Double Local-to-Unity: Inference under Nearly Nonstationary Volatility
双重局部趋近单位根：近非平稳波动率下的推断AI 解读 ↓
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双重局部趋近单位根：近非平稳波动率下的推断
- 研究问题
- 当自回归均值与波动率都高度持久、条件方差接近非平稳时，如何建立仍可用于最小二乘估计和检验的极限理论？
- 方法与思路
- 作者让自回归系数以慢于经典速率的速度趋近单位根，波动率持久参数以更慢的对数阶趋近一，并推导联合渐近理论。
- 材料陈述的贡献
- 摘要称，轻度平稳情形得到正态极限，轻度爆炸情形的自标准化统计量得到柯西极限，且这些极限不依赖波动率的详细形式。
- 阅读关注点
- 应核对两类局部化速率、平均波动率尺度和自标准化统计量的定义，以及极限分布不变性所需的规则性条件。
- 阅读边界
- 仅据 arXiv 预印本摘要，不能确认同行评审或期刊接收；规则性条件和统计量构造未展开，不能据此判断有限样本稳健性。
原文摘要
This article develops a moderate-deviation limit theory for autoregressive models with jointly persistent mean and volatility dynamics. The autoregressive coefficient is allowed to drift toward unity slower than the classical 1/n rate, while the volatility persistence parameter also converges to one at an even slower, logarithmic order, so that the conditional variance process is itself nearly nonstationary and its unconditional moments may diverge. This double localization allows the variance process to be nearly nonstationary and to evolve slowly, as observed in financial data and during asset price bubble episodes. Under standard regularity conditions, we establish consistency and distributional limits for the OLS estimator of the autoregressive coefficient that remains valid in the presence of highly persistent stochastic volatility. We show that the effective normalization for least squares inference is governed by an average volatility scale, and we derive martingale limit theorems for the OLS estimator under joint drift and volatility dynamics. In a mildly stationary regime (where the autoregressive root approaches one from below), the OLS estimator is asymptotically normal. In a mildly explosive regime (where the root approaches one from above), an OLS based self normalized statistic converges to a Cauchy limit. Strikingly, in both regimes, the limiting laws of our statistics are invariant to the detailed specification of the volatility process, even though the conditional variance is itself nearly nonstationary. Overall, the results extend moderate-deviation asymptotics to settings with drifting volatility persistence, unify local to unity inference with nearly nonstationary stochastic volatility, and deliver practically usable volatility robust statistics for empirical work in settings approaching instability and exhibiting bubbles.