Kohärente Vorhersage und Risikoanalyse für Zähldatenprozesse

 
Dreijähriges Projekt, gefördert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 394832307.

Projektziele:

Das Projekt befasst sich mit der kohärenten Prognose für Zähldatenprozesse. Zur Generierung einer kohärenten Prognose werden jeweils ein modellbasierter und ein approximativer Prognoseansatz systematisch verglichen: „Liefert das modellbasierte Prognoseverfahren einen messbaren Mehrwert für die Vorhersage von Zähldaten?“. Hierbei sind u.a. geeignete Kriterien zur Bemessung und zum effizienten Vergleich der Prognosegüte zu spezifizieren. Schließlich soll auch der Einfluss geschätzter Verteilungsparameter untersucht werden.

Extreme Quantile werden auch im Bereich der Risikoanalyse benötigt. Ziel ist die Vorhersage eines Risikos mittels prognostizierter Quantile und daraus abgeleiteter Risikomaße (auch unter Schätzunsicherheit), ferner die Bemessung der „Güte“ der Risikoprognose („Wie genau sind Prognose und prognostiziertes Risiko?“).

Für den modellbasierten Ansatz soll eine Vielzahl an Modellen herangezogen werden und somit die kohärente Prognose in einer bisher nicht dagewesenen Breite beleuchten. Neben INAR(𝑝)-Modellen mit allgemeineren Randverteilungen (etwa zur Berücksichtigung der in der Realität weit verbreiteten Phänomene der Überdispersion und Nullinflation) und Modellen für den Fall eines endlichen Trägers der Form {0,…,n} sollen insbesondere Modelle einbezogen werden, die jenseits ARMA-artiger Autokorrelationsstrukturen angesiedelt sind. Hierbei ist u.a. an Regressionsansätze gedacht, mit denen beispielsweise Trend und Saisonalität berücksichtigt werden können.

Projektlaufzeit:

Oktober 2018 – September 2021.

Projektresultate:

• 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: Plot-Appendix.zip.
   
• 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: Supplement-Material.zip.
   
• Weiß, C.H., Homburg, A., Alwan, L.C., Frahm, G., Göb, R. (2022):
  Efficient Accounting for Estimation Uncertainty in Coherent Forecasting of Count Processes.
  Journal of Applied Statistics 49(8), pp. 1957-1978 (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: RiskPredCountTS_suppl.zip.
   
• Homburg, A., Weiß, C.H., Alwan, L.C., Frahm, G., Göb, R. (2023):
  PMF-Forecasting for Count Processes: A Comprehensive Performance Analysis.
  Theory and Applications of Time Series Analysis and Forecasting: Selected Contributions from ITISE 2021, Contributions to Statistics, Springer, pp. 79-90 (direct access).
   
  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: Full simulation results PmfPredCountTS_suppl.zip.
   
  Corresponding plenary talk „On PMF-Forecasting for Count Processes“ at the 7th International conference on Time Series and Forecasting (ITISE 2021), Granada, 19. – 21. Juli, 2021, see the video and the slides.