【计量经济学】时间序列分析笔记 Multiequation Time Series Models

时间序列复习 chapter 6

密码 2022年6月11日-6月13日

Chapter 6 Multiequation Time Series Models

不同变量之间的关系

Autoregressive Distributed Lag (ADL) Model

自回归分布滞后模型

• When additional predictors and their lags are added to an autoregression, the result is an autoregressive distributed lag model.
• The autoregressive distributed lag model with $p$ lags of $y_t$ and $q$ lags of $x_t$, denoted $\mathrm{ADL}(p,q)$, is
  $$\begin{array}{rl}& y_t={\varphi }_0+{\varphi }_1{y}_{t-1}+{\varphi }_2{y}_{t-2}+\cdots +{\varphi }_p{y}_{t-p}\\ & +{\gamma }_1{x}_{t-1}+{\gamma }_2{x}_{t-2}+\cdots +{\gamma }_q{x}_{t-q}+{ϵ}_{yt}\\ & \text{ or }\Phi (L)y_t={\varphi }_0+\Gamma (L)x_{t-1}+{ϵ}_{yt}\end{array}$$
  where $\Phi (L)=\left(1-{\varphi }_{1}L-\cdots -{\varphi }_p{L}^p\right)$ and $\Gamma (L)={\gamma }_1+{\gamma }_{2}L+\cdots +{\gamma }_q{L}^{q-1}$ are polynomials in “L”.

More generally, we could include the contemporaneous value of $x_t$
$$\begin{array}{c}{y}_t={\varphi }_0+{\varphi }_1{y}_{t-1}+{\varphi }_2{y}_{t-2}+\cdots +{\varphi }_p{y}_{t-p}\\ +{\gamma }_0{x}_t+{\gamma }_1{x}_{t-1}+{\gamma }_2{x}_{t-2}+\cdots +{\gamma }_q{x}_{t-q}+{ϵ}_{yt}\\ \text{ or }\Phi (L)y_t={\varphi }_0+\Psi (L)x_t+{ϵ}_{yt}\\ \Phi (L)=\left(1-{\varphi }_{1}L-\cdots -{\varphi }_p{L}^p\right)\text{ and }\Psi (L)={\gamma }_0+{\gamma }_{1}L+\cdots +{\gamma }_q{L}^q\end{array}$$
(1) $\left\{x_t\right\}$ is exogenous that evolves independently of $\left\{y_t\right\}$ :
$$x_t={\delta }_0+{\delta }_1{x}_{t-1}+\cdots +{\delta }_r{x}_{t-r}+{ϵ}_{xt}\text{ or }D(L)x_t={\delta }_0+{ϵ}_{xt}$$
where $\left\{{ϵ}_{xt}\right\}$ is independent of $\left\{{ϵ}_{yt}\right\}$.
(2) $\left\{y_t\right\}$ and $\left\{x_t\right\}$ are stationary.

如果都是t-1期的变量可以用来forecast

包含t期的可以用来估计动态因果效应

Cross-correlation Function (CCF)

• The cross-correlation between $y_t$ and $x_{t-k}$ is defined to be
  $${\rho }_{yx}(k)\equiv \frac{\mathrm{cov}\left(y_t,x_{t-k}\right)}{\sqrt{\mathrm{var}\left(y_t\right)\mathrm{var}\left(x_t\right)}}.$$
• Under stationarity condition, ${\rho }_{yx}(k)$ is invariant to $t$.
• Unlike autocorrelation, ${\rho }_{yx}(k)\ne {\rho }_{yx}(-k)$ in general.
• Plotting each value of ${\rho }_{yx}(k)(k\ge 0)$ yields the cross-correlation function (CCF) or cross-correlogram.
• The examination of the sample CCF provides the same type of information as the sample ACF in an ARMA model.

使用Yule-Walker 的方式来计算CCF

ADL model

when $X_t$ is a white noise

1. ${\rho }_{yx}(k)$ is zero until the first nonzero element of $X_t$前的系数
2. a spike at lag d indicates $X_{t-d}$ directly affects $y_t$
3. the decay pattern for ${\rho }_{yx}(k)$ is determined by AR part of $y_t$

Estimation: OLS or MLE

Model diagnostics :

the estimation residuals should behave as a white noise process and are uncorrelated with $\left\{x_t,x_{t-1},\cdots \right\}$.

