Statistical Papers

, Volume 60, Issue 1, pp 123–146 | Cite as

A generalized least squares estimation method for the autoregressive conditional duration model

  • Wanbo LuEmail author
  • Rui Ke
Regular Article


A generalized least squares estimation method with inequality constraints for the autoregressive conditional duration model is proposed in this paper. The estimation procedure includes three stages. The final generalized least-squares estimator is consistent and \(\sqrt{T}\)—asymptotically normal distributed. Our estimator has the advantage over the often used quasi-maximum likelihood estimator in which it easily implemented and does not require the choice of initial values for the iterative optimization procedure. A large number of simulation studies confirm our theoretical results and suggest that the proposed estimator is more robust compared to quasi-maximum likelihood estimator. An application to IBM volume duration shows that the performance of the proposed estimation is better than quasi-maximum likelihood estimation in forecasting.


Autoregressive conditional duration model Generalized least squares estimator Quasi-maximum likelihood estimator Monte Carlo simulation 

Mathematics Subject Classification

62F12 62M10 



Wanbo Lu’s research is sponsored by the National Science Foundation of China (71101118) and the Program for New Century Excellent Talents in University (NCET-13-0961) and the Fundamental Research Funds for the Central Universities (JBK150501, JBK16062) in China. Rui Ke’s research is sponsored by the Fundamental Research Funds for the Central Universities (JBK1507102) in China.


  1. Allen D, Chan F, McAleer M, Peiris S (2008) Finite sample properties of the QMLE for the Log-ACD model: application to Australian stocks. J Economet 147(1):163–185MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alonso AM, Daniel P, Romo J (2006) Introducing model uncertainty by moving blocks bootstrap. Stat Pap 47(2):167–179MathSciNetCrossRefzbMATHGoogle Scholar
  3. An HZ, Zhao CG, Hannan EJ (1982) Autocorrelation, autoregression and autoregressive approximation. Ann Stat 10(3):926–936MathSciNetCrossRefzbMATHGoogle Scholar
  4. Barczy M, Espany M, Pap G, Scotto M, Silva ME (2012) Additive outliers in INAR(1) models. Stat Pap 53(4):935–949MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bauwens L, Giot P (2001) Econometric modelling of stock market intraday activity. Springer Science & Business Media, New YorkCrossRefzbMATHGoogle Scholar
  6. Brockwell PJ, Davis RA (2013) Time series: theory and methods. Springer Science & Business Media, New YorkzbMATHGoogle Scholar
  7. Bühlmann P (1997) Sieve bootstrap for time series. Bernoulli 3(2):123–148MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chen C, Liu LM (1993) Joint estimation of model parameters and outlier effects in time series. J Am Stat Assoc 88(421):284–297zbMATHGoogle Scholar
  9. Chiang MH, Wang LM (2012) Additive outlier detection and estimation for the logarithmic autoregressive conditional duration model. Commun Stat Simul Comput 41(3):287–301MathSciNetCrossRefzbMATHGoogle Scholar
  10. Diebold FX (2012) Empirical modeling of exchange rate dynamics. Springer Science & Business Media, New YorkzbMATHGoogle Scholar
  11. Engle RF (2000) The econometrics of ultra-high-frequency data. Econometrica 68(1):1–22CrossRefzbMATHGoogle Scholar
  12. Engle RF (2002) New frontiers for ARCH models. J Appl Economet 17(5):425–446CrossRefGoogle Scholar
  13. Engle RF, Russell JR (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66(5):1127–1162MathSciNetCrossRefzbMATHGoogle Scholar
  14. Francq C, Zakoian JM (2011) GARCH models: structure, statistical inference and financial applications. Wiley, LondonGoogle Scholar
  15. Gonçalves S, Kilian L (2007) Asymptotic and bootstrap inference for AR (\(\infty \)) processes with conditional heteroskedasticity. Economet Rev 26(6):609–641MathSciNetCrossRefzbMATHGoogle Scholar
  16. Grammig J, Maurer KO (2000) Non-monotonic hazard functions and the autoregressive conditional duration model. Economet J 3(1):16–38CrossRefzbMATHGoogle Scholar
  17. Greene WH (2007) Econometric analysis, 6th edn. Pearson Prentice Hall, Upper Saddle RiverGoogle Scholar
  18. Hannan EJ, Deistler M (1988) The statistical theory of linear systems. Wiley, New YorkzbMATHGoogle Scholar
  19. Hautsch N (2011) Econometrics of financial high-frequency data. Springer Science & Business Media, New YorkzbMATHGoogle Scholar
  20. Koreisha S, Pukkila T (1990) A generalized least squares approach for estimation of autoregressive moving average models. J Time Ser Anal 11(2):139–151MathSciNetCrossRefzbMATHGoogle Scholar
  21. Liu W, Wang H, Chen M (2011) Least absolute deviation estimation of autoregressive conditional duration model. Acta Math Appl Sin 27(2):243–254MathSciNetCrossRefzbMATHGoogle Scholar
  22. Liew CK (1976) Inequality constrained least-squares estimation. J Am Stat Assoc 71(355):746–751MathSciNetCrossRefzbMATHGoogle Scholar
  23. Ng S, Perron P (1995) Unit root tests in ARMA models with data-dependent methods for the selection of the truncation lag. J Am Stat Assoc 90(429):268–281MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wahlberg B (1989) Estimation of autoregressive moving, average models via high-order autoregressive approximations. J Time Ser Anal 10(3):283–299MathSciNetCrossRefzbMATHGoogle Scholar
  25. Werner HJ (1990) On inequality constrained generalized least-squares estimation. Linear Algebra Appl 127:379–392MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of StatisticsSouthwestern University of Finance and EconomicsChengduChina

Personalised recommendations