弱相依非线性时间序列模型拟似然估计量的均匀高斯近似

中文摘要 AI 辅助

我们研究仅指定条件期望的半参数时间序列模型类的估计与推断，该模型类的条件期望是已知连接函数作用于过去观测值与协变量的线性组合。该模型类在单一框架内涵盖计数、二元、有界及条件异方差响应，参数通过基于一阶条件矩的拟似然估计方程进行估计。在平稳性及通过函数相依测度表达的弱相依条件下，我们建立两个结果：其一，利用针对弱相依序列的Fuk--Nagaev不等式，证明估计量以概率$1-o(n^{-1/2})$落在真实值的收缩邻域内；其二，结合针对弱相依序列的Berry--Esseen界与高斯反集中论证以控制线性展开的余项，得到估计量线性投影的均匀Berry--Esseen界（对所有投影方向均成立）。从该投影界出发，我们推导了具有明确覆盖误差的学生化置信区间，以及针对参数线性假设的保守Bonferroni检验。在实际数据分析中，我们将双有界数据的Beta自回归模型扩展至由分布函数逆给出的任意连接函数，并将其应用于大盘科技股收益率的10对已实现相关性分析，使用纳斯达克（Nasdaq）与道琼斯（Dow Jones）指数收益率作为协变量。

英文摘要

We study estimation and inference for a semiparametric class of time series models that specify only the conditional expectation, which is a known link function applied to a linear combination of past observations and covariates. The class covers count, binary, bounded and conditionally heteroskedastic responses within a single formulation, and the parameter is estimated by a quasi-likelihood estimating equation based on the first conditional moment. Under stationarity and a weak-dependence condition expressed through the functional dependence measure, we establish two results. First, using a Fuk--Nagaev inequality for weakly dependent sequences, we show that the estimator is localized in a shrinking neighbourhood of the true value with probability $1-o(n^{-1/2})$. Second, combining a Berry--Esseen bound for weakly dependent sequences with a Gaussian anti-concentration argument to control the remainder of the linear expansion, we obtain a Berry--Esseen bound for linear projections of the estimator, uniform over projection directions. From the projected bound we derive studentized confidence intervals with explicit coverage error and a conservative Bonferroni test for linear hypotheses on the parameters. For real data analysis, we extend the Beta autoregression for double-bounded data to an arbitrary link given by the inverse of a distribution function, and apply it to ten pairwise realized correlations of large-cap technology-stock returns, using Nasdaq and Dow~Jones index returns as covariates.

发表机构

• Université du Québec à Montréal, Département de Mathématiques（蒙特利尔大学数学系）
• Vanderbilt University, Department of Mathematics（范德堡大学数学系）

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