp(−sx)
(9)
i=1
where θ = (ω, α1 , . . . , αp , β1 , . . . , βq )0 and s = (s1 , . . . , sk )0 denotes the additional parameters required by the density function, f (xi , λ). Then the Maximum Likelihood Estimator (MLE), λˆ , of λ is given by
λˆ = arg max l(λ). λ∈Λ
(5)
However, the functional form of the likelihood function depends on the distribution of ηi . Mathematically, ηi can follow any distribution function, F (x), such that P (x < 0) = 0. Some of the most popular choices for the distribution of ηi and their density functions are as follows:
where m denotes the location parameter, s denotes the scale parameter, g and k denote the additional scale parameters, and 0 (k) is the gamma function, such that
0 (k) =
∞
Z
sk−1 exp(−s)ds. 0
The log-likelihood function in each case is given by Eq. (4) with f (xi , λ) replaced by the appropriate density function. In practice, the true distribution of ηi is seldom known, such ˆ , as defined in Eq. (5), will be the Quasi-MLE (QMLE) rather thatλ than the MLE. Engle and Russell (1997) suggested using the ˆ , Bollerslev and Wooldridge (1992) robust covariance matrix, H (λ)
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
167
Table 3a True Distribution: Exponential; Sample Size: 500; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Weibull
0.199 0.198 0.323 0.053 0.036 0.106 3.042
G. Gamma
0.201 0.200 0.312 0.101 0.029 0.079 3.129
Exponential
0.199 0.190 0.755 −0.159 0.101 0.588 5.174
Normal
0.199 0.198 0.308 0.094 0.028 0.105 3.172
0.199 0.198 0.323 0.053 0.036 0.106 3.042
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.098 −0.067
−0.027 −0.004
−0.508 −0.117
−0.085 −0.067
−0.096 −0.035
2.906
3.437
−5.763
−5.518
1.053
1.087
−0.390
−0.261
3.769 148.485 0.000
8.159
−86.026
4.257
3.135
−5.317
−6.407
2.569
−13.918
3.566 73.285 0.000
1.101
1.076
−0.151
−0.464
424.718 2.210E+07 0.000
3.689 70.018 0.000
3.993 228.610 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.686 0.691 0.899 0.387 0.065 −0.523 3.769
0.687 0.690 0.852 0.442 0.052 −0.411 3.554
0.504 0.671 1.016 −1.026 0.476 −2.132 6.543
0.692 0.695 0.859 0.470 0.051 −0.415 3.533
0.686 0.691 0.899 0.387 0.065 −0.523 3.769
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.086 −0.157
−0.153 −0.213
−9.452 −0.218
−0.065 −0.107
−0.097 −0.188
5.511
5.574
−4.221
−3.534
1.062 0.483 4.286 320.108 0.000
21.029
−5.908E+03
1.068 0.354 3.774 136.201 0.000
∂ 2l ∂λ∂λ0
−1
∂l ∂λ
∂l ∂λ
0
∂ 2l ∂λ∂λ0
−10
.
128.046
log xi = w ˜ +
(11)
log xi = φi + µi where µi = log ηi . Under the assumption that ηi follows a log-normal distribution, this implies that µi follows a normal distribution. Therefore, the log-likelihood function for (11) can be written as T 1X
2 i=1
log s2 +
log xi − φi s
2
.
(12)
Eq. (11) models the logarithm of the duration rather than duration itself. The advantage of this approach is that log xi can now be rewritten as an ARMA(r , r ) process, where r = max(p, q), as shown in the following proposition:
1.077 0.613 4.809 590.483 0.000
Proposition 1. Assuming the random variable, xi , follows the stochastic process as defined in (3), then log xi can be represented as a ARMA(r , r ) process where r = max(p, q), that is
(10)
λ=λˆ
xi = exp(φi )ηi
l(θ) = −
1.099 0.241 3.774 102.975 0.000
0.000
Although several papers have attempted to derive the moment conditions for the ACD and Log-ACD models (see, for example, Bauwens et al. (2003)), the statistical properties of the QMLE for the Log-ACD model is still unknown (see Ghysels and Jasiak (1997) and Feng et al. (2004)). Another interesting feature of the Log-ACD model is the possibility of linearising the process. Note that log xi = φi + log ηi
6.928
−4.065
−37.019 1.599E+03 3.160E+08
instead of the asymptotic covariance matrix, to obtain the variance ˆ , where for λ
ˆ =E H (λ)
5.373
−4.195
r X
δj log xi−j +
j =1
r X
θj ξi−j + ξi
where ξi ∼ iid(0, σξ2 ) and w ˜ =w+ Proof. See Appendix.
(13)
j =1
Pr
j =1
θj E (µi ) + E (µi ).
Remark 1. It is straightforward to show that the conditional likelihood for Eq. (13) is equivalent to that of (12), so that the existing structural and statistical properties for the ARMA (p, q) model can be applied directly to the model given in (11). The statistical properties if QMLE under log-normal density will be presented in the next sub-section. The finite sample performance of this estimation method will be analysed in the Section 3. 2.2. Theoretical results This section establishes the statistical properties, namely, consistency and asymptotic normality of QMLE with the lognormal density for the Log-ACD model as defined in Eq. (3). The moment structure of the Log-ACD model has been established in Bauwens et al. (2003) Consider the Log-ACD model as defined in Eq. (3). The associated log-likelihood function with the log-normal density is
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 3b True Distribution: Exponential; Sample Size: 1000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Weibull
0.200 0.200 0.296 0.115 0.025 0.027 2.997
G. Gamma
0.200 0.200 0.278 0.126 0.020 0.059 3.166
Exponential
0.198 0.196 0.648 −0.124 0.079 0.451 5.335
0.199 0.199 0.270 0.128 0.019 0.087 3.195
Normal 0.200 0.200 0.296 0.115 0.025 0.027 2.997
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.056
−0.025 0.009 3.682 −4.536 1.040 −0.157 3.303 23.731 0.000
−0.077 −0.055
−0.054
0.003 3.336 −5.457 1.014 −0.260 3.395 52.906 0.000
−0.949 −0.060 3.411 −1.304E+03 24.009 −53.585 2.908E+03 1.050E+09 0.000
3.771 −5.153 1.050 −0.109 3.565 45.539 0.000
0.018 3.272 −5.452 1.026 −0.289 3.411 62.669 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.693 0.694 0.837 0.514 0.043 −0.243 3.258
0.694 0.695 0.824 0.555 0.035 −0.248 3.375
0.585 0.685 0.982 −1.010 0.387 −3.201 12.708
0.696 0.697 0.818 0.560 0.034 −0.253 3.362
0.693 0.694 0.837 0.514 0.043 −0.243 3.258
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.069 −0.137
−0.105 −0.148
−28.731 −0.153
−0.044 −0.075
−0.079 −0.148
6.171
4.679
−3.811
−3.325
1.033 0.405 3.934 190.266 0.000
24.573
−6.122E+04
1.048 0.278 3.549 75.940 0.000
1124.960
l(λ) =
T X
6.133
−3.432
1.075 0.225 3.940 135.069 0.000
−53.946 2.934E+03 1.070E+09 0.000
defined as
5.537
−4.289
1.034 0.424 3.912 192.964 0.000
ensure the existence of the first two moments of log xi which is required by the log-normal density. li (λ)
i=1
li (λ) = log f (xi , λ) = −
− log xi −
1
log 2π −
1
2 2 1 (log xi − φi )2
log s2
(14)
s2
2
where α = (α1 , . . . , αp ) , β = (β1 , . . . , βq )0 , θ = (ω, α 0 , β 0 )0 and λ = (s2 , θ 0 )0 and α0 , β0 , θ0 , λ0 denotes the true parameters of α, β, θ , λ, respectively. The theoretical results are derived based on the following assumptions: 0
1. Λ is an open subset of the Euclidean space, Rp+q+1 , such that λ0 ∈ Λ. 2. xi is stationary and ergodic. Pp i 3. α(L) and β(L) are left co-prime, where α(L) = i=1 αi L and Pq Pp Pq i β(L) = i=1 βi L . Moreover, j=1 α0j + j=1 β0j < 1. 4. E (log ηi ) = 0 E [(log ηi )2 ] = s20 ∀i = 1, . . . , T .
Assumption 2 can be replaced by any conditions that ensure the stationarity and ergodicity of the duration process. The conditions of the existence of moment can be found in Bauwens et al. (2003). Assumption 3 ensures the invertibility of the process and identification of the parameters. Moreover, Assumption 3 is likely to be a sufficient condition for Assumption 2. If this is the case, then Assumption 2 can clearly be removed. Assumption 4 is needed to
Proposition 2. Under Assumptions (1)–(4), the Quasi Maximum Likelihood Estimate (QMLE) as defined in (14) for the Log-ACD model ˆ →p λ 0 . as defined in Eq. (3) is consistent, λ Proof. See Appendix.
Proposition 3. Under Assumptions (1)–(4), the Quasi Maximum Likelihood Estimate (QMLE) as defined in (14) for the √ Log-ACD ˆ − model as defined in Eq. (3) is asymptotically normal, T (λ λ0 ) →d N (0, A(λ0 )−1 B(λ0 )A(λ0 )−1 ) where
∂ 2 l and T →∞ ∂λ∂λ0 λ=λ0 ∂ l ∂ l −1 . B(λ0 ) = lim T E T →∞ ∂λ ∂λ0 A(λ0 ) = lim T −1 E
λ=λ0
Proof. See Appendix.
Remark 2. Notice the results in Propositions 2 and 3 apply explicitly to the Log-ACD model as defined in Eq. (3) where the standardised residual, ηi , has a unit mean. It is unclear whether these results hold for the model as defined in Eq. (2) where the standardised residuals, εi , may have mean different from unity. This should not pose a serious problem in practice, as the estimation is typically conducted for Log-ACD model as specified in Eq. (3) rather than the alternate specification as defined in Eq. (2).
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
169
Table 3c True Distribution: Exponential; Sample Size: 3000; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal
Weibull
G. Gamma
0.199 0.200 0.254 0.152 0.014 0.029 3.138
0.200 0.200 0.242 0.158 0.011 −0.006 3.159
0.198 0.196 0.489 −0.047 0.051 −0.020 6.906
Exponential 0.200 0.200 0.238 0.158 0.011 0.015 3.086
Normal 0.199 0.200 0.254 0.152 0.014 0.029 3.138
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.067 −0.028
−0.022
−0.037 −0.020
−0.060 −0.021
3.195 −3.864 0.982 −0.162 3.224 19.345 0.000
0.021 3.299 −4.201 1.021 −0.101 3.169 8.633 0.013
−0.402 −0.102
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.699 0.699 0.782 0.595 0.024 −0.126 3.220
0.698 0.698 0.766 0.621 0.020 −0.117 3.139
0.664 0.699 0.982 −1.002 0.229 −6.039 42.663
0.699 0.699 0.772 0.631 0.019 −0.103 3.120
0.699 0.699 0.782 0.595 0.024 −0.126 3.220
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.002 −0.041
−0.051 −0.078
−4.224 −0.002
−0.031 −0.057
−0.010 −0.048
4.017
3.957
−3.493
−3.684
0.983 0.229 3.216 32.093 0.000
1.010 0.138 3.078 10.294 0.006
3. Finite sample properties This section provides Monte Carlo evidence about the finite sample properties of the MLE and the QMLE, as defined in (5). The Data Generating Process (DGP) for each of the realisations is given by: xi = exp(φi )ηi
φi = 0.01 + 0.2 log xi + 0.7φi−1 . The steps for the Monte Carlo analysis in this section are as follows: 1. For each of the distributions as defined in Eqs. (6)–(9), the DGP defined above will be used to generate realisations with sample sizes of 500, 1000 and 3000. 2. The parameters of the Log-ACD models will be estimated from the realisations generated in step 1 above by maximising the log-likelihood functions, as defined in Eq. (5), based on the distributions defined in Eqs. (6)–(9) and the log-likelihood function defined in Eq. (4). 3. Repeat Steps 1–2 above 3000 times, so that there are 3000 replications. The first set of results given in Tables 2a–5c feature an analysis of the properties of the QMLE as applied to the Log-ACD model, based on a variety of probability distributions, namely Weibull, exponential, generalised gamma and Log-normal, all of which will produce the required positive standardised residuals. The results of the Monte Carlo experiments shown in the four sections of Tables
2.901
−56.303 2.748
−10.488 163.445 3.270E+06 0.000
30.755
−1.962E+03 62.552
−23.690 660.382 5.425E+07 0.000
3.486
3.218
−4.383
−3.875
1.034
0.997
−0.012
−0.162
3.206 5.357 0.069
3.183 17.272 0.000
4.255
4.051
−3.767
−3.574
1.040 0.091 3.265 12.950 0.002
0.995 0.223 3.187 29.172 0.000
2a–5c simulate the finite sample properties on the basis of samples of sizes 500, 1000, and 3000, each with 3000 simulations. Overall, both MLE and QMLE are close to their true values, even in relatively small samples. It is interesting to note that the (Q)MLE based on the log-normal density, as defined in Eq. (6), is identical to those obtained by maximising Eq. (12). This should not be surprising as Eq. (12) is a monotonic transformation of the likelihood function based on the log-normal density, as defined in Eq. (6). Although the transformation does not affect the estimates, it does affect the standard deviation of the estimates, as can be seen in Tables 2a–5c. The (Q)MLE produce seemingly unbiased estimates of the parameters. Regardless of the true underlying distribution, the (Q)MLE, by assuming the log-normal, Weibull and normal densities, seem to be robust and asymptotically normal, which is supported by the skewness and kurtosis of the t-ratios for the estimates. As the sample size increases, the skewness and kurtosis of the (Q)MLE under the log-normal, Weibull and normal densities, tend towards 0 and 3, respectively. In addition, as the sample size increases, the Jarque-Bera Lagrange multiplier statistics also generally lead to non-rejection of the null hypothesis of normality. Moreover, the convergence rates seem to be faster for MLE than for QMLE in these cases, which is not surprising as MLE should be more efficient than QMLE. However, the finite sample properties of QMLE under the assumption of the generalised gamma and exponential distributions are less than desirable in some cases. The problem with the generalised gamma distribution is partly due to the difficulties in obtaining robust and accurate numerical derivatives of the likelihood functions for purposes of maximisation. This could be improved by
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 4a True Distribution: Weibull; Sample Size: 500; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal 0.200 0.199 0.346 0.062 0.036 0.098 3.273
Weibull 0.201 0.200 0.313 0.099 0.028 0.068 3.188
G. Gamma 0.199 0.192 0.677 −0.115 0.097 0.358 4.255
Exponential 0.199 0.198 0.318 0.090 0.028 0.066 3.192
Normal 0.200 0.199 0.346 0.062 0.036 0.098 3.273
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.073 −0.033
−0.019
−0.084 −0.066
−0.072 −0.031
3.204 −4.960 1.057 −0.409 3.867 175.721 0.000
0.010 3.505 −4.821 1.055 −0.209 3.290 31.991 0.000
−0.464 −0.070
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.684 0.689 0.910 0.340 0.066 −0.561 4.166
0.687 0.689 0.860 0.442 0.052 −0.420 3.720
0.494 0.665 1.017 −1.026 0.485 −2.112 6.367
0.692 0.695 0.856 0.477 0.050 −0.385 3.589
0.684 0.689 0.910 0.340 0.066 −0.561 4.166
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.113 −0.186
−0.166 −0.228
−15.061 −0.255
−0.071 −0.105
−0.124 −0.202
5.338
5.053
−3.655
−4.089
1.057 0.553 4.232 338.632 0.000
1.048 0.339 3.693 116.267 0.000
specifying the analytical derivatives of the likelihood function in the optimisation routine. The problem with the exponential distribution is more basic. In many cases, the exponential distribution simply does not have the flexibility to approximate the true underlying distribution which leads to poor finite sample properties, even though the QMLE with exponential density may be asymptotically normal. These Monte Carlo results suggest that the choice of density to determine the likelihood function is important. The density should be sufficiently flexible to provide a good approximation to a wide range of distributions, but also sufficiently accurate so that it does not induce unnecessary numerical difficulties. 4. Asymmetric Log-ACD model The Log-ACD model, as defined in Eq. (3), assumes that the conditional duration is affected by the previous duration but not by the direction of the previous price change. In other words, the model assumes that a positive price change has the same impact on the duration for the next price movement as does a negative price change. It is important to note that the movements in both bid and ask prices contain important information regarding the overall performance of the stock. Thus, it would not seem to be reasonable to assume that the frequency of price changes is unaffected by the direction of previous price changes. There are a wide variety of models available which extend the basic ACD model in various ways. We propose a couple of straightforward methods in this paper, providing two different versions of asymmetric ACD models. Bauwens and Giot in their (2006) survey refer to ‘first generation ACD’ models. These
4.638
−35.321
4.459
3.262
−4.277
−5.119
2.069
1.063
1.066
−3.820
−0.156
−0.385
40.970 1.855E+05 0.000
54.451
−2.324E+04 431.422
−52.679 2.833E+03 9.910E+08 0.000
3.471 39.332 0.000
3.650 125.741 0.000
5.047
4.977
−3.978
−3.819
1.077 0.290 3.829 126.670 0.000
1.059 0.553 4.112 303.965 0.000
begin with the linear ACD model of Engle and Russell (1998) and the logarithmic ACD model of Bauwens and Giot (2000), plus further extensions into augmented ACD classes by Fernandes and Grammig (2006), plus Hautsch’s (2006) semi-parametric model. As mentioned previously, all these models have their GARCH counterparts. The ‘second generation’ ACD models consist of regime-switching ACD models and mixture models; and the mixture can involve both the error distribution and the dynamic component. Zhang et al. (2001) develop a threshold ACD model (TACD) whilst Meitz and TerÄasvirta (2006) develop a class of smooth transition ACD models (STACD) which generalise the linear and logarithmic models in a particular way. Our two extensions belong to the first generation of models and the models are similar in spirit to that of Bauwens et al. (2003). They model the direction of the price change between consecutive trades by means of a competing risks model; in which the direction of the price change is triggered by a Bernoulli process, leading to a form of asymmetric ACD model. The basic linear ACD models only capture the duration between market events and do not include information given by the price process, which is of great importance in the context of market microstructure issues and the process of information impounding. The combination of information given by the price process and the duration between market events is an important extension. This was first addressed by Engle (2000) who suggested an ACD–GARCH model. The durations were captured by a marginal ACD model, and the volatility of returns was modelled by a GARCH process, conditionally on the duration. Russell and Engle (2005) combine an ACD model to capture durations and a generalised linear model on
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
171
Table 4b True Distribution: Weibull; Sample Size: 1000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.199 0.199 0.300 0.123 0.024 0.156 3.287
Weibull
G. Gamma
0.200 0.200 0.276 0.126 0.019 0.077 3.119
0.198 0.194 0.729 −0.095 0.077 0.486 6.375
Exponential 0.199 0.199 0.268 0.127 0.019 0.099 3.031
Normal 0.199 0.199 0.300 0.123 0.024 0.156 3.287
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.065 −0.046
−0.026 −0.006
−0.473 −0.106
−0.057 −0.035
−0.068 −0.039
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.693 0.694 0.835 0.447 0.044 −0.391 3.835
0.694 0.694 0.798 0.537 0.034 −0.206 3.254
0.583 0.683 0.990 −1.016 0.382 −3.140 12.415
0.696 0.697 0.797 0.565 0.033 −0.218 3.122
0.693 0.694 0.835 0.447 0.044 −0.391 3.835
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.075 −0.128
−0.107 −0.185
−7.711 −0.180
−0.054 −0.096
−0.079 −0.147
3.470
3.745
−4.220
−3.767
7.347
−54.412
1.004
1.016
2.512
1.032
1.010
−0.122
−7.789
−0.019
−0.220
3.495 45.215 0.000
3.169 10.916 0.004
5.540
4.246
−3.830
−3.705
1.021 0.362 3.905 167.001 0.000
120.361 1.743E+06 0.000
27.102
−7.481E+03
1.026 0.296 3.274 52.889 0.000
144.215
−47.162 2.422E+03 7.290E+08 0.000
4.1. Model 1: Asymmetric Log-ACD using an indicator function (ALACDI) The first approach to accommodate any asymmetric effects is similar to that of the Glosten et al. (1992) GJR model for capturing asymmetric effects in conditional volatility models. Let Di be the indicator function, such that Di =
0, 1,
1pi ≥ 0 1pi < 0
3.366
−4.204
−0.172
conditional probabilities of the price process in their autoregressive conditional multinomial (ACM) model. For this reason, two asymmetric Log-ACD models are proposed to capture the asymmetric properties of the conditional duration. Model 1 uses an indicator function to capture asymmetric effects in a similar manner to that of the Glosten et al. (1992) GJR model for capturing asymmetric effects in models of conditional volatility. Model 2 accommodates asymmetric effects by using dummy variables. Although both models are intended to capture asymmetric effects, the interpretation of the two models is quite different. Thus, they should be viewed as complementary rather than competing models. In addition, it is important to note that Models 1 and 2 can be rewritten as Threshold Autoregressive Moving Average (TARMA) and ARMAX models, respectively. Hence, the structural and statistical properties of the proposed models can be easily established using existing theoretical results, which will facilitate the testing of various hypotheses regarding market microstructure.
