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Wolfgang Schmid

Publications and source records attributed to Wolfgang Schmid.

At least 19 recordsLinked to original sources
arXiv2026

Comparative Analysis of Spatiotemporal Volatility Models: An Empirical Study on Financial Network Series

Various spatiotemporal and network GARCH models have recently been proposed to capture volatility interactions, such as the transmission of market risk across financial networks. These approaches rely heavily on the specification of the adjacency or spatiotemporal weight matrix, for which several alternatives exist in the literature. This paper evaluates the out-of-sample forecasting performance of a range of spatiotemporal volatility models and multivariate GARCH benchmarks under nine alternative network specifications. The empirical analysis uses daily data for 16 sectorally diversified S&P 500 stocks from 22 December 1998 to 20 October 2024. A one-step-ahead forecasting framework is implemented, and models are assessed using BIC, RMSFE, and MAFE, with forecasts evaluated against a single realised volatility proxy based on squared log-returns. The nine spatial weight matrices reflect diverse economic and statistical relationships, including Granger-filtered and EGARCH-based spillovers. Results show that some spatiotemporal models outperform standard GARCH benchmarks in out-of-sample forecasting accuracy. Notably, the Dynamic Spatiotemporal ARCH model achieves the lowest RMSFE and MAFE across all network specifications at minimal computational cost. Pairwise Diebold-Mariano tests confirm significant differences in predictive accuracy. These findings underscore the value of incorporating spatial structure into volatility modelling as a parsimonious and interpretable alternative for financial network analysis.

stat.AP↗
arXiv2025

Exponential Spatiotemporal GARCH Model with Asymmetric Volatility Spillovers

This paper introduces a spatiotemporal exponential generalised autoregressive conditional heteroscedasticity (spatiotemporal E-GARCH) model, extending traditional spatiotemporal GARCH models by incorporating asymmetric volatility spillovers, while also generalising the time-series E-GARCH model to a spatiotemporal setting with instantaneous, potentially asymmetric volatility spillovers across space. The model allows for both temporal and spatial dependencies in volatility dynamics, capturing how financial shocks propagate across time, space, and network structures. We establish the theoretical properties of the model, deriving stationarity conditions and moment existence results. For estimation, we propose a quasi-maximum likelihood (QML) estimator and assess its finite-sample performance through Monte Carlo simulations. Empirically, we apply the model to financial networks, specifically analysing volatility spillovers in stock markets. We compare different network structures and analyse asymmetric effects in instantaneous volatility interactions.

stat.AP↗
arXiv2025

The set of minimal distances of the monoid of plus-minus weighted zero-sum sequences and applications to the characterization problem

Recently a systematic investigation of monoids of sequences of plus-minus weighted zero-sum sequences had been started, which is among others motivated by applications to monoids of norms of algebraic integers. In the current paper these investigations are continued. The focus is on the set of minimal distances of these monoids, which is an important arithmetical invariant. Applications to the characterization problem are discussed as well.

math.CO↗
arXiv2024

The geometry of intersecting codes and applications to additive combinatorics and factorization theory

Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicites) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted $2$-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.

math.CO↗
arXiv2023

Spatial and Spatiotemporal Volatility Models: A Review

Spatial and spatiotemporal volatility models are a class of models designed to capture spatial dependence in the volatility of spatial and spatiotemporal data. Spatial dependence in the volatility may arise due to spatial spillovers among locations; that is, if two locations are in close proximity, they can exhibit similar volatilities. In this paper, we aim to provide a comprehensive review of the recent literature on spatial and spatiotemporal volatility models. We first briefly review time series volatility models and their multivariate extensions to motivate their spatial and spatiotemporal counterparts. We then review various spatial and spatiotemporal volatility specifications proposed in the literature along with their underlying motivations and estimation strategies. Through this analysis, we effectively compare all models and provide practical recommendations for their appropriate usage. We highlight possible extensions and conclude by outlining directions for future research.

