Testing for the extent of instability in nearly unstable processes

This article 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 $\rho \left(A_n\right)<1$ satisfying $\rho \left(A_n\right)\to 1$, 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 ${\mathscr{H}}_0$ : ‘$\alpha ={\alpha }_0$’ against ${\mathscr{H}}_1$ : ‘$\alpha >{\alpha }_0$’ when $\rho \left(A_n\right)$ lies in an inner $O\left(n^{-\alpha }\right)$-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 $-1$.