4.277
−4.065
(15)
3.334 14.056 0.001
3.479 52.571 0.000
4.346
5.468
−4.063
−3.477
1.049 0.177 3.402 35.659 0.000
1.023 0.437 3.961 209.904 0.000
where 1pi = pi − pi−1 and pi denotes the price level at the ith significant price change. The Asymmetric Log-ACD model can be defined as
φi = $ +
r s X X (αj + γj Di−j ) log xi−j + βj φi−j . j =1
(16)
j =1
Note that if γj = 0, ∀j, then Eq. (16) reduces to the Log-ACD model, as defined in Eq. (3). Considering the special case, r = s = 1, the short run persistence of the conditional duration is α1 if the previous price change is positive, while the short run persistence is α1 + γ1 if the previous price change is negative. Using a similar argument as presented in the previous section, Eq. (16) can be rewritten as a Threshold Autoregressive Moving Average (TARMA) model. However, unlike the standard TAR model in the non-linear time series literature, where the threshold value is a parameter to be estimated, the threshold value in this case is fixed at 0. More importantly, the structural and statistical properties developed for the TAR model can then be applied directly to Eq. (16) (see Tong (1983), Chan and Tong (1986), Hansen (1994) and Caner and Hansen (2001) for further details regarding the structural and statistical properties of TAR models). 4.2. Model 2: Asymmetric Log-ACD model using exogenous variables (ALACDX) The second approach to accommodate asymmetric effects is to augment the original Log-ACD model by including some exogenous
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 4c True Distribution: Weibull; Sample Size: 3000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.200 0.200 0.259 0.150 0.014 −0.027 3.312
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.049 −0.017
−0.025
−0.049 −0.026
−0.042 −0.007
3.571 −3.896 0.973 −0.235 3.511 60.101 0.000
0.002 3.884 −3.649 1.011 −0.115 3.286 16.804 0.000
−0.528 −0.074
βˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.698 0.698 0.777 0.589 0.024 −0.190 3.360
ˆ t (β) Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
Weibull 0.200 0.200 0.247 0.158 0.011 0.005 3.321
j =1
αj log xi−j +
3.388
−260.682 6.350
−31.615 1.178E+03 1.730E+08
Exponential 0.200 0.200 0.243 0.161 0.011 0.037 3.246
Normal 0.200 0.200 0.259 0.150 0.014 −0.027 3.312
4.010
3.465
−3.659
−3.835
1.045
0.983
−0.027
−0.226
0.000
Weibull
G. Gamma
Exponential
Normal
0.698 0.698 0.760 0.608 0.019 −0.141 3.273
0.664 0.698 0.956 −1.002 0.224 −6.160 44.543
0.699 0.699 0.759 0.621 0.019 −0.116 3.211
0.698 0.698 0.777 0.589 0.024 −0.190 3.360
Lognormal
Weibull
G. Gamma
Exponential
Normal
−0.036 −0.072
−0.045 −0.100
−9.444 −0.030
−0.005 −0.041
−0.045 −0.086
3.893
3.412
−3.924
−4.128
0.976 0.199 3.297 30.895 0.000
p X
0.199 0.197 0.497 −0.028 0.051 0.143 7.024
3.420 22.403 0.000
1.007 0.131 3.144 11.147 0.004
variables. In this case, the model can be defined as
φi = $ +
G. Gamma
q X j =1
βj φi−j +
r X j =1
δj Di−j +
s X
λj Xji ,(17)
j =1
where Di is the indicator function, as defined in Eq. (15), and Xji denotes the value of the jth exogenous variable at the ith price change. In our subsequent empirical analysis we adopt bid and ask volume as the exogenous variables. Blume et al. (1994) investigate the informational role of volume traded and show that it is potentially useful for technical analysis. In their model, volume statistics provide information about information precision that cannot be deduced from price statistics alone. Volume may convey additional information to price but the link between information asymmetry and volume traded is not necessarily linear. Kyle (1985) was one of the first to develop a model whereby a single trader, presumed to have a monopoly on information, places orders over time so as to maximize his trading profit before the information becomes common knowledge. Barclay and Warner (1993) find that informed traders concentrate their orders on medium-sized trades. They examined the proportion of a stock’s cumulative price change that occurs in each trade-size category using transactions data for a sample of New York Stock Exchange (NYSE) firms. The stealth trading hypothesis suggests that if privately informed traders concentrate their trades in medium sizes, and if stockprice movements are due mainly to private information revealed through these investors’ trades, then most of a stock’s cumulative price change will take place on medium-size trades. Their findings
15.979
−1.305E+04 263.956
−43.741 2.063E+03 5.310E+08 0.000
3.336 39.691 0.000
3.823
3.835
−4.365
−4.122
1.047 0.117 3.355 22.624 0.000
0.983 0.190 3.216 23.824 0.000
supported the stealth trading hypothesis. This suggests there will not necessarily be a simple relationship between volume and price duration. Note that it is straightforward to show, using similar arguments to those presented in the previous section, that Eq. (17) can be rewritten as an ARMAX model. Hence, the structural and statistical properties developed for the ARMAX model can be applied directly to Eq. (17) without any modification. As mentioned previously, the interpretations of the two models are quite different. Model 1 suggests positive and negative price movements have different effects on the short run persistence of conditional durations. However, Model 2 suggests that the unconditional expectation of duration is different for positive and negative price movements. Thus, these models accommodate two different asymmetric effects on the conditional duration. The empirical performance of the two models will be examined in the next section. 5. Data The two asymmetric models were applied to eight shares listed on the Australian Stock Exchange (ASX) using tick-bytick data for the period 1/7/2003–1/10/2003. The eight shares are Commonwealth Bank of Australia (CBA), BHP Billiton (BHP), QANTAS Airways (QAN), Coles-Myer Limited (CML), Telstra (TLS), Australia and New Zealand Bank (ANZ), Woolworths (WOW) and Woodside Petroleum (WPL). These eight companies cover a wide range of industries and service areas that include mining, energy and retail industries, telecommunications and the banking sector.
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
173
Table 5a True Distribution: Generalised Gamma; Sample Size: 500; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal 0.245 0.244 0.415 0.088 0.036 0.070 3.470
Weibull 0.252 0.250 0.399 0.121 0.038 0.256 3.190
G. Gamma 0.201 0.193 0.716 −0.100 0.078 0.737 4.849
Exponential 0.197 0.197 0.328 0.097 0.031 0.075 3.170
Normal 0.245 0.244 0.415 0.088 0.036 0.070 3.470
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
1.187 1.201 4.751 −2.848 0.951 −0.194 3.403 39.091 0.000
1.461 1.481 4.760 −2.201 1.008 −0.165 3.173 17.281 0.000
−0.292 −0.102
−0.709 −0.086
1.039 1.075 4.660 −3.061 0.967 −0.213 3.375 40.213 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.555 0.566 0.790 −0.120 0.090 −1.248 6.858
0.531 0.549 0.762 −0.132 0.104 −1.103 5.122
0.657 0.689 1.012 −0.911 0.182 −2.427 13.906
0.690 0.694 0.858 −0.045 0.060 −1.698 18.558
0.555 0.566 0.790 −0.120 0.090 −1.248 6.858
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−1.497 −1.483
−1.674 −1.637
−0.426 −0.109
−0.380 −0.063
−1.373 −1.380
1.691
1.690
−4.947
−5.060
0.714
0.739
−0.095
−0.116
3.587 47.601 0.000
3.881 103.801 0.000
The data were provided by our industry research partner SIRCA (Securities Industry Research Centre of the Asia Pacific). SIRCA is a not-for-profit research consortium of 26 universities drawn from Australia and New Zealand and a number of industry partners, including the Australian Stock Exchange (ASX), the Sydney Futures Exchange (SFE), and Reuters. This research draws upon SIRCA’s ASX intra-day data which captures all the transactions occurring on the ASX via the Stock Exchange Automated Trading System (SEATS). The data possess a wealth of information, including the date and time (to the nearest hundredth of a second that the trade took place), price information including details of the bid and ask prices, volumes, order flow (disclosed and undisclosed), value and volumes of trades, broker IDs and order flags. This paper applies the various Log-ACD models discussed above to eight Australian companies, representing five different industries in Australia. Australia and New Zealand Banking Group Limited (ANZ) and Commonwealth Bank of Australia (CBA) are selected to represent the banking industry of Australia. BHP Billiton Limited (BHP) and Woodside Petroleum Limited (WPL) are selected to represent the Mining and Energy Industry of Australia. Coles Myer Ltd (CML) and Woolworths Limited (WOW) are selected to represent the Retail Industry of Australia. Telstra Corporation Limited (TLS) and QANTAS Airways Limited (QAN) are selected to represent the Telecommunications and Transportation Industries for Australia, respectively. The summary statistics and their sample sizes are given in Table 1. The calculation of duration follows Engle and Russell (1998) and the data are further filtered by the cubic spline method, as suggested in Engle and Russell (1998), to remove the time-of-day effects.
4.123
−34.876 1.662
−4.435 72.141 6.072E+05 0.000
3.485
−1.593E+03 29.140
−54.429 2.974E+03 1.100E+09 0.000
19.800
7.352
−975.384
−926.874
18.781
−47.391 2.429E+03 7.370E+08 0.000
17.031
−53.737 2.922E+03 1.070E+09 0.000
1.993
−4.399 0.760 0.052 3.505 33.183 0.000
6. Empirical results Tables 6a–13d contain the parameter estimates of Models (3), (16) and (17) for the eight companies. These estimates are presented in three dimensional graphical format in Figs. 1–8. For all companies, each model was estimated four times with different distributional assumptions, namely the log-normal, Weibull, exponential and normal distributions, as discussed in Section 2. The generalised gamma distribution is omitted due to its computational difficulties, as outlined in Section 3. Additional exogenous variables, namely the bid and ask volumes, are also included in the ALACDX model to examine the impact of traded volumes in the previous price change on the duration of the subsequent price change. In addition, the Ljung-Box Q-statistics and the BDS statistics are calculated using the standardised residuals in each case. The results support serial independence in the standard residuals, which provide evidence to support the consistency of the QMLE. However, the Kolmogorov–Smirnov test provides strong evidence to reject the null hypothesis of the assumed distribution in each case, which indicates that the underlying distribution is unlikely to be the correct distribution in each case. However, the existence of some outliers and extreme observations may be the cause of these test outcomes, and would be an interesting area for further research. 6.1. Banking industry As shown in Tables 6a–6d and 7a–7d, the α and β estimates are positive and significant for both ANZ and CBA for each of the four
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Table 5b True Distribution: Generalised Gamma; Sample Size: 1000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.228 0.228 0.323 0.133 0.024 −0.014 3.124
t (α) ˆ
Lognormal
Weibull
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
1.064 1.099 4.593 −2.583 0.913 −0.224 3.306 36.841 0.000
1.338 1.368 4.364 −3.175 0.939 −0.386 3.526 109.242 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.624 0.627 0.773 0.305 0.050 −0.560 4.622
0.603 0.609 0.779 0.196 0.063 −0.895 5.342
0.682 0.697 0.948 −0.807 0.115 −2.261 18.333
0.695 0.696 0.810 0.539 0.039 −0.208 3.218
0.624 0.627 0.773 0.305 0.050 −0.560 4.622
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−1.322 −1.339
−1.489 −1.507
0.125 −0.042 24.475 −12.072 1.513 2.131 30.083 9.396E+04 0.000
−0.041 −0.078
−1.175 −1.195
Weibull 0.236 0.236 0.329 0.136 0.027 0.148 3.062
2.525
3.291
−4.009
−4.548
0.777 0.296 3.685 102.459 0.000
0.751 0.436 4.538 390.573 0.000
G. Gamma
Exponential
Normal
0.199 0.199 0.282 0.129 0.022 −0.021 3.122
0.228 0.228 0.323 0.133 0.024 −0.014 3.124
G. Gamma
Exponential
Normal
−0.166 −0.038
−0.090 −0.022
0.911 0.939 4.326 −3.000 0.942 −0.231 3.297 37.769 0.000
0.202 0.198 0.630 0.037 0.057 0.826 5.670
4.436
−15.705
8.236
−4.733
1.303
1.081
−1.245
−0.083
11.167 9.113E+03 0.000
4.469E+00 272.988 0.000
8.921
3.171
−3.260
−3.859
1.071 0.512 5.005 633.435 0.000
0.833 0.346 3.751 130.204 0.000
Fig. 1. Estimates for ANZ. Fig. 2. Estimates for CBA.
estimated models. Figs. 1 and 2 present these results in graphical format. These results suggest that past durations are helpful in predicting the duration before the next price change. However, there is no evidence to support asymmetric effects on duration from price changes as both the γ and δ estimates are insignificant for both banks. Interestingly, λ1 and λ2 estimates are both positive and significant in both cases, indicating that the traded bid and ask volumes at the last price change have a positive impact on duration. In addition, the inclusion of the bid and ask volumes also
has a negative news impact on the β estimates. This would suggest that the correct specification of the model is crucial for obtaining valid inferences and diagnostic checks. Moreover, the problem of omitted variables could have important implications for the interpretation of the various models. The long run persistence of past duration to future conditional duration would also appear to be substantially lower when the bid and ask volumes are included
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
175
Table 5c True Distribution: Generalised Gamma; Sample Size: 3000; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal 0.211 0.210 0.269 0.166 0.014 0.117 3.072
Weibull 0.216 0.215 0.280 0.170 0.014 0.173 3.114
G. Gamma 0.200 0.198 0.389 −0.032 0.035 0.480 5.586
Exponential 0.199 0.199 0.257 0.159 0.012 0.105 3.247
Normal 0.211 0.210 0.269 0.166 0.014 0.117 3.072
αˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
0.693 0.696 4.533 −2.432 0.925 −0.024 3.058 0.714 0.700
1.000 1.052 4.689 −2.484 0.911 −0.205 3.361 37.345 0.000
−0.131 −0.087
−1.313 −0.007
0.571 0.563 4.397 −2.606 0.941 −0.001 2.996 0.003 0.999
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.674 0.675 0.745 0.555 0.025 −0.262 3.235
0.663 0.665 0.738 0.515 0.028 −0.448 3.665
0.694 0.701 0.943 −0.969 0.070 −5.598 115.596
0.699 0.700 0.766 0.598 0.021 −0.225 3.412
0.674 0.675 0.745 0.555 0.025 −0.262 3.235
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.910 −0.930
−1.167 −1.220
−1.485
−0.773 −0.790
2.168 −4.276 0.871 0.063 3.170 5.572 0.062
2.211 −4.392 0.808 0.300 3.664 100.241 0.000
0.001 0.007 6.764 −254.489 4.789 −50.028 2.659E+03 8.830E+08 0.000
3.331
−9.468 1.109
−0.615 5.542 996.989 0.000
11.135
−3.706E+03 67.664
−54.724 2.997E+03 1.120E+09 0.000
0.000 4.298 −4.434E+03 80.962 −54.728 2.997E+03 1.120E+09 0.000
2.446
−4.119 0.898 0.058 3.086 2.575 0.276
Fig. 3. Estimates for BHP. Fig. 4. Estimates for WPL.
in the analysis. The figures reveal that the estimates have different values for the two banks and the differences are most pronounced in the asymmetric models that include bid and ask volumes. 6.2. Mining and energy industry Tables 8a–8d and 9a–9d contain the estimates for BHP and WPL, which represent the mining and energy industry of Australia, and
again these results are presented graphically in Figs. 3 and 4. As in the case of the banking industry, the α and β estimates are positive and significant for both BHP and WPL, indicating that past durations contain important information about the future duration of price changes. Although there is no evidence for asymmetric effects on duration from price change, traded bid and ask volumes appear to have significant and positive effects on the conditional duration as λ1 and λ2 are both positive and significant. Again, the
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 6a Estimates of the Log-ACD Model for ANZ. ANZ (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.156 [−8.344]
−0.627 [−9.315]
−0.156 [−8.118]
0.0800 [12.052] 0.695 [22.716]
0.066 [10.742] 0.648 [18.016]
0.265 [0.817] 0.049 [7.038] 0.588 [14.112]
α β
0.08 [12.016] 0.695 [22.271]
Table 6b Estimates of the ALACDX Model for ANZ. ANZ (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.153 [−7.752] 0.080 [11.721] 0.693 [21.731] −0.007 [−0.438]
−0.626 [−9.336]
0.273 [0.812] 0.049 [7.157] 0.584 [9.697] −0.008 [−0.515]
−0.153 [−7.448] 0.080 [11.640] 0.693 [20.961] −0.007 [−0.431]
α β δ
0.067 [11.161] 0.645 [17.714] −0.010 [−0.625]
Fig. 5. Estimates for CML.
Table 6c Estimates of the ALACD Model for ANZ. ANZ (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.156 [−8.357]
−0.628 [−9.372] 0.065 [8.327] 0.647 [18.088] 0.002 [0.207]
0.265 [0.850] 0.049 [6.123] 0.588 [12.388] 0.003 [0.016]
−0.156 [−7.856]
α β γ
0.081 [9.967] 0.695 [22.774] −0.004 [−0.341]
0.081 [9.485] 0.695 [21.385] −0.004 [−0.327]
Table 6d Estimates of the ALACDX Model with Bid and Ask Volumes for ANZ. ANZ (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.617 [−1.446] 0.094 [8.293] 0.514 [2.310] −0.021 [−0.844] 0.015 [1.014] 0.026 [1.358]
−1.902 [−3.600]
3.367 [2.261] 0.051 [5.985] 0.13 [1.493] −0.028 [−1.618] 0.029 [4.502] 0.047 [6.769]
−0.617 [−1.404] 0.094 [8.347] 0.514 [2.192] −0.021 [−0.787] 0.015 [1.012] 0.026 [1.304]
α β δ λ1 λ2
0.076 [8.774] 0.263 [1.449] −0.033 [−1.646] 0.026 [2.316] 0.041 [3.263]
Fig. 6. Estimates for WOW.
Table 7a Estimates of the Log-ACD Model for CBA. CBA (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.161 [−6.715] 0.091 [11.924] 0.671 [16.666]
−0.607 [−7.482] 0.083 [12.278] 0.642 [14.490]
0.712 [1.295] 0.068 [12.416] 0.577 [6.998]
−0.161 [−6.547] 0.091 [11.853] 0.671 [16.308]
α β
Table 7b Estimates of the ALACDX Model for CBA. CBA (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.149 [−5.830]
−0.593 [−7.262]
−0.149 [−5.958]
0.091 [11.704] 0.674 [16.402] −0.019 [−1.242]
0.082 [12.403] 0.645 [14.540] −0.019 [−1.298]
0.729 [3.133] 0.068 [10.144] 0.578 [5.593] −0.016 [−0.998]
α β δ
0.091 [12.015] 0.674 [17.204] −0.019 [−1.226]
Fig. 7. Estimates for QAN.
inclusion of the bid and ask volumes has a negative impact on the β estimates. Moreover, the long run persistence of past duration to future conditional duration would appear to be lower when the bid and ask volumes are included in the analysis.
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
177
Table 8c Estimates of the ALACD Model for BHP. BHP (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.166 −[5.800]
−0.823 −[8.771]
−0.166 −[5.893]
0.073 [8.298] 0.753 [21.662] 0.008 [0.748]
0.073 [8.590] 0.674 [19.204] 0.016 [1.531]
1.41 [2.829] 0.061 [7.684] 0.532 [7.153] 0.026 [2.842]
α β γ
0.073 [8.392] 0.753 [21.985] 0.008 [0.744]
Table 8d Estimates of the ALACDX Model with Bid and Ask Volumes for BHP. BHP (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−2.645 [−11.813]
−3.641 [−9.109]
−2.645 [−12.274]
0.090 [9.672] 0.077 [1.196] −0.048 [−1.546] 0.093 [8.391] 0.093 [8.977]
0.088 [10.595] 0.179 [2.187] −0.043 [−1.860] 0.074 [6.686] 0.075 [7.597]
2.128 [0.050] 0.074 [0.000] 0.275 [0.001] −0.030 [−0.000] 0.057 [0.000] 0.053 [0.000]
α Fig. 8. Estimates for TLS.
β δ
Table 7c Estimates of the ALACD Model for CBA. CBA (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.160 [−6.996] 0.100 [10.451] 0.671 [17.557] −0.018 [−1.600]
−0.600 [−7.830] 0.090 [11.283] 0.646 [15.535] −0.016 [−1.601]
0.663 [1.443] 0.076 [9.434] 0.597 [7.226] −0.019 [−1.583]
−0.160 [−6.915] 0.100 [10.722] 0.671 [17.321] −0.018 [−1.713]
α β γ
CBA (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.569 [−4.545]
−1.091 [−5.735] 0.091 [13.525] 0.533 [8.048] −0.020 [−1.198] 0.018 [3.299] 0.020 [3.250]
0.963 [1.437] 0.076 [13.271] 0.432 [4.248] −0.018 [−1.080] 0.017 [3.700] 0.025 [3.083]
−0.569 [−4.857]
β δ λ1 λ2
0.102 [14.038] 0.560 [8.306] −0.020 [−1.120] 0.023 [3.496] 0.022 [3.343]
0.102 [13.857] 0.560 [9.230] −0.020 [−1.264] 0.023 [3.838] 0.022 [3.409]
Table 8a Estimates of the Log-ACD Model for BHP. BHP (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.164 [−5.868]
−0.821 [−8.250] 0.081 [11.844] 0.675 [18.234]
1.422 [3.200] 0.073 [10.873] 0.530 [9.428]
−0.164 [−6.263]
α β
0.077 [10.254] 0.754 [22.047]
0.077 [10.498] 0.754 [23.163]
BHP (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.149 [−5.830]
−0.593 [−7.262]
0.091 [11.704] 0.674 [16.402] −0.019 [−1.242]
0.082 [12.403] 0.645 [14.540] −0.019 [−1.298]
0.729 [3.133] 0.068 [10.144] 0.578 [5.593] −0.016 [−0.998]
−0.149 [−5.958] 0.091 [12.015] 0.674 [17.204] −0.019 [−1.226]
β δ
Table 9a Estimates of the Log-ACD Model for WPL. WPL (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.331 [−4.594]
−1.242 [−6.871] 0.094 [10.537] 0.493 [7.047]
−2.105 [−1.936] 0.077 [8.031] 0.383 [4.562]
−0.331 [−4.611] 0.095 [8.967] 0.546 [6.444]
β
0.095 [8.711] 0.546 [6.402]
Table 9b Estimates of the ALACDX Model for WPL. WPL (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.338 [−4.656]
−1.252 [−7.016] 0.094 [10.554] 0.493 [7.149] 0.019 [0.787]
−2.097 [−1.581]
−0.338 [−4.793]
0.077 [8.935] 0.384 [3.661] −0.002 [−0.060]
0.095 [8.674] 0.548 [6.569] 0.019 [0.756]
α β δ
0.095 [8.657] 0.548 [6.414] 0.019 [0.713]
Table 9c Estimates of the ALACD Model for WPL. WPL (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.340 [−4.429]
−1.277 [−6.849]
−2.104 [−1.438]
−0.34 [−4.266]
0.102 [6.724] 0.535 [5.893] −0.012 [−0.799]
0.102 [7.700] 0.480 [6.614] −0.013 [−0.873]
0.077 [5.759] 0.383 [3.653] 0.000 [0.002]
0.102 [6.669] 0.535 [5.690] −0.012 [−0.834]
α β γ
Table 8b Estimates of the ALACDX Model for BHP.
α
λ2
α
Table 7d Estimates of the ALACDX Model with Bid and Ask Volumes for CBA.
α
λ1
0.090 [11.751] 0.077 [1.236] −0.048 [−1.606] 0.093 [8.500] 0.093 [9.470]
6.3. Retail industry Tables 10a–10d and 11a–11d contain the estimates for CML and WOW, two of Australia’s largest retail corporations and Figs. 5 and 6 present the same estimates in graphical format. As for the previous results, the α and β estimates are positive and significant for both corporations, and there appears to be an asymmetric effect on the conditional duration from previous price changes as
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 9d Estimates of the ALACDX Model with Bid and Ask Volumes for WPL.
Table 10d Estimates of the ALACDX Model with Bid and Ask Volumes for CML.
WPL (Price)
Log-Normal
Weibull
Exponential
Normal
CML (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.839 [−4.406]
−1.832 [−6.220] 0.099 [11.756] 0.407 [4.688] 0.018 [0.753] 0.015 [2.363] 0.030 [3.633]
−3.917 [−1.284]
−0.839 [−4.133]
ω
−1.409 [−5.335]
0.103 [10.397] 0.425 [3.895] 0.016 [0.588] 0.017 [2.253] 0.032 [3.112]
4.847 [2.804] 0.048 [3.230] 0.039 [0.253] −0.065 [−1.447] 0.052 [3.414] 0.046 [4.643]
−1.409 [−5.238]
0.08 [9.491] 0.29 [2.222] 0.001 [0.027] 0.013 [2.271] 0.028 [3.199]
−2.82 [−6.192] 0.085 [6.715] 0.188 [1.508] −0.079 [−2.125] 0.053 [3.614] 0.051 [4.638]
α β δ λ1 λ2
0.103 [10.550] 0.425 [3.997] 0.016 [0.528] 0.017 [2.311] 0.032 [3.265]
Table 10a Estimates of the Log-ACD Model for CML.
