Nonlinear Least Squares Estimation of Log-ACD Models

  • Zhao Chen
  • Wei Liu
  • Christina Dan Wang
  • Wu-qing Wu
  • Yao-hua Wu


This paper studies a nonlinear least squares estimation method for the logarithmic autoregressive conditional duration (Log-ACD) model. We establish the strong consistency and asymptotic normality for our estimator under weak moment conditions suitable for applications involving heavy-tailed distributions. We also discuss inference for the Log-ACD model and Log-ACD models with exogenous variables. Our results can be easily translated to study Log-GARCH models. Both simulation study and real data analysis are conducted to show the usefulness of our results.


Log-ACD model nonlinear least squares estimation Log-GARCH model heavy-tail 

2000 MR Subject Classification



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  1. [1]
    Allen, D., Chan, F., McAleer, M., Peiris, S. Finite Sample Properties of the QMLE for the Log-ACD Model: Application to Australian Stocks. Working paper, 2006zbMATHGoogle Scholar
  2. [2]
    Amemiya, T. Advanced Econometrics. Harvard University Press, Cambridge, Massachusetts,1985Google Scholar
  3. [3]
    Bauwens, L., Giot, P. The Logarithmic ACD Model: An Application to the Bid-ask Quote Process of Three NYSE Stocks. Annales D’Economie et de Statistique, 60: 117–145 (2000)CrossRefGoogle Scholar
  4. [4]
    Bauwens, L., Giot, P. Econometric Modelling of Stock Market Intraday Activity. Boston: Kluwer Academic Publishers, 2001CrossRefzbMATHGoogle Scholar
  5. [5]
    Bauwens, L., Galli, F. and Giot, P. The Moments of Log-ACD Models, CORE Discussion Paper. Available at SSRN: or DOI: 10.2139/ssrn.375180, 2003Google Scholar
  6. [6]
    Billingsley, P. Convergence of Probability Measures. New York: Wiley, 1999CrossRefzbMATHGoogle Scholar
  7. [7]
    Dufour, A., Engle, R. F. The ACD Model: Predictability of the Time between Consecutive Trades. Discussion papers in Finance. Zurich: ISMA Centre, 59Google Scholar
  8. [8]
    Engle, R., Russell, J. Autoregressive Conditional Duration: A New Model for Irregularly Spaced Data. Econometrica, 66(5): 1127–1162 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Geweke, J. Modeling the Persistence of Conditional Variances: A Comment. Econometric Reviews, 5: 57–61 (1986)CrossRefGoogle Scholar
  10. [10]
    Pantula, S.G. Modeling the Persistence of Conditional Variances: A Comment. Econometric Reviews, 5: 71–74 (1986)CrossRefGoogle Scholar
  11. [11]
    Straumann, D. Estimation in Conditionally Heteroscedastic Time Series Models, Lecture Notes in Statistics. Heidelberg: Springer, 2005zbMATHGoogle Scholar
  12. [12]
    Zhang, M.Y., Russell, J.R., Tsay, R.S. A Nonlinear Autoregressive Conditional Duration Model with Applications to Financial Transaction Data. Journal of Econometrics, 104: 179–207 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Zhao Chen
    • 1
  • Wei Liu
    • 2
  • Christina Dan Wang
    • 3
  • Wu-qing Wu
    • 4
  • Yao-hua Wu
    • 5
  1. 1.School of Data ScienceFudan UniversityShanghaiChina
  2. 2.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.New York University ShanghaiShanghaiChina
  4. 4.School of BusinessRenmin University of ChinaBeijingChina
  5. 5.Department of Statistics and FinanceUniversity of Science and Technology of ChinaHefeiChina

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