arXiv subjects

Fabian Krüger

Publications and source records attributed to Fabian Krüger.

14 recordsLinked to original sources

arXiv2026

A Kernel Score Perspective on Forecast Disagreement and the Linear Pool

This paper generalizes several results on linear pooling from squared error loss to all kernel scores. The latter are a rich family of scoring rules that covers point and distribution forecasts for univariate and multivariate, discrete and continuous settings. Its members include the Continuous Ranked Probability Score for univariate distribution forecasting and the Energy Score for multivariate distribution forecasting. Our results indicate that forecast disagreement (measured as the average pairwise divergence of all component distributions) has important implications for the linear pool's performance. The results are useful for understanding and designing linear pools in general combination settings. In particular, they motivate using the linear pool (as opposed to other combination formulas) and yield a novel condition under which equal combination weights are optimal under a given kernel scoring rule.

econ.EM↗

arXiv2025

Simple Macroeconomic Forecast Distributions for the G7 Economies

We present a simple method for predicting the distribution of output growth and inflation in the G7 economies. The method is based on point forecasts published by the International Monetary Fund (IMF), as well as robust statistics from the empirical distribution of the IMF's past forecast errors while imposing coherence of prediction intervals across horizons. We show that the technique yields calibrated prediction intervals and performs similar to, or better than, more complex time series models in terms of statistical loss functions. We provide a simple website with graphical illustrations of our forecasts, as well as time-stamped data files that document their real-time character.

stat.AP↗

arXiv2025

Simplifying Random Forests' Probabilistic Forecasts

Since their introduction by Breiman, Random Forests (RFs) have proven to be useful for both classification and regression tasks. The RF prediction of a previously unseen observation can be represented as a weighted sum of all training sample observations. This nearest-neighbor-type representation is useful, among other things, for constructing forecast distributions (Meinshausen, 2006). In this paper, we consider simplifying RF-based forecast distributions by sparsifying them. That is, we focus on a small subset of $k$ nearest neighbors while setting the remaining weights to zero. This simplification, which we refer to as `Top$k$', greatly improves the interpretability of RF predictions. It can be applied to any forecasting task without re-training existing RF models. In empirical experiments, we document that the simplified predictions can be similar to or exceed the original ones in terms of forecasting performance. We explore the statistical sources of this finding via a stylized analytical model of RFs. The model suggests that simplification is particularly promising if the unknown true forecast distribution contains many small weights that are estimated imprecisely.

stat.AP↗

arXiv2024

Prediction intervals for economic fixed-event forecasts

The fixed-event forecasting setup is common in economic policy. It involves a sequence of forecasts of the same (`fixed') predictand, so that the difficulty of the forecasting problem decreases over time. Fixed-event point forecasts are typically published without a quantitative measure of uncertainty. To construct such a measure, we consider forecast postprocessing techniques tailored to the fixed-event case. We develop regression methods that impose constraints motivated by the problem at hand, and use these methods to construct prediction intervals for gross domestic product (GDP) growth in Germany and the US.

econ.EM↗

arXiv2023

Score-based calibration testing for multivariate forecast distributions

Calibration tests based on the probability integral transform (PIT) are routinely used to assess the quality of univariate distributional forecasts. However, PIT-based calibration tests for multivariate distributional forecasts face various challenges. We propose two new types of tests based on proper scoring rules, which overcome these challenges. They arise from a general framework for calibration testing in the multivariate case, introduced in this work. The new tests have good size and power properties in simulations and solve various problems of existing tests. We apply the tests to forecast distributions for macroeconomic and financial time series data.

econ.EM↗

arXiv2023

Direction Augmentation in the Evaluation of Armed Conflict Predictions

In many forecasting settings, there is a specific interest in predicting the sign of an outcome variable correctly in addition to its magnitude. For instance, when forecasting armed conflicts, positive and negative log-changes in monthly fatalities represent escalation and de-escalation, respectively, and have very different implications. In the ViEWS forecasting challenge, a prediction competition on state-based violence, a novel evaluation score called targeted absolute deviation with direction augmentation (TADDA) has therefore been suggested, which accounts for both for the sign and magnitude of log-changes. While it has a straightforward intuitive motivation, the empirical results of the challenge show that a no-change model always predicting a log-change of zero outperforms all submitted forecasting models under the TADDA score. We provide a statistical explanation for this phenomenon. Analyzing the properties of TADDA, we find that in order to achieve good scores, forecasters often have an incentive to predict no or only modest log-changes. In particular, there is often an incentive to report conservative point predictions considerably closer to zero than the forecaster's actual predictive median or mean. In an empirical application, we demonstrate that a no-change model can be improved upon by tailoring predictions to the particularities of the TADDA score. We conclude by outlining some alternative scoring concepts.

stat.AP↗

arXiv2023

Learning to forecast: The probabilistic time series forecasting challenge

We report on a course project in which students submit weekly probabilistic forecasts of two weather variables and one financial variable. This real-time format allows students to engage in practical forecasting, which requires a diverse set of skills in data science and applied statistics. We describe the context and aims of the course, and discuss design parameters like the selection of target variables, the forecast submission process, the evaluation of forecast performance, and the feedback provided to students. Furthermore, we describe empirical properties of students' probabilistic forecasts, as well as some lessons learned on our part.

stat.OT↗

arXiv2022

From point forecasts to multivariate probabilistic forecasts: The Schaake shuffle for day-ahead electricity price forecasting

