Consistency and asymptotic normality in a class of nearly unstable processes

This paper deals with inference in a class of stable but nearly-unstable processes. Autoregressive processes are considered, which the bridge between stability and instability is expressed by time-varying companion matrix $$A_{n}$$ spectral radius $$\rho (A_{n}) < 1$$ satisfying \rightarrow . framework particularly suitable to understand unit root issues focusing on inner boundary circle. Consistency established for empirical covariance OLS estimation together asymptotic normality under appropriate hypotheses when A, limit $$A_n$$ , has real spectrum, particular case deduced A also contains complex eigenvalues. The process integrated either one (located at 1 or $$-1$$ ), even two roots located Finally, set simulations illustrate behavior OLS. results essentially proved $$L^2$$ computations theory triangular arrays martingales.