In this paper, we study a regression model with a break in trend regressor, in which the model errors are assumed to be mildly integrated. To be precise, we suppose the model errors are generated by an AR(1) process with the autoregressive coefficient \(\rho _{T}=1+{c}/{k_{T}}\), where T is the sample size, c is a negative constant, and \(\{k_T, T\in {\mathbb {N}}\}\) is a sequence of positive constants diverging to infinity such that \(k_T=o(T)\). We estimate the break date/break fraction and other parameters in the model using the least squares method. The asymptotic properties, including the consistency, rates of convergence as well as the limiting distributions, of the estimates are examined. The results derived in this paper bridge the findings in Perron and Zhu (Journal of Econometrics 129:65–119, 2005) who estimated the break date/break fraction in trend regressor under I(0) and I(1) model errors. We also show that the phase transition for the estimation error of the least squares estimate of the break date occurs when \(k_{T}\) has the same order of magnitude as \(T^{1/2}\). Monte Carlo simulations and an empirical study are given to illustrate the finite-sample performance of estimates.

在本文中，我们研究了一个带有趋势回归量中断的回归模型，其中假设模型误差被温和地整合。准确地说，我们假设模型误差是由具有自回归系数\(\rho _{T}=1+{c}/{k_{T}}\)的 AR(1) 过程生成的，其中T是样本大小，c是一个负常数，而\(\{k_T, T\in {\mathbb {N}}\}\)是一个发散到无穷大的正常数序列，使得\(k_T=o(T)\ ). 我们使用最小二乘法估计模型中的中断日期/中断分数和其他参数。检查了估计的渐近特性，包括一致性、收敛速度以及极限分布。本文得出的结果与 Perron 和 Zhu (Journal of Econometrics 129:65–119, 2005) 的发现相吻合，他们估计了I (0) 和I (1) 模型误差下趋势回归量中的断裂日期/断裂分数。我们还表明，当\(k_{T}\)与\(T^{1/2}\)具有相同的数量级时，发生中断日期最小二乘估计误差的相变. 给出了蒙特卡罗模拟和实证研究来说明估计的有限样本性能。