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 $-1$.