1 格兰格因果性

1.1 介绍

考虑两个时间序列之间的因果性。这里的因果性指的是时间顺序上的关系，如果\(X_{t-1}, X_{t-2}, \dots\)对\(Y_t\)有作用，而\(Y_{t-1}, Y_{t-2}, \dots\)对\(X_t\)没有作用，则称\(\{X_t \}\)是\(\{ Y_t \}\)的格兰格原因，而\(\{ Y_t \}\)不是\(\{ X_t \}\)的格兰格原因。如果\(X_{t-1}, X_{t-2}, \dots\)对\(Y_t\)有作用， \(Y_{t-1}, Y_{t-2}, \dots\)对\(X_t\)也有作用，则在没有进一步信息的情况下无法确定两个时间序列的因果性关系。

注意这种因果性与采样频率有关系，在日数据或者月度数据中能发现的领先——滞后性质的因果关系，到年度数据可能就以及混杂在以前变成同步的关系了。

1.2 格兰格因果性的定义

设\(\{ \xi_t \}\)为一个时间序列， \(\{ \boldsymbol{\eta}_t \}\)为向量时间序列，记 \[\begin{aligned} \bar{\boldsymbol{\eta}}_t =& \{ \boldsymbol{\eta}_{t-1}, \boldsymbol{\eta}_{t-2}, \dots \} \end{aligned}\]

记 \(\text{Pred}(\xi_t | \bar{\boldsymbol{\eta}}_t)\)为基于 \(\boldsymbol{\eta}_{t-1}, \boldsymbol{\eta}_{t-2}, \dots\) 对\(\xi_t\)作的最小均方误差无偏预报，其解为条件数学期望\(E(\xi_t | \boldsymbol{\eta}_{t-1}, \boldsymbol{\eta}_{t-2}, \dots)\)，在一定条件下可以等于\(\xi_t\)在\(\boldsymbol{\eta}_{t-1}, \boldsymbol{\eta}_{t-2}, \dots\)张成的线性Hilbert空间的投影（比如，\((\xi_t, \boldsymbol{\eta}_t)\)为平稳正态多元时间序列），即最优线性预测。直观理解成基于过去的\(\{\boldsymbol{\eta}_{t-1}, \boldsymbol{\eta}_{t-2}, \dots \}\)的信息对当前的\(\xi_t\)作的最优预测。

令一步预测误差为 \[ \varepsilon(\xi_t | \bar{\boldsymbol{\eta}}_t) = \xi_t - \text{Pred}(\xi_t | \bar{\boldsymbol{\eta}}_t) \] 令一步预测误差方差，或者均方误差，为 \[ \sigma^2(\xi_t | \bar{\boldsymbol{\eta}}_t) = \text{Var}(\varepsilon_t(\xi_t | \bar{\boldsymbol{\eta}}_t) ) = E \left[ \xi_t - \text{Pred}(\xi_t | \bar{\boldsymbol{\eta}}_t) \right]^2 \]