α β δ λ1 λ2
0.113 [8.444] 0.262 [2.566] −0.098 [−2.295] 0.049 [3.530] 0.052 [4.026]
0.113 [7.866] 0.262 [2.400] −0.098 [−2.329] 0.049 [3.526] 0.052 [4.238]
Table 11a Estimates of the Log-ACD Model for WOW.
CML (ACD)
Log-Normal
Weibull
Exponential
Normal
WOW (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.374 [−5.911]
−1.342 [−7.223]
−0.374 [−5.842]
ω
−0.935 [−8.970]
−1.115 [−0.008]
−0.247 [−7.166]
0.082 [6.634] 0.431 [5.786]
0.105 [7.428] 0.471 [6.132]
α
−0.247 [−7.114]
0.105 [7.424] 0.471 [6.117]
0.361 [1.058] 0.050 [4.203] 0.348 [4.534]
0.119 [12.298] 0.600 [13.194]
0.114 [12.823] 0.574 [13.074]
0.100 [0.082] 0.451 [0.055]
0.119 [12.496] 0.600 [13.400]
α β
Table 10b Estimates of the ALACDX Model for CML.
β
Table 11b Estimates of the ALACDX Model for WOW.
CML (Price)
Log-Normal
Weibull
Exponential
Normal
WOW (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.348 [−5.989]
−1.348 [−7.161]
−0.348 [−5.881]
ω
−0.91 [−8.772]
−1.224 [−1.262]
−0.222 [−6.322]
0.083 [6.995] 0.416 [5.423] −0.062 [−1.728]
0.106 [7.717] 0.455 [6.332] −0.083 [−2.145]
α
−0.222 [−6.381]
0.106 [8.253] 0.455 [6.231] −0.083 [−2.064]
0.475 [0.961] 0.051 [4.754] 0.317 [3.601] −0.038 [−1.150]
0.121 [12.571] 0.592 [12.939] −0.061 [−2.408]
0.115 [12.984] 0.568 [12.860] −0.076 [−3.249]
0.101 [8.618] 0.441 [5.167] −0.09 [−4.037]
0.121 [12.619] 0.592 [12.837] −0.061 [−2.375]
α β δ
Table 10c Estimates of the ALACD Model for CML.
β δ
Table 11c Estimates of the ALACD Model for WOW.
CML (AACD)
Log-Normal
Weibull
Exponential
Normal
WOW (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.373 [−5.567]
−1.358 [−7.300]
−0.373 [−5.704]
ω
0.105 [6.318] 0.472 [5.779] −0.002 [−0.061]
0.079 [5.093] 0.424 [5.658] 0.008 [0.352]
0.406 [1.201] 0.041 [2.606] 0.325 [3.798] 0.021 [0.963]
0.105 [6.350] 0.472 [5.953] −0.002 [−0.062]
α
−0.249 [−6.455] 0.117 [10.440] 0.597 [11.925] 0.006 [0.446]
−0.952 [−8.706]
−1.258 [−1.184]
0.107 [10.171] 0.567 [12.347] 0.017 [1.165]
0.086 [5.383] 0.426 [3.722] 0.033 [1.589]
−0.249 [−6.990] 0.117 [10.673] 0.597 [12.900] 0.006 [0.449]
α β γ
the δ estimates are negative and significant in all cases except one, namely the δ estimate is not significant for CML under the exponential distribution. This would suggest that the presence of asymmetric effects would be industry dependent. In addition, the coefficients of the bid and ask volumes continue to be positive and significant. Again the Figures reveal that the large differences in estimates occur in the context of the asymmetric model after the inclusion of volume statistics. 6.4. Transport and telecommunication industries
β γ
Table 11d Estimates of the ALACDX model with bid and ask volumes for WOW. WOW (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.703 [−4.799]
−1.387 [−7.047]
−2.228 [−6.627]
0.129 [13.924] 0.512 [8.622] −0.062 [−2.368] 0.019 [2.597] 0.029 [3.367]
0.12 [14.515] 0.516 [9.824] −0.075 [−3.141] 0.016 [2.260] 0.025 [3.481]
0.104 [8.917] 0.391 [3.627] −0.084 [−3.010] 0.019 [2.334] 0.031 [3.497]
−0.703 [−4.953] 0.129 [14.204] 0.512 [8.946] −0.062 [−2.334] 0.019 [2.704] 0.029 [3.472]
α β δ λ1
Tables 12a–12d contain the parameter estimates for QAN and Tables 13a–13d contain the parameter estimates for TLS, which represent the air and telecommunications industries for Australia, respectively. These results are presented graphically in Figs. 7 and 8. Interestingly, both δ and γ estimates are significant, but with opposite signs. This would suggest that a negative price shock has lower short run persistence on the conditional duration but has a positive impact on future conditional duration. In other words,
λ2
a negative price change will lead to a longer duration before the next price change, but the impact will not last as long as a positive price change. Again, bid and ask volumes play important roles in determining future duration as λ1 and λ2 are both positive and
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185 Table 12a Estimates of the Log-ACD model for QAN.
179
Table 13b Estimates of the ALACDX model for TLS.
QAN (ACD)
Log-Normal
Weibull
Exponential
Normal
TLS (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.544 [−7.084]
−1.777 [−8.759]
−0.674 [−1.186]
−0.544 [−6.759]
ω
0.111 [6.906] 0.335 [4.255]
0.09 [6.279] 0.34 [4.867]
0.057 [4.748] 0.368 [4.436]
0.111 [6.507] 0.335 [4.058]
α
−0.319 [−3.731]
−1.517 [−6.314]
−0.319 [−3.398]
0.128 [7.013] 0.606 [8.379] −0.058 [−1.047]
0.142 [8.906] 0.553 [8.762] −0.061 [−1.109]
1.670 [2.987] 0.128 [7.332] 0.555 [9.304] −0.033 [−0.620]
α β
β δ
0.128 [6.977] 0.606 [8.006] −0.058 [−0.953]
Table 12b Estimates of the ALACDX model for QAN. QAN (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.644 [−7.688]
−1.822 [−8.699]
−0.838 [−1.794]
−0.644 [−7.584]
0.108 [6.194] 0.347 [4.464] 0.201 [3.411]
0.088 [5.604] 0.354 [4.815] 0.157 [3.328]
0.056 [3.825] 0.379 [4.271] 0.116 [2.445]
0.108 [6.180] 0.347 [4.325] 0.201 [3.744]
α β δ
Table 13c Estimates of the ALACD model for TLS. TLS (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.352 [−4.444]
−1.562 [−6.537]
−0.352 [−4.424]
0.128 [5.980] 0.602 [8.465] 0.003 [0.120]
0.132 [6.110] 0.549 [8.599] 0.022 [0.805]
1.617 [4.053] 0.103 [4.976] 0.555 [9.934] 0.049 [2.034]
α β γ
0.128 [5.791] 0.602 [8.455] 0.003 [0.120]
Table 12c Estimates of the ALACD model for QAN. QAN (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.583 [−6.722]
−1.897 [−8.037]
−0.916 [−1.368]
0.157 [6.235] 0.295 [3.353] −0.083 [−2.515]
0.135 [5.904] 0.295 [3.560] −0.077 [−2.492]
0.096 [4.824] 0.309 [2.872] −0.064 [−2.342]
−0.583 [−6.634] 0.157 [6.321] 0.295 [3.261] −0.083 [−2.528]
α β γ
Table 13d Estimates of the ALACDX model with bid and ask volumes for TLS. TLS (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−2.020 [−5.122]
−2.921 [−7.461]
−2.02 [−5.413]
0.122 [7.025] 0.484 [5.735] 0.006 [0.082] 0.042 [3.108] 0.087 [3.985]
0.128 [8.361] 0.518 [8.353] −0.018 [−0.290] 0.037 [3.116] 0.070 [4.144]
0.796 [1.674] 0.120 [6.777] 0.563 [8.840] −0.022 [−0.402] 0.029 [2.829] 0.031 [2.247]
α β δ λ1
Table 12d Estimates of the ALACDX model with bid and ask volumes for QAN. QAN (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−1.688 [−6.488]
−2.670 [−8.927]
0.100 [5.771] 0.310 [4.074] 0.170 [2.481] 0.046 [2.529] 0.045 [3.011]
0.081 [5.350] 0.333 [4.908] 0.123 [2.251] 0.038 [2.770] 0.034 [2.821]
−1.988 [−2.588] 0.051 [3.826] 0.330 [4.441] 0.083 [1.814] 0.031 [2.893] 0.021 [1.966]
−1.688 [−6.596] 0.100 [5.768] 0.310 [4.350] 0.170 [2.726] 0.046 [2.893] 0.045 [3.039]
α β δ λ1 λ2
Table 13a Estimates of the Log-ACD model for TLS. TLS (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.352 [−4.441]
−1.569 [−8.080]
0.129 [7.612] 0.602 [8.537]
0.144 [9.630] 0.547 [9.946]
1.680 [4.660] 0.129 [9.406] 0.553 [9.621]
−0.352 [−4.451] 0.129 [7.427] 0.602 [8.489]
α β
significant for QAN. However, the results for TLS are qualitatively identical to those of the banking industry. No evidence is found for asymmetric effects on duration from price changes, but λ1 and λ2 are both positive and significant, which indicates the importance of the bid and ask volumes in predicting the duration for the next price change.
λ2
0.122 [7.391] 0.484 [5.826] 0.006 [0.092] 0.042 [3.342] 0.087 [4.246]
6.5. Bootstrapped confidence intervals Section 2 showed that the QMLE based on the log-normal density is consistent and asymptotically normal for the Log-ACD model. It is, however, unclear whether the same results hold when different densities are used to construct the QMLE. Moreover, although the Monte Carlo results presented in Section 3 provided support for the theoretical results, the usefulness of the asymptotic results in the finite sample is still unclear from the empirical perspective. Therefore, this section compares the bootstrapped confidence intervals and the confidence intervals constructed by using the robust standard errors (asymptotic confidence intervals) of the estimates obtained in the empirical section. As reported in the empirical section, the diagnostic tests indicated that the standardised residuals of the Log-ACD model under different distributional assumptions are IID in all cases but the residuals did not follow any of the assumed distributions. Therefore the bootstrapping techniques can be used to provide an accurate approximation to the underlying distribution of QMLE in the finite sample. Thus, comparing the bootstrapped confidence intervals and the asymptotic confidence intervals will provide insight into the applicability of asymptotic normality in the finite sample. The bootstrapped confidence intervals are constructed by using the percentile method as proposed in Efron and Tibshirani (1993). Tables 14a–14d contain the estimates of the Log-ACD model, their bootstrapped confidence intervals and the asymptotic confidence intervals for the eight companies using 4 different
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 14a Confidence intervals of QMLE with log-normal density. ANZ
Estimates
CI-Lower
CI-Upper
BCI-Lower
BCI-Upper
ω α β
−0.156
−0.193
−0.120
−0.181
−0.121
0.080 0.695
0.067 0.635
0.093 0.754
0.068 0.652
CBA
Estimates −0.161 0.091 0.671
CI-Lower −0.208 0.076 0.591
CI-Upper −0.113 0.106 0.751
BCI-Lower −0.192 0.081 0.620
BHP
Estimates −0.164 0.077 0.754
CI-Lower −0.219 0.062 0.687
CI-Upper −0.110 0.091 0.821
BCI-Lower −0.204 0.067 0.709
Estimates −0.331 0.095 0.546
CI-Lower −0.472 0.074 0.379
CI-Upper −0.190 0.117 0.714
BCI-Lower −0.414 0.082 0.453
Estimates −0.374 0.105 0.471
CI-Lower −0.498 0.077 0.320
CI-Upper −0.250 0.132 0.622
BCI-Lower −0.495 0.085 0.317
Estimates −0.247 0.119 0.600
CI-Lower −0.315 0.100 0.511
CI-Upper −0.179 0.138 0.689
BCI-Lower −0.294 0.105 0.539
Estimates −0.352 0.129 0.602
CI-Lower −0.507 0.096 0.464
CI-Upper −0.197 0.163 0.740
BCI-Lower −0.483 0.104 0.485
Estimates −0.544 0.111 0.335
CI-Lower −0.695 0.079 0.181
CI-Upper −0.394 0.142 0.489
BCI-Lower −0.791 0.083 0.112
ω α β ω α β
WPL
ω α β
CML
ω α β
WOW
ω α β
TLS
ω α β
QAN
ω α β
CI Width
BCI Width
0.089 0.753
0.073 0.026 0.120
0.061 0.021 0.102
BCI-Upper −0.133 0.103 0.721
CI Width 0.095 0.031 0.160
BCI Width 0.060 0.022 0.102
Asymmetry −0.004 0.001 −0.001
BCI-Upper
CI Width 0.110 0.029 0.134
BCI Width 0.073 0.020 0.084
−0.007 −0.000 −0.006
CI Width 0.282 0.043 0.334
BCI Width 0.150 0.027 0.179
CI Width 0.248 0.055 0.302
BCI Width 0.212 0.043 0.280
−0.028
CI Width 0.136 0.038 0.178
BCI Width 0.096 0.028 0.126
Asymmetry 0.002 −0.000 0.003
CI Width 0.311 0.067 0.276
BCI Width 0.233 0.054 0.213
−0.020
0.137 0.531
CI Width 0.301 0.063 0.308
BCI Width 0.417 0.054 0.419
−0.076 −0.002 −0.027
CI Width
BCI Width
Asymmetry
0.264 0.024 0.141
0.099 0.026 0.157
CI Width 0.315 0.026 0.172
BCI Width 0.073 0.026 0.124
Asymmetry 0.841 0.004 −0.009
CI Width 0.390 0.027 0.145
BCI Width 0.094 0.058 0.117
Asymmetry 1.169 0.038 −0.007
CI Width 0.708 0.035 0.274
BCI Width 0.176 0.032 0.218
Asymmetry 1.721 −0.000 0.005
CI Width 0.728 0.049 0.292
BCI Width 0.342 0.048 0.412
Asymmetry 1.833 0.001 0.032
CI Width 0.409 0.035 0.172
BCI Width 0.100 0.040 0.136
Asymmetry 1.326 0.011 −0.002
CI Width 0.761 0.058 0.216
BCI Width 0.197 0.279 0.315
Asymmetry 2.347 0.228 −0.137
CI Width 0.795 0.056 0.274
BCI Width 0.485 0.071 0.536
Asymmetry 2.468 0.013 0.041
−0.132 0.087 0.793 BCI-Upper
−0.264 0.109 0.632 BCI-Upper
−0.284 0.128 0.597 BCI-Upper
−0.198 0.133 0.664 BCI-Upper
−0.250 0.158 0.698 BCI-Upper
−0.374
Asymmetry 0.011
−0.003 0.016
Asymmetry
Asymmetry
−0.017 0.001
−0.007 Asymmetry
−0.030 0.003
Asymmetry
−0.029 0.003 Asymmetry
Table 14b Confidence intervals of QMLE with Weibull density. ANZ
Estimates
CI-Lower
CI-Upper
BCI-Lower
BCI-Upper
ω α β
−0.627
−0.759
−0.495
−0.244
−0.145
0.066 0.648
0.054 0.577
0.079 0.718
0.058 0.570
0.083 0.727
CBA
Estimates −0.607 0.082 0.643
CI-Lower −0.764 0.069 0.557
CI-Upper −0.449 0.096 0.729
BCI-Lower −0.223 0.071 0.576
Estimates −0.821 0.081 0.675
CI-Lower −1.016 0.067 0.602
CI-Upper −0.626 0.094 0.748
BCI-Lower −0.283 0.071 0.613
Estimates −1.242 0.094 0.493
CI-Lower −1.596 0.077 0.356
CI-Upper −0.888 0.112 0.631
BCI-Lower −0.469 0.078 0.387
Estimates −1.342 0.082 0.431
CI-Lower −1.706 0.058 0.285
CI-Upper −0.978 0.107 0.577
BCI-Lower −0.596 0.059 0.241
Estimates −0.935 0.114 0.574
CI-Lower −1.140 0.097 0.488
CI-Upper −0.731 0.132 0.660
BCI-Lower −0.322 0.100 0.505
Estimates −1.569 0.144 0.547
CI-Lower −1.949 0.114 0.440
CI-Upper −1.188 0.173 0.655
BCI-Lower −0.494 0.118 0.321
Estimates −1.777 0.090 0.340
CI-Lower −2.175 0.062 0.203
CI-Upper −1.379 0.118 0.477
BCI-Lower −0.786 0.061 0.093
ω α β
BHP
ω α β
WPL
ω α β
CML
ω α β
WOW
ω α β
TLS
ω α β
QAN
ω α β
BCI-Upper
−0.150 0.097 0.700 BCI-Upper
−0.189 0.128 0.730 BCI-Upper
−0.293 0.110 0.605 BCI-Upper
−0.255 0.107 0.653 BCI-Upper
−0.223 0.140 0.641 BCI-Upper
−0.296 0.397 0.637 BCI-Upper
−0.301 0.132 0.628
0.864 0.008 0.002
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
181
Table 14c Confidence intervals of QMLE with exponential density. ANZ
ω α β CBA
ω α β
BHP
ω α β
WPL
Estimates 0.265 0.049 0.588
CI-Lower
−0.370
CI-Upper
BCI-Lower
BCI-Upper
−0.337
−0.172
0.037 0.450
0.062 0.708
0.036 0.506
0.900 0.063 0.669
Estimates 0.710 0.068 0.578
CI-Lower −0.555 0.057 0.425
CI-Upper 1.974 0.079 0.730
BCI-Lower −0.312 0.058 0.466
Estimates 1.422 0.073 0.530
CI-Lower 0.551 0.060 0.420
CI-Upper 2.294 0.086 0.640
BCI-Lower −0.508 0.061 0.400
Estimates
CI-Upper 0.026 0.095 0.548
BCI-Lower −0.603 0.062 0.270
ω α β
−2.105 0.077 0.383
CI-Lower −4.236 0.058 0.219
CML
Estimates 0.361 0.050 0.348
CI-Lower −0.308 0.027 0.198
CI-Upper 1.031 0.074 0.498
BCI-Lower −1.038 0.029 −0.204
Estimates
CI-Upper 24.958 2.469 16.611
BCI-Lower −0.473 0.082 0.359
ω α β
WOW
ω α β
−1.115 0.100 0.451
CI-Lower −27.188 −2.270 −15.709
TLS
Estimates 1.680 0.129 0.553
CI-Lower 0.974 0.102 0.441
CI-Upper 2.387 0.155 0.666
BCI-Lower −0.601 0.105 0.406
Estimates
CI-Lower −1.788 0.033 0.205
CI-Upper 0.440 0.080 0.530
BCI-Lower −1.078 0.023 −0.153
CI-Lower −0.194 0.067 0.633
CI-Upper −0.119 0.093 0.756
BCI-Lower −0.192 0.070 0.638
CI-Lower −0.208 0.076 0.592
CI-Upper −0.113 0.106 0.750
BCI-Lower −0.190 0.080 0.624
CI-Lower −0.216 0.062 0.691
CI-Upper −0.113 0.091 0.818
BCI-Lower −0.203 0.065 0.710
CI-Lower −0.471 0.074 0.380
CI-Upper −0.190 0.116 0.712
BCI-Lower −0.404 0.081 0.459
CI-Lower −0.500 0.077 0.321
CI-Upper −0.249 0.132 0.622
BCI-Lower −0.495 0.085 0.326
CI-Lower −0.314 0.101 0.512
CI-Upper −0.179 0.138 0.688
BCI-Lower −0.295 0.106 0.541
CI-Lower −0.507 0.095 0.463
CI-Upper −0.197 0.164 0.741
BCI-Lower −0.488 0.100 0.497
CI-Lower −0.702 0.077 0.173
CI-Upper −0.387 0.144 0.496
BCI-Lower −0.784 0.082 0.081
ω α β
QAN
ω α β
−0.674 0.057 0.368
BCI-Upper
−0.189 0.079 0.659 BCI-Upper
−0.302 0.087 0.626 BCI-Upper
−0.357 0.111 0.551 BCI-Upper
−0.198 0.071 0.726 BCI-Upper
−0.306 0.118 0.559 BCI-Upper
−0.319 0.170 0.638 BCI-Upper
−0.215 0.094 0.738
CI width
BCI width
1.271 0.027 0.163
0.165 0.025 0.258
Asymmetry
CI Width 2.529 0.022 0.305
BCI Width 0.123 0.021 0.194
CI Width 1.743 0.026 0.220
BCI Width 0.206 0.026 0.226
CI Width 4.262 0.037 0.329
BCI Width 0.246 0.049 0.281
Asymmetry 3.249 0.020 0.054
CI Width 1.340 0.047 0.301
BCI Width 0.839 0.041 0.930
−1.959 −0.000 −0.174
CI Width 52.1460 4.739 32.320
BCI Width 0.167 0.036 0.200
Asymmetry 1.452 0.001 0.015
CI Width 1.413 0.054 0.226
BCI Width 0.282 0.065 0.231
CI Width 2.229 0.047 0.325
BCI Width 0.863 0.071 0.892
CI Width 0.075 0.026 0.122
BCI Width 0.067 0.022 0.107
CI Width 0.094 0.030 0.158
BCI Width 0.061 0.024 0.103
Asymmetry 0.003 0.002 0.010
CI Width 0.103 0.029 0.128
BCI Width 0.073 0.022 0.087
−0.004 −0.000 −0.002
CI Width 0.281 0.042 0.332
BCI Width 0.148 0.030 0.174
Asymmetry 0.002 0.002 −0.000
CI Width 0.251 0.055 0.301
BCI Width 0.221 0.043 0.267
CI Width 0.135 0.037 0.175
BCI Width 0.095 0.027 0.120
CI Width 0.310 0.068 0.278
BCI Width 0.252 0.057 0.203
−0.020 −0.002 −0.007
CI Width 0.316 0.067 0.323
BCI Width 0.466 0.055 0.479
−0.013 −0.002 −0.027
−1.039 0.001
−0.017 Asymmetry
−1.921 0.001
−0.031 Asymmetry
−3.655 0.002
−0.034
Asymmetry
Asymmetry
−4.281 0.018
−0.063 Asymmetry 0.054 0.004 −0.150
Table 14d Confidence intervals of QMLE with normal density. ANZ
Estimates
ω α β
−0.156
CBA
ω α β
BHP
ω α β
WPL
ω α β
CML
ω α β
WOW
ω α β
TLS
ω α β
QAN
ω α β
0.080 0.695 Estimates
−0.161 0.091 0.671 Estimates
−0.164 0.077 0.754 Estimates
−0.331 0.095 0.546 Estimates
−0.374 0.105 0.471 Estimates
−0.247 0.119 0.600 Estimates
−0.352 0.129 0.602 Estimates
−0.544 0.111 0.335
BCI-Upper
−0.125 0.092 0.745 BCI-Upper
−0.129 0.104 0.727 BCI-Upper
−0.130 0.088 0.797 BCI-Upper
−0.256 0.111 0.633 BCI-Upper
−0.273 0.127 0.593 BCI-Upper
−0.201 0.133 0.661 BCI-Upper
−0.236 0.157 0.700 BCI-Upper
−0.318 0.137 0.561
Asymmetry
−0.004 0.003
−0.005
Asymmetry
Asymmetry
−0.019 0.003
−0.024 Asymmetry
−0.003 −0.000 0.002 Asymmetry
Asymmetry
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
densities, namely, log-normal, Weibull, exponential and normal. The Generalised Gamma distribution is not used in this section due to the computational difficulties as mentioned in the previous section. The widths of both confidence intervals are also presented for purposes of comparison. In addition, asymptotic confidence intervals assume symmetry whereas bootstrapped confidence intervals do not impose such an assumption; therefore a measure of asymmetry is also presented in Tables 14a–14d to examine the symmetric property of the confidence intervals in finite samples. Let θˆ denote the QMLE of θ and let θL and θU denote the lower and upper bounds of the confidence interval for θˆ , respectively. A simple measure of asymmetry can be calculated as
τ = (θU − θˆ ) − (θˆ − θL ). Obviously, if the confidence interval is symmetric then τ = 0; if τ > 0 (τ < 0) then the confidence interval is positively (negatively) skewed. As shown in Tables 14a and 14d, the asymptotic confidence intervals and the bootstrapped confidence intervals from the log-normal and normal densities are reasonably symmetric. In addition, the bootstrapped confidence intervals are subsets of the asymptotic confidence intervals. This demonstrates efficiency gain in the estimates from the bootstrapping method. However, in the cases of Weibull and Exponential, the symmetric assumption does not hold in some cases, especially for the estimates of ω and β as shown in Tables 14b and 14c. The presence of asymmetry leads to the bootstrapped confidence intervals being wider than the asymptotic confidence intervals. This suggests that the asymptotic confidence intervals may provide poor approximation in finite samples when Weibull and Exponential distribution are used for QMLE.
