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. In this vein the process we consider has a companion matrix with spectral radius satisfying , a situation that we describe as `nearly unstable'. The question we investigate is the following: given an observed path supposed to come from a nearly-unstable process, is it possible to test for the `extent of instability', \textit{i.e.} to test how close we are to the unit root? In this regard, we develop a strategy to evaluate and to test for against when lies in an inner -neighborhood of the unity, for some . Empirical evidence is given (on simulations and real time series) about the advantages of the flexibility induced by such a procedure compared to the usual unit root tests and their binary responses. As a by-product, we also build a symmetric procedure for the usually left out situation where the dominant root lies around .
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