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Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

A2 - Nonstationarity

Classic time series theory mainly deals with stationary processes. However, for real life processes, this assumption can be too restrictive; it does not allow the incorporation of structural changes or parameter shifts in complex systems. An extension of classic methods for stationary time series is challenging because of the asymptotic nature of available methods and results. Recent developments in probability theory and statistics has opened up new perspectives in the non-asymptotic analysis of nonstationary time series. In particular, the switching regime and smooth transition models can be treated in a unified way.



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

Zongwu Cai: His main interests are Econometrics, Quantitative Finance, Nonlinear Time Series. His research includes work in Theoretical and applied econometrics, quantitative finance and risk management, nonparametric curve estimation problems, nonlinear and nonstationary time series, panel data analysis.


Exemplary PhD-Theses

  1. Multivariate ranks and signs for change-point detection

Change-point detection for complex systems described by multivariate data is especially challenging if no information by the distribution before and after the structural change is available. In the univariate situation, the nonparametric methods based on ranks and signs are known to be powerful and effective. The possibility of extending the rank-based methods to the multivariate case has been discussed over the years. Only recently, Chernozhukov et al. (2015) offered an approach to define  multivariate ranks and signs based on the optimal transport of the original data to the uniform sample on the unit ball. A great advantage of this transform is pivotal under the null hypothesis of homogeneity. This fact opens the door to design a nonparametric procedure for a nonparametric change-point detection which does not require any prior information on the data distribution before and after a change-point.


  1. Pattern recognition and multiscale change-point detection

Parametric change-point detection is based on parameter estimation in a rolling window. If two estimates from neighbouring windows differ significantly, this indicates a possible structural change there. The main issue by this procedure is quantifying when the difference become significant. Another issue is the choice of window length: small windows lead to high variability of the estimators, large windows increases the delay and the location error in change-point detection. This thesis systematically studies the effect of a change-point pattern. It appears that a usual parameter shift yields a triangle pattern, a change of the slope or smooth transition result in a trapezoid type pattern. Pattern recognition allows for  the decrease of the variability of the change point detector and helps to  understand the type of structural change. The thesis includes a rigorous study of the change-point pattern phenomena and its application to econometric and financial data.


  1. Hidden Markov models (HMM) modelling and multivariate clustering

HMM are frequently used to describe the regime switching in dynamical stochastic systems. The main challenge in such modelling is to identify the number of states (regimes) of the systems. The methods, like cross-validation, appear to be unstable and unreliable in many situations because the model is not identifiable when too many states are used. Under natural assumptions, one can associate a hidden state with a cluster of the stationary (marginal) distribution of the observed system. Therefore, the problem of recovering the number of states of the system can be solved by clustering the marginal distribution. The thesis will focus on developing an Adaptive Weights Clustering method for automatic clustering of the high dimensional distribution using the ideas from Polzehl and Spokoiny (2000, 2006). Further, this procedure will be combined with HMM method to model the dynamics of a complex system.



  • Chernozhukov V, Galichon A, Hallin M, Henry M (2015) Monge-Kantorovich depth, quantiles, ranks, and signs. Working Papers ECARES 2015-02.

  • Polzehl J, Spokoiny V (2000) Adaptive weights smoothing with applications to image restoration. Journal of the Royal Statistical Society, Ser. B, 62, 2, 335–354.

  • Polzehl J, Spokoiny V (2006) Propagation-Separation Approach for Local Likelihood Estimation. Probability Theory and Related Fields, 135 (3), 335–362, DOI: 10.1007/s00440-005-0464-1