Acknowledgments The authors are grateful for the financial support of the Australian Research Council and the data provided by their industry partner SIRCA. Appendix Proof of Proposition 1. From the second equation in (4)
φi = w +
αj log xi−j +
j =1
= w+
q X
βj φi−j
j=1
r X
αj log xi−j +
j =1
r X
βj φi−j
j=1
where αp+1 = · · · = αq = 0, if q > p and βq+1 = · · · = βp = 0, if q < p. Notice that µi = log xi − φi , so the last equation above can be rewritten as follows: r X
log xi = w +
r X
αj log xi−j +
j =1
r X
αj log xi−j +
j =1
r X
αj log xi−j +
j =1
−
r X
βj φi−j + µi
j =1
r X
= w+
βj φi−j + log xi − φi
j=1
r X
= w+
βj log xi−j +
j =1
r X
βj φi−j
j =1
βj log xi−j + µi
j =1 r X
= w+
7. Concluding remarks The paper examined the finite sample properties of the Quasi Maximum Likelihood Estimator (QMLE) of the Logarithmic Autoregressive Conditional Duration (Log-ACD) model. The structural and statistical properties of the Log-ACD model were examined by establishing the relationship between the Log-ACD model and the Autoregressive–Moving Average (ARMA) model. The theoretical results developed in the paper were evaluated using Monte Carlo experiments for four different types of distributions: Weibull, exponential, generalised gamma and Lognormal, all of which produced the required positive standardised residuals. Two asymmetric Log-ACD models were then developed to capture any asymmetric properties of the conditional duration. The objective was to capture any asymmetric or ‘leverage’ type behaviour in the conditional expected duration. The results suggest that the conditional expected duration is not only persistent but also reacts to information shock in asset returns, in the form of positive versus negative price movements. It is frequently argued that trading activity and asset return volatility are correlated with the intensity of market information flow. This means that trading becomes more intensive as the information flow intensifies. This means that increases in information flow will tend to be associated with shorter durations. It has also been suggested that investors trading on information might try to hide the fact that they have information by trading in small parcels. Our analysis of volume effects appears to be consistent with this, in that bid and ask volumes appear to be positively correlated with price durations. Our results are consistent with the argument that the intensity of information disclosure impacts on both price durations and trading volumes.
p X
(αj + βj ) log xi−j −
j =1
r X
βj (log xi − φi−j ) + µi ,
j =1
and hence, log xi can be expressed as, log xi = w +
r X
δj log xi−j +
r X
θj µi−j + µi ,
j=1
j =1
where δj = αj + βj and θj = −βj . In addition, if E (µi ) 6= 0, then the last equation can be rewritten as log xi = w +
r X
θj E (µi ) + E (µi ) +
j =1
+
r X
r X
δj log xi−j
j =1
θj (µi−j − E (µi )) + (µi − E (µi ))
j =1
=w ˜ +
r X
δj log xi−j +
j =1
r X
θj ξi−j + ξi
(18)
j =1
where ξi = µi − E (µi ), hence, ξi ∼ iid(0, σξ2 ), and w ˜ = w+
Pr
j =1
θj E (µi ) + E (µi ).
Proof of Proposition 2. It is sufficient to verify the following conditions for Theorem 4.1.2 in Amemiya (1985). A1. Λ is an open subset of the Euclidean Space Rp+q+1 , such that λ0 ∈ λ. A2. l(λ) is a measurable function of xi for all i and for all λ ∈ Λ. Moreover, ∂ l/∂λexists and is continuous in an open neighbourhood of λ0 . A3. T −1 l(λ) converges to a non-stochastic function in probability uniformly in λ at a open neighbourhood of λ0 and the nonstochastic function attains a strict local maximum at λ0 .
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
A1 is satisfied automatically by Assumption 1. For A2, notice that
∂l ∂l 2 = ∂s ∂l ∂λ ∂θ
183
Now, under Assumption 4
∂ 2 li −E ∂(s2 )2
∂ 2 li ∂(s2 )2
=
1
(log ηi )2 − 1
2s6
and under Assumption 2,and 4
where
T ∂l T 1 X (log xi − φi )2 = − + ∂ s2 2s2 2 i=1 s4
lim supλ∈Λ
and
T →∞
P T therefore, limT →∞ supλ∈Λ i=1 li (λ) − E [li (λ)] = 0, a.s.
Now, consider the maximisation problem in Box 1. Notice R1 (s2 ) < 0, ∀s2 since ∀x > 0, m(x) = log x − x ≤ 0 and m(x)is maximised at x = 1, therefore, s2 = s20 . Moreover, R2 (θ , s2 ) ≥ 0 ∀θ, s2 and therefore the solution to the maximisation problem is the θ such that R2 (θ , s2 ) = 0, which will occur if and only if φi (θ0 ) = φi (θ ) for all i which implies θ = θ0 . Therefore the likelihood function converges into a non-stochastic function which attains a strict local maximum at λ0 . These imply Conditions A1 to p
ˆ → λ0 . This A3 are satisfied under Assumptions (1)–(4) and hence λ completes the proof. Proof of Proposition 3. Given the result in Proposition 2, it is sufficient to verify the following conditions from Theorem 4.1.3 in Amemiya (1985) ∂2l
B1. ∂λ∂λ0 exists and is continuous in an open, convex neighbourhood of θ 0 .
λ=λT
∂2l ∂λ∂λ0
λ=λ0
First, consider the following first and second order partial derivatives ∀i = 1, . . . , T ;
∂ li 1 1 ∂ li log xi − φi ∂φi = − 2 + 4 (log xi − φi )2 ; = ∂ s2 2s 2s ∂θ s2 ∂θ 2 ∂ li 1 1 = 4 − 6 (log xi − φi )2 ; ∂(s2 )2 2s 2s ∂ li ∂ li 1 log xi − φi ∂ 2 φi ∂ 2 li = − + ; ∂θ∂θ 0 ∂θ ∂θ 0 s2 s2 ∂θ ∂θ 0 2 ∂ li 1 ∂φi = − 2 (log xi − φi ) 0 . 2 0 ∂ s ∂θ s ∂θ Under Assumption 1, B1 is satisfied by examining the derivatives above directly. For B2, it is sufficient to verify that lim supλ∈Λ
T 1 X
T i =1
−E
∂ 2 li −E ∂λ∂λ0 λT
! ∂ 2 li
=0 ∂λ∂λ0 λ0
2
Furthermore, under Assumption 2, 3, and 4
∂ 2 li −E ∂θ ∂θ 0
∂ 2 li ∂θ ∂θ 0
∂φi ∂φi ∂φi ∂φi 1 = E − ∂θ ∂θ 0 ∂θ ∂θ 0 s2 log xi − φi ∂ 2 φi + . s2 ∂θ ∂θ 0
Notice that lim
T →∞
T 1 X ∂φi ∂φi
T i=1 ∂θ ∂θ 0
−E
∂φi ∂φi ∂θ ∂θ 0
a.s.
and lim
T →∞
T 1 X log xi − φi ∂ 2 φi
∂θ ∂θ 0
s2
T i =1
=E
log ηi ∂ 2 φi
∂ 2 li ∂ 2 li 2 2 = ∂(s2 ) ∂ li ∂λ∂λ0 ∂θ ∂ s2
∂ 2 li ∂ s2 ∂θ 0 . ∂ 2 li ∂θ ∂θ 0
∂θ ∂θ 0 2 1 ∂ φi = 2 E (log ηi ) E s ∂θ ∂θ 0 = 0 a.s. s2
Therefore,
! ∂ 2 li ∂ 2 li
lim supλ∈Λ −E
T →∞ T i=1 ∂θ ∂θ 0 θT ∂θ ∂θ 0 θ0
T ∂φ ∂φ ∂φ ∂φ 1 1 X i i i i = lim supλ∈Λ E − T →∞ T i =1 ∂θ ∂θ 0 ∂θ ∂θ 0 s2
log xi − φi ∂ 2 φi +
s2 ∂θ ∂θ 0
X ! T
1 ∂φi ∂φi 1 ∂φi ∂φi
− = lim supλ∈Λ 2 E T →∞
s ∂θ ∂θ 0 T ∂θ ∂θ 0 i=1
T
1 X log ηi ∂ 2 φi +
T i=1 s2 ∂θ ∂θ 0
T 1 X
= 0 a.s. Finally, notice that
where
= limT →∞ supλ∈Λ (log ηi ) − 1
T i=1 2s6
# " T
1 X (log ηi )2
− 1 = 0 a.s. = lim supλ∈Λ 6 T →∞
2s i=1 T
in probability for any
sequence λT such that plim λT = λ0 . ∂l B3. T −1/2 ∂λ |λ=λ0 → N (0, B(λ0 )) where B(λ0 ) = limT →∞ T −1 ∂l ∂l E ( ∂λ ∂λ0 )|λ=λ0 .
T →∞
converges to a finite non-singular matrix
A(λ0 ) = limT →∞ T −1 E
T i=1 ∂(s2 )2
∂ 2 li ∂(s2 )2
∂l Therefore, under Assumptions (1)–(3), ∂λ exists and is continuous for all λ ∈ Λ and l(λ) is a measurable function of xi for all i and for all λ ∈ Λ. For A3, let gi (λ) = li (λ)− E [li (λ)], it is obvious that E [gi (λ)] = 0. Under Assumptions (1) and (2) E supλ∈Λ gi (λ) < ∞ and
2
T 1 X 1
T X ∂l log xi − φi ∂φi = . ∂θ s2 ∂θ i =1
∂ l B2. T −1 ∂λ∂λ 0
T 1 X ∂ 2 li
∂ 2 li −E ∂ s2 ∂θ 0
and therefore
∂ 2 li ∂ s2 ∂θ 0
=−
1 s4
log ηi
∂φi ∂θ 0
184
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
max l(λ) − l(λ0 ) = max E [li (λ) − li (λ0 )] λ∈Λ
λ∈Λ
1
= max E λ∈Λ
2
1
= max E λ∈Λ
= max λ∈Λ
1
2
log
log
log
2
s20 s2 s20 s2
s20 s2
− −
1
(log xi − φi (θ ))2 s2
2 1
(log xi − φi (θ0 ))2 s20 s2
2
s0
−
s2
(φi (θ0 ) − φi (θ ))2 + (log ηi )2 + 2 log ηi (φi (θ0 ) − φi (θ ))
2
−
−
−
(log ηi )2
s20
1 E (φi (θ0 ) − φi (θ ))2 2
s2
by Assumption 4
= max R1 (s2 ) − R2 (θ, s2 ) λ∈Λ
where R1 (s ) = 2
1 2
log
s20
s2
s20
!
− s2
and R2 (θ , s2 ) =
2 1 E (φi (θ0 )−φi (θ )) 2 s2
Box 1.
!
T
2 1 X ∂ 2 l i ∂ li −E lim supλ∈Λ
T →∞ T i=1 ∂ s2 ∂θ 0 λT ∂ s2 ∂θ 0 λ0
T ∂φi 1 X 1
− 4 log ηi 0 = lim supλ∈Λ T →∞ T i =1 s ∂θ
T
1 X log ηi ∂φi
= lim supλ∈Λ − 4
T →∞
s i=1 T ∂θ 0
1 ∂φi
− = supλ∈Λ E log η i
s4 ∂θ 0
1 ∂φi
− = supλ∈Λ E log η E ( ) i
s4 ∂θ 0 = 0. Hence, B2 is satisfied. For B3, let ST =
T X
c0
i=1
∂ li ∂λ
where c is a constant vector with the same dimension as λ. Notice that ST is a martingale array with respect to the information set, It −1 and therefore
E
ST2 T
= c0E
∂ li ∂ li c>0 ∂λ ∂λ0
by the given assumptions. Using the Central Limit Theorem of Stout (1974), T 1 X ∂ li
T 1/2 i=1 ∂λ
d
− → N (0, B(λ0 ))
where B (λ0 )
=
limT →∞ T1 E
∂l ∂l ∂λ ∂λ0 λ=λ0
. Therefore B3 is
satisfied and hence all the conditions in Amemiya (1985) have been established. This completes the proof. References Amemiya, T., 1985. Advanced Econometrics. Harvard University Press, Cambridge, Mass. Barclay, M.J., Warner, J.B, 1993. Stealth trading and volatility: Which trades move prices? Journal of Financial Economics 34, 281–306. Bauwens, L., Giot, P., 2000. The logarithmic ACD model: An application to the bidask quote process of three NYSE stocks. Annales D’Economie et de Statistique 60, 117–145. Bauwens, L., Galli, F., Giot, P., 2003. The moments of Log-ACD models. CORE Discussion Paper. Available at SSRN: http://ssrn.com/abstract=375180 or doi:10.2139/ssrn.375180.
Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307–327. Bollerslev, T., Wooldridge, J.M., 1992. Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews 11, 143–172. Blume, L., Easley, D., O’Hara, M., 1994. Market statistics and technical analysis: The role of volume. Journal of Finance 49, 153–181. Caner, M., Hansen, B.E., 2001. Threshold auto-regression with a unit root. Econometrica 69, 1555–1596. Chan, K.S., Tong, H., 1986. On estimating thresholds in autoregressive models. Journal of Time Series Analysis 7, 179–190. Efron, B., Tibshirani, R.J., 1993. An Introduction to the Bootstrap. Chapman and Hall, New York. Engle, R.F., 2000. The econometrics of ultra-high-frequency data. Econometrica 68 (1), 1–22. Engle, R.F., Russell, J.R., 1994. Forecasting transaction rates: The autoregressive conditional duration model. National Bureau of Economic Research Working Paper, No 4966. Engle, R.F., Russell, J.R., 1997. Forecasting the frequency of changes in quoted foreign exchange prices with the autoregressive conditional duration model. Journal of Empirical Finance 4, 187–212. Engle, R.F., Russell, J.R., 1998. Autoregressive conditional duration: A new model for irregularly spaced transactions data. Econometrica 66, 1127–1162. Dufour, A., Engle, R.F., 2000. Time and the price impact of a trade. Journal of Finance 55, 2467–2498. Fernandes, M., Grammig, J., 2006. A family of autoregressive conditional duration models. Journal of Econometrics 130, 1–23. Feng, D., Jiang, G.J., Song, P.X.K., 2004. Stochastic conditional duration models with ‘‘leverage effect’’ for financial transaction data. Journal of Financial Econometrics 2, 390–421. Glosten, L.R., Jagannathan, R., Runkle, D.E., 1992. On the relation between the expected value and volatility of the nominal excess return on stocks. Journal of Finance 46, 1779–1801. Ghysels, E., Jasiak, J., 1997. GARCH for irregularly spaced financial data: The ACD GARCH model. Working Paper. Department of Economics, Pennsylvania State University. Hautsch, N., 2006. Testing the conditional mean function of autoregressive conditional duration models. Working Paper. Department of Economics, University of Copenhagen. Hansen, B.E., 1994. Autoregressive conditional density estimation. International Economic Review 35, 705–730. Hujer, R., Kokot, S., Vuletic, S., 2003. Comparison of MSACD models. Working Paper. Faculty of Economics and Business Administration, Johann Wolfgang GoetheUniversity. Kyle, A., 1985. Continuous auctions and insider trading. Econometrica 53, 1315–1335. Lee, S.-W., Hansen, B.E., 1994. Asymptotic theory for the GARCH(1,1) quasimaximum likelihood estimator. Econometric Theory 10, 29–52. Ling, S., McAleer, M., 2003. Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 278–308. McAleer, M., 2005. Automated inference and learning in modelling financial volatility. Econometric Theory 21, 232–261. Meitz, M., TerÄasvirta, T., 2006. Evaluating models of autoregressive conditional duration. Journal of Business & Economic Statistics 24, 104–124. Nelson, D.B., 1991. Conditional heteroscedasticity in asset returns: A new approach. Econometrica 59, 347–370. Russell, J.R., Engle, R.F., 2005. A discrete-state continuous-time model of financial transactions prices and times: The autoregressive conditional multinomialautoregressive conditional duration model. Journal of Business & Economic Statistics 23, 166–180.
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Contents lists available at ScienceDirect
Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom
Finite sample properties of the QMLE for the Log-ACD model: Application to Australian stocks David Allen a,∗ , Felix Chan b , Michael McAleer c , Shelton Peiris d a
School of Accounting, Finance and Economics, Edith Cowan University, Australia
b
School of Economics and Finance, Curtin University of Technology, Australia
c
School of Economics and Commerce, University of Western Australia, Australia
d
Department of Mathematics and Statistics, University of Sydney, Australia
article
info
Article history: Available online 20 September 2008 Keywords: Conditional duration Asymmetry ACD Log-ACD Monte Carlo simulation
a b s t r a c t This paper concerns the properties of the Quasi Maximum Likelihood Estimator (QMLE) of the Logarithmic Autoregressive Conditional Duration (Log-ACD) model. Proofs of consistency and asymptotic normality of QMLE for the Log-ACD model with log-normal density are presented. This is an important issue as the Log-ACD is used widely for testing various market microstructure models and effects. Knowledge of the distribution of the QMLE is crucial for purposes of valid inference and diagnostic checking. The theoretical results developed in the paper are evaluated using Monte Carlo experiments. The experimental results also provide insights into the finite sample properties of the Log-ACD model under different distributional assumptions. Finally, this paper presents two extensions to the Log-ACD model to accommodate asymmetric effects. The usefulness of these novel models will be evaluated empirically using data from Australian stocks. © 2008 Elsevier B.V. All rights reserved.
