Conference Agenda

When panel data exhibit cross-sectional dependence, particular care is required, as cross-dependence may be induced by omitting relevant variables. If these variables correlate with the regressors, rendering them endogenous, sophisticated approaches such as the CCE approach or the PC estimator are recommended. These approaches may however be difficult to implement or build on strong assumptions. Therefore, if regressor endogeneity can reasonably be excluded, it is common to resort to simpler estimators in conjunction with panel-robust standard errors.
In practice, it is furthermore not uncommon for cross-dependence to be of various strength, ranging from weak (e.g.\ in spatial data models) to strong (when unobserved factors load onto all units).
We quantify the impact the strength of cross-dependence has on the within estimator and on the associated (panel-robust) tests when unobserved factors in errors are not confounding. Using theoretical justifications and Monte Carlo simulation we find that cross-dependence alone in the errors is not sufficient to impact the fixed-effects estimator or its standard errors. Rather, it is the interaction of error and regressor cross-dependence that causes problems such as reduced convergence rates and distorted significance tests in finite samples.

We consider a structural matrix-autregressive (SMAR) model to conduct impulse response analysis for structural shocks to matrix-valued time series. The MAR model of order $p$ offers a parsimonious and interpretable framework for these time series, thus addressing issues of high-dimensionality in corresponding vector-autoregressive (VAR) models. To interpret the dynamics, we resort to impulse response analysis as a popular tool from the SVAR context. Its conclusions rely on the valid identification of structural shocks that are mutually contemporaneously uncorrelated and interpretable. In contrast to the existing literature, the proposed SMAR model enables the identification of multiple structural shocks. To address the restrictive nature of the single-term MAR($p$) model, we discuss the extension to a multi-term SMAR($p$) model as a compromise between the single-term SMAR and the (unrestricted) SVAR model, trading off parsimony against flexibility. We discuss its identification, focusing in particular on issues that arise due to the typical Kronecker-product structure of the coefficient matrices in the MAR framework. Further, we discuss estimation and inference in the general multi-term SMAR($p$) model, including a bootstrap method to compute confidence bands for the impulse response curves. In this context, a key point concerns model misspecification and the use of MAR models to approximate more general SVAR data generating processes. Finally, we demonstrate the performance and practical use of our approach by Monte Carlo simulations and a real data application.

Vector Multiplicative Error Models (vMEMs) provide a flexible framework for modeling multivariate non-negative time series. Within this framework, each variable is expressed as the product of its conditional mean—modeled as a function of past observations—and a positive innovation with unit expectation. Consequently, the model can capture dynamic cross-dependencies and have proven useful in applications such as modeling durations, volatilities, and trading volumes. This contribution focuses on goodness-of-fit (GOF) tests for vMEMs, aiming to assess whether the model structure and the assumed innovation distribution adequately reflect the properties of the observed data. We propose a GOF test statistic and derive its asymptotic distribution under the null hypothesis. The performance of a bootstrap version of the test is illustrated through Monte Carlo simulations.

A novel time series framework is proposed which addresses all relevant empirical properties of a time series, making it an essentially universal model. More specifically, the dynamics in all conditional moments of a suitable continuous or discrete distribution are modeled jointly and without the need to make restrictive assumptions about the functional form of the link functions. Furthermore, all considered explanatory variables are allowed to exhibit nonlinear and potentially time-varying effects on the conditional moments. This can be achieved by employing a simple feedforward neural network with a single hidden layer and an output for each conditional moment (parameter). In contrast to many (deep) neural network approaches, the proposed model is stochastically interpretable and allows for the calculation of standard errors, and in particular, confidence intervals. Many conventional time series frameworks such as (integer-valued) GARCH can be interpreted as simplified special cases of the proposed model. Several empirical applications are presented to illustrate the capabilities and the implementation.

