Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data

  • Helton Saulo
  • Jeremias Leão
  • Víctor Leiva
  • Robert G. Aykroyd
Regular Article


Modern financial markets now record the precise time of each stock trade, along with price and volume, with the aim of analysing the structure of the times between trading events—leading to a big data problem. In this paper, we propose and compare two Birnbaum–Saunders autoregressive conditional duration models specified in terms of time-varying conditional median and mean durations. These models provide a novel alternative to the existing autoregressive conditional duration models due to their flexibility and ease of estimation. Diagnostic tools are developed to allow goodness-of-fit assessment and to detect departures from assumptions, including the presence of outliers and influential cases. These diagnostic tools are based on the parameter estimates using residual analysis and the Cook distance for global influence, and different perturbation schemes for local influence. A thorough Monte Carlo study is presented to evaluate the performance of the maximum likelihood estimators, and the forecasting ability of the models is assessed using the traditional and density forecast evaluation techniques. The Monte Carlo study suggests that the parameter estimators are asymptotically unbiased, consistent and normally distributed. Finally, a full analysis of a real-world financial transaction data set, from the German DAX in 2016, is presented to illustrate the proposed approach and to compare the fitting and forecasting performances with existing models in the literature. One case related to the duration time is identified as potentially influential, but its removal does not change resulting inferences demonstrating the robustness of the proposed approach. Fitting and forecasting performances favor the proposed models and, in particular, the median-based approach gives additional protection against outliers, as expected.


Big data Birnbaum–Saunders distribution Forecasting ability Influence diagnostics Likelihood-based methods Monte Carlo simulation R software 



The authors thank the Editors and reviewers for their constructive comments on an earlier version of this manuscript. The research was partially supported by CNPq and CAPES Grants from the Brazilian Government and by FONDECYT 1160868 Grant from the Chilean Government.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Helton Saulo
    • 1
    • 2
  • Jeremias Leão
    • 3
  • Víctor Leiva
    • 4
    • 5
  • Robert G. Aykroyd
    • 6
  1. 1.Institute of Mathematics and StatisticsUniversidade Federal de GoiásGoiâniaBrazil
  2. 2.Department of StatisticsUniversidade de BrasíliaBrasíliaBrazil
  3. 3.Department of StatisticsUniversidade Federal do AmazonasManausBrazil
  4. 4.School of Industrial EngineeringPontificia Universidad Católica de ValparaísoValparaísoChile
  5. 5.Faculty of AdministrationAccounting and Economics, Universidad Federal de GoiásGoiâniaBrazil
  6. 6.Department of StatisticsUniversity of LeedsLeedsUK

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