1. Introduction An accurate description of the dynamics of duration between stock price changes has important implications and applications for the analysis of financial markets. Engle and Russell (1997) proposed the Autoregressive Conditional Duration (ACD) model, which assumes that the duration between price changes follows a process similar to that of Bollerslev’s (1986) Generalised Autoregressive Conditional Heteroscedasticity (GARCH) model. Both models are based on dynamic time series processes in the underlying variables. Due to the structural similarity between the GARCH and ACD models, Engle and Russell (1997) provided a proof of consistency and asymptotic normality for the QMLE of the ACD model following the approach of Lee and Hansen (1994), arguing that the theoretical results could be applied directly to the ACD model. Based on this result, researchers have subsequently proposed numerous extensions to the ACD model in a similar manner to the extensions of the GARCH model. These extensions include Bauwens and Giot’s (2000) Logarithmic ACD (Log-ACD) model, and Dufour and Engle’s (2000) Box-Cox ACD (BCACD) model and Exponential ACD (EXPACD) model, Zhang et al.’s (2001)
∗ Corresponding address: School of Accounting, Finance and Economics, Edith Cowan University, Faculty of Business and Law, 6027 Joondalup, WA, Australia. E-mail address: [email protected] (D. Allen). 0304-4076/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2008.09.020
Threshold ACD (TACD) model, and Hujer et al.’s (2003) Markov Switching ACD (MSACD) model. For a discussion of the structural and statistical properties of a variety of univariate and multivariate conditional volatility models, see McAleer (2005). This paper is concerned with the asymptotic and finite sample properties of the Quasi Maximum Likelihood Estimator (QMLE) for the Log-ACD model. The motivation of this paper is two-fold. First, Engle and Russell (1997) derived the asymptotic properties of the ACD model based on the results of Lee and Hansen (1994) for the GARCH model. However, such asymptotic properties are not yet available for the volatility counterpart of the Log-ACD model, namely Nelson’s (1991) Exponential GARCH (EGARCH) model. Therefore, the distribution of the QMLE for Log-ACD is still unknown, which is particularly important as the ACD model is often used for testing hypotheses about the market microstructure. Thus, the distribution of QMLE is crucial for purposes of valid inference and diagnostic checking. Second, in the GARCH literature the standardised residuals are often assumed to be normally distributed. The QMLE based on the normal density has been proven to be consistent and asymptotically normal under fairly general conditions by Ling and McAleer (2003). However, the assumption of normality cannot be applied to ACD (or Log-ACD) models as the standardised residuals of these models are required to be positive. The Weibull, exponential, generalised gamma and log-normal are four of the most widely used probability density functions (pdf). A natural question is the nature of the statistical
164
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 1 Summary statistics for eight companies on the Australian stock exchange. Statistics
ANZ
CBA
BHP
WPL
CML
WOW
TLS
QAN
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Observations
80.1535 36 18485 1 232.1636 51.39532 3739.022 16384
57.55475 27 18316 1 168.9323 78.25914 8343.062 16384
100.3229 32 18560 1 312.1668 33.61695 1721.953 14854
133.8996 52 18191 1 367.0297 29.96407 1363.638 11130
287.579 109 18229 1 604.7731 11.50926 261.3184 5104
136.1386 53 18220 1 407.2284 26.17681 990.3336 11105
461.3966 123 18302 1 973.3416 6.607736 78.29639 3172
463.057 152 15798 1 915.73 6.082173 64.82925 3122
Table 2a True Distribution: Lognormal; Sample Size: 500; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal 0.214 0.214 0.347 0.060 0.035 0.025 3.179
Weibull 0.205 0.204 0.353 0.035 0.039 0.104 3.127
G. Gamma 0.181 0.191 0.393 −0.028 0.067 −0.788 3.688
Exponential 0.198 0.196 0.379 0.011 0.047 0.176 3.221
Normal 0.214 0.214 0.347 0.060 0.035 0.025 3.179
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
0.369 0.408 3.739 −4.598 1.027 −0.196 3.374 36.600 0.000
0.066 0.121 3.412 −5.903 1.107 −0.290 3.443 66.504 0.000
−7.656 −0.084
−0.142 −0.079
0.185 0.232 3.517 −4.578 1.060 −0.208 3.278 31.216 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.646 0.654 0.868 0.084 0.071 −0.868 5.620
0.669 0.676 0.882 0.154 0.076 −0.738 4.617
0.482 0.675 0.940 −1.026 0.544 −2.241 6.223
0.681 0.692 0.961 −0.076 0.099 −1.991 13.953
0.646 0.654 0.868 0.084 0.071 −0.868 5.620
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.686 −0.730
−0.296 −0.374
−134.088 −0.215
−1.482 −0.089
−0.478 −0.530
4.133
5.433
−3.784
−3.815
0.956 0.323 3.764 124.676 0.000
1.083 0.384 3.582 115.953 0.000
properties of QMLE for the (Log-)ACD model based on a variety of alternative probability distributions. This paper will present the proofs of consistency and asymptotic normality of QMLE for the Log-ACD model with log-normal density. The theoretical results developed in the paper are evaluated using Monte Carlo experiments. The experimental results also provide insights into the finite sample properties of the Log-ACD model under different distributional assumptions. Moreover, the relationship between the Log-ACD and the Autoregressive–Moving Average (ARMA) model will also be discussed in details. As the structural and statistical properties of the ARMA process are well established, the properties for the ARMA model will also apply to the Log-ACD model. Moreover, the statistical properties of the ARMA model with exogenous variables (ARMAX) will also apply to the Log-ACD model with exogenous variables. This is particularly important as ACD models with exogenous variables are often used for purposes of testing hypotheses about mar-
3.242
−1.620E+04 299.533
−52.959 2857.967 1.018E+09 0.000
14.251
−1.131E+05 3.316E+03 −29.428 906.741 1.023E+08 0.000
6.725
−15.993 1.253
−1.079 13.049 1.317E+04 0.000
9.021
−3.973E+03 72.677
−54.546 2.981E+03 1.108E+09 0.000
4.223
−3.691 1.018 0.279 3.517 72.233 0.000
ket microstructures. Therefore, these properties are crucial for ensuring valid inferences and diagnostic checking. The second part of the paper will propose two alternative methods for accommodating asymmetric effects. The Log-ACD model assumes that the duration between price movements is affected only by the previous duration but not by the direction of the price change. However, since the market frequently has a different attitude to price rises as compared with price falls, it is important to examine how the direction of the price movement affects the future duration. Although many alternative asymmetric ACD models have been proposed in the literature, including Zhang et al.’s (2001) Threshold ACD (TACD) model and Hujer et al.’s (2003) Markov Switching ACD (MSACD) model, these models often lack structural and/or statistical properties and can be difficult to estimate. Moreover, using these models for purposes of testing hypotheses about market microstructure is not always straightforward. However, the
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
165
Table 2b True Distribution: Lognormal; Sample Size: 1000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.207 0.207 0.302 0.123 0.025 0.108 3.173
Weibull
G. Gamma
0.202 0.203 0.305 0.119 0.028 0.082 2.981
Exponential
0.190 0.196 0.351 −0.022 0.051 −1.694 7.362
0.199 0.198 0.317 0.100 0.033 0.127 3.046
Normal 0.207 0.207 0.302 0.123 0.025 0.108 3.173
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
0.275 0.302 3.816 −4.003 1.023 −0.130 3.241 15.799 0.000
0.035 0.101 3.297 −4.052 1.081 −0.241 3.149 31.736 0.000
−9.491 −0.101
−0.152 −0.065
0.128 0.158 3.609 −4.363 1.056 −0.150 3.196 16.067 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.675 0.677 0.819 0.496 0.045 −0.311 3.312
0.686 0.689 0.828 0.465 0.050 −0.397 3.453
0.601 0.692 0.879 −1.010 0.383 −3.792 15.789
0.693 0.697 0.868 0.370 0.059 −0.559 4.044
0.675 0.677 0.819 0.496 0.045 −0.311 3.312
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.493 −0.539
−0.181 −0.258
−85.761 −0.120
0.013 −0.026 5.251 −4.328E+00 1.162 0.317 3.735E+00 117.927 0.000
−0.324 −0.383
4.437
5.454
−3.571
−3.381
0.989 0.324 3.755 123.900 0.000
3.785
3.489
−2.484E+04 453.635
2.215
−54.683 2.993E+03 1.120E+09
−33.697 1.564E+03 3.050E+08
0.000
0.000
5.834
−1.862E+05 3.471E+03 −51.878 2.765E+03 9.550E+08
1.073 0.403 3.633 131.321 0.000
0.000
methods proposed in this paper are simple and straightforward to implement in practice. Moreover, the structural and statistical properties of the Threshold Autoregressive (TAR) model and ARMAX model can be applied directly to the two new asymmetric models that are proposed in this paper. The empirical performance of the models will be examined using tick-by-tick data for eight Australian shares that are traded on the Australia Stock Exchange (ASX). The paper is organised as follows. Section 2 discusses the LogACD model and the distributions that are most frequently used for obtaining the QMLE. A novel method of estimation is also proposed. The statistical properties of QMLE for the Log-ACD model with log-normal density will be analysed in details. Section 3 provides the Monte Carlo experiments and numerical results for the finite sample properties. Section 4 presents two new models for accommodating asymmetric effects. The data are discussed in Section 5. Empirical examples and estimates are given in Section 6. Finally, Section 7 contains some concluding comments.
4.825
−3.478 1.042 0.324 3.635 102.968 0.000
where xi is the duration and εi is the independently and identically distributed (iid) innovation. The connection between the structures of the ACD and GARCH models is obvious. √ Considering yi = xi (which always holds as duration is always positive), then yi essentially follows a GARCH (p,q) process. As xi is positive for all i, it is natural to assume that ψi and εi are both positive. Mathematically, εi can follow any distribution F (x) with probability P (x < 0) = 0. A sufficient condition to ensure the positivity of ψi is ω > 0, αi > 0 ∀i = 1, . . . , p and βi ≥ 0 ∀i = 1, . . . , q. This condition is identical to that of the GARCH model for ensuring that the conditional variance is positive. However, this sufficient condition may be too restrictive in some cases. Bauwens and Giot (2000) resolved this issue by proposing the Logarithmic ACD (Log-ACD) model as follows: xi = exp(ϕi )εi
ϕi = ω +
p X
2. Model specifications and theoretical results
αj log xi−j +
q X
j =1
βj ϕi−j .
(2)
j=1
Note that E (εi ) = ν > 0, so that
2.1. ACD model Engle and Russell (1994) proposed the Autoregressive Conditional Duration (ACD) model as follows: x i = ψ i εi ,
εi ∼ iid p q X X ψi = ω + αj xi−j + βj ψi−j , j=1
−103.182
j=1
(1)
E (xi | Ii−1 ) = ν exp(ϕi ), where Ii is the information set available up to the ith price change. Let exp(φi ) = ν exp(ϕi ), with
φi = $ +
p X j =1
αj log xi−j +
q X j =1
βj φi−j ,
166
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 2c True Distribution: Lognormal; Sample Size: 3000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.203 0.203 0.260 0.158 0.014 0.058 2.973
Weibull
G. Gamma
0.201 0.201 0.262 0.145 0.016 0.070 3.073
Exponential
0.189 0.198 0.279 −0.005 0.048 −2.971 11.900
Normal
0.200 0.199 0.269 0.132 0.019 0.085 3.197
0.203 0.203 0.260 0.158 0.014 0.058 2.973
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
0.189 0.178 3.546 −3.234 1.010 −0.014 2.953 0.381 0.826
0.029 0.063 3.288 −4.130 1.057 −0.139 3.131 11.858 0.003
−96.143 −0.058
−0.062 −0.014
0.092 0.094 3.338 −3.551 1.008 −0.078 2.974 3.135 0.209
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.691 0.691 0.770 0.587 0.025 −0.150 3.134
0.695 0.696 0.792 0.567 0.027 −0.160 3.238
0.603 0.696 0.807 −1.004 0.390 −3.834 15.818
0.697 0.698 0.811 0.550 0.032 −0.212 3.365
0.691 0.691 0.770 0.587 0.025 −0.150 3.134
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.331 −0.367
−0.129 −0.171
−114.526 −0.080
−0.025 −0.018
−0.207 −0.256
3.565
4.636
−3.992
−4.136
1.004 0.108 3.043 6.067 0.048
3.780
φi = $ +
j =1
βj φi−j ,
1.051 0.274 3.381 55.757 0.000
3.954
−3.795
1.297
1.004 0.198 3.161 22.945 0.000
−6.329 178.028 3.849E+06 0.000
1. Lognormal distribution:
(3)
1
√
xs 2π
exp −
1
log(x) − m
2
2 !
s
.
(6)
2. Weibull distribution:
j =1
ε
T X
6.207
−34.982
0.000
where ηi = νi . Eq. (3) is more convenient for purposes of obtaining the QMLE as E (ηi ) = 1 and hence avoids potential identification problems. The parameters in model (3) can be estimated by the Maximum Likelihood method. Let l(θ ) be the log-likelihood function with parameter vector λ = (θ 0 , s0 )0 so that l(λ) =
6.687 1.785E+03 0.000
4.425
f1 (x) =
αj log xi−j +
1.133
−0.415
−1.073E+05 2.461E+03 −35.410 1.395E+03 2.430E+08
xi = exp(φi )ηi q X
−11.657
0.000
where $ = ω + log ν . Then E (xi | Ii−1 ) = exp(φi ) and Eq. (2) can be rewritten as p X
3.875
−2.562E+05 4.691E+03 −54.278 2.962E+03 1.100E+09
f2 (x) =
g
x−m
s
g −1
s
exp −
x−m s
g
.
(7)
3. Generalised Gamma distribution: f3 (x) =
g
0 (k)s
x kg −1 s
x g
exp −
s
.
(8)
4. Exponential distribution: log f (xi , λ),
(4)
f4 (x) = s exp(−sx)
(9)
i=1
where θ = (ω, α1 , . . . , αp , β1 , . . . , βq )0 and s = (s1 , . . . , sk )0 denotes the additional parameters required by the density function, f (xi , λ). Then the Maximum Likelihood Estimator (MLE), λˆ , of λ is given by
λˆ = arg max l(λ). λ∈Λ
(5)
However, the functional form of the likelihood function depends on the distribution of ηi . Mathematically, ηi can follow any distribution function, F (x), such that P (x < 0) = 0. Some of the most popular choices for the distribution of ηi and their density functions are as follows:
where m denotes the location parameter, s denotes the scale parameter, g and k denote the additional scale parameters, and 0 (k) is the gamma function, such that
0 (k) =
∞
Z
sk−1 exp(−s)ds. 0
The log-likelihood function in each case is given by Eq. (4) with f (xi , λ) replaced by the appropriate density function. In practice, the true distribution of ηi is seldom known, such ˆ , as defined in Eq. (5), will be the Quasi-MLE (QMLE) rather thatλ than the MLE. Engle and Russell (1997) suggested using the ˆ , Bollerslev and Wooldridge (1992) robust covariance matrix, H (λ)
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
167
Table 3a True Distribution: Exponential; Sample Size: 500; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Weibull
0.199 0.198 0.323 0.053 0.036 0.106 3.042
G. Gamma
0.201 0.200 0.312 0.101 0.029 0.079 3.129
Exponential
0.199 0.190 0.755 −0.159 0.101 0.588 5.174
Normal
0.199 0.198 0.308 0.094 0.028 0.105 3.172
0.199 0.198 0.323 0.053 0.036 0.106 3.042
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.098 −0.067
−0.027 −0.004
−0.508 −0.117
−0.085 −0.067
−0.096 −0.035
2.906
3.437
−5.763
−5.518
1.053
1.087
−0.390
−0.261
3.769 148.485 0.000
8.159
−86.026
4.257
3.135
−5.317
−6.407
2.569
−13.918
3.566 73.285 0.000
1.101
1.076
−0.151
−0.464
424.718 2.210E+07 0.000
3.689 70.018 0.000
3.993 228.610 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.686 0.691 0.899 0.387 0.065 −0.523 3.769
0.687 0.690 0.852 0.442 0.052 −0.411 3.554
0.504 0.671 1.016 −1.026 0.476 −2.132 6.543
0.692 0.695 0.859 0.470 0.051 −0.415 3.533
0.686 0.691 0.899 0.387 0.065 −0.523 3.769
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.086 −0.157
−0.153 −0.213
−9.452 −0.218
−0.065 −0.107
−0.097 −0.188
5.511
5.574
−4.221
−3.534
1.062 0.483 4.286 320.108 0.000
21.029
−5.908E+03
1.068 0.354 3.774 136.201 0.000
∂ 2l ∂λ∂λ0
−1
∂l ∂λ
∂l ∂λ
0
∂ 2l ∂λ∂λ0
−10
.
128.046
log xi = w ˜ +
(11)
log xi = φi + µi where µi = log ηi . Under the assumption that ηi follows a log-normal distribution, this implies that µi follows a normal distribution. Therefore, the log-likelihood function for (11) can be written as T 1X
2 i=1
log s2 +
log xi − φi s
2
.
(12)
Eq. (11) models the logarithm of the duration rather than duration itself. The advantage of this approach is that log xi can now be rewritten as an ARMA(r , r ) process, where r = max(p, q), as shown in the following proposition:
1.077 0.613 4.809 590.483 0.000
Proposition 1. Assuming the random variable, xi , follows the stochastic process as defined in (3), then log xi can be represented as a ARMA(r , r ) process where r = max(p, q), that is
(10)
λ=λˆ
xi = exp(φi )ηi
l(θ) = −
1.099 0.241 3.774 102.975 0.000
0.000
Although several papers have attempted to derive the moment conditions for the ACD and Log-ACD models (see, for example, Bauwens et al. (2003)), the statistical properties of the QMLE for the Log-ACD model is still unknown (see Ghysels and Jasiak (1997) and Feng et al. (2004)). Another interesting feature of the Log-ACD model is the possibility of linearising the process. Note that log xi = φi + log ηi
6.928
−4.065
−37.019 1.599E+03 3.160E+08
instead of the asymptotic covariance matrix, to obtain the variance ˆ , where for λ
ˆ =E H (λ)
5.373
−4.195
r X
δj log xi−j +
j =1
r X
θj ξi−j + ξi
where ξi ∼ iid(0, σξ2 ) and w ˜ =w+ Proof. See Appendix.
(13)
j =1
Pr
j =1
θj E (µi ) + E (µi ).
Remark 1. It is straightforward to show that the conditional likelihood for Eq. (13) is equivalent to that of (12), so that the existing structural and statistical properties for the ARMA (p, q) model can be applied directly to the model given in (11). The statistical properties if QMLE under log-normal density will be presented in the next sub-section. The finite sample performance of this estimation method will be analysed in the Section 3. 2.2. Theoretical results This section establishes the statistical properties, namely, consistency and asymptotic normality of QMLE with the lognormal density for the Log-ACD model as defined in Eq. (3). The moment structure of the Log-ACD model has been established in Bauwens et al. (2003) Consider the Log-ACD model as defined in Eq. (3). The associated log-likelihood function with the log-normal density is
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 3b True Distribution: Exponential; Sample Size: 1000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Weibull
0.200 0.200 0.296 0.115 0.025 0.027 2.997
G. Gamma
0.200 0.200 0.278 0.126 0.020 0.059 3.166
Exponential
0.198 0.196 0.648 −0.124 0.079 0.451 5.335
0.199 0.199 0.270 0.128 0.019 0.087 3.195
Normal 0.200 0.200 0.296 0.115 0.025 0.027 2.997
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.056
−0.025 0.009 3.682 −4.536 1.040 −0.157 3.303 23.731 0.000
−0.077 −0.055
−0.054
0.003 3.336 −5.457 1.014 −0.260 3.395 52.906 0.000
−0.949 −0.060 3.411 −1.304E+03 24.009 −53.585 2.908E+03 1.050E+09 0.000
3.771 −5.153 1.050 −0.109 3.565 45.539 0.000
0.018 3.272 −5.452 1.026 −0.289 3.411 62.669 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.693 0.694 0.837 0.514 0.043 −0.243 3.258
0.694 0.695 0.824 0.555 0.035 −0.248 3.375
0.585 0.685 0.982 −1.010 0.387 −3.201 12.708
0.696 0.697 0.818 0.560 0.034 −0.253 3.362
0.693 0.694 0.837 0.514 0.043 −0.243 3.258
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.069 −0.137
−0.105 −0.148
−28.731 −0.153
−0.044 −0.075
−0.079 −0.148
6.171
4.679
−3.811
−3.325
1.033 0.405 3.934 190.266 0.000
24.573
−6.122E+04
1.048 0.278 3.549 75.940 0.000
1124.960
l(λ) =
T X
6.133
−3.432
1.075 0.225 3.940 135.069 0.000
−53.946 2.934E+03 1.070E+09 0.000
defined as
5.537
−4.289
1.034 0.424 3.912 192.964 0.000
ensure the existence of the first two moments of log xi which is required by the log-normal density. li (λ)
i=1
li (λ) = log f (xi , λ) = −
− log xi −
1
log 2π −
1
2 2 1 (log xi − φi )2
log s2
(14)
s2
2
where α = (α1 , . . . , αp ) , β = (β1 , . . . , βq )0 , θ = (ω, α 0 , β 0 )0 and λ = (s2 , θ 0 )0 and α0 , β0 , θ0 , λ0 denotes the true parameters of α, β, θ , λ, respectively. The theoretical results are derived based on the following assumptions: 0
1. Λ is an open subset of the Euclidean space, Rp+q+1 , such that λ0 ∈ Λ. 2. xi is stationary and ergodic. Pp i 3. α(L) and β(L) are left co-prime, where α(L) = i=1 αi L and Pq Pp Pq i β(L) = i=1 βi L . Moreover, j=1 α0j + j=1 β0j < 1. 4. E (log ηi ) = 0 E [(log ηi )2 ] = s20 ∀i = 1, . . . , T .
Assumption 2 can be replaced by any conditions that ensure the stationarity and ergodicity of the duration process. The conditions of the existence of moment can be found in Bauwens et al. (2003). Assumption 3 ensures the invertibility of the process and identification of the parameters. Moreover, Assumption 3 is likely to be a sufficient condition for Assumption 2. If this is the case, then Assumption 2 can clearly be removed. Assumption 4 is needed to
Proposition 2. Under Assumptions (1)–(4), the Quasi Maximum Likelihood Estimate (QMLE) as defined in (14) for the Log-ACD model ˆ →p λ 0 . as defined in Eq. (3) is consistent, λ Proof. See Appendix.
Proposition 3. Under Assumptions (1)–(4), the Quasi Maximum Likelihood Estimate (QMLE) as defined in (14) for the √ Log-ACD ˆ − model as defined in Eq. (3) is asymptotically normal, T (λ λ0 ) →d N (0, A(λ0 )−1 B(λ0 )A(λ0 )−1 ) where
∂ 2 l and T →∞ ∂λ∂λ0 λ=λ0 ∂ l ∂ l −1 . B(λ0 ) = lim T E T →∞ ∂λ ∂λ0 A(λ0 ) = lim T −1 E
λ=λ0
Proof. See Appendix.
Remark 2. Notice the results in Propositions 2 and 3 apply explicitly to the Log-ACD model as defined in Eq. (3) where the standardised residual, ηi , has a unit mean. It is unclear whether these results hold for the model as defined in Eq. (2) where the standardised residuals, εi , may have mean different from unity. This should not pose a serious problem in practice, as the estimation is typically conducted for Log-ACD model as specified in Eq. (3) rather than the alternate specification as defined in Eq. (2).
