Inference in heavy-tailed non-stationary multivariate time series
厚尾非平稳多元时间序列中的推断AI 解读 ↓
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厚尾非平稳多元时间序列中的推断
- 研究问题
- 当非平稳多元序列可能厚尾、甚至方差不存在时，如何在未知尾指数的情况下识别共同随机趋势数量并估计趋势？
- 方法与思路
- 作者利用样本二阶矩矩阵的特征值增长差异，构造随机化统计量与序贯检验估计趋势数，并以主成分估计共同趋势。
- 材料陈述的贡献
- 摘要称趋势数估计具有一致性，趋势估计在可逆线性变换意义下误差相对趋势更小，并讨论创新尺度异质性的扩展。
- 阅读关注点
- 核对特征值增长差异成立的过程条件、随机化与序贯检验实施方式，以及趋势估计一致性的定义和异质尺度扩展范围。
- 阅读边界
- 所据为 arXiv 预印本，不能确认同行评审或期刊接收；摘要未给出完整过程条件、检验调参规则及小样本表现的数值。
原文摘要
We study inference on the common stochastic trends in a non-stationary, $N$-variate time series $y_{t}$, in the possible presence of heavy tails. We propose a novel methodology which does not require any knowledge or estimation of the tail index, or even knowledge as to whether certain moments (such as the variance) exist or not, and develop an estimator of the number of stochastic trends $m$ based on the eigenvalues of the sample second moment matrix of $y_{t}$. We study the rates of such eigenvalues, showing that the first $m$ ones diverge, as the sample size $T$ passes to infinity, at a rate faster by $O\left(T \right)$ than the remaining $N-m$ ones, irrespective of the tail index. We thus exploit this eigen-gap by constructing, for each eigenvalue, a test statistic which diverges to positive infinity or drifts to zero according to whether the relevant eigenvalue belongs to the set of the first $m$ eigenvalues or not. We then construct a randomised statistic based on this, using it as part of a sequential testing procedure, ensuring consistency of the resulting estimator of $m$. We also discuss an estimator of the common trends based on principal components and show that, up to a an invertible linear transformation, such estimator is consistent in the sense that the estimation error is of smaller order than the trend itself. Finally, we also consider the case in which we relax the standard assumption of \textit{i.i.d.} innovations, by allowing for heterogeneity of a very general form in the scale of the innovations. A Monte Carlo study shows that the proposed estimator for $m$ performs particularly well, even in samples of small size. We complete the paper by presenting four illustrative applications covering commodity prices, interest rates data, long run PPP and cryptocurrency markets.