Testing for the extent of instability in nearly unstable processes

This paper deals with unit root issues in time series analysis. It has been known for a long time that unit root tests may be flawed when a series although stationary has a root close to unity. That motivated recent papers dedicated to autoregressive processes where the bridge between stability and instability is expressed by means of time-varying coefficients. The process we consider has a companion matrix \(A_{n}\) with spectral radius \(\rho(A_{n}) < 1\) satisfying \(\rho(A_{n}) \rightarrow 1\), a situation described as `nearly-unstable'. The question we investigate is: given an observed path supposed to come from a nearly-unstable process, is it possible to test for the `extent of instability', i.e. to test how close we are to the unit root? In this regard, we develop a strategy to evaluate \(\alpha\) and to test for \(\mathcal{H}_0 : ``\alpha = \alpha_0"\) against \(\mathcal{H}_1 : ``\alpha > \alpha_0"\) when \(\rho(A_{n})\) lies in an inner \(O(n^{-\alpha})\)-neighborhood of the unity, for some \(0 < \alpha < 1\). Empirical evidence is given about the advantages of the flexibility induced by such a procedure compared to the common unit root tests. We also build a symmetric procedure for the usually left out situation where the dominant root lies around \(-1\).