econ.EM↗
arXiv2022

Evaluating Active Learning Heuristics for Sequential Diagnosis

Given a malfunctioning system, sequential diagnosis aims at identifying the root cause of the failure in terms of abnormally behaving system components. As initial system observations usually do not suffice to deterministically pin down just one explanation of the system's misbehavior, additional system measurements can help to differentiate between possible explanations. The goal is to restrict the space of explanations until there is only one (highly probable) explanation left. To achieve this with a minimal-cost set of measurements, various (active learning) heuristics for selecting the best next measurement have been proposed. We report preliminary results of extensive ongoing experiments with a set of selection heuristics on real-world diagnosis cases. In particular, we try to answer questions such as "Is some heuristic always superior to all others?", "On which factors does the (relative) performance of the particular heuristics depend?" or "Under which circumstances should I use which heuristic?"

cs.AI↗
arXiv2021

Generalized Spatial and Spatiotemporal ARCH Models

In time-series analyses, particularly for finance, generalized autoregressive conditional heteroscedasticity (GARCH) models are widely applied statistical tools for modelling volatility clusters (i.e., periods of increased or decreased risk). In contrast, it has not been considered to be of critical importance until now to model spatial dependence in the conditional second moments. Only a few models have been proposed for modelling local clusters of increased risks. In this paper, we introduce a novel spatial GARCH process in a unified spatial and spatiotemporal GARCH framework, which also covers all previously proposed spatial ARCH models, exponential spatial GARCH, and time-series GARCH models. In contrast to previous spatiotemporal and time series models, this spatial GARCH allows for instantaneous spill-overs across all spatial units. For this common modelling framework, estimators are derived based on a non-linear least-squares approach. Eventually, the use of the model is demonstrated by a Monte Carlo simulation study and by an empirical example that focuses on real estate prices from 1995 to 2014 across the ZIP-Code areas of Berlin. A spatial autoregressive model is applied to the data to illustrate how locally varying model uncertainties (e.g., due to latent regressors) can be captured by the spatial GARCH-type models.

stat.ME↗
arXiv2020

On the incomparability of systems of sets of lengths

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. We consider the system $\mathcal L (H)$ of all sets of lengths of $H$ and study when $\mathcal L (H)$ contains or is contained in a system $\mathcal L (H')$ of a Krull monoid $H'$ with finite class group $G'$, prime divisors in all classes and Davenport constant $\mathsf D (G')=\mathsf D (G)$. Among others, we show that if $G$ is either cyclic of order $m \ge 7$ or an elementary $2$-group of rank $m-1 \ge 6$, and $G'$ is any group which is non-isomorphic to $G$ but with Davenport constant $\mathsf D (G')=\mathsf D (G)$, then the systems $\mathcal L (H)$ and $\mathcal L (H')$ are incomparable.

math.AC↗
arXiv2020

Spatial and Spatiotemporal GARCH Models -- A Unified Approach

In time-series analyses, particularly for finance, generalized autoregressive conditional heteroscedasticity (GARCH) models are widely applied statistical tools for modelling volatility clusters (i.e., periods of increased or decreased risk). In contrast, it has not been considered to be of critical importance until now to model spatial dependence in the conditional second moments. Only a few models have been proposed for modelling local clusters of increased risks. In this paper, we introduce novel spatial GARCH and exponential GARCH processes in a unified spatial and spatiotemporal GARCH-type model, which also covers all previously proposed spatial ARCH models as well as time-series GARCH models. For this common modelling framework, estimators are derived based on nonlinear least squares and on the maximum-likelihood approach. In addition to the theoretical contributions of this paper, we suggest a model selection strategy that is verified by a series of Monte Carlo simulation studies. Eventually, the use of the unified model is demonstrated by an empirical example that focuses on real estate prices from 1995 to 2014 across the ZIP-Code areas of Berlin. A spatial autoregressive model is applied to the data to illustrate how locally varying model uncertainties can be captured by the spatial GARCH-type models.

