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Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

A3 - Bootstrapping methods for time series

The classical bootstrap methods and results are designed for independent data. Bootstrap validation for time series is very challenging and the available results are not as sharp as for independent data. The existing methods include block-bootstrap and subsampling, and rely on mixing properties of the considered data. The parametric bootstrap for time series mainly uses that the innovations are independent. The research within the IRTG will focus on developing new methods and approaches in bootstrapping multivariate and nonstationary time series.

Coordination

Vladimir Spokoiny: His research interests are mathematical statistics and econometrics (time series, dimension reduction, error-in-variable and instrumental regression) with various applications.

Wolfgang Karl 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.

 

Exemplary PhD-Theses

  1. Unit root problem and bootstrap

Many econometric phenomena like co-integration or interactions can be explained via the unit-root behaviour of the corresponding multivariate autoregressive time series. It is well known that the asymptotic inference and statistical properties for stationary and unit-root directions in the parameter space, see e.g. Lai and Wei (1982). This makes the empirical and theoretical analysis very complicated with a striking difference between regimes: one has to identify and clearly separate unit-root directions. The thesis aims at developing a unified approach to unit root analysis based a non-asymptotic Fisher expansion of the corresponding log-likelihood and equally applies to stationary and unit root regimes. The further goal is to design and validate a bootstrap procedure for the unit-root inference.

 

  1. Estimation and inference on nonparametric diffusion using Gaussian process priors

Many econometric and financial models are described as SDE, examples are given by famous Black-Scholes, Vasicek, etc. Extensions to generalized Black-Scholes stochastic volatility models require the estimation of the functions of drift and diffusion from fully or discretely observed realizations of the process. Statistical methods are complicated and very involved from algorithmic and theoretical viewpoints because the log-likelihood is given via stochastic integrals and cannot be computed exactly. Some numerical approximation can be used to assess the log-likelihood function. Existing computational methods for Bayesian inference include a variety of Markov chain Monte Carlo (MCMC) methods, sequential Monte Carlo (SMC) methods, Kalman- type methods, maximum a posteriori (MAP) approaches based on classic variational methods, and variational Gaussian process approximations (VGPA). The thesis will focus on understanding the impact of the prior given by a Gaussian process of certain regularity. The Bayesian inference and Bayesian model selection (prior choice) are the ultimate goals of the thesis.

 

References

  • Brockwell P J, Davis R A (1991) Time series: Theory and methods, Second Edition, New York, Springer.
  • Lai T L, Wei C Z (1982) Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann. Statist., 10, 154–166, DOI: 10.1214/aos/1176345697.
  • Lütkepohl H (1993) Introduction to Multiple Time Series Analysis (2nd ed.). Springer- Verlag, Berlin.