Modeling price risks is crucial for economic decision making in energy markets. Besides the risk of a single price, the dependence structure of multiple prices is often relevant. We therefore propose a generic and easy-to-implement method for creating multivariate probabilistic forecasts based on univariate point forecasts of day-ahead electricity prices. While each univariate point forecast refers to one of the day's 24 hours, the multivariate forecast distribution models dependencies across hours. The proposed method is based on simple copula techniques and an optional time series component. We illustrate the method for five benchmark data sets recently provided by Lago et al. (2020). Furthermore, we demonstrate an example for constructing realistic prediction intervals for the weighted sum of consecutive electricity prices, as, e.g., needed for pricing individual load profiles.

econ.EM↗

arXiv2020

Predictive Inference Based on Markov Chain Monte Carlo Output

In Bayesian inference, predictive distributions are typically in the form of samples generated via Markov chain Monte Carlo (MCMC) or related algorithms. In this paper, we conduct a systematic analysis of how to make and evaluate probabilistic forecasts from such simulation output. Based on proper scoring rules, we develop a notion of consistency that allows to assess the adequacy of methods for estimating the stationary distribution underlying the simulation output. We then provide asymptotic results that account for the salient features of Bayesian posterior simulators, and derive conditions under which choices from the literature satisfy our notion of consistency. Importantly, these conditions depend on the scoring rule being used, such that the choices of approximation method and scoring rule are intertwined. While the logarithmic rule requires fairly stringent conditions, the continuous ranked probability score (CRPS) yields consistent approximations under minimal assumptions. These results are illustrated in a simulation study and an economic data example. Overall, mixture-of-parameters approximations which exploit the parametric structure of Bayesian models perform particularly well. Under the CRPS, the empirical distribution function is a simple and appealing alternative option.

stat.ME↗

arXiv2019

Generic Conditions for Forecast Dominance

Recent studies have analyzed whether one forecast method dominates another under a class of consistent scoring functions. While the existing literature focuses on empirical tests of forecast dominance, little is known about the theoretical conditions under which one forecast dominates another. To address this question, we derive a new characterization of dominance among forecasts of the mean functional. We present various scenarios under which dominance occurs. Unlike existing results, our results allow for the case that the forecasts' underlying information sets are not nested, and allow for uncalibrated forecasts that suffer, e.g., from model misspecification or parameter estimation error. We illustrate the empirical relevance of our results via data examples from finance and economics.

stat.ME↗

arXiv2018

Forecast dominance testing via sign randomization

We propose randomization tests of whether forecast 1 outperforms forecast 2 across a class of scoring functions. This hypothesis is of applied interest: While the prediction context often prescribes a certain class of scoring functions, it is typically hard to motivate a specific choice on statistical or substantive grounds. We investigate the asymptotic behavior of the test statistics under mild conditions, avoiding the need to assume particular dynamic properties of forecasts and realizations. The properties of the one-sided tests depend on a corresponding version of Anderson's inequality, which we state as a conjecture of independent interest. Numerical experiments and a data example indicate that the tests have good size and power properties in practically relevant situations.

math.ST↗

arXiv2018

Evaluating probabilistic forecasts with scoringRules

Probabilistic forecasts in the form of probability distributions over future events have become popular in several fields including meteorology, hydrology, economics, and demography. In typical applications, many alternative statistical models and data sources can be used to produce probabilistic forecasts. Hence, evaluating and selecting among competing methods is an important task. The scoringRules package for R provides functionality for comparative evaluation of probabilistic models based on proper scoring rules, covering a wide range of situations in applied work. This paper discusses implementation and usage details, presents case studies from meteorology and economics, and points to the relevant background literature.

stat.CO↗

arXiv2017

Murphy Diagrams: Forecast Evaluation of Expected Shortfall

Motivated by the Basel 3 regulations, recent studies have considered joint forecasts of Value-at-Risk and Expected Shortfall. A large family of scoring functions can be used to evaluate forecast performance in this context. However, little intuitive or empirical guidance is currently available, which renders the choice of scoring function awkward in practice. We therefore develop graphical checks (Murphy diagrams) of whether one forecast method dominates another under a relevant class of scoring functions, and propose an associated hypothesis test. We illustrate these tools with simulation examples and an empirical analysis of S&P 500 and DAX returns.

q-fin.RM↗

arXiv2015

Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations, and Forecast Rankings

In the practice of point prediction, it is desirable that forecasters receive a directive in the form of a statistical functional, such as the mean or a quantile of the predictive distribution. When evaluating and comparing competing forecasts, it is then critical that the scoring function used for these purposes be consistent for the functional at hand, in the sense that the expected score is minimized when following the directive. We show that any scoring function that is consistent for a quantile or an expectile functional, respectively, can be represented as a mixture of extremal scoring functions that form a linearly parameterized family. Scoring functions for the mean value and probability forecasts of binary events constitute important examples. The quantile and expectile functionals along with the respective extremal scoring functions admit appealing economic interpretations in terms of thresholds in decision making. The Choquet type mixture representations give rise to simple checks of whether a forecast dominates another in the sense that it is preferable under any consistent scoring function. In empirical settings it suffices to compare the average scores for only a finite number of extremal elements. Plots of the average scores with respect to the extremal scoring functions, which we call Murphy diagrams, permit detailed comparisons of the relative merits of competing forecasts.

math.ST↗