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
169
Table 3c True Distribution: Exponential; Sample Size: 3000; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal
Weibull
G. Gamma
0.199 0.200 0.254 0.152 0.014 0.029 3.138
0.200 0.200 0.242 0.158 0.011 −0.006 3.159
0.198 0.196 0.489 −0.047 0.051 −0.020 6.906
Exponential 0.200 0.200 0.238 0.158 0.011 0.015 3.086
Normal 0.199 0.200 0.254 0.152 0.014 0.029 3.138
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.067 −0.028
−0.022
−0.037 −0.020
−0.060 −0.021
3.195 −3.864 0.982 −0.162 3.224 19.345 0.000
0.021 3.299 −4.201 1.021 −0.101 3.169 8.633 0.013
−0.402 −0.102
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.699 0.699 0.782 0.595 0.024 −0.126 3.220
0.698 0.698 0.766 0.621 0.020 −0.117 3.139
0.664 0.699 0.982 −1.002 0.229 −6.039 42.663
0.699 0.699 0.772 0.631 0.019 −0.103 3.120
0.699 0.699 0.782 0.595 0.024 −0.126 3.220
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.002 −0.041
−0.051 −0.078
−4.224 −0.002
−0.031 −0.057
−0.010 −0.048
4.017
3.957
−3.493
−3.684
0.983 0.229 3.216 32.093 0.000
1.010 0.138 3.078 10.294 0.006
3. Finite sample properties This section provides Monte Carlo evidence about the finite sample properties of the MLE and the QMLE, as defined in (5). The Data Generating Process (DGP) for each of the realisations is given by: xi = exp(φi )ηi
φi = 0.01 + 0.2 log xi + 0.7φi−1 . The steps for the Monte Carlo analysis in this section are as follows: 1. For each of the distributions as defined in Eqs. (6)–(9), the DGP defined above will be used to generate realisations with sample sizes of 500, 1000 and 3000. 2. The parameters of the Log-ACD models will be estimated from the realisations generated in step 1 above by maximising the log-likelihood functions, as defined in Eq. (5), based on the distributions defined in Eqs. (6)–(9) and the log-likelihood function defined in Eq. (4). 3. Repeat Steps 1–2 above 3000 times, so that there are 3000 replications. The first set of results given in Tables 2a–5c feature an analysis of the properties of the QMLE as applied to the Log-ACD model, based on a variety of probability distributions, namely Weibull, exponential, generalised gamma and Log-normal, all of which will produce the required positive standardised residuals. The results of the Monte Carlo experiments shown in the four sections of Tables
2.901
−56.303 2.748
−10.488 163.445 3.270E+06 0.000
30.755
−1.962E+03 62.552
−23.690 660.382 5.425E+07 0.000
3.486
3.218
−4.383
−3.875
1.034
0.997
−0.012
−0.162
3.206 5.357 0.069
3.183 17.272 0.000
4.255
4.051
−3.767
−3.574
1.040 0.091 3.265 12.950 0.002
0.995 0.223 3.187 29.172 0.000
2a–5c simulate the finite sample properties on the basis of samples of sizes 500, 1000, and 3000, each with 3000 simulations. Overall, both MLE and QMLE are close to their true values, even in relatively small samples. It is interesting to note that the (Q)MLE based on the log-normal density, as defined in Eq. (6), is identical to those obtained by maximising Eq. (12). This should not be surprising as Eq. (12) is a monotonic transformation of the likelihood function based on the log-normal density, as defined in Eq. (6). Although the transformation does not affect the estimates, it does affect the standard deviation of the estimates, as can be seen in Tables 2a–5c. The (Q)MLE produce seemingly unbiased estimates of the parameters. Regardless of the true underlying distribution, the (Q)MLE, by assuming the log-normal, Weibull and normal densities, seem to be robust and asymptotically normal, which is supported by the skewness and kurtosis of the t-ratios for the estimates. As the sample size increases, the skewness and kurtosis of the (Q)MLE under the log-normal, Weibull and normal densities, tend towards 0 and 3, respectively. In addition, as the sample size increases, the Jarque-Bera Lagrange multiplier statistics also generally lead to non-rejection of the null hypothesis of normality. Moreover, the convergence rates seem to be faster for MLE than for QMLE in these cases, which is not surprising as MLE should be more efficient than QMLE. However, the finite sample properties of QMLE under the assumption of the generalised gamma and exponential distributions are less than desirable in some cases. The problem with the generalised gamma distribution is partly due to the difficulties in obtaining robust and accurate numerical derivatives of the likelihood functions for purposes of maximisation. This could be improved by
170
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 4a True Distribution: Weibull; Sample Size: 500; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal 0.200 0.199 0.346 0.062 0.036 0.098 3.273
Weibull 0.201 0.200 0.313 0.099 0.028 0.068 3.188
G. Gamma 0.199 0.192 0.677 −0.115 0.097 0.358 4.255
Exponential 0.199 0.198 0.318 0.090 0.028 0.066 3.192
Normal 0.200 0.199 0.346 0.062 0.036 0.098 3.273
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.073 −0.033
−0.019
−0.084 −0.066
−0.072 −0.031
3.204 −4.960 1.057 −0.409 3.867 175.721 0.000
0.010 3.505 −4.821 1.055 −0.209 3.290 31.991 0.000
−0.464 −0.070
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.684 0.689 0.910 0.340 0.066 −0.561 4.166
0.687 0.689 0.860 0.442 0.052 −0.420 3.720
0.494 0.665 1.017 −1.026 0.485 −2.112 6.367
0.692 0.695 0.856 0.477 0.050 −0.385 3.589
0.684 0.689 0.910 0.340 0.066 −0.561 4.166
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.113 −0.186
−0.166 −0.228
−15.061 −0.255
−0.071 −0.105
−0.124 −0.202
5.338
5.053
−3.655
−4.089
1.057 0.553 4.232 338.632 0.000
1.048 0.339 3.693 116.267 0.000
specifying the analytical derivatives of the likelihood function in the optimisation routine. The problem with the exponential distribution is more basic. In many cases, the exponential distribution simply does not have the flexibility to approximate the true underlying distribution which leads to poor finite sample properties, even though the QMLE with exponential density may be asymptotically normal. These Monte Carlo results suggest that the choice of density to determine the likelihood function is important. The density should be sufficiently flexible to provide a good approximation to a wide range of distributions, but also sufficiently accurate so that it does not induce unnecessary numerical difficulties. 4. Asymmetric Log-ACD model The Log-ACD model, as defined in Eq. (3), assumes that the conditional duration is affected by the previous duration but not by the direction of the previous price change. In other words, the model assumes that a positive price change has the same impact on the duration for the next price movement as does a negative price change. It is important to note that the movements in both bid and ask prices contain important information regarding the overall performance of the stock. Thus, it would not seem to be reasonable to assume that the frequency of price changes is unaffected by the direction of previous price changes. There are a wide variety of models available which extend the basic ACD model in various ways. We propose a couple of straightforward methods in this paper, providing two different versions of asymmetric ACD models. Bauwens and Giot in their (2006) survey refer to ‘first generation ACD’ models. These
4.638
−35.321
4.459
3.262
−4.277
−5.119
2.069
1.063
1.066
−3.820
−0.156
−0.385
40.970 1.855E+05 0.000
54.451
−2.324E+04 431.422
−52.679 2.833E+03 9.910E+08 0.000
3.471 39.332 0.000
3.650 125.741 0.000
5.047
4.977
−3.978
−3.819
1.077 0.290 3.829 126.670 0.000
1.059 0.553 4.112 303.965 0.000
begin with the linear ACD model of Engle and Russell (1998) and the logarithmic ACD model of Bauwens and Giot (2000), plus further extensions into augmented ACD classes by Fernandes and Grammig (2006), plus Hautsch’s (2006) semi-parametric model. As mentioned previously, all these models have their GARCH counterparts. The ‘second generation’ ACD models consist of regime-switching ACD models and mixture models; and the mixture can involve both the error distribution and the dynamic component. Zhang et al. (2001) develop a threshold ACD model (TACD) whilst Meitz and TerÄasvirta (2006) develop a class of smooth transition ACD models (STACD) which generalise the linear and logarithmic models in a particular way. Our two extensions belong to the first generation of models and the models are similar in spirit to that of Bauwens et al. (2003). They model the direction of the price change between consecutive trades by means of a competing risks model; in which the direction of the price change is triggered by a Bernoulli process, leading to a form of asymmetric ACD model. The basic linear ACD models only capture the duration between market events and do not include information given by the price process, which is of great importance in the context of market microstructure issues and the process of information impounding. The combination of information given by the price process and the duration between market events is an important extension. This was first addressed by Engle (2000) who suggested an ACD–GARCH model. The durations were captured by a marginal ACD model, and the volatility of returns was modelled by a GARCH process, conditionally on the duration. Russell and Engle (2005) combine an ACD model to capture durations and a generalised linear model on
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
171
Table 4b True Distribution: Weibull; Sample Size: 1000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.199 0.199 0.300 0.123 0.024 0.156 3.287
Weibull
G. Gamma
0.200 0.200 0.276 0.126 0.019 0.077 3.119
0.198 0.194 0.729 −0.095 0.077 0.486 6.375
Exponential 0.199 0.199 0.268 0.127 0.019 0.099 3.031
Normal 0.199 0.199 0.300 0.123 0.024 0.156 3.287
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.065 −0.046
−0.026 −0.006
−0.473 −0.106
−0.057 −0.035
−0.068 −0.039
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.693 0.694 0.835 0.447 0.044 −0.391 3.835
0.694 0.694 0.798 0.537 0.034 −0.206 3.254
0.583 0.683 0.990 −1.016 0.382 −3.140 12.415
0.696 0.697 0.797 0.565 0.033 −0.218 3.122
0.693 0.694 0.835 0.447 0.044 −0.391 3.835
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.075 −0.128
−0.107 −0.185
−7.711 −0.180
−0.054 −0.096
−0.079 −0.147
3.470
3.745
−4.220
−3.767
7.347
−54.412
1.004
1.016
2.512
1.032
1.010
−0.122
−7.789
−0.019
−0.220
3.495 45.215 0.000
3.169 10.916 0.004
5.540
4.246
−3.830
−3.705
1.021 0.362 3.905 167.001 0.000
120.361 1.743E+06 0.000
27.102
−7.481E+03
1.026 0.296 3.274 52.889 0.000
144.215
−47.162 2.422E+03 7.290E+08 0.000
4.1. Model 1: Asymmetric Log-ACD using an indicator function (ALACDI) The first approach to accommodate any asymmetric effects is similar to that of the Glosten et al. (1992) GJR model for capturing asymmetric effects in conditional volatility models. Let Di be the indicator function, such that Di =
0, 1,
1pi ≥ 0 1pi < 0
3.366
−4.204
−0.172
conditional probabilities of the price process in their autoregressive conditional multinomial (ACM) model. For this reason, two asymmetric Log-ACD models are proposed to capture the asymmetric properties of the conditional duration. Model 1 uses an indicator function to capture asymmetric effects in a similar manner to that of the Glosten et al. (1992) GJR model for capturing asymmetric effects in models of conditional volatility. Model 2 accommodates asymmetric effects by using dummy variables. Although both models are intended to capture asymmetric effects, the interpretation of the two models is quite different. Thus, they should be viewed as complementary rather than competing models. In addition, it is important to note that Models 1 and 2 can be rewritten as Threshold Autoregressive Moving Average (TARMA) and ARMAX models, respectively. Hence, the structural and statistical properties of the proposed models can be easily established using existing theoretical results, which will facilitate the testing of various hypotheses regarding market microstructure.
4.277
−4.065
(15)
3.334 14.056 0.001
3.479 52.571 0.000
4.346
5.468
−4.063
−3.477
1.049 0.177 3.402 35.659 0.000
1.023 0.437 3.961 209.904 0.000
where 1pi = pi − pi−1 and pi denotes the price level at the ith significant price change. The Asymmetric Log-ACD model can be defined as
φi = $ +
r s X X (αj + γj Di−j ) log xi−j + βj φi−j . j =1
(16)
j =1
Note that if γj = 0, ∀j, then Eq. (16) reduces to the Log-ACD model, as defined in Eq. (3). Considering the special case, r = s = 1, the short run persistence of the conditional duration is α1 if the previous price change is positive, while the short run persistence is α1 + γ1 if the previous price change is negative. Using a similar argument as presented in the previous section, Eq. (16) can be rewritten as a Threshold Autoregressive Moving Average (TARMA) model. However, unlike the standard TAR model in the non-linear time series literature, where the threshold value is a parameter to be estimated, the threshold value in this case is fixed at 0. More importantly, the structural and statistical properties developed for the TAR model can then be applied directly to Eq. (16) (see Tong (1983), Chan and Tong (1986), Hansen (1994) and Caner and Hansen (2001) for further details regarding the structural and statistical properties of TAR models). 4.2. Model 2: Asymmetric Log-ACD model using exogenous variables (ALACDX) The second approach to accommodate asymmetric effects is to augment the original Log-ACD model by including some exogenous
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Table 4c True Distribution: Weibull; Sample Size: 3000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.200 0.200 0.259 0.150 0.014 −0.027 3.312
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.049 −0.017
−0.025
−0.049 −0.026
−0.042 −0.007
3.571 −3.896 0.973 −0.235 3.511 60.101 0.000
0.002 3.884 −3.649 1.011 −0.115 3.286 16.804 0.000
−0.528 −0.074
βˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.698 0.698 0.777 0.589 0.024 −0.190 3.360
ˆ t (β) Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
Weibull 0.200 0.200 0.247 0.158 0.011 0.005 3.321
j =1
αj log xi−j +
3.388
−260.682 6.350
−31.615 1.178E+03 1.730E+08
Exponential 0.200 0.200 0.243 0.161 0.011 0.037 3.246
Normal 0.200 0.200 0.259 0.150 0.014 −0.027 3.312
4.010
3.465
−3.659
−3.835
1.045
0.983
−0.027
−0.226
0.000
Weibull
G. Gamma
Exponential
Normal
0.698 0.698 0.760 0.608 0.019 −0.141 3.273
0.664 0.698 0.956 −1.002 0.224 −6.160 44.543
0.699 0.699 0.759 0.621 0.019 −0.116 3.211
0.698 0.698 0.777 0.589 0.024 −0.190 3.360
Lognormal
Weibull
G. Gamma
Exponential
Normal
−0.036 −0.072
−0.045 −0.100
−9.444 −0.030
−0.005 −0.041
−0.045 −0.086
3.893
3.412
−3.924
−4.128
0.976 0.199 3.297 30.895 0.000
p X
0.199 0.197 0.497 −0.028 0.051 0.143 7.024
3.420 22.403 0.000
1.007 0.131 3.144 11.147 0.004
variables. In this case, the model can be defined as
φi = $ +
G. Gamma
q X j =1
βj φi−j +
r X j =1
δj Di−j +
s X
λj Xji ,(17)
j =1
where Di is the indicator function, as defined in Eq. (15), and Xji denotes the value of the jth exogenous variable at the ith price change. In our subsequent empirical analysis we adopt bid and ask volume as the exogenous variables. Blume et al. (1994) investigate the informational role of volume traded and show that it is potentially useful for technical analysis. In their model, volume statistics provide information about information precision that cannot be deduced from price statistics alone. Volume may convey additional information to price but the link between information asymmetry and volume traded is not necessarily linear. Kyle (1985) was one of the first to develop a model whereby a single trader, presumed to have a monopoly on information, places orders over time so as to maximize his trading profit before the information becomes common knowledge. Barclay and Warner (1993) find that informed traders concentrate their orders on medium-sized trades. They examined the proportion of a stock’s cumulative price change that occurs in each trade-size category using transactions data for a sample of New York Stock Exchange (NYSE) firms. The stealth trading hypothesis suggests that if privately informed traders concentrate their trades in medium sizes, and if stockprice movements are due mainly to private information revealed through these investors’ trades, then most of a stock’s cumulative price change will take place on medium-size trades. Their findings
15.979
−1.305E+04 263.956
−43.741 2.063E+03 5.310E+08 0.000
3.336 39.691 0.000
3.823
3.835
−4.365
−4.122
1.047 0.117 3.355 22.624 0.000
0.983 0.190 3.216 23.824 0.000
supported the stealth trading hypothesis. This suggests there will not necessarily be a simple relationship between volume and price duration. Note that it is straightforward to show, using similar arguments to those presented in the previous section, that Eq. (17) can be rewritten as an ARMAX model. Hence, the structural and statistical properties developed for the ARMAX model can be applied directly to Eq. (17) without any modification. As mentioned previously, the interpretations of the two models are quite different. Model 1 suggests positive and negative price movements have different effects on the short run persistence of conditional durations. However, Model 2 suggests that the unconditional expectation of duration is different for positive and negative price movements. Thus, these models accommodate two different asymmetric effects on the conditional duration. The empirical performance of the two models will be examined in the next section. 5. Data The two asymmetric models were applied to eight shares listed on the Australian Stock Exchange (ASX) using tick-bytick data for the period 1/7/2003–1/10/2003. The eight shares are Commonwealth Bank of Australia (CBA), BHP Billiton (BHP), QANTAS Airways (QAN), Coles-Myer Limited (CML), Telstra (TLS), Australia and New Zealand Bank (ANZ), Woolworths (WOW) and Woodside Petroleum (WPL). These eight companies cover a wide range of industries and service areas that include mining, energy and retail industries, telecommunications and the banking sector.
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
173
Table 5a True Distribution: Generalised Gamma; Sample Size: 500; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal 0.245 0.244 0.415 0.088 0.036 0.070 3.470
Weibull 0.252 0.250 0.399 0.121 0.038 0.256 3.190
G. Gamma 0.201 0.193 0.716 −0.100 0.078 0.737 4.849
Exponential 0.197 0.197 0.328 0.097 0.031 0.075 3.170
Normal 0.245 0.244 0.415 0.088 0.036 0.070 3.470
t (α) ˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
1.187 1.201 4.751 −2.848 0.951 −0.194 3.403 39.091 0.000
1.461 1.481 4.760 −2.201 1.008 −0.165 3.173 17.281 0.000
−0.292 −0.102
−0.709 −0.086
1.039 1.075 4.660 −3.061 0.967 −0.213 3.375 40.213 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.555 0.566 0.790 −0.120 0.090 −1.248 6.858
0.531 0.549 0.762 −0.132 0.104 −1.103 5.122
0.657 0.689 1.012 −0.911 0.182 −2.427 13.906
0.690 0.694 0.858 −0.045 0.060 −1.698 18.558
0.555 0.566 0.790 −0.120 0.090 −1.248 6.858
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−1.497 −1.483
−1.674 −1.637
−0.426 −0.109
−0.380 −0.063
−1.373 −1.380
1.691
1.690
−4.947
−5.060
0.714
0.739
−0.095
−0.116
3.587 47.601 0.000
3.881 103.801 0.000
The data were provided by our industry research partner SIRCA (Securities Industry Research Centre of the Asia Pacific). SIRCA is a not-for-profit research consortium of 26 universities drawn from Australia and New Zealand and a number of industry partners, including the Australian Stock Exchange (ASX), the Sydney Futures Exchange (SFE), and Reuters. This research draws upon SIRCA’s ASX intra-day data which captures all the transactions occurring on the ASX via the Stock Exchange Automated Trading System (SEATS). The data possess a wealth of information, including the date and time (to the nearest hundredth of a second that the trade took place), price information including details of the bid and ask prices, volumes, order flow (disclosed and undisclosed), value and volumes of trades, broker IDs and order flags. This paper applies the various Log-ACD models discussed above to eight Australian companies, representing five different industries in Australia. Australia and New Zealand Banking Group Limited (ANZ) and Commonwealth Bank of Australia (CBA) are selected to represent the banking industry of Australia. BHP Billiton Limited (BHP) and Woodside Petroleum Limited (WPL) are selected to represent the Mining and Energy Industry of Australia. Coles Myer Ltd (CML) and Woolworths Limited (WOW) are selected to represent the Retail Industry of Australia. Telstra Corporation Limited (TLS) and QANTAS Airways Limited (QAN) are selected to represent the Telecommunications and Transportation Industries for Australia, respectively. The summary statistics and their sample sizes are given in Table 1. The calculation of duration follows Engle and Russell (1998) and the data are further filtered by the cubic spline method, as suggested in Engle and Russell (1998), to remove the time-of-day effects.
4.123
−34.876 1.662
−4.435 72.141 6.072E+05 0.000
3.485
−1.593E+03 29.140
−54.429 2.974E+03 1.100E+09 0.000
19.800
7.352
−975.384
−926.874
18.781
−47.391 2.429E+03 7.370E+08 0.000
17.031
−53.737 2.922E+03 1.070E+09 0.000
1.993
−4.399 0.760 0.052 3.505 33.183 0.000
6. Empirical results Tables 6a–13d contain the parameter estimates of Models (3), (16) and (17) for the eight companies. These estimates are presented in three dimensional graphical format in Figs. 1–8. For all companies, each model was estimated four times with different distributional assumptions, namely the log-normal, Weibull, exponential and normal distributions, as discussed in Section 2. The generalised gamma distribution is omitted due to its computational difficulties, as outlined in Section 3. Additional exogenous variables, namely the bid and ask volumes, are also included in the ALACDX model to examine the impact of traded volumes in the previous price change on the duration of the subsequent price change. In addition, the Ljung-Box Q-statistics and the BDS statistics are calculated using the standardised residuals in each case. The results support serial independence in the standard residuals, which provide evidence to support the consistency of the QMLE. However, the Kolmogorov–Smirnov test provides strong evidence to reject the null hypothesis of the assumed distribution in each case, which indicates that the underlying distribution is unlikely to be the correct distribution in each case. However, the existence of some outliers and extreme observations may be the cause of these test outcomes, and would be an interesting area for further research. 6.1. Banking industry As shown in Tables 6a–6d and 7a–7d, the α and β estimates are positive and significant for both ANZ and CBA for each of the four
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Table 5b True Distribution: Generalised Gamma; Sample Size: 1000; Replication: 3000
αˆ
Lognormal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.228 0.228 0.323 0.133 0.024 −0.014 3.124
t (α) ˆ
Lognormal
Weibull
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
1.064 1.099 4.593 −2.583 0.913 −0.224 3.306 36.841 0.000
1.338 1.368 4.364 −3.175 0.939 −0.386 3.526 109.242 0.000
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.624 0.627 0.773 0.305 0.050 −0.560 4.622
0.603 0.609 0.779 0.196 0.063 −0.895 5.342
0.682 0.697 0.948 −0.807 0.115 −2.261 18.333
0.695 0.696 0.810 0.539 0.039 −0.208 3.218
0.624 0.627 0.773 0.305 0.050 −0.560 4.622
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−1.322 −1.339
−1.489 −1.507
0.125 −0.042 24.475 −12.072 1.513 2.131 30.083 9.396E+04 0.000
−0.041 −0.078
−1.175 −1.195
Weibull 0.236 0.236 0.329 0.136 0.027 0.148 3.062
2.525
3.291
−4.009
−4.548
0.777 0.296 3.685 102.459 0.000
0.751 0.436 4.538 390.573 0.000
G. Gamma
Exponential
Normal
0.199 0.199 0.282 0.129 0.022 −0.021 3.122
0.228 0.228 0.323 0.133 0.024 −0.014 3.124
G. Gamma
Exponential
Normal
−0.166 −0.038
−0.090 −0.022
0.911 0.939 4.326 −3.000 0.942 −0.231 3.297 37.769 0.000
0.202 0.198 0.630 0.037 0.057 0.826 5.670
4.436
−15.705
8.236
−4.733
1.303
1.081
−1.245
−0.083
11.167 9.113E+03 0.000
4.469E+00 272.988 0.000
8.921
3.171
−3.260
−3.859
1.071 0.512 5.005 633.435 0.000
0.833 0.346 3.751 130.204 0.000
Fig. 1. Estimates for ANZ. Fig. 2. Estimates for CBA.
estimated models. Figs. 1 and 2 present these results in graphical format. These results suggest that past durations are helpful in predicting the duration before the next price change. However, there is no evidence to support asymmetric effects on duration from price changes as both the γ and δ estimates are insignificant for both banks. Interestingly, λ1 and λ2 estimates are both positive and significant in both cases, indicating that the traded bid and ask volumes at the last price change have a positive impact on duration. In addition, the inclusion of the bid and ask volumes also
has a negative news impact on the β estimates. This would suggest that the correct specification of the model is crucial for obtaining valid inferences and diagnostic checks. Moreover, the problem of omitted variables could have important implications for the interpretation of the various models. The long run persistence of past duration to future conditional duration would also appear to be substantially lower when the bid and ask volumes are included
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
175
Table 5c True Distribution: Generalised Gamma; Sample Size: 3000; Replication: 3000
αˆ Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Lognormal 0.211 0.210 0.269 0.166 0.014 0.117 3.072
Weibull 0.216 0.215 0.280 0.170 0.014 0.173 3.114
G. Gamma 0.200 0.198 0.389 −0.032 0.035 0.480 5.586
Exponential 0.199 0.199 0.257 0.159 0.012 0.105 3.247
Normal 0.211 0.210 0.269 0.166 0.014 0.117 3.072
αˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
0.693 0.696 4.533 −2.432 0.925 −0.024 3.058 0.714 0.700
1.000 1.052 4.689 −2.484 0.911 −0.205 3.361 37.345 0.000
−0.131 −0.087
−1.313 −0.007
0.571 0.563 4.397 −2.606 0.941 −0.001 2.996 0.003 0.999
βˆ
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
0.674 0.675 0.745 0.555 0.025 −0.262 3.235
0.663 0.665 0.738 0.515 0.028 −0.448 3.665
0.694 0.701 0.943 −0.969 0.070 −5.598 115.596
0.699 0.700 0.766 0.598 0.021 −0.225 3.412
0.674 0.675 0.745 0.555 0.025 −0.262 3.235
ˆ t (β)
Lognormal
Weibull
G. Gamma
Exponential
Normal
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque–Bera Probability
−0.910 −0.930
−1.167 −1.220
−1.485
−0.773 −0.790
2.168 −4.276 0.871 0.063 3.170 5.572 0.062
2.211 −4.392 0.808 0.300 3.664 100.241 0.000
0.001 0.007 6.764 −254.489 4.789 −50.028 2.659E+03 8.830E+08 0.000
3.331
−9.468 1.109
−0.615 5.542 996.989 0.000
11.135
−3.706E+03 67.664
−54.724 2.997E+03 1.120E+09 0.000
0.000 4.298 −4.434E+03 80.962 −54.728 2.997E+03 1.120E+09 0.000
2.446
−4.119 0.898 0.058 3.086 2.575 0.276
Fig. 3. Estimates for BHP. Fig. 4. Estimates for WPL.
in the analysis. The figures reveal that the estimates have different values for the two banks and the differences are most pronounced in the asymmetric models that include bid and ask volumes. 6.2. Mining and energy industry Tables 8a–8d and 9a–9d contain the estimates for BHP and WPL, which represent the mining and energy industry of Australia, and
again these results are presented graphically in Figs. 3 and 4. As in the case of the banking industry, the α and β estimates are positive and significant for both BHP and WPL, indicating that past durations contain important information about the future duration of price changes. Although there is no evidence for asymmetric effects on duration from price change, traded bid and ask volumes appear to have significant and positive effects on the conditional duration as λ1 and λ2 are both positive and significant. Again, the
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 6a Estimates of the Log-ACD Model for ANZ. ANZ (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.156 [−8.344]
−0.627 [−9.315]
−0.156 [−8.118]
0.0800 [12.052] 0.695 [22.716]
0.066 [10.742] 0.648 [18.016]
0.265 [0.817] 0.049 [7.038] 0.588 [14.112]
α β
0.08 [12.016] 0.695 [22.271]
Table 6b Estimates of the ALACDX Model for ANZ. ANZ (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.153 [−7.752] 0.080 [11.721] 0.693 [21.731] −0.007 [−0.438]
−0.626 [−9.336]
0.273 [0.812] 0.049 [7.157] 0.584 [9.697] −0.008 [−0.515]
−0.153 [−7.448] 0.080 [11.640] 0.693 [20.961] −0.007 [−0.431]
α β δ
0.067 [11.161] 0.645 [17.714] −0.010 [−0.625]
Fig. 5. Estimates for CML.