stat.ME↗
arXiv2020

Statistical inference for the EU portfolio in high dimensions

In this paper, using the shrinkage-based approach for portfolio weights and modern results from random matrix theory we construct an effective procedure for testing the efficiency of the expected utility (EU) portfolio and discuss the asymptotic behavior of the proposed test statistic under the high-dimensional asymptotic regime, namely when the number of assets $p$ increases at the same rate as the sample size $n$ such that their ratio $p/n$ approaches a positive constant $c\in(0,1)$ as $n\to\infty$. We provide an extensive simulation study where the power function and receiver operating characteristic curves of the test are analyzed. In the empirical study, the methodology is applied to the returns of S\&P 500 constituents.

q-fin.PM↗
arXiv2019

Direct zero-sum problems for certain groups of rank three

We determine the exact value of the $η$-constant and the multiwise Davenport constants for finite abelian groups of rank three having the form $G \simeq C_2 \oplus C_{n_2} \oplus C_{n_3}$ with $2 \mid n_2 \mid n_3$. Moreover, we determine the Erdős-Ginzburg-Ziv constant of these groups under the assumption that $n_2/2$ has Property D or $n_2 = n_3$.

math.NT↗
arXiv2019

Tests for the weights of the global minimum variance portfolio in a high-dimensional setting

In this study, we construct two tests for the weights of the global minimum variance portfolio (GMVP) in a high-dimensional setting, namely, when the number of assets $p$ depends on the sample size $n$ such that $\frac{p}{n}\to c \in (0,1)$ as $n$ tends to infinity. In the case of a singular covariance matrix with rank equal to $q$ we assume that $q/n\to \tilde{c}\in(0, 1)$ as $n\to\infty$. The considered tests are based on the sample estimator and on the shrinkage estimator of the GMVP weights. We derive the asymptotic distributions of the test statistics under the null and alternative hypotheses. Moreover, we provide a simulation study where the power functions and the receiver operating characteristic curves of the proposed tests are compared with other existing approaches. We observe that the test based on the shrinkage estimator performs well even for values of $c$ close to one.

q-fin.ST↗
arXiv2019

A characterization of class groups via sets of lengths

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u\_1 \cdot \ldots \cdot u\_k$, with irreducibles $u\_1, \ldots u\_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible $k$ is called the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ depends only on the class group $G$. In the present paper we study the inverse question asking whether or not the system $\mathcal L (H)$ is characteristic for the class group. Consider a further Krull monoid $H'$ with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are isomorphic (apart from two well-known pairings).

math.AC↗
arXiv2018

On the Harborth constant of $C_3 \oplus C_{3n}$

For a finite abelian group $(G,+, 0)$ the Harborth constant $\mathsf{g}(G)$ is the smallest integer $k$ such that each squarefree sequence over $G$ of length $k$, equivalently each subset of $G$ of cardinality at least $k$, has a subsequence of length $\exp(G)$ whose sum is $0$. In this paper, it is established that $\mathsf{g}(G)= 3n + 3$ for prime $n \neq 3$ and $\mathsf{g}(C_3 \oplus C_9)= 13$.

math.CO↗
arXiv2018

Bayesian mean-variance analysis: Optimal portfolio selection under parameter uncertainty

The paper solves the problem of optimal portfolio choice when the parameters of the asset returns distribution, like the mean vector and the covariance matrix are unknown and have to be estimated by using historical data of the asset returns. The new approach employs the Bayesian posterior predictive distribution which is the distribution of the future realization of the asset returns given the observable sample. The parameters of the posterior predictive distributions are functions of the observed data values and, consequently, the solution of the optimization problem is expressed in terms of data only and does not depend on unknown quantities. In contrast, the optimization problem of the traditional approach is based on unknown quantities which are estimated in the second step leading to a suboptimal solution. We also derive a very useful stochastic representation of the posterior predictive distribution whose application leads not only to the solution of the considered optimization problem, but provides the posterior predictive distribution of the optimal portfolio return used to construct a prediction interval. A Bayesian efficient frontier, a set of optimal portfolios obtained by employing the posterior predictive distribution, is constructed as well. Theoretically and using real data we show that the Bayesian efficient frontier outperforms the sample efficient frontier, a common estimator of the set of optimal portfolios known to be overoptimistic.

q-fin.ST↗