Extreme Spectra of Var Models and Orders of Near-Cointegration

In this paper, we study the spectral properties of a bivariate vector autoregressive VAR(p) model when a root z_0 = rho_e-super-i Lambda _0 of the determinant of the model's characteristic matrix Phi(z) approaches the unit circle, the border of non-stationarity. Let Phi_xx(z), Phi_xy(z), Phi_yx(z), Phi_yy(z) be the polynomial elements of Phi(z). We show that, depending on the relation of the order of z_0 as root of det(Phi(z)) with the orders of z_0 as root of Phi_ij(z), (i,j is an element of [x,y]), the two marginal spectra may tend to infinity at Lambda _0, while the coherence may tend to unity at Lambda _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. Copyright 2005 Blackwell Publishing Ltd.