, Volume 81, Issue 6, pp 619–651 | Cite as

Goodness-of-fit testing of a count time series’ marginal distribution

  • Christian H. WeißEmail author


Popular goodness-of-fit tests like the famous Pearson test compare the estimated probability mass function with the corresponding hypothetical one. If the resulting divergence value is too large, then the null hypothesis is rejected. If applied to i. i. d. data, the required critical values can be computed according to well-known asymptotic approximations, e. g., according to an appropriate \(\chi ^2\)-distribution in case of the Pearson statistic. In this article, an approach is presented of how to derive an asymptotic approximation if being concerned with time series of autocorrelated counts. Solutions are presented for the case of a fully specified null model as well as for the case where parameters have to be estimated. The proposed approaches are exemplified for (among others) different types of CLAR(1) models, INAR(p) models, discrete ARMA models and Hidden-Markov models.


Count time series Goodness-of-fit test Estimated parameters Asymptotic approximation Quadratic-form distribution 

Mathematics Subject Classification

60G10 62F03 62F05 62M10 



The author thanks the Editor, the Associate Editor and the referees for carefully reading the article and for their comments, which greatly improved the article. The iceberg order data of Sect. 3.4 were kindly made available to the author by the Deutsche Börse. Prof. Dr. Joachim Grammig, University of Tübingen, is to be thanked for processing of it to make it amenable to data analysis. I am also very grateful to Prof. Dr. Robert Jung, University of Hohenheim, for his kind support to get access to the data.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsHelmut Schmidt UniversityHamburgGermany

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