Frahm, Gabriel

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) as 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.

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.

We propose the outperformance probability as a new performance measure, which can be used in order to compare a strategy with a specified benchmark, and develop the basic statistical properties of its maximum-likelihood estimator in a Brownian-motion framework. The given results are used to investigate the question of whether mutual funds are able to beat the S&P 500 or the Russell 1000. Most mutual funds that are taken into consideration are, in fact, able to beat the market. We argue that one should refer to differential returns when comparing a strategy with a given benchmark and not compare both the strategy and the benchmark with the money-market account. This explains why mutual funds often appear to underperform the market, but this conclusion is fallacious.