• 如果残差不是白噪音，那么$y_t$前的系数不是adequate
• 如果残差与$X_t$序列相关，那么$X_t$前的系数不是adequate

1. estimate equation:ARDL
   • options: constant conditional variance
     • ordinary 同方差为常数
     • white 异方差
     • HAC 各个时期之间存在关系
2. AIC/SBC 确定lag
3. 系数的显著性检验：Wald test-coefficient restrictions
4. 重新estimate
5. model diagnostics
   • $y_t$前的系数： residual diagnostics — correlogram-Q-statistics 残差是否为白噪音
   • $X_t$前的系数：residual and spread as a group :CCF 残差是否与$X_t$序列相关

Wald test-coefficient restrictions ???

$\left\{x_t\right\}$ 内生怎么办呢？

Structural Vector Autoregression(VAR)

We treat both $y_t$ and $x_t$ as endogenous variables:
$$\begin{array}{rl}& y_t-b_{12}^{(0)}{x}_t=c_{10}+b_{11}^{(1)}{y}_{t-1}+b_{12}^{(1)}{x}_{t-1}+{ϵ}_{yt}\\ & x_t-b_{21}^{(0)}{y}_t=c_{20}+b_{21}^{(1)}{y}_{t-1}+b_{22}^{(1)}{x}_{t-1}+{ϵ}_{xt}\end{array}\left[\begin{array}{c}{ϵ}_{yt}\\ {ϵ}_{xt}\end{array}\right]\stackrel{i.i.d}{\sim }N\left(0,\left[\begin{array}{cc}{\sigma }_y^2& 0\\ 0& {\sigma }_x^2\end{array}\right]\right)$$

• It is a first-order structural vector autoregression(VAR), that incorporates feedback, because $y_t$ and $x_t$ are allowed to affect each other.
• $\left\{{ϵ}_{yt},{ϵ}_{xt}\right\}$ are called the structural shocks. If $b_{12}^{(0)}\ne 0,{ϵ}_{xt}$ has an indirect contemporaneous effect on $y_t$; and if $b_{21}^{(0)}\ne 0,{ϵ}_{yt}$ has an indirect contemporaneous effect on $x_t$.

Equivalently, we can write
$$\begin{array}{rl}& \left[\begin{array}{lr}1& -b_{12}^{(0)}\\ -b_{21}^{(0)}& 1\end{array}\right]\left[\begin{array}{l}{y}_t\\ x_t\end{array}\right]=\left[\begin{array}{l}{c}_{10}\\ c_{20}\end{array}\right]+\left[\begin{array}{cc}{b}_{11}^{(1)}& b_{12}^{(1)}\\ b_{21}^{(1)}& b_{22}^{(1)}\end{array}\right]\left[\begin{array}{l}{y}_{t-1}\\ x_{t-1}\end{array}\right]+\left[\begin{array}{c}{ϵ}_{yt}\\ {ϵ}_{xt}\end{array}\right]\\ & \text{ or }{B}_0{z}_t=c+B_1{z}_{t-1}+{ϵ}_t,{ϵ}_t\stackrel{i.i.d}{\sim }N\left(0,\left[\begin{array}{cc}{\sigma }_y^2& 0\\ 0& {\sigma }_x^2\end{array}\right]\right).\end{array}$$
不能使用ols了，因为simultaneous equation bias.

We can then pre-multiply both sides by

伴随矩阵：主队掉，副取反
$$\begin{array}{r}{B}_0^{-1}=\frac{1}{1-b_{12}^{(0)}{b}_{21}^{(0)}}\left[\begin{array}{cc}1& b_{12}^{(0)}\\ b_{21}^{(0)}& 1\end{array}\right]\text{ to give }\\ z_t=B_0^{-1}c+B_0^{-1}{B}_1{z}_{t-1}+B_0^{-1}{ϵ}_t.\end{array}$$
a reduced-form model
$${\Phi }_0\equiv B_0^{-1}c,{\Phi }_1\equiv B_0^{-1}{B}_1,a_t\equiv B_0^{-1}{ϵ}_t,\Sigma \equiv B_0^{-1}\left[\begin{array}{cc}{\sigma }_y^2& 0\\ 0& {\sigma }_x^2\end{array}\right]B_0^{-1\prime }$$

$$z_t={\Phi }_0+{\Phi }_1{z}_{t-1}+a_t,a_t=\left[\begin{array}{l}{a}_{1t}\\ a_{2t}\end{array}\right]\stackrel{i.i.d}{\sim }N(0,\Sigma ).$$

