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