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Mathias Vetter

Publications and source records attributed to Mathias Vetter.

At least 19 recordsLinked to original sources
arXiv2026

On covariation estimation for multivariate continuous It\^o semimartingales with noise in non-synchronous observation schemes

This paper presents a Hayashi-Yoshida type estimator for the covariation matrix of continuous It\^o semimartingales observed with noise. The coordinates of the multivariate process are assumed to be observed at highly frequent non-synchronous points. The estimator of the covariation matrix is designed via a certain combination of the local averages and the Hayashi-Yoshida estimator. Our method does not require any synchronization of the observation scheme (as e.g. previous tick method or refreshing time method) and it is robust to some dependence structure of the noise process. We show the associated central limit theorem for the proposed estimator and provide a feasible asymptotic result. Our proofs are based on a blocking technique and a stable convergence theorem for semimartingales. Finally, we show simulation results for the proposed estimator to illustrate its finite sample properties.

econ.EM↗
arXiv2024

Testing for unspecified periodicities in binary time series

Given independent random variables $Y_1, \ldots, Y_n$ with $Y_i \in \{0,1\}$ we test the hypothesis whether the underlying success probabilities $p_i$ are constant or whether they are periodic with an unspecified period length of $r \ge 2$. The test relies on an auxiliary integer $d$ which can be chosen arbitrarily, using which a new time series of length $d$ is constructed. For this new time series, the test statistic is derived according to the classical $g$ test by Fisher. Under the null hypothesis of a constant success probability it is shown that the test keeps the level asymptotically, while it has power for most alternatives, i.e. typically in the case of $r \ge 3$ and where $r$ and $d$ have common divisors.

math.ST↗
arXiv2024

On goodness-of-fit testing for self-exciting point processes

Despite the wide usage of parametric point processes in theory and applications, a sound goodness-of-fit procedure to test whether a given parametric model is appropriate for data coming from a self-exciting point processes has been missing in the literature. In this work, we establish a bootstrap-based goodness-of-fit test which empirically works for all kinds of self-exciting point processes (and even beyond). In an infill-asymptotic setting we also prove its asymptotic consistency, albeit only in the particular case that the underlying point process is inhomogeneous Poisson.

math.ST↗
arXiv2022

On the estimation of the jump activity index in the case of random observation times

We propose a nonparametric estimator of the jump activity index $β$ of a pure-jump semimartingale $X$ driven by a $β$-stable process when the underlying observations are coming from a high-frequency setting at irregular times. The proposed estimator is based on an empirical characteristic function using rescaled increments of $X$, with a limit which depends in a complicated way on $β$ and the distribution of the sampling scheme. Utilising an asymptotic expansion we derive a consistent estimator for $β$ and prove an associated central limit theorem.

math.ST↗
arXiv2020

A universal approach to estimate the conditional variance in semimartingale limit theorems

The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard task, in particular when the underlying process contains jumps. For this reason, several authors have recently discussed methods to automatically estimate the conditional variance, i.e. they build a consistent estimator from the original statistics, but computed at various different time scales. Their methods work in several situations, but are essentially restricted to the case of continuous paths always. The aim of this work is to present a new method to consistently estimate the conditional variance which works regardless of whether the underlying process is continuous or has jumps. We will discuss the case of power variations in detail and give insight to the heuristics behind the approach.

math.PR↗
arXiv2018

Multiscale change point detection for dependent data

In this paper we study the theoretical properties of the simultaneous multiscale change point estimator (SMUCE) proposed by Frick et al. (2014) in regression models with dependent error processes. Empirical studies show that in this case the change point estimate is inconsistent, but it is not known if alternatives suggested in the literature for correlated data are consistent. We propose a modification of SMUCE scaling the basic statistic by the long run variance of the error process, which is estimated by a difference-type variance estimator calculated from local means from different blocks. For this modification we prove model consistency for physical dependent error processes and illustrate the finite sample performance by means of a simulation study.

math.ST↗
arXiv2018

Laws of large numbers for Hayashi-Yoshida-type functionals

In high-frequency statistics and econometrics sums of functionals of increments of stochastic processes are commonly used and statistical inference is based on the asymptotic behaviour of these sums as the mesh of the observation times tends to zero. Inspired by the famous Hayashi-Yoshida estimator for the quadratic covariation process based on two asynchronously observed stochastic processes we investigate similar sums based on increments of two asynchronously observed stochastic processes for general functionals. We find that our results differ from corresponding results in the setting of equidistant and synchronous observations which has been well studied in the literature. Further we observe that in the setting of asynchronous observations the asymptotic behaviour is not only determined by the nature of the functional but also depends crucially on the asymptotics of the observation scheme.

math.PR↗
arXiv2017

The null hypothesis of common jumps in case of irregular and asynchronous observations

This paper proposes novel tests for the absence of jumps in a univariate semimartingale and for the absence of common jumps in a bivariate semimartingale. Our methods rely on ratio statistics of power variations based on irregular observations, sampled at different frequencies. We develop central limit theorems for the statistics under the respective null hypotheses and apply bootstrap procedures to assess the limiting distributions. Further we define corrected statistics to improve the finite sample performance. Simulations show that the test based on our corrected statistic yields good results and even outperforms existing tests in the case of regular observations.