• Eq.(7) is a first-order vector autoregression or $\mathrm{VAR}(1)$, that can be estimated by OLS.
• In the econometric literature, the $\mathrm{VAR}(1)$ model is also called a reduced-form model because it does not show explicitly the concurrent dependence between $y_t$ and $x_t$.
• In general, $a_{1t}$ and $a_{2t}$ are correlated.
• $\mathrm{cov}\left(a_{1t},a_{2t}\right)=0$, if $b_{12}^{(0)}=b_{21}^{(0)}=0$.
• $\Sigma$是一个对称矩阵，所以参数在这里会减少

Weak Stationarity

The first moments and the second moments of $z_t$ are time-invariant.
$$\begin{array}{c}\text{ Both }E\left(z_t\right)=\left[\begin{array}{c}E\left(y_t\right)\\ E\left(x_t\right)\end{array}\right]\equiv \mu \text{ and }\\ \mathrm{var}\left(z_t\right)=\left[\begin{array}{cc}\mathrm{var}\left(y_t\right)& \mathrm{cov}\left(y_t,x_t\right)\\ \mathrm{cov}\left(x_t,y_t\right)& \mathrm{var}\left(x_t\right)\end{array}\right]\equiv {\Gamma }_0=\left[\begin{array}{cc}{\Gamma }_{11}(0)& {\Gamma }_{12}(0)\\ {\Gamma }_{21}(0)& {\Gamma }_{22}(0)\end{array}\right]\end{array}$$
are time invariant.

Lag-k cross-covariance matrices of $z_t$ :
$$\begin{array}{rl}{\Gamma }_k& =\left[\begin{array}{ll}{\Gamma }_{11}(k)& {\Gamma }_{12}(k)\\ {\Gamma }_{21}(k)& {\Gamma }_{22}(k)\end{array}\right]\equiv \mathrm{cov}\left(z_t,z_{t-k}\right)\\ & =\left[\begin{array}{cc}\mathrm{cov}\left(y_t,y_{t-k}\right)& \mathrm{cov}\left(y_t,x_{t-k}\right)\\ \mathrm{cov}\left(x_t,y_{t-k}\right)& \mathrm{cov}\left(x_t,x_{t-k}\right)\end{array}\right]\end{array}$$

• For a weakly stationary series, ${\Gamma }_k$ is invariant to $t$.
• ${\Gamma }_k$ is NOT symmetric if $k\ne 0$. Consider ${\Gamma }_1$ :

Let the diagonal matrix D be
$$D\equiv \left[\begin{array}{cc}\mathrm{std}\left(y_t\right)& 0\\ 0& \mathrm{std}\left(x_t\right)\end{array}\right]=\left[\begin{array}{cc}\sqrt{{\Gamma }_{11}(0)}& 0\\ 0& \sqrt{{\Gamma }_{22}(0)}\end{array}\right]$$
The concurrent cross-correlation matrix:
$${\rho }_0\equiv \mathrm{corr}\left(z_t,z_t\right)=\left[\begin{array}{cc}1& \mathrm{corr}\left(y_t,x_t\right)\\ \mathrm{corr}\left(x_t,y_t\right)& 1\end{array}\right]=D^{-1}{\Gamma }_0{D}^{-1}$$
Lag-k cross-correlation matrix (CCM) :
$$\begin{array}{rl}{\rho }_k& =\left[\begin{array}{ll}{\rho }_{11}(k)& {\rho }_{12}(k)\\ {\rho }_{21}(k)& {\rho }_{22}(k)\end{array}\right]\equiv \mathrm{corr}\left(z_t,z_{t-k}\right)\\ & =\left[\begin{array}{ll}\mathrm{corr}\left(y_t,y_{t-k}\right)& \mathrm{corr}\left(y_t,x_{t-k}\right)\\ \mathrm{corr}\left(x_t,y_{t-k}\right)& \mathrm{corr}\left(x_t,x_{t-k}\right)\end{array}\right]=D^{-1}{\Gamma }_k{D}^{-1}\end{array}$$
Estimate CCM : Sample Cross-Correlation Matrices

把前面的公式里的值用估计量替代

• The cross-covariance matrix ${\Gamma }_k$ can be estimated by ${\Gamma }_k=\frac{1}{T}{\sum }_{t=k+1}^T\left(z_t-\bar{z}\right){\left(z_{t-k}-\bar{z}\right)}^{\prime }$, for $k\ge 0$, where $\bar{z}=\left({\sum }_{t=1}^T{z}_t\right)/T$ is the vector of sample means.
• The cross-correlation matrix ${\rho }_k$ is estimated by ${\rho }_k=D^{-1}{\Gamma }_k{D}^{-1}$, for $k\ge 0$, where $D$ is the $2\times 2$ diagonal matrix of the sample standard deviations of the component series.