Table 6c Estimates of the ALACD Model for ANZ. ANZ (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.156 [−8.357]
−0.628 [−9.372] 0.065 [8.327] 0.647 [18.088] 0.002 [0.207]
0.265 [0.850] 0.049 [6.123] 0.588 [12.388] 0.003 [0.016]
−0.156 [−7.856]
α β γ
0.081 [9.967] 0.695 [22.774] −0.004 [−0.341]
0.081 [9.485] 0.695 [21.385] −0.004 [−0.327]
Table 6d Estimates of the ALACDX Model with Bid and Ask Volumes for ANZ. ANZ (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.617 [−1.446] 0.094 [8.293] 0.514 [2.310] −0.021 [−0.844] 0.015 [1.014] 0.026 [1.358]
−1.902 [−3.600]
3.367 [2.261] 0.051 [5.985] 0.13 [1.493] −0.028 [−1.618] 0.029 [4.502] 0.047 [6.769]
−0.617 [−1.404] 0.094 [8.347] 0.514 [2.192] −0.021 [−0.787] 0.015 [1.012] 0.026 [1.304]
α β δ λ1 λ2
0.076 [8.774] 0.263 [1.449] −0.033 [−1.646] 0.026 [2.316] 0.041 [3.263]
Fig. 6. Estimates for WOW.
Table 7a Estimates of the Log-ACD Model for CBA. CBA (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.161 [−6.715] 0.091 [11.924] 0.671 [16.666]
−0.607 [−7.482] 0.083 [12.278] 0.642 [14.490]
0.712 [1.295] 0.068 [12.416] 0.577 [6.998]
−0.161 [−6.547] 0.091 [11.853] 0.671 [16.308]
α β
Table 7b Estimates of the ALACDX Model for CBA. CBA (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.149 [−5.830]
−0.593 [−7.262]
−0.149 [−5.958]
0.091 [11.704] 0.674 [16.402] −0.019 [−1.242]
0.082 [12.403] 0.645 [14.540] −0.019 [−1.298]
0.729 [3.133] 0.068 [10.144] 0.578 [5.593] −0.016 [−0.998]
α β δ
0.091 [12.015] 0.674 [17.204] −0.019 [−1.226]
Fig. 7. Estimates for QAN.
inclusion of the bid and ask volumes has a negative impact on the β estimates. Moreover, the long run persistence of past duration to future conditional duration would appear to be lower when the bid and ask volumes are included in the analysis.
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
177
Table 8c Estimates of the ALACD Model for BHP. BHP (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.166 −[5.800]
−0.823 −[8.771]
−0.166 −[5.893]
0.073 [8.298] 0.753 [21.662] 0.008 [0.748]
0.073 [8.590] 0.674 [19.204] 0.016 [1.531]
1.41 [2.829] 0.061 [7.684] 0.532 [7.153] 0.026 [2.842]
α β γ
0.073 [8.392] 0.753 [21.985] 0.008 [0.744]
Table 8d Estimates of the ALACDX Model with Bid and Ask Volumes for BHP. BHP (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−2.645 [−11.813]
−3.641 [−9.109]
−2.645 [−12.274]
0.090 [9.672] 0.077 [1.196] −0.048 [−1.546] 0.093 [8.391] 0.093 [8.977]
0.088 [10.595] 0.179 [2.187] −0.043 [−1.860] 0.074 [6.686] 0.075 [7.597]
2.128 [0.050] 0.074 [0.000] 0.275 [0.001] −0.030 [−0.000] 0.057 [0.000] 0.053 [0.000]
α Fig. 8. Estimates for TLS.
β δ
Table 7c Estimates of the ALACD Model for CBA. CBA (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.160 [−6.996] 0.100 [10.451] 0.671 [17.557] −0.018 [−1.600]
−0.600 [−7.830] 0.090 [11.283] 0.646 [15.535] −0.016 [−1.601]
0.663 [1.443] 0.076 [9.434] 0.597 [7.226] −0.019 [−1.583]
−0.160 [−6.915] 0.100 [10.722] 0.671 [17.321] −0.018 [−1.713]
α β γ
CBA (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.569 [−4.545]
−1.091 [−5.735] 0.091 [13.525] 0.533 [8.048] −0.020 [−1.198] 0.018 [3.299] 0.020 [3.250]
0.963 [1.437] 0.076 [13.271] 0.432 [4.248] −0.018 [−1.080] 0.017 [3.700] 0.025 [3.083]
−0.569 [−4.857]
β δ λ1 λ2
0.102 [14.038] 0.560 [8.306] −0.020 [−1.120] 0.023 [3.496] 0.022 [3.343]
0.102 [13.857] 0.560 [9.230] −0.020 [−1.264] 0.023 [3.838] 0.022 [3.409]
Table 8a Estimates of the Log-ACD Model for BHP. BHP (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.164 [−5.868]
−0.821 [−8.250] 0.081 [11.844] 0.675 [18.234]
1.422 [3.200] 0.073 [10.873] 0.530 [9.428]
−0.164 [−6.263]
α β
0.077 [10.254] 0.754 [22.047]
0.077 [10.498] 0.754 [23.163]
BHP (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.149 [−5.830]
−0.593 [−7.262]
0.091 [11.704] 0.674 [16.402] −0.019 [−1.242]
0.082 [12.403] 0.645 [14.540] −0.019 [−1.298]
0.729 [3.133] 0.068 [10.144] 0.578 [5.593] −0.016 [−0.998]
−0.149 [−5.958] 0.091 [12.015] 0.674 [17.204] −0.019 [−1.226]
β δ
Table 9a Estimates of the Log-ACD Model for WPL. WPL (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.331 [−4.594]
−1.242 [−6.871] 0.094 [10.537] 0.493 [7.047]
−2.105 [−1.936] 0.077 [8.031] 0.383 [4.562]
−0.331 [−4.611] 0.095 [8.967] 0.546 [6.444]
β
0.095 [8.711] 0.546 [6.402]
Table 9b Estimates of the ALACDX Model for WPL. WPL (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.338 [−4.656]
−1.252 [−7.016] 0.094 [10.554] 0.493 [7.149] 0.019 [0.787]
−2.097 [−1.581]
−0.338 [−4.793]
0.077 [8.935] 0.384 [3.661] −0.002 [−0.060]
0.095 [8.674] 0.548 [6.569] 0.019 [0.756]
α β δ
0.095 [8.657] 0.548 [6.414] 0.019 [0.713]
Table 9c Estimates of the ALACD Model for WPL. WPL (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.340 [−4.429]
−1.277 [−6.849]
−2.104 [−1.438]
−0.34 [−4.266]
0.102 [6.724] 0.535 [5.893] −0.012 [−0.799]
0.102 [7.700] 0.480 [6.614] −0.013 [−0.873]
0.077 [5.759] 0.383 [3.653] 0.000 [0.002]
0.102 [6.669] 0.535 [5.690] −0.012 [−0.834]
α β γ
Table 8b Estimates of the ALACDX Model for BHP.
α
λ2
α
Table 7d Estimates of the ALACDX Model with Bid and Ask Volumes for CBA.
α
λ1
0.090 [11.751] 0.077 [1.236] −0.048 [−1.606] 0.093 [8.500] 0.093 [9.470]
6.3. Retail industry Tables 10a–10d and 11a–11d contain the estimates for CML and WOW, two of Australia’s largest retail corporations and Figs. 5 and 6 present the same estimates in graphical format. As for the previous results, the α and β estimates are positive and significant for both corporations, and there appears to be an asymmetric effect on the conditional duration from previous price changes as
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 9d Estimates of the ALACDX Model with Bid and Ask Volumes for WPL.
Table 10d Estimates of the ALACDX Model with Bid and Ask Volumes for CML.
WPL (Price)
Log-Normal
Weibull
Exponential
Normal
CML (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.839 [−4.406]
−1.832 [−6.220] 0.099 [11.756] 0.407 [4.688] 0.018 [0.753] 0.015 [2.363] 0.030 [3.633]
−3.917 [−1.284]
−0.839 [−4.133]
ω
−1.409 [−5.335]
0.103 [10.397] 0.425 [3.895] 0.016 [0.588] 0.017 [2.253] 0.032 [3.112]
4.847 [2.804] 0.048 [3.230] 0.039 [0.253] −0.065 [−1.447] 0.052 [3.414] 0.046 [4.643]
−1.409 [−5.238]
0.08 [9.491] 0.29 [2.222] 0.001 [0.027] 0.013 [2.271] 0.028 [3.199]
−2.82 [−6.192] 0.085 [6.715] 0.188 [1.508] −0.079 [−2.125] 0.053 [3.614] 0.051 [4.638]
α β δ λ1 λ2
0.103 [10.550] 0.425 [3.997] 0.016 [0.528] 0.017 [2.311] 0.032 [3.265]
Table 10a Estimates of the Log-ACD Model for CML.
α β δ λ1 λ2
0.113 [8.444] 0.262 [2.566] −0.098 [−2.295] 0.049 [3.530] 0.052 [4.026]
0.113 [7.866] 0.262 [2.400] −0.098 [−2.329] 0.049 [3.526] 0.052 [4.238]
Table 11a Estimates of the Log-ACD Model for WOW.
CML (ACD)
Log-Normal
Weibull
Exponential
Normal
WOW (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.374 [−5.911]
−1.342 [−7.223]
−0.374 [−5.842]
ω
−0.935 [−8.970]
−1.115 [−0.008]
−0.247 [−7.166]
0.082 [6.634] 0.431 [5.786]
0.105 [7.428] 0.471 [6.132]
α
−0.247 [−7.114]
0.105 [7.424] 0.471 [6.117]
0.361 [1.058] 0.050 [4.203] 0.348 [4.534]
0.119 [12.298] 0.600 [13.194]
0.114 [12.823] 0.574 [13.074]
0.100 [0.082] 0.451 [0.055]
0.119 [12.496] 0.600 [13.400]
α β
Table 10b Estimates of the ALACDX Model for CML.
β
Table 11b Estimates of the ALACDX Model for WOW.
CML (Price)
Log-Normal
Weibull
Exponential
Normal
WOW (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.348 [−5.989]
−1.348 [−7.161]
−0.348 [−5.881]
ω
−0.91 [−8.772]
−1.224 [−1.262]
−0.222 [−6.322]
0.083 [6.995] 0.416 [5.423] −0.062 [−1.728]
0.106 [7.717] 0.455 [6.332] −0.083 [−2.145]
α
−0.222 [−6.381]
0.106 [8.253] 0.455 [6.231] −0.083 [−2.064]
0.475 [0.961] 0.051 [4.754] 0.317 [3.601] −0.038 [−1.150]
0.121 [12.571] 0.592 [12.939] −0.061 [−2.408]
0.115 [12.984] 0.568 [12.860] −0.076 [−3.249]
0.101 [8.618] 0.441 [5.167] −0.09 [−4.037]
0.121 [12.619] 0.592 [12.837] −0.061 [−2.375]
α β δ
Table 10c Estimates of the ALACD Model for CML.
β δ
Table 11c Estimates of the ALACD Model for WOW.
CML (AACD)
Log-Normal
Weibull
Exponential
Normal
WOW (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.373 [−5.567]
−1.358 [−7.300]
−0.373 [−5.704]
ω
0.105 [6.318] 0.472 [5.779] −0.002 [−0.061]
0.079 [5.093] 0.424 [5.658] 0.008 [0.352]
0.406 [1.201] 0.041 [2.606] 0.325 [3.798] 0.021 [0.963]
0.105 [6.350] 0.472 [5.953] −0.002 [−0.062]
α
−0.249 [−6.455] 0.117 [10.440] 0.597 [11.925] 0.006 [0.446]
−0.952 [−8.706]
−1.258 [−1.184]
0.107 [10.171] 0.567 [12.347] 0.017 [1.165]
0.086 [5.383] 0.426 [3.722] 0.033 [1.589]
−0.249 [−6.990] 0.117 [10.673] 0.597 [12.900] 0.006 [0.449]
α β γ
the δ estimates are negative and significant in all cases except one, namely the δ estimate is not significant for CML under the exponential distribution. This would suggest that the presence of asymmetric effects would be industry dependent. In addition, the coefficients of the bid and ask volumes continue to be positive and significant. Again the Figures reveal that the large differences in estimates occur in the context of the asymmetric model after the inclusion of volume statistics. 6.4. Transport and telecommunication industries
β γ
Table 11d Estimates of the ALACDX model with bid and ask volumes for WOW. WOW (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.703 [−4.799]
−1.387 [−7.047]
−2.228 [−6.627]
0.129 [13.924] 0.512 [8.622] −0.062 [−2.368] 0.019 [2.597] 0.029 [3.367]
0.12 [14.515] 0.516 [9.824] −0.075 [−3.141] 0.016 [2.260] 0.025 [3.481]
0.104 [8.917] 0.391 [3.627] −0.084 [−3.010] 0.019 [2.334] 0.031 [3.497]
−0.703 [−4.953] 0.129 [14.204] 0.512 [8.946] −0.062 [−2.334] 0.019 [2.704] 0.029 [3.472]
α β δ λ1
Tables 12a–12d contain the parameter estimates for QAN and Tables 13a–13d contain the parameter estimates for TLS, which represent the air and telecommunications industries for Australia, respectively. These results are presented graphically in Figs. 7 and 8. Interestingly, both δ and γ estimates are significant, but with opposite signs. This would suggest that a negative price shock has lower short run persistence on the conditional duration but has a positive impact on future conditional duration. In other words,
λ2
a negative price change will lead to a longer duration before the next price change, but the impact will not last as long as a positive price change. Again, bid and ask volumes play important roles in determining future duration as λ1 and λ2 are both positive and
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185 Table 12a Estimates of the Log-ACD model for QAN.
179
Table 13b Estimates of the ALACDX model for TLS.
QAN (ACD)
Log-Normal
Weibull
Exponential
Normal
TLS (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.544 [−7.084]
−1.777 [−8.759]
−0.674 [−1.186]
−0.544 [−6.759]
ω
0.111 [6.906] 0.335 [4.255]
0.09 [6.279] 0.34 [4.867]
0.057 [4.748] 0.368 [4.436]
0.111 [6.507] 0.335 [4.058]
α
−0.319 [−3.731]
−1.517 [−6.314]
−0.319 [−3.398]
0.128 [7.013] 0.606 [8.379] −0.058 [−1.047]
0.142 [8.906] 0.553 [8.762] −0.061 [−1.109]
1.670 [2.987] 0.128 [7.332] 0.555 [9.304] −0.033 [−0.620]
α β
β δ
0.128 [6.977] 0.606 [8.006] −0.058 [−0.953]
Table 12b Estimates of the ALACDX model for QAN. QAN (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−0.644 [−7.688]
−1.822 [−8.699]
−0.838 [−1.794]
−0.644 [−7.584]
0.108 [6.194] 0.347 [4.464] 0.201 [3.411]
0.088 [5.604] 0.354 [4.815] 0.157 [3.328]
0.056 [3.825] 0.379 [4.271] 0.116 [2.445]
0.108 [6.180] 0.347 [4.325] 0.201 [3.744]
α β δ
Table 13c Estimates of the ALACD model for TLS. TLS (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.352 [−4.444]
−1.562 [−6.537]
−0.352 [−4.424]
0.128 [5.980] 0.602 [8.465] 0.003 [0.120]
0.132 [6.110] 0.549 [8.599] 0.022 [0.805]
1.617 [4.053] 0.103 [4.976] 0.555 [9.934] 0.049 [2.034]
α β γ
0.128 [5.791] 0.602 [8.455] 0.003 [0.120]
Table 12c Estimates of the ALACD model for QAN. QAN (AACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.583 [−6.722]
−1.897 [−8.037]
−0.916 [−1.368]
0.157 [6.235] 0.295 [3.353] −0.083 [−2.515]
0.135 [5.904] 0.295 [3.560] −0.077 [−2.492]
0.096 [4.824] 0.309 [2.872] −0.064 [−2.342]
−0.583 [−6.634] 0.157 [6.321] 0.295 [3.261] −0.083 [−2.528]
α β γ
Table 13d Estimates of the ALACDX model with bid and ask volumes for TLS. TLS (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−2.020 [−5.122]
−2.921 [−7.461]
−2.02 [−5.413]
0.122 [7.025] 0.484 [5.735] 0.006 [0.082] 0.042 [3.108] 0.087 [3.985]
0.128 [8.361] 0.518 [8.353] −0.018 [−0.290] 0.037 [3.116] 0.070 [4.144]
0.796 [1.674] 0.120 [6.777] 0.563 [8.840] −0.022 [−0.402] 0.029 [2.829] 0.031 [2.247]
α β δ λ1
Table 12d Estimates of the ALACDX model with bid and ask volumes for QAN. QAN (Price)
Log-Normal
Weibull
Exponential
Normal
ω
−1.688 [−6.488]
−2.670 [−8.927]
0.100 [5.771] 0.310 [4.074] 0.170 [2.481] 0.046 [2.529] 0.045 [3.011]
0.081 [5.350] 0.333 [4.908] 0.123 [2.251] 0.038 [2.770] 0.034 [2.821]
−1.988 [−2.588] 0.051 [3.826] 0.330 [4.441] 0.083 [1.814] 0.031 [2.893] 0.021 [1.966]
−1.688 [−6.596] 0.100 [5.768] 0.310 [4.350] 0.170 [2.726] 0.046 [2.893] 0.045 [3.039]
α β δ λ1 λ2
Table 13a Estimates of the Log-ACD model for TLS. TLS (ACD)
Log-Normal
Weibull
Exponential
Normal
ω
−0.352 [−4.441]
−1.569 [−8.080]
0.129 [7.612] 0.602 [8.537]
0.144 [9.630] 0.547 [9.946]
1.680 [4.660] 0.129 [9.406] 0.553 [9.621]
−0.352 [−4.451] 0.129 [7.427] 0.602 [8.489]
α β
significant for QAN. However, the results for TLS are qualitatively identical to those of the banking industry. No evidence is found for asymmetric effects on duration from price changes, but λ1 and λ2 are both positive and significant, which indicates the importance of the bid and ask volumes in predicting the duration for the next price change.
