J. Ital. Statist. Soc. (1998) 3, pp. 297-306 TESTS OF INTEGRATION IN CIRCULAR AUTOREGRESSIVE MODELS Paolo Paruolo* Universita dell'Insubria, Varese, Italy Summary Two likelihood-ratio tests of the hypothesis of integration of order I in Gaussian circular vector autoregressive models (CAR) are derived. The tests have non-standard limit distributions which can be expressed as functionals of Brownian motion. Inference on the unit roots can be decomposed and addressed in independent subsystems with 1 or 2 components. Keywords: Spatial autoregressiveprocesses, integration, cointegration, circular matrices. 1. Introduction Vector autoregressive processes (AR) in discrete time are often used to construct models for space-time series, see e.g. Niu and Tiao (1995) and references therein. We consider AR processes that can describe phenomena taking place in adjacent zones on a circle, or on a curve homotopic to a circle; such type of observations are typical e.g. of stratospheric ozone concentrations measured on contiguous regions of equal latitude and area, as the ones analyzed in Niu and Tiao (1995). Within the class of AR processes we characterize the subclass of processes which are invariant with respect to rotations of the zones, that is with respect to the choice of the reference system; such processes are called circular AR processes, CAR. Some characteristics of the CAR class are described in Section 2. In this paper we consider inference on the number of unit roots at frequency zero for integrated CAR processes of order one. In Section 3 we summarize the parametric restrictions that characterize CAR processes integrated of order one, while in Section 4 an equivalent representation of the CAR processes is intro- * Address for correspondence: Facolta di Economia,Universitadell'lnsubria, ViaRavasi, 2 - 1-21100, Varese,Italy. E-mail: pparuolo@mail.eco.uninsubria.it Partial financial support from MURST grants ex 60% is gratefully acknowledged, This paper written while the Author was working at the Statistics Department, University of Bologna. The paper was presented at the XXXIX Scientific Meeting of the Italian Statistical Society held in Sorrento, April 1998. 297