Stationarity Conditions for VAR(1)

Iteration from $z_t$ back to $z_0$ yields
$$z_t=\sum _{j=0}^{t-1}{\Phi }_1^j{\Phi }_0+{\Phi }_1^t{z}_0+\sum _{j=0}^{t-1}{\Phi }_1^j{a}_{t-j}$$
where ${\left({\Phi }_1\right)}^0=I_2$ - the identity matrix. Continuing to iterate backward another $\mathrm{n}$ periods, we obtain
$$z_t=\sum _{j=0}^{t+n-1}{\Phi }_1^j{\Phi }_0+{\Phi }_1^{t+n}{z}_{-n}+\sum _{j=0}^{t+n-1}{\Phi }_1^j{a}_{t-j}$$
Stability requires that ${\lim }_{n\to \infty }{\Phi }_1^n=0$, where ${\Phi }_1=\left[\begin{array}{ll}{\varphi }_{11}& {\varphi }_{12}\\ {\varphi }_{21}& {\varphi }_{22}\end{array}\right]$.

• Eigenvalues of ${\Phi }_1$ are all smaller than 1 in absolute value.
• Roots of $\det \left({\Phi }_1-\alpha I\right)=0$ are all smaller than 1 in modulus.
• Roots of $\det \left(I-{\Phi }_1\lambda \right)=0$ are all larger than 1 in modulus.
• Roots of $\left(1-{\varphi }_{11}\lambda \right)\left(1-{\varphi }_{22}\lambda \right)-{\varphi }_{12}{\varphi }_{21}{\lambda }^2=0$ lie outside the unit circle.

stable+发生很久或者一直在均衡方程可以得到stationary

$$\begin{gathered} z_t={\Phi }_0+{\Phi }_1{z}_{t-1}+a_t,a_t=\left[\begin{array}{l}{a}_{1t}\\ a_{2t}\end{array}\right]\stackrel{i.i.d}{\sim }N(0,\Sigma ). \\ {z}_t=\sum _{j=0}^{t-1}{\Phi }_1^j{\Phi }_0+{\Phi }_1^t{z}_0+\sum _{j=0}^{t-1}{\Phi }_1^j{a}_{t-j} \end{gathered}$$
If $z_t$ is stationary

• $\mu =E\left(z_t\right)={\left(I-{\Phi }_1\right)}^{-1}{\Phi }_0$
• ${\Gamma }_0=\mathrm{var}\left(z_t\right)={\sum }_{i=0}^{\infty }{\Phi }_1^i\Sigma {\left({\Phi }_1^i\right)}^{\prime }$ (in Eq. (12), let $n\to \infty )$.
• ${\Gamma }_k={\Phi }_1{\Gamma }_{k-1}$, for $k\ge 1$.
• ${\rho }_k={\rm Y}{\rho }_{k-1}$, for $k\ge 1$, where ${\rm Y}=D^{-1}{\Phi }_{1}D$.

estimate each equation separately by OLS or seemingly unrelated regression (SUR).

• SUR：似不相关回归
  • 一类为“联立方程组”(simultaneous equations)，即不同方程间存在内在联系，一个方程的解释变量是 另一方程的被解释变量。
  • 另一类为“似不相关回归”(Seemingly Unrelated Regression Estimation，简记 SUR 或 SURE)，即各方程的变量之间没有内在 联系，但各方程的扰动项之间存在相关性

预测相当于向前迭代，求解相当于向后迭代

Identification of Structural VAR : Cholesky Decomposition

• If we could identify $B_0$, then we can retrieve $c$ and $B_1$.
• Sim’s recursive ordering : one of the coefficients on the contemporaneous terms is zero (e.g. $b_{12}^{(0)}=0$ ), i.e. $B_0$ and $B_0^{-1}$ are lower triangular matrices with unit diagonal elements.
• Cholesky decomposition : $\Sigma =LG{L}^{\prime }$, where $L$ is a lower triangular matrix with unit diagonal elements and $\mathrm{G}$ is a diagonal matrix.
• Then we have $B_0=L^{-1},c=L^{-1}{\Phi }_0$, $B_1=L^{-1}{\Phi }_1,{ϵ}_t=L^{-1}{a}_t$, and $\left[\begin{array}{cc}{\sigma }_y^2& 0\\ 0& {\sigma }_x^2\end{array}\right]=G$.