Research on count time series has grown substantially, leading to the development of numerous models designed to capture key characteristics such as trends, seasonality, overdispersion, outliers, and complex dependence structures. Despite these advances, the evaluation of such models remains challenging due to the limited availability of real-world count time series. This scarcity often forces researchers to illustrate new methods using only a few datasets, which restricts systematic comparison and hinders robust performance assessment. Addressing this gap is essential for advancing methodological development and ensuring practical applicability in diverse domains.
In this presentation, we focus on the generation of synthetic count time series that can exhibit a wide range of controlled behaviors. Our approach relies on Mixture INAR (MixINAR) models and a set of descriptive features designed to summarize key properties of count data. We first discuss which features are most suitable for characterizing count time series and then show how they can be used to guide the simulation of diverse collections of synthetic data. By comparing the simulated series with commonly used benchmark datasets, we illustrate how realistic and flexible the generated time series can be. Overall, this work provides a practical way to create representative collections of count time series, supporting more thorough testing and comparison of models in this area.

This work is financed by National Funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) within the project TSP2Net, with reference 2023.13039.PEX, https://doi.org/10.54499/2023.13039.PEX

INAR and INGARCH-type processes are widely used approaches to model time series of counts. In this talk, I will speak about a class of generalized INARMA (integer-valued autoregressive) models which contains both of the aforementioned types of models as special cases. Notably I will outline a generalization of the INAR model which parallels the extension of the INARCH to the INGARCH process. Special attention is given to inference questions. These include maximum likelihood, moment-based and Gaussian quasi-likelihood techniques for parameter estimation. Moreover, I will discuss various testing problems. The developed methods are illustrated in simulation studies and a data example on childhood diseases in the German state of Bavaria.

We introduce a new framework for the theoretical analysis of learning algorithms called self-regularization. In a nutshell, self-regularized learning algorithms guarantee implicitly that they produce sufficiently regular prediction functions. Central examples of self-regularized learning algorithms include gradient descent and regularized empirical risk minimization. We establish a general theory for the statistical analysis of self-regularized algorithms which in many cases yields minmax-optimal learning rates.

Max Schölpple, Ingo Steinwart

Institut für Stochastik und Anwendungen, Universität Stuttgart, Pfaffenwaldring 57,
70569 Stuttgart

Fuk-Nagaev and Rosenthal-type inequalities are proven for the sums of independent random matrices, focusing on the situation when the norms of the matrices possess finite moments of only low orders. The bounds depend on the intrinsic dimensional characteristics, such as the effective rank, as opposed to the dimension of the ambient space. The advantages of such results are illustrated in several applications, including new moment inequalities for sample covariance matrices and the corresponding eigenvectors of heavy-tailed random vectors.

Authors:

Moritz Jirak, Stanislav Minsker, Yiqiu Shen, Martin Wahl

The spectrum of the Laplace-Beltrami operator encodes essential geometric information about a smooth manifold. In practice, the manifold is unknown, but supports a finite sample of random points. It is then standard to approximate its spectrum by the spectrum of the resulting graph Laplacian. When the manifold is bounded, it is known that the graph Laplacian eigen-converges to the Neumann Laplacian. However, finite sample results, such as convergence rates, are still lacking, and are at the center of this talk.

A new approach for nonparametric estimation of quantile density functions based on
local polynomial estimation is presented. Estimation of the quantile density is important
because it appears for example in the expression for the asymptotic variance of empirical
and kernel type estimators of the quantile function.

The new approach uses a local polynomial regression on (F_n(X_i), Q_n(F_n(X_i))), where F_n
and Q_n represent the empirical distribution function and the empirical quantile function
respectively.

The new approach has more advantageous properties at the boundary than classical quan-
tile density estimators. We present a result on the asymptotic normality of the proposed
estimator. In this context we also get the leading bias term and the asymptotic variance,
which we use to compare the new estimator to classical estimators, especially focusing on
the boundary.

Keywords: asymptotic normality, bias rates, boundary adaptation, empirical quan-
tile function, nonparametric function estimator

There is a great variety of statistical models expressing relations between response variables of interest and explanatory variables, ranging from classical conditional mean regression to fully distributional regression models. We are particularly interested in expressing regression models by means of copulas which are a valuable tool to separate marginal distributions and dependencies. New goodness-of-fit tests and new measures of deviation can be developed based on such copula representations. These tests are desirable since regression models often impose parametric or semiparametric assumptions to overcome the curse of dimensionality, running a risk of misspecification. We present a new goodness-of-fit test for the classical mean regression model. More importantly, we also introduce a new measure of deviation between the true regression function and the imposed parametric assumption. By self-normalization, we develop pivotal inference for this measure including tests for relevant hypotheses. These inference tools are illustrated via simulated and empirical data.

