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 with spectral radius satisfying , 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 and to test for : ‘’ against : ‘’ when lies in an inner ‐neighborhood of the unity, for some . 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 .