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Extreme Spectra of Var Models and Orders of Near‐Cointegration
Author(s) -
Ioannidis E. E.,
Chronis G. A.
Publication year - 2005
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.2004.00408.x
Subject(s) - mathematics , autoregressive model , unit root , cointegration , bivariate analysis , series (stratigraphy) , unit circle , order (exchange) , univariate , polynomial , combinatorics , pure mathematics , mathematical analysis , statistics , multivariate statistics , paleontology , finance , economics , biology
. In this paper, we study the spectral properties of a bivariate vector autoregressive VAR( p ) model when a root z 0 = ρ 0 e i λ 0 of the determinant of the model's characteristic matrix Φ( z ) approaches the unit circle, the border of non‐stationarity. Let Φ xx ( z ), Φ xy ( z ), Φ yx ( z ), Φ yy ( z ) be the polynomial elements of Φ( z ). We show that, depending on the relation of the order of z 0 as root of det(Φ( z )) with the orders of z 0 as root of Φ ij ( z ), ( i , j ∈ { x , y }), the two marginal spectra may tend to infinity at λ 0 , while the coherence may tend to unity at λ 0 . We investigate the conditions under which any of the above will occur, in detail. In the specific case where z 0 →1, the marginal series will be near‐integrated of certain orders of near‐integration, while there will eventually exist a linear combination of them with a lower order of near‐integration. We study the possible combinations of their orders of near‐integration. Finally, we develop a strategy with the help of which one may define a VAR( p ) model with pre‐specified extreme spectral features and give some examples. Beyond the benefits of this latter for VAR model simulation, the analysis has, moreover, implications concerning the adequacy of VAR model fitting.