Rank-based association measures, including Spearman’s rho, Gini’s gamma and Spearman’s footrule, are well established in continuous settings, but become problematic when ties are present. We investigate these measures in context of zero-inflated data, where continuous random variables have an increased probability mass at zero and there is a substantial number of ties. Such data is commonly found in fields such as insurance, health care and weather forecasting. Traditional rank-based estimators exhibit a large bias in these settings. To overcome this problem, we derive new formulations of the association measures and propose plug-in estimators. In a simulation study, we show that these outperform state-of-the-art estimators. Additionally, we make the estimator interpretable by deriving its achievable bounds.

In this paper, we develop nonlinear higher-order shrinkage estimators for both covariance and precision matrices. Our framework applies to settings in which the sample size n is either larger or smaller than p, the dimensionality of the data-generating process. The proposed estimators incorporate higher-order moments up to an arbitrary order and therefore encompass linear shrinkage estimators as special cases. We derive recursive representations of these higher-order nonlinear shrinkage estimators using partial exponential Bell polynomials.

Through simulation studies, the proposed methods are compared with the oracle nonlinear shrinkage estimator and are shown to be particularly effective in settings where no closed-form expressions for nonlinear shrinkage estimators are available. The theoretical derivations rely on mild assumptions on the underlying model, including the existence of fourth moments and a bounded spectrum of the true population covariance matrix. The finite-sample performance of the proposed estimators is evaluated in an extensive simulation study and benchmarked against existing approaches. Our main finding is that the higher-order shrinkage estimators can outperform well-established nonlinear shrinkage methods, particularly when the concentration ratio p/n is large.

We develop methodology and theory for the detection of a phase transition in a time-series of high-dimensional random matrices. In the model we study, at each time point $ t = 1,2,\ldots $, we observe a deformed Wigner matrix $ \mathbf{M}_t $, where the unobservable deformation represents a latent signal. This signal is detectable only in the supercritical regime, and our objective is to detect the transition to this regime in real time, as new matrix--valued observations arrive.
Our approach is based on a partial sum process of extremal eigenvalues of $\mathbf{M}_t$, and its theoretical analysis combines state-of-the-art tools from random-matrix-theory and Gaussian approximations. The resulting detector is self-normalized, which ensures appropriate scaling for convergence and a pivotal limit,,without any additional parameter estimation. Simulations show excellent performance for varying dimensions. Applications to pollution monitoring and social interactions in primates illustrate the usefulness of our approach.

Random geometric graphs provide a fundamental model for spatially embedded networks, yet their spectral fluctuations remain poorly understood. In this talk, I will present the first rigorous results on Gaussian fluctuations of linear eigenvalue statistics for such graphs. Specifically, we establish central limit theorems for quantities of the form $\mathrm{Tr}[\phi(A)]$, where $A$ denotes the adjacency matrix and $\phi$ belongs to a broad class of test functions, including non-polynomial functions. In the polynomial setting, we go further and prove a quantitative central limit theorem with an explicit rate of convergence to the limiting Gaussian distribution. I will also discuss extensions of these results to other canonical spatial networks, such as $k$-nearest neighbor graphs and relative neighborhood graphs. Together, these results highlight new mechanisms governing spectral fluctuations in random spatial structures and reveal a delicate interplay between geometry, local dependence, and spectral behavior.

The talk is based on joint work with Christian Hirsch (Aarhus) and Kyeongsik Nam (Seoul).

In this talk, we consider a two-sample smooth test for testing the equality of multivariate distributions. Dependency between the two samples is allowed for. For instance, the data can be mixing. The asymptotic distribution under the null hypothesis is derived, and consistency of the two-sample smooth test for dependent samples is shown.

