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.

 

Projektziele:

Das anlaufende IFF-Projekt befasst sich mit der Modelldiagnostik für Zähldatenzeitreihen, siehe auch das folgende Poster für Hintergrundinformationen. Ziele des Projekts sind eine Untersuchung von Anwendbarkeit und Performanz vorhandener diagnostischer Werkzeuge sowie die Entwicklung neuartiger Diagnoseverfahren. Hierzu sollen zunächst exemplarisch die sog. Pearson-Residuen eingehend analysiert und theoretisch fundiert werden. Dabei sollen für möglichst verschiedene Modellklassen analytische Ausdrücke für die asymptotische Verteilung von Mittelwert und Varianz der Pearson-Residuen gefunden werden, was wiederum neuartige Signifikanztests ermöglicht; die Performanz der entwickelten Tests soll dann mittels Simulationen untersucht und alle Ergebnisse in einem Arbeitspapier zusammengefasst werden.

Parallel dazu soll während der IFF-Phase ein bei der DFG zu beantragendes weiterführendes Projekt ausführlich vorbereitet werden. Dazu zählen die Vorbereitung von Ansätzen für neue diagnostische Tests sowie das Design der angestrebten Simulationsstudien. Einen weiteren Schwerpunkt stellt das Thema Bootstrap dar: für existierende Goodness-of-Fit-, Dispersions- und Nullinflationstests sind Untersuchungen nur innerhalb spezieller Modellfamilien möglich, da die jeweilige asymptotische Verteilung der Teststatistiken auf ebendiesen Modellannahmen beruht. Ein geeigneter Bootstrap-Ansatz würde die Chance bieten, die Tests auch auf andere Modellfamilien zu übertragen; aber wann ist dies möglich, und welche Performanz ergibt sich? Während der IFF-Phase soll ein Plan erarbeitet werden, wie welche Bootstrap-Verfahren zur Modelldiagnostik für Zähldatenzeitreihen entwickelt werden könnten.

Projektlaufzeit:

Juli 2018 – Juni 2019.

Publikationen:

 

 

HSU

Letzte Änderung: 12. February 2021