λ2
0.122 [7.391] 0.484 [5.826] 0.006 [0.092] 0.042 [3.342] 0.087 [4.246]
6.5. Bootstrapped confidence intervals Section 2 showed that the QMLE based on the log-normal density is consistent and asymptotically normal for the Log-ACD model. It is, however, unclear whether the same results hold when different densities are used to construct the QMLE. Moreover, although the Monte Carlo results presented in Section 3 provided support for the theoretical results, the usefulness of the asymptotic results in the finite sample is still unclear from the empirical perspective. Therefore, this section compares the bootstrapped confidence intervals and the confidence intervals constructed by using the robust standard errors (asymptotic confidence intervals) of the estimates obtained in the empirical section. As reported in the empirical section, the diagnostic tests indicated that the standardised residuals of the Log-ACD model under different distributional assumptions are IID in all cases but the residuals did not follow any of the assumed distributions. Therefore the bootstrapping techniques can be used to provide an accurate approximation to the underlying distribution of QMLE in the finite sample. Thus, comparing the bootstrapped confidence intervals and the asymptotic confidence intervals will provide insight into the applicability of asymptotic normality in the finite sample. The bootstrapped confidence intervals are constructed by using the percentile method as proposed in Efron and Tibshirani (1993). Tables 14a–14d contain the estimates of the Log-ACD model, their bootstrapped confidence intervals and the asymptotic confidence intervals for the eight companies using 4 different
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D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
Table 14a Confidence intervals of QMLE with log-normal density. ANZ
Estimates
CI-Lower
CI-Upper
BCI-Lower
BCI-Upper
ω α β
−0.156
−0.193
−0.120
−0.181
−0.121
0.080 0.695
0.067 0.635
0.093 0.754
0.068 0.652
CBA
Estimates −0.161 0.091 0.671
CI-Lower −0.208 0.076 0.591
CI-Upper −0.113 0.106 0.751
BCI-Lower −0.192 0.081 0.620
BHP
Estimates −0.164 0.077 0.754
CI-Lower −0.219 0.062 0.687
CI-Upper −0.110 0.091 0.821
BCI-Lower −0.204 0.067 0.709
Estimates −0.331 0.095 0.546
CI-Lower −0.472 0.074 0.379
CI-Upper −0.190 0.117 0.714
BCI-Lower −0.414 0.082 0.453
Estimates −0.374 0.105 0.471
CI-Lower −0.498 0.077 0.320
CI-Upper −0.250 0.132 0.622
BCI-Lower −0.495 0.085 0.317
Estimates −0.247 0.119 0.600
CI-Lower −0.315 0.100 0.511
CI-Upper −0.179 0.138 0.689
BCI-Lower −0.294 0.105 0.539
Estimates −0.352 0.129 0.602
CI-Lower −0.507 0.096 0.464
CI-Upper −0.197 0.163 0.740
BCI-Lower −0.483 0.104 0.485
Estimates −0.544 0.111 0.335
CI-Lower −0.695 0.079 0.181
CI-Upper −0.394 0.142 0.489
BCI-Lower −0.791 0.083 0.112
ω α β ω α β
WPL
ω α β
CML
ω α β
WOW
ω α β
TLS
ω α β
QAN
ω α β
CI Width
BCI Width
0.089 0.753
0.073 0.026 0.120
0.061 0.021 0.102
BCI-Upper −0.133 0.103 0.721
CI Width 0.095 0.031 0.160
BCI Width 0.060 0.022 0.102
Asymmetry −0.004 0.001 −0.001
BCI-Upper
CI Width 0.110 0.029 0.134
BCI Width 0.073 0.020 0.084
−0.007 −0.000 −0.006
CI Width 0.282 0.043 0.334
BCI Width 0.150 0.027 0.179
CI Width 0.248 0.055 0.302
BCI Width 0.212 0.043 0.280
−0.028
CI Width 0.136 0.038 0.178
BCI Width 0.096 0.028 0.126
Asymmetry 0.002 −0.000 0.003
CI Width 0.311 0.067 0.276
BCI Width 0.233 0.054 0.213
−0.020
0.137 0.531
CI Width 0.301 0.063 0.308
BCI Width 0.417 0.054 0.419
−0.076 −0.002 −0.027
CI Width
BCI Width
Asymmetry
0.264 0.024 0.141
0.099 0.026 0.157
CI Width 0.315 0.026 0.172
BCI Width 0.073 0.026 0.124
Asymmetry 0.841 0.004 −0.009
CI Width 0.390 0.027 0.145
BCI Width 0.094 0.058 0.117
Asymmetry 1.169 0.038 −0.007
CI Width 0.708 0.035 0.274
BCI Width 0.176 0.032 0.218
Asymmetry 1.721 −0.000 0.005
CI Width 0.728 0.049 0.292
BCI Width 0.342 0.048 0.412
Asymmetry 1.833 0.001 0.032
CI Width 0.409 0.035 0.172
BCI Width 0.100 0.040 0.136
Asymmetry 1.326 0.011 −0.002
CI Width 0.761 0.058 0.216
BCI Width 0.197 0.279 0.315
Asymmetry 2.347 0.228 −0.137
CI Width 0.795 0.056 0.274
BCI Width 0.485 0.071 0.536
Asymmetry 2.468 0.013 0.041
−0.132 0.087 0.793 BCI-Upper
−0.264 0.109 0.632 BCI-Upper
−0.284 0.128 0.597 BCI-Upper
−0.198 0.133 0.664 BCI-Upper
−0.250 0.158 0.698 BCI-Upper
−0.374
Asymmetry 0.011
−0.003 0.016
Asymmetry
Asymmetry
−0.017 0.001
−0.007 Asymmetry
−0.030 0.003
Asymmetry
−0.029 0.003 Asymmetry
Table 14b Confidence intervals of QMLE with Weibull density. ANZ
Estimates
CI-Lower
CI-Upper
BCI-Lower
BCI-Upper
ω α β
−0.627
−0.759
−0.495
−0.244
−0.145
0.066 0.648
0.054 0.577
0.079 0.718
0.058 0.570
0.083 0.727
CBA
Estimates −0.607 0.082 0.643
CI-Lower −0.764 0.069 0.557
CI-Upper −0.449 0.096 0.729
BCI-Lower −0.223 0.071 0.576
Estimates −0.821 0.081 0.675
CI-Lower −1.016 0.067 0.602
CI-Upper −0.626 0.094 0.748
BCI-Lower −0.283 0.071 0.613
Estimates −1.242 0.094 0.493
CI-Lower −1.596 0.077 0.356
CI-Upper −0.888 0.112 0.631
BCI-Lower −0.469 0.078 0.387
Estimates −1.342 0.082 0.431
CI-Lower −1.706 0.058 0.285
CI-Upper −0.978 0.107 0.577
BCI-Lower −0.596 0.059 0.241
Estimates −0.935 0.114 0.574
CI-Lower −1.140 0.097 0.488
CI-Upper −0.731 0.132 0.660
BCI-Lower −0.322 0.100 0.505
Estimates −1.569 0.144 0.547
CI-Lower −1.949 0.114 0.440
CI-Upper −1.188 0.173 0.655
BCI-Lower −0.494 0.118 0.321
Estimates −1.777 0.090 0.340
CI-Lower −2.175 0.062 0.203
CI-Upper −1.379 0.118 0.477
BCI-Lower −0.786 0.061 0.093
ω α β
BHP
ω α β
WPL
ω α β
CML
ω α β
WOW
ω α β
TLS
ω α β
QAN
ω α β
BCI-Upper
−0.150 0.097 0.700 BCI-Upper
−0.189 0.128 0.730 BCI-Upper
−0.293 0.110 0.605 BCI-Upper
−0.255 0.107 0.653 BCI-Upper
−0.223 0.140 0.641 BCI-Upper
−0.296 0.397 0.637 BCI-Upper
−0.301 0.132 0.628
0.864 0.008 0.002
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
181
Table 14c Confidence intervals of QMLE with exponential density. ANZ
ω α β CBA
ω α β
BHP
ω α β
WPL
Estimates 0.265 0.049 0.588
CI-Lower
−0.370
CI-Upper
BCI-Lower
BCI-Upper
−0.337
−0.172
0.037 0.450
0.062 0.708
0.036 0.506
0.900 0.063 0.669
Estimates 0.710 0.068 0.578
CI-Lower −0.555 0.057 0.425
CI-Upper 1.974 0.079 0.730
BCI-Lower −0.312 0.058 0.466
Estimates 1.422 0.073 0.530
CI-Lower 0.551 0.060 0.420
CI-Upper 2.294 0.086 0.640
BCI-Lower −0.508 0.061 0.400
Estimates
CI-Upper 0.026 0.095 0.548
BCI-Lower −0.603 0.062 0.270
ω α β
−2.105 0.077 0.383
CI-Lower −4.236 0.058 0.219
CML
Estimates 0.361 0.050 0.348
CI-Lower −0.308 0.027 0.198
CI-Upper 1.031 0.074 0.498
BCI-Lower −1.038 0.029 −0.204
Estimates
CI-Upper 24.958 2.469 16.611
BCI-Lower −0.473 0.082 0.359
ω α β
WOW
ω α β
−1.115 0.100 0.451
CI-Lower −27.188 −2.270 −15.709
TLS
Estimates 1.680 0.129 0.553
CI-Lower 0.974 0.102 0.441
CI-Upper 2.387 0.155 0.666
BCI-Lower −0.601 0.105 0.406
Estimates
CI-Lower −1.788 0.033 0.205
CI-Upper 0.440 0.080 0.530
BCI-Lower −1.078 0.023 −0.153
CI-Lower −0.194 0.067 0.633
CI-Upper −0.119 0.093 0.756
BCI-Lower −0.192 0.070 0.638
CI-Lower −0.208 0.076 0.592
CI-Upper −0.113 0.106 0.750
BCI-Lower −0.190 0.080 0.624
CI-Lower −0.216 0.062 0.691
CI-Upper −0.113 0.091 0.818
BCI-Lower −0.203 0.065 0.710
CI-Lower −0.471 0.074 0.380
CI-Upper −0.190 0.116 0.712
BCI-Lower −0.404 0.081 0.459
CI-Lower −0.500 0.077 0.321
CI-Upper −0.249 0.132 0.622
BCI-Lower −0.495 0.085 0.326
CI-Lower −0.314 0.101 0.512
CI-Upper −0.179 0.138 0.688
BCI-Lower −0.295 0.106 0.541
CI-Lower −0.507 0.095 0.463
CI-Upper −0.197 0.164 0.741
BCI-Lower −0.488 0.100 0.497
CI-Lower −0.702 0.077 0.173
CI-Upper −0.387 0.144 0.496
BCI-Lower −0.784 0.082 0.081
ω α β
QAN
ω α β
−0.674 0.057 0.368
BCI-Upper
−0.189 0.079 0.659 BCI-Upper
−0.302 0.087 0.626 BCI-Upper
−0.357 0.111 0.551 BCI-Upper
−0.198 0.071 0.726 BCI-Upper
−0.306 0.118 0.559 BCI-Upper
−0.319 0.170 0.638 BCI-Upper
−0.215 0.094 0.738
CI width
BCI width
1.271 0.027 0.163
0.165 0.025 0.258
Asymmetry
CI Width 2.529 0.022 0.305
BCI Width 0.123 0.021 0.194
CI Width 1.743 0.026 0.220
BCI Width 0.206 0.026 0.226
CI Width 4.262 0.037 0.329
BCI Width 0.246 0.049 0.281
Asymmetry 3.249 0.020 0.054
CI Width 1.340 0.047 0.301
BCI Width 0.839 0.041 0.930
−1.959 −0.000 −0.174
CI Width 52.1460 4.739 32.320
BCI Width 0.167 0.036 0.200
Asymmetry 1.452 0.001 0.015
CI Width 1.413 0.054 0.226
BCI Width 0.282 0.065 0.231
CI Width 2.229 0.047 0.325
BCI Width 0.863 0.071 0.892
CI Width 0.075 0.026 0.122
BCI Width 0.067 0.022 0.107
CI Width 0.094 0.030 0.158
BCI Width 0.061 0.024 0.103
Asymmetry 0.003 0.002 0.010
CI Width 0.103 0.029 0.128
BCI Width 0.073 0.022 0.087
−0.004 −0.000 −0.002
CI Width 0.281 0.042 0.332
BCI Width 0.148 0.030 0.174
Asymmetry 0.002 0.002 −0.000
CI Width 0.251 0.055 0.301
BCI Width 0.221 0.043 0.267
CI Width 0.135 0.037 0.175
BCI Width 0.095 0.027 0.120
CI Width 0.310 0.068 0.278
BCI Width 0.252 0.057 0.203
−0.020 −0.002 −0.007
CI Width 0.316 0.067 0.323
BCI Width 0.466 0.055 0.479
−0.013 −0.002 −0.027
−1.039 0.001
−0.017 Asymmetry
−1.921 0.001
−0.031 Asymmetry
−3.655 0.002
−0.034
Asymmetry
Asymmetry
−4.281 0.018
−0.063 Asymmetry 0.054 0.004 −0.150
Table 14d Confidence intervals of QMLE with normal density. ANZ
Estimates
ω α β
−0.156
CBA
ω α β
BHP
ω α β
WPL
ω α β
CML
ω α β
WOW
ω α β
TLS
ω α β
QAN
ω α β
0.080 0.695 Estimates
−0.161 0.091 0.671 Estimates
−0.164 0.077 0.754 Estimates
−0.331 0.095 0.546 Estimates
−0.374 0.105 0.471 Estimates
−0.247 0.119 0.600 Estimates
−0.352 0.129 0.602 Estimates
−0.544 0.111 0.335
BCI-Upper
−0.125 0.092 0.745 BCI-Upper
−0.129 0.104 0.727 BCI-Upper
−0.130 0.088 0.797 BCI-Upper
−0.256 0.111 0.633 BCI-Upper
−0.273 0.127 0.593 BCI-Upper
−0.201 0.133 0.661 BCI-Upper
−0.236 0.157 0.700 BCI-Upper
−0.318 0.137 0.561
Asymmetry
−0.004 0.003
−0.005
Asymmetry
Asymmetry
−0.019 0.003
−0.024 Asymmetry
−0.003 −0.000 0.002 Asymmetry
Asymmetry
182
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
densities, namely, log-normal, Weibull, exponential and normal. The Generalised Gamma distribution is not used in this section due to the computational difficulties as mentioned in the previous section. The widths of both confidence intervals are also presented for purposes of comparison. In addition, asymptotic confidence intervals assume symmetry whereas bootstrapped confidence intervals do not impose such an assumption; therefore a measure of asymmetry is also presented in Tables 14a–14d to examine the symmetric property of the confidence intervals in finite samples. Let θˆ denote the QMLE of θ and let θL and θU denote the lower and upper bounds of the confidence interval for θˆ , respectively. A simple measure of asymmetry can be calculated as
τ = (θU − θˆ ) − (θˆ − θL ). Obviously, if the confidence interval is symmetric then τ = 0; if τ > 0 (τ < 0) then the confidence interval is positively (negatively) skewed. As shown in Tables 14a and 14d, the asymptotic confidence intervals and the bootstrapped confidence intervals from the log-normal and normal densities are reasonably symmetric. In addition, the bootstrapped confidence intervals are subsets of the asymptotic confidence intervals. This demonstrates efficiency gain in the estimates from the bootstrapping method. However, in the cases of Weibull and Exponential, the symmetric assumption does not hold in some cases, especially for the estimates of ω and β as shown in Tables 14b and 14c. The presence of asymmetry leads to the bootstrapped confidence intervals being wider than the asymptotic confidence intervals. This suggests that the asymptotic confidence intervals may provide poor approximation in finite samples when Weibull and Exponential distribution are used for QMLE.
Acknowledgments The authors are grateful for the financial support of the Australian Research Council and the data provided by their industry partner SIRCA. Appendix Proof of Proposition 1. From the second equation in (4)
φi = w +
αj log xi−j +
j =1
= w+
q X
βj φi−j
j=1
r X
αj log xi−j +
j =1
r X
βj φi−j
j=1
where αp+1 = · · · = αq = 0, if q > p and βq+1 = · · · = βp = 0, if q < p. Notice that µi = log xi − φi , so the last equation above can be rewritten as follows: r X
log xi = w +
r X
αj log xi−j +
j =1
r X
αj log xi−j +
j =1
r X
αj log xi−j +
j =1
−
r X
βj φi−j + µi
j =1
r X
= w+
βj φi−j + log xi − φi
j=1
r X
= w+
βj log xi−j +
j =1
r X
βj φi−j
j =1
βj log xi−j + µi
j =1 r X
= w+
7. Concluding remarks The paper examined the finite sample properties of the Quasi Maximum Likelihood Estimator (QMLE) of the Logarithmic Autoregressive Conditional Duration (Log-ACD) model. The structural and statistical properties of the Log-ACD model were examined by establishing the relationship between the Log-ACD model and the Autoregressive–Moving Average (ARMA) model. The theoretical results developed in the paper were evaluated using Monte Carlo experiments for four different types of distributions: Weibull, exponential, generalised gamma and Lognormal, all of which produced the required positive standardised residuals. Two asymmetric Log-ACD models were then developed to capture any asymmetric properties of the conditional duration. The objective was to capture any asymmetric or ‘leverage’ type behaviour in the conditional expected duration. The results suggest that the conditional expected duration is not only persistent but also reacts to information shock in asset returns, in the form of positive versus negative price movements. It is frequently argued that trading activity and asset return volatility are correlated with the intensity of market information flow. This means that trading becomes more intensive as the information flow intensifies. This means that increases in information flow will tend to be associated with shorter durations. It has also been suggested that investors trading on information might try to hide the fact that they have information by trading in small parcels. Our analysis of volume effects appears to be consistent with this, in that bid and ask volumes appear to be positively correlated with price durations. Our results are consistent with the argument that the intensity of information disclosure impacts on both price durations and trading volumes.
p X
(αj + βj ) log xi−j −
j =1
r X
βj (log xi − φi−j ) + µi ,
j =1
and hence, log xi can be expressed as, log xi = w +
r X
δj log xi−j +
r X
θj µi−j + µi ,
j=1
j =1
where δj = αj + βj and θj = −βj . In addition, if E (µi ) 6= 0, then the last equation can be rewritten as log xi = w +
r X
θj E (µi ) + E (µi ) +
j =1
+
r X
r X
δj log xi−j
j =1
θj (µi−j − E (µi )) + (µi − E (µi ))
j =1
=w ˜ +
r X
δj log xi−j +
j =1
r X
θj ξi−j + ξi
(18)
j =1
where ξi = µi − E (µi ), hence, ξi ∼ iid(0, σξ2 ), and w ˜ = w+
Pr
j =1
θj E (µi ) + E (µi ).
Proof of Proposition 2. It is sufficient to verify the following conditions for Theorem 4.1.2 in Amemiya (1985). A1. Λ is an open subset of the Euclidean Space Rp+q+1 , such that λ0 ∈ λ. A2. l(λ) is a measurable function of xi for all i and for all λ ∈ Λ. Moreover, ∂ l/∂λexists and is continuous in an open neighbourhood of λ0 . A3. T −1 l(λ) converges to a non-stochastic function in probability uniformly in λ at a open neighbourhood of λ0 and the nonstochastic function attains a strict local maximum at λ0 .
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
A1 is satisfied automatically by Assumption 1. For A2, notice that
∂l ∂l 2 = ∂s ∂l ∂λ ∂θ
183
Now, under Assumption 4
∂ 2 li −E ∂(s2 )2
∂ 2 li ∂(s2 )2
=
1
(log ηi )2 − 1
2s6
and under Assumption 2,and 4
where
T ∂l T 1 X (log xi − φi )2 = − + ∂ s2 2s2 2 i=1 s4
lim supλ∈Λ
and
T →∞
P T therefore, limT →∞ supλ∈Λ i=1 li (λ) − E [li (λ)] = 0, a.s.
Now, consider the maximisation problem in Box 1. Notice R1 (s2 ) < 0, ∀s2 since ∀x > 0, m(x) = log x − x ≤ 0 and m(x)is maximised at x = 1, therefore, s2 = s20 . Moreover, R2 (θ , s2 ) ≥ 0 ∀θ, s2 and therefore the solution to the maximisation problem is the θ such that R2 (θ , s2 ) = 0, which will occur if and only if φi (θ0 ) = φi (θ ) for all i which implies θ = θ0 . Therefore the likelihood function converges into a non-stochastic function which attains a strict local maximum at λ0 . These imply Conditions A1 to p
ˆ → λ0 . This A3 are satisfied under Assumptions (1)–(4) and hence λ completes the proof. Proof of Proposition 3. Given the result in Proposition 2, it is sufficient to verify the following conditions from Theorem 4.1.3 in Amemiya (1985) ∂2l
B1. ∂λ∂λ0 exists and is continuous in an open, convex neighbourhood of θ 0 .
λ=λT
∂2l ∂λ∂λ0
λ=λ0
First, consider the following first and second order partial derivatives ∀i = 1, . . . , T ;
∂ li 1 1 ∂ li log xi − φi ∂φi = − 2 + 4 (log xi − φi )2 ; = ∂ s2 2s 2s ∂θ s2 ∂θ 2 ∂ li 1 1 = 4 − 6 (log xi − φi )2 ; ∂(s2 )2 2s 2s ∂ li ∂ li 1 log xi − φi ∂ 2 φi ∂ 2 li = − + ; ∂θ∂θ 0 ∂θ ∂θ 0 s2 s2 ∂θ ∂θ 0 2 ∂ li 1 ∂φi = − 2 (log xi − φi ) 0 . 2 0 ∂ s ∂θ s ∂θ Under Assumption 1, B1 is satisfied by examining the derivatives above directly. For B2, it is sufficient to verify that lim supλ∈Λ
T 1 X
T i =1
−E
∂ 2 li −E ∂λ∂λ0 λT
! ∂ 2 li
=0 ∂λ∂λ0 λ0
2
Furthermore, under Assumption 2, 3, and 4
∂ 2 li −E ∂θ ∂θ 0
∂ 2 li ∂θ ∂θ 0
∂φi ∂φi ∂φi ∂φi 1 = E − ∂θ ∂θ 0 ∂θ ∂θ 0 s2 log xi − φi ∂ 2 φi + . s2 ∂θ ∂θ 0
Notice that lim
T →∞
T 1 X ∂φi ∂φi
T i=1 ∂θ ∂θ 0
−E
∂φi ∂φi ∂θ ∂θ 0
a.s.
and lim
T →∞
T 1 X log xi − φi ∂ 2 φi
∂θ ∂θ 0
s2
T i =1
=E
log ηi ∂ 2 φi
∂ 2 li ∂ 2 li 2 2 = ∂(s2 ) ∂ li ∂λ∂λ0 ∂θ ∂ s2
∂ 2 li ∂ s2 ∂θ 0 . ∂ 2 li ∂θ ∂θ 0
∂θ ∂θ 0 2 1 ∂ φi = 2 E (log ηi ) E s ∂θ ∂θ 0 = 0 a.s. s2
Therefore,
! ∂ 2 li ∂ 2 li
lim supλ∈Λ −E
T →∞ T i=1 ∂θ ∂θ 0 θT ∂θ ∂θ 0 θ0
T ∂φ ∂φ ∂φ ∂φ 1 1 X i i i i = lim supλ∈Λ E − T →∞ T i =1 ∂θ ∂θ 0 ∂θ ∂θ 0 s2
log xi − φi ∂ 2 φi +
s2 ∂θ ∂θ 0
X ! T
1 ∂φi ∂φi 1 ∂φi ∂φi
− = lim supλ∈Λ 2 E T →∞
s ∂θ ∂θ 0 T ∂θ ∂θ 0 i=1
T
1 X log ηi ∂ 2 φi +
T i=1 s2 ∂θ ∂θ 0
T 1 X
= 0 a.s. Finally, notice that
where
= limT →∞ supλ∈Λ (log ηi ) − 1
T i=1 2s6
# " T
1 X (log ηi )2
− 1 = 0 a.s. = lim supλ∈Λ 6 T →∞
2s i=1 T
in probability for any
sequence λT such that plim λT = λ0 . ∂l B3. T −1/2 ∂λ |λ=λ0 → N (0, B(λ0 )) where B(λ0 ) = limT →∞ T −1 ∂l ∂l E ( ∂λ ∂λ0 )|λ=λ0 .
T →∞
converges to a finite non-singular matrix
A(λ0 ) = limT →∞ T −1 E
T i=1 ∂(s2 )2
∂ 2 li ∂(s2 )2
∂l Therefore, under Assumptions (1)–(3), ∂λ exists and is continuous for all λ ∈ Λ and l(λ) is a measurable function of xi for all i and for all λ ∈ Λ. For A3, let gi (λ) = li (λ)− E [li (λ)], it is obvious that E [gi (λ)] = 0. Under Assumptions (1) and (2) E supλ∈Λ gi (λ) < ∞ and
2
T 1 X 1
T X ∂l log xi − φi ∂φi = . ∂θ s2 ∂θ i =1
∂ l B2. T −1 ∂λ∂λ 0
T 1 X ∂ 2 li
∂ 2 li −E ∂ s2 ∂θ 0
and therefore
∂ 2 li ∂ s2 ∂θ 0
=−
1 s4
log ηi
∂φi ∂θ 0
184
D. Allen et al. / Journal of Econometrics 147 (2008) 163–185
max l(λ) − l(λ0 ) = max E [li (λ) − li (λ0 )] λ∈Λ
λ∈Λ
1
= max E λ∈Λ
2
1
= max E λ∈Λ
= max λ∈Λ
1
2
log
log
log
2
s20 s2 s20 s2
s20 s2
− −
1
(log xi − φi (θ ))2 s2
2 1
(log xi − φi (θ0 ))2 s20 s2
2
s0
−
s2
(φi (θ0 ) − φi (θ ))2 + (log ηi )2 + 2 log ηi (φi (θ0 ) − φi (θ ))
2
−
−
−
(log ηi )2
s20
1 E (φi (θ0 ) − φi (θ ))2 2
s2
by Assumption 4
= max R1 (s2 ) − R2 (θ, s2 ) λ∈Λ
where R1 (s ) = 2
1 2
log
s20
s2
s20
!
− s2
and R2 (θ , s2 ) =
2 1 E (φi (θ0 )−φi (θ )) 2 s2
Box 1.
!
T
2 1 X ∂ 2 l i ∂ li −E lim supλ∈Λ
T →∞ T i=1 ∂ s2 ∂θ 0 λT ∂ s2 ∂θ 0 λ0
T ∂φi 1 X 1
− 4 log ηi 0 = lim supλ∈Λ T →∞ T i =1 s ∂θ
T
1 X log ηi ∂φi
= lim supλ∈Λ − 4
T →∞
s i=1 T ∂θ 0
1 ∂φi
− = supλ∈Λ E log η i
s4 ∂θ 0
1 ∂φi
− = supλ∈Λ E log η E ( ) i
s4 ∂θ 0 = 0. Hence, B2 is satisfied. For B3, let ST =
T X
c0
i=1
∂ li ∂λ
where c is a constant vector with the same dimension as λ. Notice that ST is a martingale array with respect to the information set, It −1 and therefore
E
ST2 T
= c0E
∂ li ∂ li c>0 ∂λ ∂λ0
by the given assumptions. Using the Central Limit Theorem of Stout (1974), T 1 X ∂ li
T 1/2 i=1 ∂λ
d
− → N (0, B(λ0 ))
where B (λ0 )
=
limT →∞ T1 E
∂l ∂l ∂λ ∂λ0 λ=λ0
. Therefore B3 is
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