math.ST↗
arXiv2017

Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes

In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection and the localisation of gradual changes in the jump characteristic of a discretely observed Ito semimartingale. We propose a new measure of time variation for the jump behaviour of the process. The statistical uncertainty of a corresponding estimate is analyzed by deriving new results on the weak convergence of a sequential empirical tail integral process and a corresponding multiplier bootstrap procedure.

math.ST↗
arXiv2016

Testing for simultaneous jumps in case of asynchronous observations

This paper proposes a novel test for simultaneous jumps in a bivariate Itô semimartingale when observation times are asynchronous and irregular. Inference is built on a realized correlation coefficient for the jumps of the two processes which is estimated using bivariate power variations of Hayashi-Yoshida type without an additional synchronization step. An associated central limit theorem is shown whose asymptotic distribution is assessed using a bootstrap procedure. Simulations show that the test works remarkably well in comparison with the much simpler case of regular observations.

math.ST↗
arXiv2016

Nonparametric change-point analysis of volatility

This work develops change-point methods for statistics of high-frequency data. The main interest is in the volatility of an Itô semi-martingale, the latter being discretely observed over a fixed time horizon. We construct a minimax-optimal test to discriminate continuous paths from paths comprising volatility jumps. This is embedded into a more general theory to infer the smoothness of volatilities. In a high-frequency framework we prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we develop methods to infer changes in the Hurst parameter of fractional volatility processes. A simulation study demonstrates the practical value in finite-sample applications.

math.ST↗
arXiv2015

Estimation of integrated volatility of volatility with applications to goodness-of-fit testing

In this paper, we are concerned with nonparametric inference on the volatility of volatility process in stochastic volatility models. We construct several estimators for its integrated version in a high-frequency setting, all based on increments of spot volatility estimators. Some of those are positive by construction, others are bias corrected in order to attain the optimal rate $n^{-1/4}$. Associated central limit theorems are proven which can be widely used in practice, as they are the key to essentially all tools in model validation for stochastic volatility models. As an illustration we give a brief idea on a goodness-of-fit test in order to check for a certain parametric form of volatility of volatility.

math.ST↗
arXiv2015

Weak convergence of the empirical truncated distribution function of the Lévy measure of an Itō semimartingale

Given an Itō semimartingale with a time-homogeneous jump part observed at high frequency, we prove weak convergence of a normalized truncated empirical distribution function of the Lévy measure to a Gaussian process. In contrast to competing procedures, our estimator works for processes with a non-vanishing diffusion component and under simple assumptions on the jump process.

math.ST↗
arXiv2015

On U- and V-statistics for discontinuous Ito semimartingales

In this paper we examine the asymptotic theory for U-statistics and V-statistics of discontinuous Ito semimartingales that are observed at high frequency. For different types of kernel functions we show laws of large numbers and associated stable central limit theorems. In most of the cases the limiting process will be conditionally centered Gaussian. The structure of the kernel function determines whether the jump and/or the continuous part of the semimartingale contribute to the limit.

math.PR↗
arXiv2014

Nonparametric tests for detecting breaks in the jump behaviour of a time-continuous process

This paper is concerned with tests for changes in the jump behaviour of a time-continuous process. Based on results on weak convergence of a sequential empirical tail integral process, asymptotics of certain tests statistics for breaks in the jump measure of an Ito semimartingale are constructed. Whenever limiting distributions depend in a complicated way on the unknown jump measure, empirical quantiles are obtained using a multiplier bootstrap scheme. An extensive simulation study shows a good performance of our tests in finite samples.

stat.ME↗
arXiv2013

A test for stationarity based on empirical processes

In this paper we investigate the problem of testing the assumption of stationarity in locally stationary processes. The test is based on an estimate of a Kolmogorov-Smirnov type distance between the true time varying spectral density and its best approximation through a stationary spectral density. Convergence of a time varying empirical spectral process indexed by a class of certain functions is proved, and furthermore the consistency of a bootstrap procedure is shown which is used to approximate the limiting distribution of the test statistic. Compared to other methods proposed in the literature for the problem of testing for stationarity the new approach has at least two advantages: On one hand, the test can detect local alternatives converging to the null hypothesis at any rate $g_T\to0$ such that $g_TT^{1/2}\to \infty$, where $T$ denotes the sample size. On the other hand, the estimator is based on only one regularization parameter while most alternative procedures require two. Finite sample properties of the method are investigated by means of a simulation study, and a comparison with several other tests is provided which have been proposed in the literature.

math.ST↗
arXiv2013

Nonparametric inference on Lévy measures and copulas

In this paper nonparametric methods to assess the multivariate Lévy measure are introduced. Starting from high-frequency observations of a Lévy process $\mathbf{X}$, we construct estimators for its tail integrals and the Pareto-Lévy copula and prove weak convergence of these estimators in certain function spaces. Given n observations of increments over intervals of length $Δ_n$, the rate of convergence is $k_n^{-1/2}$ for $k_n=nΔ_n$ which is natural concerning inference on the Lévy measure. Besides extensions to nonequidistant sampling schemes analytic properties of the Pareto-Lévy copula which, to the best of our knowledge, have not been mentioned before in the literature are provided as well. We conclude with a short simulation study on the performance of our estimators and apply them to real data.

math.ST↗