In this study, we derive the limiting distribution of the least squares estimator (LSE) and the localized LSE for mildly explosive autoregressive models with locally stationary disturbance and verify that it is Cauchy as in the iid case. We also investigate the limiting distribution of two types of Dickey–Fuller unit root tests, designed for detecting a bubble period in economic time series data, and show that these tests are consistent. To evaluate the methods, we conduct a simulation study and carry out a data analysis using time series data on bitcoin prices.

中文翻译：

---

具有局部平稳扰动的轻度爆炸过程的渐近特性

在这项研究中，我们推导出最小二乘估计量 (LSE) 和局部 LSE 的极限分布，用于具有局部平稳扰动的轻度爆炸性自回归模型，并验证它在 iid 情况下是柯西。我们还研究了两种类型的 Dickey-Fuller 单位根检验的极限分布，这些检验旨在检测经济时间序列数据中的泡沫期，并表明这些检验是一致的。为了评估这些方法，我们进行了模拟研究并使用比特币价格的时间序列数据进行数据分析。