Model Diagnostics for Count Time Series

Project partner:


Two-years project, funded by Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 437270842.

Project aims:

Time Series of counts arise in many different situations in economics and related fields. They can have various forms with respect to their dependence structure or their marginal distribution. As classical models for real-valued time series are not able to maintain the discrete nature of count data, a great many of models tailor-made for time series of counts have been developed. An adequate modelling of count data processes is important, to do forecasting, to monitor the subsequent process of the time series to reveal structural changes as soon as possible, or just to obtain a better understanding of the underlying count data process.

The planned research project on model diagnostics for time series of counts comprises three central steps of model building: model identification, model selection and model validation. While a large number of methods have been proposed for real-valued (continuous) time series and are available since a long time, corresponding approaches for discrete-valued time series are far less developed. The existing methods are scarce and most of them are available only in rudimentary form (e.g., as heuristic application guidelines), and more rigorous, theory-based methods rely on restrictive model assumptions or focus on isolated characteristics of the process as, e.g., dispersion. Corresponding issues do also hold for goodness-of-fit tests: while numerous goodness-of-fit tests for continuous-valued time series have been proposed that are not only capable to test for specific models but also whole model classes, the applicability of available goodness-of-fit tests for time series of counts is restricted, e.g., to parametric assumptions.

The planned research project features two complimentary lines of attack for model diagnosis in time series of counts. On the one hand, we aim to develop parametric methods for model diagnosis for time series of counts, which take into account various characteristics of the underlying distribution and/or dependence pattern. Further, diagnostic tools developed and widely applied for real-valued time series shall be made applicable also to time series of counts by using suitable parametric bootstrap implementations. On the other hand, we aim to develop goodness-of-fit tests based on joint distributions that are capable to consistently distinguish between different model classes. For the implementation, but also to allow for a broader applicability of the above-mentioned diagnostic tools, suitable semi-parametric bootstrap methods for time series of counts shall be developed and employed for model diagnostics. For the proposed methods, we want to investigate in detail the performance and the applicability by elaborate comparative simulations studies and applications to real data sets relevant in economic sciences.

Project duration:

October 2020 – September 2022.

Project results:

  • The first parametric methods for model diagnosis to be developed focus on count time series with a Poisson marginal distribution. The aim is to develop statistical tests that use the Stein-Chen identity. As a corresponding preparatory work, we wrote the following article:
    Weiß, C.H., Aleksandrov, B. (2020):
    Computing (Bivariate) Poisson Moments using Stein–Chen Identities.
    Accepted for publication in The American Statistician.
    Abstract: The (bivariate) Poisson distribution is the most common distribution for (bivariate) count random variables. The univariate Poisson distribution is characterized by the famous Stein–Chen identity. We demonstrate that this identity allows to derive even sophisticated moment expressions in such a simple way that the corresponding computations can be presented in an introductory Statistics class. Then, we newly derive different types of Stein–Chen identity for the bivariate Poisson distribution. These are shown to be very useful for computing joint moments, again in a surprisingly simple way. We also explain how to extend our results to the general multivariate case.
  • Aleksandrov, B., Weiß, C.H., Jentsch, C. (2020):
    Goodness-of-Fit Tests for Poisson Count Time Series based on the Stein–Chen Identity.
    Working paper, in preparation.
    Abstract: To test the null hypothesis of a Poisson marginal distribution, test statistics based on the Stein–Chen identity are proposed. For a wide class of Poisson count time series, the asymptotic distribution of different types of Stein–Chen statistics is derived, also if multiple statistics are jointly applied. The performance of the tests is analyzed with simulations, as well as the question which Stein–Chen functions should be used for which alternative. Illustrative data examples are presented, and possible extensions of the novel Stein–Chen approach are discussed as well.
  • to be continued!


Einjähriges Projekt, gefördert durch die Interne Forschungsförderung (IFF2018) der HSU Hamburg.



Die IFF-Förderung des Projektes „Modelldiagnostik für Zähldatenzeitreihen“ ermöglichte die Zwischenfinanzierung einer wissenschaftlichen Mitarbeiterstelle, welche mit einem Nachwuchswissenschaftler besetzt wurde. Im Rahmen dieser Förderung wurden für spezielle Arten von Zähldatenprozess, einen sog. Poisson-INAR(1)- und Poisson-INARCH(1)-Prozess, analytische Ausdrücke für die asymptotische Verteilung von Quadratmittel und Varianz der Pearson-Residuen hergeleitet, was wiederum neuartige Signifikanztests ermöglichte. Die Performanz der entwickelten Tests wurde mittels Simulationen untersucht und alle Ergebnisse gemeinsam mit dem geförderten Nachwuchswissenschaftler in dem Manuskript „Testing the Dispersion Structure of Count Time Series Using Pearson Residuals“ zusammengefasst. Das Manuskript wurde von den „AStA Advances in Statistical Analysis“ zur Veröffentlichung angenommen.

Mithilfe der durch die IFF finanzierten Stelle konnte ein inhaltlich wesentlich erweiterter Projektantrag unter dem gleichlautenden Titel „Modelldiagnostik für Zähldatenzeitreihen“ erarbeitet werden, der in wesentlichen Punkten auf den Inhalten des Antrags der IFF basiert. Der Projektantrag (Sachbeihilfe) wurde am 20.12.2019 von der Deutschen Forschungsgemeinschaft (DFG) bewilligt.

Einen Überblick über das IFF-Projekt bietet folgendes Poster.


Juli 2018 – Juni 2019.





Letzte Änderung: 29. April 2021