Bivariate VAR(1)

Consider the stationary two-variable $\mathrm{VAR}(1)$ model
$$z_t={\Phi }_0+{\Phi }_1{z}_{t-1}+a_t.$$
In Eq.(12), let $n\to \infty$, then we can get the $\mathrm{VMA}(\infty )$ representation
$$z_t=\mu +\sum _{s=0}^{\infty }{\Psi }_s{a}_{t-s}\text{, }$$
where ${\Psi }_s={\Phi }_1^s$ and $\mu ={\left(I-{\Phi }_1\right)}^{-1}{\Phi }_0$

Eq(12)
$$z_t=\sum _{j=0}^{t+n-1}{\Phi }_1^j{\Phi }_0+{\Phi }_1^{t+n}{z}_{-n}+\sum _{j=0}^{t+n-1}{\Phi }_1^j{a}_{t-j}$$

the impulse response functions

求脉冲响应函数：y,x对残差项shock的偏导数

Denote ${\Psi }_s=\left[\begin{array}{ll}{\psi }_{11}(s)& {\psi }_{12}(s)\\ {\psi }_{21}(s)& {\psi }_{22}(s)\end{array}\right]$

• ${\psi }_{11}(s)=\frac{\partial y_t}{\partial a_{1,t-s}},{\psi }_{12}(s)=\frac{\partial y_t}{\partial a_{2,t-s}}$.
• ${\psi }_{21}(s)=\frac{\partial x_t}{\partial a_{1,t-s}},{\psi }_{22}(s)=\frac{\partial x_t}{\partial a_{2,t-s}}$.
• The four sets of coefficients ${\psi }_{11}(s),{\psi }_{12}(s),{\psi }_{21}(s),{\psi }_{22}(s)$ are called the impulse response functions.
• A plot of ${\psi }_{ij}(s)$ as a function of $s$ is a practical way to visually represent the behavior of $\left(y_t,x_t\right)$ in response to the various shocks.

the impulse response functions of the structural shocks.

If we could further identify $a_t=B_0^{-1}{ϵ}_t$, where ${ϵ}_t$ is defined in the structural VAR. Thus
$$z_t=\mu +\sum _{s=0}^{\infty }{\Phi }_1^s{B}_0^{-1}{ϵ}_{t-s}$$
Denote ${\Pi }_s={\Phi }_1^s{B}_0^{-1}=\left[\begin{array}{ll}{\pi }_{11}(s)& {\pi }_{12}(s)\\ {\pi }_{21}(s)& {\pi }_{22}(s)\end{array}\right]$
${\pi }_{11}(s)=\frac{\partial y_t}{\partial {ϵ}_{y,t-s}},{\pi }_{12}(s)=\frac{\partial y_t}{\partial {ϵ}_{x,t-s}}$.
${\pi }_{21}(s)=\frac{\partial x_t}{\partial {ϵ}_{y,t-s}},{\pi }_{22}(s)=\frac{\partial x_t}{\partial {ϵ}_{x,t-s}}$.

• The four sets of coefficients ${\pi }_{11}(s),{\pi }_{12}(s),{\pi }_{21}(s),{\pi }_{22}(s)$ are the impulse response functions of the structural shocks.

Forecast Error Variance Decomposition

预测均方误差分解

Consider j-step-ahead forecast using the $\mathrm{VMA}(\infty )$ representation of the structural model Eq.(13) :

• $z_{t+j}=\mu +{\sum }_{s=0}^{\infty }{\Pi }_s{ϵ}_{t+j-s}$
• $E\left[z_{t+j}\mid {\mathcal{F}}_t\right]=\mu +{\sum }_{s=j}^{\infty }{\Pi }_s{ϵ}_{t+j-s}$
• j-step-ahead forecast error $e_t(j)={\sum }_{s=0}^{j-1}{\Pi }_s{ϵ}_{t+j-s}$ Therefore,
• j-step-ahead forecast error for $y_{t+j}$ is
  $$\sum _{s=0}^{j-1}{\pi }_{11}(s){ϵ}_{y,t+j-s}+\sum _{s=0}^{j-1}{\pi }_{12}(s){ϵ}_{x,t+j-s}.$$
• $j$-step-ahead forecast error variance of $y_{t+j}$ is ${\sigma }_y^2(j)={\sigma }_y^2{\sum }_{s=0}^{j-1}{\pi }_{11}^2(s)+{\sigma }_x^2{\sum }_{s=0}^{j-1}{\pi }_{12}^2(s).$

