Robust Estimation of Transition Matrices in High Dimensional Heavy-tailed Vector Autoregressive Processes

Gaussian vector autoregressive (VAR) processes have been extensively studied in the literature. However, Gaussian assumptions are stringent for heavy-tailed time series that frequently arises in finance and economics. In this paper, we develop a unified framework for modeling and estimating heavy-tailed VAR processes. In particular, we generalize the Gaussian VAR model by an elliptical VAR model that naturally accommodates heavy-tailed time series. Under this model, we develop a quantile-based robust estimator for the transition matrix of the VAR process. We show that the proposed estimator achieves parametric rates of convergence in high dimensions. This is the first work in analyzing heavy-tailed high dimensional VAR processes. As an application of the proposed framework, we investigate Granger causality in the elliptical VAR process, and show that the robust transition matrix estimator induces sign-consistent estimators of Granger causality. The empirical performance of the proposed methodology is demonstrated by both synthetic and real data. We show that the proposed estimator is robust to heavy tails, and exhibit superior performance in stock price prediction.

[1]  C. Sims.  MACROECONOMICS AND REALITY , 1977 .

[2]  C. Granger.  Testing for causality: a personal viewpoint , 1980 .

[3]  G. Simons,et al.  On the theory of elliptically contoured distributions , 1981 .

[4]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[5]  P. Rousseeuw,et al.  Alternatives to the Median Absolute Deviation , 1993 .

[6]  James D. Hamilton.  Time Series Analysis , 1994 .

[7]  R. Adler,et al.  A practical guide to heavy tails: statistical techniques and applications , 1998 .

[8]  Ruey S. Tsay,et al.  Analysis of Financial Time Series: Tsay/Analysis of Financial Time Series , 2005 .

[9]  Ruey S. Tsay,et al.  Analysis of Financial Time Series , 2005 .

[10]  Chih-Ling Tsai,et al.  Regression coefficient and autoregressive order shrinkage and selection via the lasso , 2007 .

[11]  João Ricardo Sato,et al.  Modeling gene expression regulatory networks with the sparse vector autoregressive model , 2007, BMC Systems Biology.

[12]  Helmut Ltkepohl,et al.  New Introduction to Multiple Time Series Analysis , 2007 .

[13]  Korbinian Strimmer,et al.  Learning causal networks from systems biology time course data: an effective model selection procedure for the vector autoregressive process , 2007, BMC Bioinformatics.

[14]  K. Ramanan,et al.  Concentration Inequalities for Dependent Random Variables via the Martingale Method , 2006, math/0609835.

[15]  Nan-Jung Hsu,et al.  Subset selection for vector autoregressive processes using Lasso , 2008, Comput. Stat. Data Anal..

[16]  Q. Yao,et al.  Modelling multiple time series via common factors , 2008 .

[17]  M. Mohri,et al.  Stability Bounds for Stationary φ-mixing and β-mixing Processes , 2010 .

[18]  Ali Shojaie,et al.  Discovering graphical Granger causality using the truncating lasso penalty , 2010, Bioinform..

[19]  Y. Nardi,et al.  Autoregressive process modeling via the Lasso procedure , 2008, J. Multivar. Anal..

[20]  T. Cai,et al.  A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation , 2011, 1102.2233.

[21]  Bruno Rémillard,et al.  Copula-Based Semiparametric Models for Multivariate Time Series , 2011, J. Multivar. Anal..

[22]  G. Michailidis,et al.  Estimation in High-dimensional Vector Autoregressive Models , 2013 .

[23]  Fang Han,et al.  Transition Matrix Estimation in High Dimensional Time Series , 2013, ICML.

[24]  Todd E. Clark,et al.  Large Vector Autoregressions with Asymmetric Priors , 2015 .

[25]  Han Liu,et al.  Joint estimation of multiple graphical models from high dimensional time series , 2013, Journal of the Royal Statistical Society. Series B, Statistical methodology.