考虑两个时间序列\(\{ X_t \}\)和\(\{ Y_t \}\)， \(\{(X_t, Y_t) \}\)宽平稳或严平稳。

如果 \[ \sigma^2(Y_t | \bar Y_t, \bar X_t) < \sigma^2(Y_t | \bar Y_t) \] 则称\(\{ X_t \}\)是\(\{ Y_t \}\)的格兰格原因，记作\(X_t \Rightarrow Y_t\)。这不排除\(\{ Y_t \}\)也可以是\(\{ X_t \}\)的格兰格原因。
如果\(X_t \Rightarrow Y_t\)，而且\(Y_t \Rightarrow X_t\)，则称互相有反馈关系，记作\(X_t \Leftrightarrow Y_t\)。
如果 \[ \sigma^2(Y_t | \bar Y_t, X_t, \bar X_t) < \sigma^2(Y_t | \bar Y_t, \bar X_t) \] 即除了过去的信息，增加同时刻的\(X_t\)信息后对\(Y_t\)预测有改进，则称\(\{X_t \}\)对\(\{Y_t \}\)有瞬时因果性。这时\(\{Y_t \}\)对\(\{X_t \}\)也有瞬时因果性。
如果\(X_t \Rightarrow Y_t\)，则存在最小的正整数\(m\)，使得 \[ \sigma^2(Y_t | \bar Y_t, X_{t-m}, X_{t-m-1}, \dots) < \sigma^2(Y_t | \bar Y_t, X_{t-m-1}, X_{t-m-2}, \dots) \] 称\(m\)为因果性滞后值(causality lag)。如果\(m>1\)，这意味着在已有\(Y_{t-1}, Y_{t-2}, \dots\)和\(X_{t-m}, X_{t-m-1}, \dots\)的条件下，增加\(X_{t-1}\), , \(X_{t-m+1}\)不能改进对\(Y_t\)的预测。

例1.1 设\(\{ \varepsilon_t, \eta_t \}\)是相互独立的零均值白噪声列， \(\text{Var}(\varepsilon_t)=1\), \(\text{Var}(\eta_t)=1\), 考虑 \[\begin{aligned} Y_t =& X_{t-1} + \varepsilon_t \\ X_t =& \eta_t + 0.5 \eta_{t-1} \end{aligned}\]

用\(L(\cdot|\cdot)\)表示最优线性预测，则 \[\begin{aligned} & L(Y_t | \bar Y_t, \bar X_t) \\ =& L(X_{t-1} | X_{t-1}, \dots, Y_{t-1}, \dots) + L(\varepsilon_t | \bar Y_t, \bar X_t) \\ =& X_{t-1} + 0 \\ =& X_{t-1} \\ \sigma(Y_t | \bar Y_t, \bar X_t) =& \text{Var}(\varepsilon_t) = 1 \end{aligned}\] 而 \[ Y_t = \eta_{t-1} + 0.5\eta_{t-2} + \varepsilon_t \] 有 \[\begin{aligned} \gamma_Y(0) = 2.25, \gamma_Y(1) = 0.5, \gamma_Y(k) = 0, k \geq 2 \end{aligned}\] 所以\(\{Y_t \}\)是一个MA(1)序列，设其方程为 \[ Y_t = \zeta_t + b \zeta_{t-1}, \zeta_t \sim \text{WN}(0, \sigma_\zeta^2) \] 可以解出 \[\begin{aligned} \rho_Y(1) =& \frac{\gamma_Y(1)}{\gamma_Y(0)} = \frac{2}{9} \\ b =& \frac{1 - \sqrt{1 - 4 \rho_Y^2(1)}}{2 \rho_Y(1)} \approx 0.2344 \\ \sigma_\zeta^2 =& \frac{\gamma_Y(1)}{b} \approx 2.1328 \end{aligned}\] 于是 \[\begin{aligned} \sigma(Y_t | \bar Y_t) =& \sigma_\zeta^2 = 2.1328 \end{aligned}\] 所以 \[\begin{aligned} \sigma(Y_t | \bar Y_t, \bar X_t) = 1 < 2.1328 = \sigma(Y_t | \bar Y_t) \end{aligned}\] 即\(X_t\)是\(Y_t\)的格兰格原因。

