A performance analysis of prediction intervals for count time series

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.

Efficient accounting for estimation uncertainty in coherent forecasting of count processes

“…While these numbers improve with increasing sample size T and with increasing α, they become worse with increasing μ (which is equal to the variance of the equidispersed Poisson distribution). These results regarding the effect of estimation uncertainty on the forecasts are in line with the studies by Homburg et al [7,8]. Now, the question is how to incorporate the parameter uncertainty into the coherent forecasting.…”

“…The effect of estimation uncertainty on coherent forecasts was investigated for selected models by Freeland and McCabe [5], Jung and Tremayne [12], Silva et al [21], Bisaglia and Gerolimetto [3], and more comprehensively for a large variety of count processes by Homburg et al [7,8]. The latter authors concluded that the coherent central PFs are usually only slightly affected by estimation error, whereas the non-central and the interval forecasts may suffer a lot from estimated parameters.…”

“…Motivated by such applications, there are also many papers on the forecasting of the underlying count process . In particular, coherent forecasting techniques have been investigated by many researchers, such as [ 3 , 5 , 7 , 8 , 12 , 16 , 17 , 21 ]. The term coherent forecasting was first used by Freeland and McCabe [ 5 ] to express that the computed forecasts of a count process should also be count values themselves.…”

“…[ 21 ], Bisaglia and Gerolimetto [ 3 ], and more comprehensively for a large variety of count processes by Homburg et al. [ 7 , 8 ]. The latter authors concluded that the coherent central PFs are usually only slightly affected by estimation error, whereas the non-central and the interval forecasts may suffer a lot from estimated parameters.…”

Analysis and Forecasting of Risk in Count Processes

“…Although approximate risk forecasting might appear attractive at a first glance, because of simplified computations and readily available software implementations (e. g., Chan and Nadarajah 2019), it is not clear whether the obtained risk forecasts are indeed competitive to the coherent risk forecasts or they lead to a serious misjudgment of the actual risk. The studies of Homburg et al (2019Homburg et al ( , 2020 about point and interval forecasting of count time series may serve as a warning that approximate approaches might end up in rather misleading results. Therefore, besides a detailed analysis of coherent risk forecasting for various count processes in the presence of estimation uncertainty, we also investigate the performance of approximate risk forecasting in comparison to the coherent forecasting approach.…”