Coherent Forecasting and Risk Analysis for Count Processes

Project partners:


Three-years project, funded by Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 394832307.

Project aims:

The project considers the coherent forecasting of count processes. For generating a coherent prediction, we always systematically compare a model-based and an approximative forecasting approach: “Does the model-based forecasting approach lead to a notable added value for the prediction of count data?”. For this purpose (among others), appropriate criteria for the evaluation and efficient comparison of the forecast performance are to be specified. Also the effect of estimated model parameters is to be investigated.

Extreme quantiles are also need for risk analysis. The aim is to predict a risk by a quantile forecast and corresponding deduced risk measures (also under estimation uncertainty), and also the evaluation of the “goodness” of the risk prection (“How accurate are prediction and forecasted risk?”).

For the model-based approach, a multitude of models shall be considered such that the coherent forecasting is examined in an yet unprecedented breadth. Besides INAR(𝑝) models with rather general marginal distributions (e.g., to consider the phenomena of overdispersion and zero inflation, which are quite common in real applications) and models for the case of a bounded support of the form {0,…,n}, also models beyond an ARMA-like autocorrelation structure shall be included. Here, it is intended to use regression approaches being able to deal with trend and seasonality.

Project duration:

October 2018 – September 2021.

Project results:

  • Homburg, A., Weiß, C.H., Alwan, L.C., Frahm, G., Göb, R. (2019):
    Evaluating Approximate Point Forecasting of Count Processes.
    Econometrics 7(3), 30 (open access),
    Special Issue “Discrete-Valued Time Series: Modelling, Estimation and Forecasting”.
    Abstract: In forecasting count processes, practitioners often ignore the discreteness of counts and compute forecasts based on Gaussian approximations instead. For both central and non-central point forecasts, and for various types of count processes, the performance of such approximate point forecasts is analyzed. The considered data-generating processes include different autoregressive schemes with varying model orders, count models with overdispersion or zero inflation, counts with a bounded range, and counts exhibiting trend or seasonality. We conclude that Gaussian forecast approximations should be avoided.
    Supplementary material:
  • Nik, S., Weiß, C.H. (2020):
    CLAR(1) Point Forecasting under Estimation Uncertainty.
    Statistica Neerlandica 74(4), pp. 489-516 (open access).
    Abstract: Forecast error is not only caused by the randomness of the data-generating process, but also by the uncertainty due to estimated model parameters. We investigate these different sources of forecast error for a popular type of count process, the Poisson first-order integer-valued autoregressive (INAR(1)) process. But many of our analytical derivations also hold for the more general family of conditional linear AR(1) (CLAR(1)) processes. In addition, results from a simulation study are presented, to verify and complement our asymptotic approximations.
  • Homburg, A., Weiß, C.H., Alwan, L.C., Frahm, G., Göb, R. (2021):
    A Performance Analysis of Prediction Intervals for Count Time Series.
    Journal of Forecasting 40(4), pp. 603-625 (open access).
    Abstract: One of the major motivations for the analysis and modeling of time series data is the forecasting of future outcomes. The use of interval forecasts instead of point forecasts allows us to incorporate the apparent forecast uncertainty. When forecasting count time series, one also has to account for the discreteness of the range, which is done by using coherent prediction intervals (PIs) relying on a count model. We provide a comprehensive performance analysis of coherent PIs for diverse types of count processes. We also compare them to approximate PIs that are computed based on a Gaussian approximation. Our analyses rely on an extensive simulation study. It turns out that the Gaussian approximations do considerably worse than the coherent PIs. Furthermore, special characteristics such as overdispersion, zero inflation, or trend clearly affect the PIs’ performance. We conclude by presenting two empirical applications of PIs for count time series: the demand for blood bags in a hospital, and the number of company liquidations in Germany.
    Supplementary material:
  • Weiß, C.H., Homburg, A., Alwan, L.C., Frahm, G., Göb, R. (2021):
    Efficient Accounting for Estimation Uncertainty in Coherent Forecasting of Count Processes.
    Accepted for publication in Journal of Applied Statistics (open access).
    Abstract: Coherent forecasting techniques for count processes generate forecasts that consist of count values themselves. In practice, forecasting always relies on a fitted model and so the obtained forecast values are affected by estimation uncertainty. Thus, they may differ from the true forecast values as they would have been obtained from the true data generating process. We propose a computationally efficient resampling scheme that allows to express the uncertainty in common types of coherent forecasts for count processes. The performance of the resampling scheme, which results in ensembles of forecast values, is investigated in a simulation study. A real-data example is used to demonstrate the application of the proposed approach in practice. It is shown that the obtained ensembles of forecast values can be presented in a visual way that allows for an intuitive interpretation.
  • Homburg, A., Weiß, C.H., Frahm, G., Alwan, L.C., Göb, R. (2021):
    Analysis and Forecasting of Risk in Count Processes.
    Journal of Risk and Financial Management 14(4), 182 (open access).
    Abstract: Risk measures are commonly used to prepare for a prospective occurrence of an adverse event. If we are concerned with discrete risk phenomena such as counts of natural disasters, counts of infections by a serious disease, or counts of certain economic events, then the required risk forecasts are to be computed for an underlying count process. In practice, however, the discrete nature of count data is sometimes ignored and risk forecasts are calculated based on Gaussian time series models. But even if methods from count time series analysis are used in an adequate manner, the performance of risk forecasting is affected by estimation uncertainty as well as certain discreteness phenomena. To get a thorough overview of the aforementioned issues in risk forecasting of count processes, a comprehensive simulation study was done considering a broad variety of risk measures and count time series models. It becomes clear that Gaussian approximate risk forecasts substantially distort risk assessment and, thus, should be avoided. In order to account for the apparent estimation uncertainty in risk forecasting, we use bootstrap approaches for count time series. The relevance and the application of the proposed approaches are illustrated by real data examples about counts of storm surges and counts of financial transactions.
    Supplementary material:
  • Homburg, A., Weiß, C.H., Alwan, L.C., Frahm, G., Göb, R. (2021):
    On PMF-Forecasting for Count Processes.
    Accepted for publication in Proceedings of ITISE 2021, 2021.
    Abstract: Coherent forecasting techniques account for the discrete nature of count processes. Besides point and interval forecasts, a third way for achieving coherent forecasts is to consider the full predictive probability mass function (PMF) the actual forecast value. For a large variety of count processes, the performance of PMF forecasting under estimation uncertainty is analyzed. Furthermore, also Gaussian approximate PMF forecasting is investigated. Different approaches for performance evaluation are taken into consideration, with the main focus on mean squared errors computed for either the full PMF, or its lower and upper tails, respectively. A real-world example from finance is presented for illustration.
    Supplementary material: Supplemenatry sections PmfPredCountTS_suppl.pdf, full simulation results
  • to be continued!


