On The Peña–Box Model

Abstract

a factor model which aimed to explore the possibility of using lower-dimensional series to represent or explain an observed higher-dimensional multiple time series. However, there were no statistics with distribution results with which to build the model. In this paper, we derive a statistical procedure to build the model for stationary and first-order non-stationary series. The main idea, conducted by the canonical correlation analysis between present series and non-present series, is an extension of the concept of the scalar component model proposed by Tiao and Tsay [Journal of the Royal Statistical Society B (1989) Vol. 51, pp. 157-213]. Finally, simulation studies and reanalysis of two real data sets are illustrated.All matrices, C Z (h), h " 0, are of rank r, the number of factors, and have the same null space. Hence, a (k À r) · k matrix, M 1 , exists such thatThis reminds us of the SCM(0,0), a special case of the SCM proposed by Tiao and Tsay (1989). In that paper, Z t is said to have (k À r) SCM(0,0) if a 812

A generalized time‐effect factor model and its application: recovering trend of temperature by pollen data

“…Furthermore, following the arguments given in Hu and Chou (2004), the statistic defined in Equation (5) is valid also for the first-order nonstationary series. We thus have the following theorem whose proof is given in the Appendix.…”

“…The reason is that if Z t consists of s 2-SCM(0,q), then, for i > s,Ŝ i (m) will go to infinity becauseλ 2 i converges to a positive value. Detail arguments can be found in Hu and Chou (2004). An estimated GTFM with 2-SCM structure can therefore be constructed as follows.…”

“…, Z t+m ) are zero for all m ≥ q + 1. Therefore, a statistic based on the sample canonical correlation coefficients can be constructed to identify the model following the discussions of Hu and Chou (2004). First, calculating the transformation matrix of canonical correlation coefficients between Z t and Y m,q,t ,M = [M 1 ,M 2 ] , which is composed of the eigenvectors ofββ , wherê q,t .…”

“…The proof is based on the arguments of Tiao and Tsay (1989) and Hu and Chou (2004) for verifying the SCM and the Peña-Box model, respectively. If Z t consists of s 2-SCM(0,q), a matrix, M, exists such that MZ t = (S t , N t ) , where all nontrivial linear transformation of the b-dimensional series N t are nonstationary and S t is a (…”

Identifying the time‐effect factors of multiple time series

“…To identify the model, Hu and Chou (2004) gave a procedure to test the number of factors assuming that Z t represents a Gaussian process. However, for the model to be applied to nonlinear series, e.g., ARCH and GARCH series, a consistent estimator for identifying the model is proposed.…”