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 $A_{n}$ with spectral radius $\rho(A_{n}) < 1$ satisfying $\rho(A_{n}) \rightarrow 1$, 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 $\alpha$ 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 (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 $-1$.

翻译：本文研究时间序列分析中的单位根问题。长期以来人们已知，当序列虽为平稳但根接近1时，单位根检验可能存在缺陷。这促使近期研究关注自回归过程，其中稳定性与非稳定性之间的桥梁通过时变系数来表达。在此思路下，我们考虑的过程具有伴矩阵$A_{n}$，其谱半径$\rho(A_{n}) < 1$满足$\rho(A_{n}) \rightarrow 1$，我们将此情形描述为“近非稳定”。我们探究的问题是：给定一个假设来自近非稳定过程的观测路径，能否检验其“不稳定程度”，即检验我们距离单位根有多近？为此，我们发展了一种策略来评估$\alpha$，并在$\rho(A_{n})$位于单位根的内部$O(n^{-\alpha})$-邻域内（对于某个$0 < \alpha < 1$）时，检验原假设$\mathcal{H}_0 : "\alpha = \alpha_0"$对立假设$\mathcal{H}_1 : "\alpha > \alpha_0"$。通过模拟和真实时间序列的实证证据表明，与通常的单位根检验及其二元响应相比，此方法带来的灵活性具有优势。作为副产品，我们还为通常被忽略的主根位于-1附近的情形构建了一个对称的检验程序。