Explosive strong periodic autoregression with multiplicity one

摘要

It is well known that the asymptotic distribution of the ordinary least squares estimate (OLSE) for a strong periodic autoregression (PAR) driven by an independent innovation is Gaussian under periodic stationarity and functional of the Brownian motion under periodic integration. This paper studies the asymptotic distribution of OLSE for strong pth order PAR (p) models in the explosive case with multiplicity one, i.e. when a certain autoregressive parameter lies outside both the periodic stationarity domain and its boundary, while all remaining parameters are inside the periodic stationarity region. It will be shown that the OLSE for PAR(1), scaled by a function of the monodromy parameter, converges in distribution to a random vector whose distribution reduces under the normality assumption to the standard multivariate Cauchy distribution. Furthermore, for a PAR(p) model, the OLSE, scaled by a function of the design matrix of the Gaussian model, is asymptotically Gaussian with independent components. Thus, the knife edge effect known for linear time-invariant AR models is still valid in the strong periodically time-varying case. (C) 2014 Elsevier B.V. All rights reserved.