MULTIVARIATE REDUCED-RANK NONLINEAR TIME SERIES MODELING

Panels of nonlinear time series data are increasingly collected in scientic studies, and a fundamental problem is to study the common dynamic structures of such data. We propose a new model for exploring the common dynamic structure in multivariate nonlinear time series. The basic idea is that the panel of time series are driven by an underlying low-dimensional nonlinear principal component process that is modeled as some nonlinear function of the past lags of the time series. In particular, we consider in some detail the REduced-rank Threshold AutoRegressive (RETAR) model whose nonlinear principal component process is a piecewise linear vector-valued function of past lags of the panel of time series. We propose an estimation scheme for the RETAR model and derive the large sample properties of the estimator. We illustrate the RETAR model using a modern panel of eight Canada lynx series, and demonstrate a classication of lynx series that is broadly similar to that reported by Stenseth et al. (1999), who used a dieren t approach.

[1]  Felix Schlenk,et al.  Proof of Theorem 3 , 2005 .

[2]  D. Pollard.  Convergence of stochastic processes , 1984 .

[3]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[4]  Neil H. Timm,et al.  Multivariate Reduced-Rank Regression , 1999, Technometrics.

[5]  A. Izenman.  17 Assessing dimensionality in multivariate regression , 1980 .

[6]  F. Graybill,et al.  Matrices with Applications in Statistics. , 1984 .

[7]  H. Tong,et al.  On consistent nonparametric order determination and chaos , 1992 .

[8]  H. Tong,et al.  Testing for Common Structures in a Panel of Threshold Models , 2004, Biometrics.

[9]  K. S. Lim.  A COMPARATIVE STUDY OF VARIOUS UNIVARIATE TIME SERIES MODELS FOR CANADIAN LYNX DATA , 1987 .

[10]  H. Tong,et al.  On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations , 1985, Advances in Applied Probability.

[11]  K. Chan,et al.  Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model , 1993 .

[12]  K. Chan.  A Review of Some Limit Theorems of Markov Chains and Their Applications , 1993 .

[13]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .

[14]  R. Tweedie.  Criteria for classifying general Markov chains , 1976, Advances in Applied Probability.

[15]  G. Reinsel,et al.  Multivariate Reduced-Rank Regression: Theory and Applications , 1998 .

[16]  F. G. Foster.  On the Stochastic Matrices Associated with Certain Queuing Processes , 1953 .

[17]  Gregory C. Reinsel,et al.  Reduced rank models for multiple time series , 1986 .

[18]  Howell Tong,et al.  Some Comments on the Canadian Lynx Data , 1977 .

[19]  K. S. Chan,et al.  Chaos: A Statistical Perspective , 2001 .

[20]  Howell Tong,et al.  Threshold autoregression, limit cycles and cyclical data- with discussion , 1980 .

[21]  H. Tong.  Non-linear time series. A dynamical system approach , 1990 .

[22]  Mohsen Pourahmadi,et al.  Nonparametric and non‐linear models and data mining in time series: a case‐study on the Canadian lynx data , 2008 .

[23]  P. Robinson,et al.  Generalized canonical analysis for time series , 1973 .

[24]  E. Nummelin.  General irreducible Markov chains and non-negative operators: Preface , 1984 .