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 $-1$.

翻译：本文探讨时间序列分析中的单位根问题。长期以来已知，当序列虽平稳但其根接近单位圆时，单位根检验可能存在缺陷。这促使近期研究关注通过时变系数表达稳定性与不稳定性之间过渡的自回归过程。我们所考虑过程的伴随矩阵$A_{n}$具有谱半径$\rho(A_{n}) < 1$且满足$\rho(A_{n}) \rightarrow 1$，这种情形被称为"近乎不稳定"。我们研究的问题是：给定一个假设来自近乎不稳定过程的观测路径，是否可能检验"不稳定程度"，即检验我们距离单位根有多近？对此，当$\rho(A_{n})$位于单位圆内$O(n^{-\alpha})$邻域（其中$0 < \alpha < 1$）时，我们开发了评估$\alpha$并检验$\mathcal{H}_0 : ``\alpha = \alpha_0"$相对于$\mathcal{H}_1 : ``\alpha > \alpha_0"$的策略。实证证据表明，与传统单位根检验相比，此方法具有灵活性优势。我们还针对通常被忽略的、主导根位于$-1$附近的情形构建了对称检验程序。