Research

Publications

Title: Shapley Curves: A Smoothing Perspective
Co-Authors: Georg Keilbar, Wolfgang Härdle

Abstract: This paper fills the limited statistical understanding of Shapley values as a variable importance measure from a nonparametric (or smoothing) perspective. We introduce population-level Shapley curves to measure the true variable importance, determined by the conditional expectation function and the distribution of covariates. Having defined the estimand, we derive minimax convergence rates and asymptotic normality under general conditions for the two leading estimation strategies. For finite sample inference, we propose a novel version of the wild bootstrap procedure tailored for capturing lower-order terms in the estimation of Shapley curves. Numerical studies confirm our theoretical findings, and an empirical application analyzes the determining factors of vehicle prices.

Published at Journal of Business & Economic Statistics

Software

Title: Python Library for Early Stopping Methods
Co-Authors: Laura Hucker, Bernhard Stankewitz, Eric Ziebell

Abstract: For iterative estimation procedures applied to statistical inverse problems, it is necessary to choose a suitable iteration index to avoid under- and overfitting. Classical model selection criteria can be prohibitively expensive in high dimensions. In the last few years, it has been shown for several regularisation methods that sequential early stopping can achieve statistical and computational efficiency by halting at a data-driven index depending on previous iterates only. We are in the process of implementing these residual-based stopping rules for different algorithms like Landweber, conjugate gradients and L2-boosting in our Python package “EarlyStopping”. In the future, we will include early stopping for decision trees. We demonstrate its functionality based on several simulation examples.

Work-in-progress

Title: Early Stopping for Regression Trees
Co-Author: Markus Reiss

Title: Risk Premia in the Bitcoin Market
Co-Authors: Caio Almeida, Maria Grith, Zijin Wang

Abstract: We adopt options and realized returns to offer a detailed analysis of risk premia in the Bitcoin market. First, by decomposing the index premium in the return space, we find that negative returns in the interval [-60%, -20%] explain one-third of the total Bitcoin premium (BP), directly contrasting with the S&P 500 market, where moderately negative returns explain approximately 70% of the equity premium (Beason and Schreindorfer, 2022). Then, further adopting a collection of risk-neutral densities and a new clustering algorithm, we find that risk premia vary over time depending on market volatility. In low-volatility states, the BP for positive returns is higher, the pricing kernel is steeper in the negative returns region, and variance risk premium is high. In high-volatility states, the BP attributable to positive and negative returns becomes more balanced.