Regularization for stationary multivariate time series

The complexity of multivariate time series models increases dramatically when the number of component series increases. This is a phenomenon observed in both low- and high-frequency financial data analysis. In this paper, we develop a regularization framework for multivariate time series models based on the penalized likelihood method. We show that, under certain conditions, the regularized estimators are sparse-consistent and satisfy an asymptotic normality. This framework provides a theoretical foundation for addressing the curse of dimensionality in multivariate econometric models. We illustrate the utility of our method by developing a sparse version of the full-factor multivariate GARCH model. We successfully apply this model to simulated data as well as the minute returns of the Dow Jones industrial average component stocks.

[1]  Jianqing Fan,et al.  Modelling multivariate volatilities via conditionally uncorrelated components , 2005, math/0506027.

[2]  H. Zou.  The Adaptive Lasso and Its Oracle Properties , 2006 .

[3]  J. Wooldridge,et al.  Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances , 1992 .

[4]  T. Bollerslev,et al.  Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model , 1990 .

[5]  Jianqing Fan,et al.  Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties , 2001 .

[6]  J. Norris.  Appendix: probability and measure , 1997 .

[7]  P. Bickel,et al.  Regularized estimation of large covariance matrices , 2008, 0803.1909.

[8]  J. Friedman,et al.  A Statistical View of Some Chemometrics Regression Tools , 1993 .

[9]  Pentti Saikkonen,et al.  A Multivariate Generalized Orthogonal Factor GARCH Model , 2007 .

[10]  M. Pourahmadi,et al.  Nonparametric estimation of large covariance matrices of longitudinal data , 2003 .

[11]  Adam J. Rothman,et al.  Sparse permutation invariant covariance estimation , 2008, 0801.4837.

[12]  Cun-Hui Zhang,et al.  A group bridge approach for variable selection , 2009, Biometrika.

[13]  R. Engle,et al.  Multivariate Simultaneous Generalized ARCH , 1995, Econometric Theory.

[14]  C. Hafner,et al.  ASYMPTOTIC THEORY FOR A FACTOR GARCH MODEL , 2006, Econometric Theory.

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

[16]  J. Friedman,et al.  [A Statistical View of Some Chemometrics Regression Tools]: Response , 1993 .

[17]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[18]  J. Wooldridge,et al.  A Capital Asset Pricing Model with Time-Varying Covariances , 1988, Journal of Political Economy.

[19]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[20]  Ioannis D. Vrontos,et al.  A Full-Factor Multivariate GARCH Model , 2003 .

[21]  M. Rosenblatt.  Central limit theorem for stationary processes , 1972 .

[22]  Carol Alexander,et al.  A Primer on the Orthogonal GARCH Model , 2001 .

[23]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[24]  R. Engle.  Dynamic Conditional Correlation , 2002 .

[25]  C. C. Heyde,et al.  On the central limit theorem for stationary processes , 1974 .