) t ,/ <:> PERPUSTAKAAN UNIVERSITI MALAYA Fitting Weibull ACD Models to High Frequency Transactions Data: A Semi-Parametric Approach Based on Estimating Functions By: Ng Kok Haur, David Allen and Shelton Peiris (Paper presented at the 15th International Conference on Computing in Economics and Finance held on 15-17 July 2009 at the University of Technology, Sydney, Australia)
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)t ,/<:>
PERPUSTAKAAN UNIVERSITI MALAYA
Fitting Weibull ACD Models to High FrequencyTransactions Data: A Semi-Parametric Approach
Based on Estimating Functions
By:
Ng Kok Haur, David Allenand
Shelton Peiris
(Paper presented at the 15th International Conference on Computingin Economics and Finance held on 15-17 July 2009 at the University ofTechnology, Sydney, Australia)
UNIVER51ii' or
Abstract
PERPUSTAKAAN UNIVERSITI MALAVA
Fitting Weibull ACD Models to High Frequency Transactions Data:
A Semi-parametric Approach based on Estimating Functions
K.H. Ng, University of Malaya, Kuala Lumpur, Malaysia-David Allen, Edith Cowan University, WA
Shelton Peiris, The University of Sydney, NSW
Autoregressive conditional duration (ACD) models play an important role in financial
modeling. This paper considers the estimation of the Weibull ACD model using a semi-
parametric approach based on the theory of estimating functions (EF). We apply the EF
and the maximum likelihood (ML) methods to a data set given in Tsay (2003, p203) to
compare these two methods. It is shown that the EF approach is easier to apply in
practice and gives better estimates than the MLE. Results show that the EF approach is
compatible with the ML method in parameter estimation. Furthermore, the computation
speed for the EF approach is much faster than for the MLE and therefore offers a
significant reduction of the completion time.
Keywords: Weibull distribution, Autoregression, Conditional duration, Estimatingfunction, Maximum likelihood, Standard error, Applications, Financial data, Semi-parametric, High frequency data, Transactions, Time series.
where &; is follow the standardized Weibull distribution with parameter a.To assess the performance ofML and EF methods given in Section (3.1) and (3.2)
on this two models, the standard errors were computed. Standard errors of oi.a.b,a for
the Modell are 0.0477,0.0107,0.0217 and 0.0116 respectively. The standard errors of
oi.a.b.a for the Model 2 are 0.0506,0.0114,0.0231 and 0.0223. The EF method in
general is comparable to the ML method in term of parameter estimates and standard
errors. Furthermore, we note that if we use the ML method to find the estimates, the
method needs to search for the maximum value under the maximum likelihood
procedure. One the other hand, th EF approach is just solving the simultaneous
equations to obtain the estimates. Thus, we would expect a reduction in computation time
if we use EF method instead of that based on the ML method. The reason is that the EF
method is only involved in solving the simultaneous nonlinear equations while the ML
method needs to search for the maximum value of likelihood function. It is important to
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note t~at the EF method requires 8.172 seconds in a Core 2 Duo 2.2 GHz computer to
obtain .the solution while the ML method requires 41.578 seconds.
5. Co elusion
This paper applied the EF approach in parameter estimation of Weibull ACD
models nd compared the properties with the corresponding ML estimates. Results show
that the tandard errors of the estimates using either EF or ML methods are comparable.
However, the computation time for EF method is much shorter than that of the ML
method.
Referenc s
[1] Allen, D., Chan, F., McAleer, M., Peiris, M.S. (2008), Finite sample properties oftheQMLE for the Log-ACD model: Application to Australian stocks, Journal ofEconometrics, 147, 163-185.
[2] Bauwens, L. and Giot, P. (2000), The logarithmic ACD model: An application to thebid-ask quote process of three NYSE stocks. Annales D 'Economie et de Statistique60, 117-145.
[3] Engle, R.F. and Russell, l.R. (1998), Autoregressive conditional duration: A newmodel for irregularly spaced transaction data, Econometrica, 66, 1127-1162.
[4] Engle.R.F. (1982), Autoregressive conditional heteroscedasticity with estimates ofvariance of Ll.K. inflation, Econometrica, 31,987-1008.
[5] Godambe, V.P. (1985), The foundations of finite sample estimation in stochasticprocesses, Biometrika, 72,419-428.
[6] Peiris, M.S., Allen, D. and Yang, W. (2005), Some statistical models for durationsand an application to news corporation stock prices, Mathematics and Computers inSimulation, 68, 549-556.
[7] Peiris, M.S. , K.H, Ng. and Ibrahim,M. (2007), A review of Recent Developments ofFinancial Time Series: ACD Modelling using the Estimating Function Approach, SriLankan Journal of Applied Statistics, 8:1-17.
[8] Peiris, M.S. and KH, Ng. (2007), Optimal estimation of autoregressive models withnon-stationary innovations: a simulation study. (submmitted to International Journalof Modelling and Simulation)
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[9] Tsay, R.S. (2002), Analysis offinancial time series, John Wiley & Inc.
[10] Peiris,M.S., K.B, Ng and Allen, D. (2008), On estimation of Weibull ACD modelsusing estimating functions: A simulation study. (submitted for publication)
[11] Thavaneswaran. A, Peiris, M.S.(1996), Nonparametric estimation for somenonlinear models, Statistics and Probability Letters, 28,227-233.
[12] Zhang, M.Y. Russell, J.R. and Tsay, R,S. (2001), A nonlinear autoregressiveconditional duration model with applications to financial duration data, Journal ofEconometrics, 104, 179-207.
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Figure 1: Time plots of durations for IBM stock traded in the first five trading days ofNovember 1990: the adjusted series.