Granger Causality

取阈值：首先进行回归分析，然后得到的参数，进行排序，根据极端冲击，负极端取0.1，正极端取0.9，作为相应变量的阈值

10个类别(9个是积极，正常，消极的组合，剩下一个是传统格兰特方法)

{shockprc:分类}

\[\begin{array}{l} \left(X_{t}^{+}, Y_{t}^{+}\right),\left(X_{t}^{+}, Y_{t}^{*}\right),\left(X_{t}^{+}, Y_{t}^{-}\right) \\ \left(X_{t}^{*}, Y_{t}^{*}\right),\left(X_{t}^{*}, Y_{t}^{-}\right),\left(X_{t}^{*}, Y_{t}^{+}\right) \\ \left(X_{t}^{-}, Y_{t}^{*}\right),\left(X_{t}^{-}, Y_{t}^{+}\right),\left(X_{t}^{-}, Y_{t}^{-}\right) \end{array}\]

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时间序列的ADF检验：(AIC)--HJC准则 lag length suggested by HJ--平稳性

\[\mathrm{HJC}=\ln \left(\left|\widehat{\Omega}_{j}\right|\right)+\mathrm{j}\left(\frac{n^{2} \ln T+2 n^{2} \ln (\ln T)}{2 T}\right), \mathrm{j}=0, \cdots, \mathrm{p} \]

hqc = log(det(VARCOV)) + (2/T)*(M*M*lag_guess + M)*log(log(T)) + M *(1 +log(2*pi));

{lag_length2}

input:矩阵(观测数据),lag length最小值，p为lag length最大值

output： hjclag - Lag length suggested by Hatermi-J criterion.

{hjcA} - Matrix of coefficient estimates 利用 HJC

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{VARLAGS}:VAR(2)--根据分类所得的矩阵(两列)

input:lags,矩阵$$d_{\max } \operatorname{lags}\left(\text { i.e., } p=k+d_{\max }\right)$$

output：x-根据lag输出input矩阵的一部分(T - lags) x K matrix, the last T-lags rows of var

\[Z_{t}:=\left(\begin{array}{c} 1 \\ P_{t}^{+} \\ P_{t-1}^{+} \\ \vdots \\ P_{t-p+1}^{+} \end{array}\right) \text {for } t=1 \ldots, T\]

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{rstrctvm}：input:VAR()阶数，变量数，addlags--限制系数(Granger Causality in Multi-variate Time Series
using a Time Ordered Restricted Vector Autoregressive Model)

\[H_{0}: C \beta=0$$,$C$ is a $p*n(1=np)$ 指标符矩阵$$d_{\max } \operatorname{lags}\left(\text { i.e., } p=k+d_{\max }\right) \]

----OUTPUT: 只含0/1的矩阵

Rvector1:A 1 indicates which coefficient is 限制到 zero; 0 is given 没有限制

(Rmatrix1)indicator
matrix

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{EstVar_Params}：为bootstraps提供数据

\[Y^{*}=\widehat{D} Z+\delta^{*} \]

Ahat:得到所需要的系数参数\(\widehat{A}\)

（1）首先在公式（3-37）成立的基础上，估计式（3-33）的系数，将残差保存
下来;
（2）等概率从（1）式保存的残差中抽样，并保证每一组残差样本的均值为 0，
以此方法来获得 bootstrap 样本 \(\hat{e}_{t}\);
(3)利用公式 \(Y^{*}=\hat{Y}_{t}^{+}+\hat{e}_{t},\) 通过 (1),(2) 估计得到的 \(\hat{Y}_{t}^{+}, \hat{e}_{t},\) 可以得到 bootstrap
数据;

leverages:保证恒定的方差

\[\text { at } V=\left(Y_{-1}^{\prime}, \cdots, Y_{-k}^{\prime}\right) \text { and } V_{i}=\left(Y_{i,-1}^{\prime}, \cdots, Y_{i,-k}^{\prime}\right) \]

\[l_{1}=\operatorname{diag}\left(V_{1}\left(V_{1}^{\prime} V_{1}\right)^{-1} V_{1}^{\prime}\right), \text { and } l_{2}=\operatorname{diag}\left(V\left(V^{\prime} V\right)^{-1} V^{\prime}\right) \]

\(H_{0}:\) the row \(j,\) column \(k\) element in \(A_{r}\) equals zero for \(r=1 \ldots, p\)

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{W_test：}

\[\text { Wald }=(C \beta)^{\prime}\left[C(Z Z)^{-1} \otimes S_{U} C\right]^{-1}(C \beta) \]

input:y:data matrix based on lag ,x:lagged values for y

Ahat，Rmatrix1--\(C\)

output: Wstat - vector of Wald statistics(得到的W统计量)

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{Bootstrap_Toda}

\[V=\left(Y_{-1}^{\prime}, \cdots, Y_{-k}^{\prime}\right) \text { and } V_{i}=\left(Y_{i,-1}^{\prime}, \cdots, Y_{i,-k}^{\prime}\right) \]

\[Y_{-P} \text { to be }\left(y_{1-P}, \cdots, y_{T-P}\right) \]

\(Y_{i,-P}\) to be \(Y_{-P}\) 's ith row.

\[d_{\max } \operatorname{lags}\left(\text { i.e., } p=k+d_{\max }\right) \]

\[y_{t}=\hat{\gamma}_{0}+\hat{\gamma}_{1} t+\hat{J}_{1} y_{t-1}+\cdots+\hat{J}_{k} y_{t-k}+\hat{J}_{k+1} y_{t-k-1}+\hat{J}_{k+2} y_{t-k-2}+\hat{\varepsilon}_{t} \]

The modified residual for \(y_{i t}\) is produced as：$$\hat{u}{i t}^{m}=\frac{\hat{u}{i t}}{\sqrt{1-l_{i t}}}$$

where the \(t\) th element of \(l_{i}\) is given by \(l_{i t}\) and the raw residual from the regression with \(y_{i t}\) as the
dependent variable is given by \(\hat{u}_{i t}\)

using the previously 确定的最佳 lag order for the original data and an assumed order of integration

假定的整合顺序

DPG(OLS)

\[H_{j}=\left[\begin{array}{cccccc} I_{n} & I_{n} & I_{n} & \cdots & I_{n} & I_{n} \\ 0 & I_{n} & I_{n} & \cdots & I_{n} & I_{n} \\ 0 & 0 & I_{n} & \cdots & I_{n} & I_{n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & I_{n} & I_{n} \\ 0 & 0 & 0 & \cdots & 0 & I_{n} \end{array}\right]\]

The resulting set of bootstrapped Wald statistics is referred to as the bootstrapped Wald distribution[Toda and Yamamoto (1995) methodology]

yhat = Xhat(2:size(Xhat,1),2:(numvars+1));
Xhat = Xhat(1:size(Xhat,1)-1,:);
[AhatTU,unneededleverage] = estvar_params(yhat,Xhat,0,0,order,addlags);%estvar_params
Wstat = W_test(yhat,Xhat,AhatTU,Rmatrix1);

:根据显著值分类0.01/0.005/0.1

    onepct_index = bootsimmax - floor(bootsimmax/100);
    fivepct_index = bootsimmax - floor(bootsimmax/20);
    tenpct_index = bootsimmax - floor(bootsimmax/10);

OUTPUT:
Wcriticalvals - matrix of critical values for Wald statistics（W统计量的临界值--分位点）

WW\(\leftarrow\)[得到的W统计量，W统计量的临界值]

利用循环做不同列的对比，取0/1--代表是否具有格兰特因果关系