This paper deals with inference in a class of stable but nearly-unstable processes. Autoregressive processes are considered, in which the bridge between stability and instability is expressed by a time-varying companion matrix $A_{n}$ with spectral radius $\rho(A_{n}) < 1$ satisfying $\rho(A_{n}) \rightarrow 1$. This framework is particularly suitable to understand unit root issues by focusing on the inner boundary of the unit circle. Consistency is established for the empirical covariance and the OLS estimation together with asymptotic normality under appropriate hypotheses when $A$, the limit of $A_n$, has a real spectrum, and a particular case is deduced when $A$ also contains complex eigenvalues. The asymptotic process is integrated with either one unit root (located at 1 or $-1$), or even two unit roots located at 1 and $-1$. Finally, a set of simulations illustrate the asymptotic behavior of the OLS. The results are essentially proved by $L^2$ computations and the limit theory of triangular arrays of martingales.

翻译：本文研究一类稳定但近乎不稳定过程的推断问题。考虑自回归过程，其中稳定性与不稳定性的桥梁由时变伴随矩阵$A_{n}$表示，其谱半径$\rho(A_{n}) < 1$且满足$\rho(A_{n}) \rightarrow 1$。该框架通过聚焦单位圆的内边界，特别适用于理解单位根问题。在$A$（$A_n$的极限）具有实谱的适当假设下，建立了经验协方差和OLS估计的一致性及渐近正态性，并推导出当$A$包含复特征值时的特例。渐近过程整合了一个单位根（位于1或-1），甚至两个分别位于1和-1的单位根。最后，通过一系列仿真实验展示了OLS的渐近行为。研究结果主要通过$L^2$计算和三角阵列鞅的极限理论证明。