Bilinear (p, 0, 1, 1) Model and its Consistent Identification

Jian Liu and Zhao-Guo Chen
Additional contact information
Jian Liu: University of British Columbia and Institute of Applied Mathematics, Academy of Science
Zhao-Guo Chen: University of British Columbia and Institute of Applied Mathematics, Academy of Science

A chapter in Computing Science and Statistics, 1992, pp 443-445 from Springer

Abstract: Abstract This paper is concerned with the BL(p, 0, 1, 1) model $$\sum_{j=0}^{p}a_jx(t-j)=e(t)+bx(t-1)e(t-1), \left\{e(t)\right\}$$ are iid, $$Ee(t)=0,Ee(t)^2=\sigma ^2$$ A concept of weak irreducibility is introduced. It is shown that the bilinear model can be expressed as an ARMA(p, 1) model in the sense of weak stationarity provided that weak irreducibility is satisfied. Using this expression we prove that the moment estimates of a j are unique and consistent. If, in addition, e(t) is JV(0, σ 2) or its first few order moment structure are known a prior, using up to the third order moments, b and σ 2 can also be estimated consistently. Furthermore, these estimates are shown to be asymptotically normal and have a convergence rate of O(ln ln N/N)1/2, where N is the number of observations. In contrast, the usual MLE estimates of the parameters have not been justified to have such asymptotic properties without assuming the finiteness of any order moments of the bilinear series and their infinite linear combinations. Finally, based on the ARMA expression, a consistent indentification procedure for the order p is proposed. Though the model discussed here is somewhat specialized, we hope that the ideas demonstrated here could highlight the possibility of making consistent inference without first solving the higher order moment problem.

Keywords: Order Moment; Moment Estimate; Bilinear System; Nonlinear Time Series; Bilinear Model (search for similar items in EconPapers)
Date: 1992
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-2856-1_72

Ordering information: This item can be ordered from
http://www.springer.com/9781461228561

DOI: 10.1007/978-1-4612-2856-1_72

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().