A3 - Bootstrapping methods for time series
Efron’s bootstrap method was originally introduced in 1979 for approximating the error distribution of plug-in estimates for smooth real-valued functionals of probability distributions on the real line based on an independent and identically distributed data sample. Starting in the late 1980s and the 1990s, a variety of modifications of the original bootstrap idea has been discussed to cope with dependent data and especially time series structures. Among other approaches, the most prominent examples are
- the block bootstrap (cf. the monograph by Lahiri, 2003)
- the Markovian bootstrap (Rajarshi, 1990)
- the nonparametric residual bootstrap (typically carried out as wild bootstrap, cf. Wu (1986), Härdle and Mammen, 1993)
- the autoregressive sieve bootstrap (Kreiss, 1988)
- several frequency domain bootstrap techniques (Hurvich and Zeger (1987), Franke and Härdle (1992), Janas and Dahlhaus (1994), Kreiss and Paparoditis, 2003)
These methods are designed for analyzing the error distribution of estimators for smooth functionals of low-dimensional probability distributions driving the corresponding time series, cf., e. g., Bühlmann (1997) for a class of test statistics falling under that scope.
Especially for the bootstrap methods in the time domain, however, a necessary prerequisite is (at least weak) stationarity of the underlying data-generating process. The problem of non-stationarities in multivariate time series analyses is a modern research topic (cf., e. g., von Bünau et al., 2009) and will be addressed in connection with several real-life applications from the various other projects described in this proposal, for instance in neuroeconomics or finance. Our general aim in this project is to approximate the joint distribution of estimators of the parameters of dynamic time series models by resampling methods, taking into account the (in general time-inhomogeneous) dependency structure between coordinates of the vector of observables. To this end, we will methodologically follow the modern approach of “companion processes” as termed in Kreiss and Paparoditis (2011). The companion process of a given time series model with respect to a certain bootstrap technique is defined as the limiting process of the resampled data points when the sample size tends to infinity and has the following convenient property: “As long as the asymptotic distribution of the statistic of interest does not change. If we switch from the underlying process to its companion […] then the […] bootstrap approach asymptotically works.” (Kreiss and Paparoditis, 2011) In other words, the scientific hypothesis of interest can be used to determine the degree of modeling sophistication by means of the companion process approach.
Another issue that will receive particular attention is the so-called “curse of dimensionality”, i. e., that the number of parameters for (time series) models grows fast with the dimension of the data space to be considered, making standard estimators that are suitable for the low-dimensional case singular or unstable when applied to high-dimensional data. One way to deal with this problem is considering techniques from the field of simultaneous statistical inference and multiple testing. For instance, the monographs by Westfall and Young (1993) and Dudoit and van der Laan (2007) are considered with resampling techniques for multiple testing problems in high dimensions. The main mathematical challenge in this part of the project consists in appropriately taking into account the dynamic of the stochastic system under consideration in the calibration step for the critical thresholds of the multiple testing procedures.
Coordination
- Thorsten Dickhaus: Thorsten Dickhaus is an expert in multiple testing theories, especially for high-dimensional data and with respect to control of the false discovery rate. He has also worked on low intensity bootstrap methods for problems involving non-standard test statistics. His expertise in applications comprises planning and evaluation of biometrical and epidemiological studies, analysis of genetics data, statistical neuro-science, and neuroeconomics.
- Yingxing Li: Her main research interests are Nonparametric and Semiparametric Regression, Dimension Reduction and functional data analysis.
- Wolfgang Härdle: His main interests are non- and semiparametrics statistics and econometrics. His research includes work in nonparametric modelling, local adaptive models, reduction techniques, stationary models, quantile regression.
- Haiqiang Chen: His main interests are Financial Econometrics, Time Series Econometrics and Financial Economics. His research includes work on financial time series, structural change points detections.