反之， \(X_t\)是MA(1)序列，有 \[ \eta_t = \frac{1}{1 + 0.5 B} X_t = \sum_{j=0}^\infty (-0.5)^j X_{t-j} \] 其中\(B\)是推移算子（滞后算子）。于是 \[\begin{aligned} L(X_t | \bar X_t) =& L(\eta_t | \bar X_t) + 0.5 L(\eta_{t-1} | \bar X_t) \\ =& 0.5 \sum_{j=0}^\infty (-0.5)^j X_{t-1-j} \\ =& - \sum_{j=1}^\infty (-0.5)^j X_{t-j} \\ \sigma(X_t | \bar X_t) =& \text{Var}(X_t - L(X_t | \bar X_t)) \\ =& \text{Var}(\eta_t) = 1 \end{aligned}\] 而 \[\begin{aligned} L(X_t | \bar X_t, \bar Y_t) =& L(\eta_t | \bar X_t, \bar Y_t) + 0.5 L(\eta_{t-1} | \bar X_t, \bar Y_t) \\ =& 0 + 0.5 L(\sum_{j=0}^\infty (-0.5)^j X_{t-1-j} | \bar X_t, \bar Y_t) \\ =& -\sum_{j=1}^\infty (-0.5)^j X_{t-j} \\ =& L(X_t | \bar X_t) \end{aligned}\] 所以\(Y_t\)不是\(X_t\)的格兰格原因。

考虑瞬时因果性。 \[\begin{aligned} L(Y_t | \bar X_t, \bar Y_t, X_t) =& X_{t-1} + 0 (\text{注意}\varepsilon_t\text{与}\{X_s, \forall s\}\text{不相关} \\ =& L(Y_t | \bar X_t, \bar Y_t) \end{aligned}\] 所以\(X_t\)不是\(Y_t\)的瞬时格兰格原因。

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例1.2 在例1.1中，如果模型改成 \[\begin{aligned} Y_t =& X_{t} + \varepsilon_t \\ X_t =& \eta_t + 0.5 \eta_{t-1} \end{aligned}\] 有怎样的结果？

这时 \[ Y_t = \varepsilon_t + \eta_t + 0.5 \eta_{t-1} \] 仍有 \[\begin{aligned} \gamma_Y(0) = 2.25, \gamma_Y(1) = 0.5, \gamma_Y(k) = 0, k \geq 2 \end{aligned}\] 所以\(Y_t\)还服从MA(1)模型 \[ Y_t = \zeta_t + b \zeta_{t-1}, b \approx 0.2344, \sigma^2_\zeta \approx 2.1328 \]

\[\begin{aligned} L(Y_t | \bar Y_t, \bar X_t) =& L(X_t | \bar Y_t, \bar X_t) + 0 \\ =& L(\eta_t | \bar Y_t, \bar X_t) + 0.5 L(\eta_{t-1} | \bar Y_t, \bar X_t) \\ =& 0 + 0.5 L(\sum_{j=0}^\infty (-0.5)^j X_{t-1-j} | \bar Y_t, \bar X_t) \\ =& - \sum_{j=1}^\infty (-0.5)^j X_{t-j} \\ =& X_t - \eta_t \\ \sigma(Y_t | \bar Y_t, \bar X_t) =& \text{Var}(\varepsilon_t + \eta_t) = 2 \end{aligned}\] 而 \[ \sigma(Y_t | \bar Y_t) = \sigma^2_\zeta \approx 2.1328 > \sigma(Y_t | \bar Y_t, \bar X_t) = 2 \] 所以\(X_t\)是\(Y_t\)的格兰格原因。

反之， \[\begin{aligned} L(X_t | \bar X_t, \bar Y_t) =& - \sum_{j=1}^\infty (-0.5)^j X_{t-j} \\ =& L(X_t | \bar X_t) \end{aligned}\] 所以\(Y_t\)不是\(X_t\)的格兰格原因。

考虑瞬时因果性。 \[\begin{aligned} L(Y_t | \bar X_t, \bar Y_t, X_t) =& X_{t} + 0 (\text{注意}\varepsilon_t\text{与}\{X_s, \forall s\}\text{不相关} \\ =& X_t \\ \sigma(Y_t | \bar X_t, \bar Y_t, X_t) =& \text{Var}(\varepsilon) \\ =& 1 < 2 = \sigma(Y_t | \bar X_t, \bar Y_t) \end{aligned}\] 所以\(X_t\)是\(Y_t\)的瞬时格兰格原因。

\[\begin{aligned} [aaa] \end{aligned}\]