Investigating Causal Relations by Econometric Models and Cross-spectral Methods(通过计量经济学模型和交叉谱方法研究因果关系)
论文背景
作者
C.W.J.Granger
2003年诺贝尔经济学奖得主, 把经济学变为可以定量研究的学科
提出问题
- 相关不是因果，那能否把相关关系(类似正反馈过程)拆解为因果关系（因为A所以B）
- 问题：已知两个随机过程的自回归形式，如何判断两个变量是否为因果关系。
解决思路
- 使用SPECTRAL METHODS，将随机变量的相关关系 –> 随机变量的相关关系功率谱(傅里叶变化, 时域 -> 频域, 波动 -> 正弦波的叠加，频率上的表示 -> 线性 -> 自回归模型)
- 查看Cross-spectral(随机变量的相关关系 -> 反馈过程 -> 因果关系)
主要内容
定义
- SPECTRAL METHODS (谱方法)
- FEEDBACK MODELS (反馈)
- CAUSALITY(因果)
实例
- TWO-VARIABLE MODELS (两个变量)
- THREE-VARIABLES MODELS (三个变量)
主要结论
- 在进行随机过程中的谱表示之后，进行的反馈机制可以分解为两个因果关系，其交叉谱可以看作是两个交叉谱之和，且其中每部分都与反馈过程的一个单向因果相关。
- 瞬时因果关系(?)
论文detail
定义
(1) spectral reresentation
$X_t$ is a stationary time series with mean zero, there are two basic spectral reresentation：
(i) Cramer representation \(X_t = \int_{-\pi}^\pi e^{it\omega}dz_x(\omega)\)
where \(Z_x(\omega)\) is a complex random process with uncorrelated increments.
(ii) the spectral reprentationof the covariance sequence
\[\mu_\tau^{xx} = E[X_t\bar{X}_{t - \tau}] = \int_{-\pi}^\pi e^{it\omega}dF_x(\omega)\](2) 交叉谱(Cross-spectral)
(有点没理解)
the cross spectrum \(Cr_(\omega)\) between \(X_t\) and \(Y_t\) ia a complex fuction of $\omega$ and arises both from
\[E[dz_x(\omega)\overline{dz_y(\omega)}] = 0 = Cr(\omega)d\omega\]and
\[\mu_\tau^{xy} =E[X_t\overline{Y_{t-\tau}}] = \int_{-\pi}^\pi e^{it\omega}Cr(\omega)d\omega\](3) the coherence and the phase
the coherence:
\[C(\omega) = \frac{|Cr(\omega)|^2}{f_x(\omega)f_y(\omega)}\]the phase:
\[\phi(\omega) = tan^{-1}\frac{Imaginary Part Of Cr(\omega)}{Real Part Of Cr(\omega)}\]measures the phase difference between corresponding frequency components
(4) causul model 自回归模型
causal model \(A_0X_t = \sum_{j=1}^m A_jX_{t-j} + \epsilon_t\)
当前时刻的行为是以往的线性组合
simple causal models：$A_0$不存在
Instantaneous Causality(瞬时因果关系)
$A_0$存在，当前这一秒决定自身
two varible case: \(X_t + b_0Y_t= \sum_{j=1}^m a_jX_{t-j} + \sum_{j=1}^m b_jY_{t-j} + \epsilon_t^{'}\)
\[Y_t + c_0X_t= \sum_{j=1}^m c_jX_{t-j} + \sum_{j=1}^m d_jY_{t-j} + \epsilon_t^{''}\](5) Causality(因果)
\(\sigma^2(X|U) < \sigma^2(X|\overline{U - Y})\)
除去Y后对X进行预测/计算，误差增大，那就说Y是X的一个因
(6) Feedback
if
\(\sigma^2(X|\bar{U}) < \sigma^2(X|\overline{U - Y})\)
\(\sigma^2(Y|\bar{U}) < \sigma^2(Y|\overline{U - X})\)
we say that feedback is occurring.
feedback is said to occur when $X_t$ is causing $Y_t$ and also $Y_t$ is causing $X_t$
TWO-VARIABLE MODELS
对两个变量的causul model进行Cramer representation以及变量替换。
\[X_t = \sum_{j=1}^m a_jX_{t-j} + \sum_{j=1}^m b_jY_{t-j} + \epsilon_t\] \[Y_t = \sum_{j=1}^m c_jX_{t-j} + \sum_{j=1}^m d_jY_{t-j} + \eta_t\]In terms of time shift operator $U-UX_t = X_{t-1}$
rewrite the equations: \(X_t = a(U)X_t + b(U)Y_t + \epsilon_t\)
\[Y_t = c(U)X_t + d(U)Y_t + \eta_t\]where x(U)s are power series in U i.e.,\(a(U) = \sum_{j = 1}^{m}\)
Using the Cramer representation of the series, i.e., \(X_t = \int_{-\pi}^\pi e^{it\omega}dZ_x(\omega)\)
\(a(U)X_t = \int_{-\pi}^{\pi}e^{it\omega}a(e^{-i\omega})dZ_x(\omega)\) the equation can be written:
\(\int_{-\pi}^{\pi}e^{it\omega}[(1-a(e^{-i\omega}))dZ_x(\omega) - b(e^{-i\omega}dZ_y(\omega) - dZ_\epsilon)(\omega)] = 0\) Same for y
the cross spectrum
\(Cr(\omega) = \frac{1}{2\pi\triangle}[(1-d)\overline{x}\sigma_\epsilon^2 + (1-\overline{a}b\sigma_\eta^2)]\)
can be written as the sum of two components
\(Cr(\omega)=C_1(\omega) + C_2(\omega)\)
where \(C_1(\omega) = \frac{\sigma_\epsilon^2}{2\pi\triangle}(1-d)\bar{c}\)
and \(C_2(\omega) = \frac{\sigma_\eta^2}{2\pi\triangle}(1-\bar{a})b\)
\[\triangle = |(1-a)(1-d) - bc|^2\]如果$Y_t$没有导致$X_t$，那么 b = 0, $C_2(\omega)$ vanishes, vice versa.
So the cross spectrum can be decomposed into the sum of two components - one which depends on the causality of X by Y and the other on the causality of Y by X.
consider a special case:
\(X_t = bY_{t-1} + \epsilon_t\) \(Y_t = cX_{t-2} + \eta_t\)
To-do List
- 手动推导一下二变量情况
- 定义的细节部分rewrite
- 时间序列的自回归模型(VAR)
