格兰杰因果关系 Granger causality

基本原则

1. 自变量发生在因变量之前。
2. 这一自变量对于被其影响的因变量的未来值有着独特的信息。

$\displaystyle{ \mathbb{P}[Y(t+1) \in A\mid \mathcal{I}(t)] \neq \mathbb{P}[Y(t+1) \in A\mid \mathcal{I}_{-X}(t)], }$

格兰杰因果分析方法

数学表述

$\displaystyle{ y }$$\displaystyle{ x }$均为平稳时间序列。为了检验$\displaystyle{ x }$不是$\displaystyle{ y }$的格兰杰原因这一零假设，首先要$\displaystyle{ y }$的适当滞后值，以包含在$\displaystyle{ y }$的单变量自回归中：

$\displaystyle{ y_t = a_0 + a_1y_{t-1} + a_2y_{t-2} + \cdots + a_my_{t-m} + \text{error}_t. }$

$\displaystyle{ y_t = a_0 + a_1y_{t-1} + a_2y_{t-2} + \cdots + a_my_{t-m} + b_px_{t-p} + \cdots + b_qx_{t-q} + \text{error}_t. }$

多元分析

$\displaystyle{ X(t) = \sum_{\tau=1}^L A_{\tau}X(t-\tau) + \varepsilon(t), }$

局限性

1. 采样不够频繁或过于频繁;
2. 非线性因果关系
3. 时间序列的非平稳性和非线性
4. 理性预期的存在。

参考文献

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