Einjähriges Projekt, gefördert durch die Interne Forschungsförderung (IFF2016) der HSU Hamburg.


Die einjährige IFF-Förderung des Projektes ermöglichte die Zwischenfinanzierung einer Stelle, welche mit einer Nachwuchswissenschaftlerin besetzt wurde. Im Rahmen dieser Förderung wurden exemplarisch für Poisson-INAR(1)-Prozesse Konzepte entwickelt und Codes in der Programmiersprache R implementiert, mit denen die exakte ℎ-Schritt-Vorhersageverteilung und diverse Gauß-AR(1)-Approximationen berechnet werden können. Insbesondere wurden diverse Kenngrößen und grafische Werkzeuge zur Bewertung der Güte der Approximation implementiert. Ein Teil der Resultate wurde von der geförderten Nachwuchswissenschaftlerin in einem Arbeitspapier zusammengefasst. Auch konnten diverse reale Datensätze gesammelt werden, welche die praktische Bedeutung der Vorhersageproblematik illustrieren.

Mithilfe der IFF-Stelle konnte ein inhaltlich wesentlich erweiterter Projektantrag unter dem gleichlautenden Titel „Kohärente Vorhersage und Risikoanalyse für Zähldatenprozesse“ erarbeitet werden, welcher die kohärente Prognose und Risikoanalyse in einer bisher nicht dagewesenen Breite beleuchten soll. Dieser Projektantrag wurde am 05.04.2018 von der Deutschen Forschungsgemeinschaft (DFG) bewilligt.

Einen Überblick über das IFF-Projekt bietet folgendes Poster.


Juli 2016 – Juni 2017.





Letzte Änderung: 2. June 2021