Satterthwaite (1941, 1946) proposed a very simple approximation to the distribution of linear combinations of Chi-squared random variables. It can be used in univariate time series analysis to approximate the distribution of the sample variance and the periodogram of Gaussian time series; we provide Wasserstein bounds and rates of convergence of the approximation towards the true distribution. Similarly, Tan & Gupta (1983) proposed an approximation to the distribution of linear combinations of Wishart random matrices. This, however, has not yet been applied to the framework of multivariate time series: we take advantage of a special case of the matrix normal distribution to propose a feasible approximation to the distribution of the sample covariance matrix of Gaussian time series.

The ordinal structure of long-range dependent time series is analyzed. To this end, so-called ordinal patterns are used, which describe the relative position of consecutive data points. Two estimators are provided for the probabilities of ordinal patterns and we prove limit theorems in different settings, namely for funtions of Hermite Rank 1 and 2. In the second setting, a Rosenblatt distribution in the limit is encountered. In the context of fractional Gaussian noise, the limit distribution is derived for an estimation of the Hurst parameter H if it is higher than 3/4. Thus, the theorems complement results for lower values of H, which can be found in the literature.

The use of ordinal patterns (OPs) for analyzing the dependence structure of univariate and continuously distributed processes has gained popularity in recent years. Here, we go one step further and consider the transcripts being computed from successive OPs in the time series. Transcripts constitute a kind of "difference" between successive OPs and thus naturally relate to two algebraic distances between OPs, the Cayley and Kendall distance. We transform the original time series into a sequence of transcripts or distances, respectively, and derive important stochastic properties thereof. We show that these properties differ substantially between different types of original process. This motivates to develop various statistics based on transcripts and algebraic distances in order to investigate the dependence structure of the original process. In particular, we derive the asymptotic distribution of these statistics under the null hypothesis of serial independence, which is then used to develop nonparametric tests for serial dependence. A simulation study shows that these novel dependence tests have appealing power properties, often outperforming the former OP-based dependence tests. We conclude with a real-world data example, where we illustrate the application and interpretaion of the proposed approaches in practice.

We study the problem of detecting changes in the marginal distributions of a multivariate time series with a novel CUSUM-type detector statistic based on the (maximum-) sliced-Wasserstein distance. This projection-based approach has two appealing properties. Firstly, unlike the family of Wasserstein distances, it does not suffer from the curse of dimensionality. And secondly, by means of the Kantorovich duality, asymptotic properties of the so-defined detector statistic can be derived from results for (sequential) empirical processes for nonstationary time series. This talk presents new weak limit theorems for sequential empirical processes under the functional dependence measure and their application to the given testing problem. Practical implications, limitations and possible extensions are discussed.

The statistical properties of empirical entropic optimal transport (empirical EOT) have attracted great interest, as this quantity has been shown to be useful for complex data analysis, among other reasons due to its computational efficiency. In several applications, it has been realized that in addition to the optimal value, also the EOT plan carries important information. For example, in cell biology, colocalization analysis based on the EOT plan has been introduced as a measure for quantification of spatial proximity of different protein assemblies. Despite recent progress in the analysis of its risk properties, a precise understanding of its statistical fluctuations to make it accessible for inference remains elusive to some extent. We derive asymptotic weak convergence result for a large class of functionals of the EOT plan, in which the colocalization process is included. As an application, we obtain uniform confidence bands for colocalization curves and bootstrap consistency. Our theory is supported by simulation studies and is illustrated by real world data analysis from mitochondrial protein colocalization.

This paper introduces and analyzes a framework that accommodates general heterogeneity in regression modeling. It demonstrates that regression models with fixed or time-varying parameters can be estimated using OLS and time-varying OLS methods, respectively, across a broad class of regressors and noise processes not covered by existing theory. The proposed setting facilitates the development of asymptotic theory and the estimation of robust standard errors. The resulting robust confidence interval estimators accommodate substantial heterogeneity in both regressors and noise. The robust standard error estimates coincide with White’s (1980) heteroskedasticity-consistent estimator but apply under much broader conditions, including models with missing data. The methods are computationally simple and perform well in Monte Carlo simulations, making them highly suitable for empirical applications. The paper also provides a brief empirical illustration.