We could decompose the j-step-ahead forecast error variance into the proportions due to each structural shock:

• $\frac{{\sigma }_y^2{\sum }_{s=0}^{j-1}{\pi }_{11}^2(s)}{{\sigma }_y^2(j)}$ due to shocks in the $\left\{{ϵ}_{yt}\right\}$ sequence.
• $\frac{{\sigma }_x^2{\sum }_{s=0}^{j-1}{\pi }_{12}^2(s)}{{\sigma }_y^2(j)}$ due to shocks in the $\left\{{ϵ}_{xt}\right\}$ sequence.
  The j-step-ahead forecast error variance of $x_{t+j}$ could be analyzed similarly.

k-dimensional VAR§ Models

The k-variable $p$-lag vector autoregressive model has the form
$$z_t={\Phi }_0+{\Phi }_1{z}_{t-1}+\cdots +{\Phi }_p{z}_{t-p}+a_t,p\ge 1$$
where ${\Phi }_0$ is a $k$-dimensional vector, ${\Phi }_j$ are $k\times k$ matrices, and $\left\{a_t\right\}$ is a sequence of serially uncorrelated random vectors with mean zero and covariance matrix $\Sigma$.

• $k^{2}p$ coefficients plus $k$ intercept terms

• using the lag operator

$$\left(I_k-{\Phi }_{1}L-\cdots -{\Phi }_p{L}^p\right)z_t={\Phi }_0+a_t$$
​ where $I_k$ is the $k\times k$ identity matrix.

• ==Stationarity condition :==the roots of $\det \left(I_k-{\Phi }_1\lambda -\cdots -{\Phi }_p{\lambda }^p\right)=0$ lie outside the unit circle.

讨论：

• 在VAR模型中所包含的变量可以根据相关的经济或金融理论进行选择，从而有助于相互预测。

• 为了获取系统中的重要信息，必须避免过度参数化和自由度损失问题。

• 如果滞后长度太小，模型就会被错误地指定；如果它太大，模型的自由度就会丢失。

VAR Models : Order Specification

• Fit $\mathrm{VAR}(\mathrm{p})$ models with orders $p=0,1,\cdots ,p_{\max }$ and choose the value of $p$ which minimizes some model selection criteria.
• Under VAR§ model, the residual is $a{t}^{(p)}$.
• The $\mathrm{M}\mathrm{L}$ estimator of $\Sigma$ is ${\Sigma }_p=\frac{1}{T}{\sum }_{t=p+1}^T{a_t}^{(p)}{\left[{a_t}^{(p)}\right]}^{\prime }$.

$$\begin{array}{rl}& AIC(p)=\log \left(\left|{\Sigma }_p\right|\right)+\frac{2\left(k^{2}p+k\right)}{T}\\ & BIC(p)=\log \left(\left|{\Sigma }_p\right|\right)+\frac{\left(k^{2}p+k\right)\log (T)}{T}\end{array}$$

Granger Causality Tests

One of the main uses of VAR models is forecasting.

The following intuitive notion of a variable’s forecasting ability is due to Granger (1969).

• If $z_2$ does not improve the forecasting performance of $z_1$, then $z_2$ does not Granger-cause $z_1$.
• If $z_2$ improves the forecasting accuracy of $z_1$, then $z_2$ is said to Granger-cause $z_1$.
• The notion of Granger-causality does not imply true causality. It only implies forecasting ability.

实际操作结果

• In a bivariate $\mathrm{VAR}(\mathrm{p})$ model, $z_2$ fails to Granger-cause $z_1$ if all of the $p$ VAR coefficient matrices ${\Phi }_1,\cdots ,{\Phi }_p$ are lower triangular:
  $$\begin{array}{rl}\left(\begin{array}{l}{z}_{1t}\\ z_{2t}\end{array}\right)& =\left(\begin{array}{l}{\varphi }_{10}\\ {\varphi }_{20}\end{array}\right)+\left(\begin{array}{cc}{\varphi }_{11}^1& 0\\ {\varphi }_{21}^1& {\varphi }_{22}^1\end{array}\right)\left(\begin{array}{c}{z}_{1,t-1}\\ z_{2,t-1}\end{array}\right)+\cdots \\ & +\left(\begin{array}{cc}{\varphi }_{11}^p& 0\\ {\varphi }_{21}^p& {\varphi }_{22}^p\end{array}\right)\left(\begin{array}{c}{z}_{1,t-p}\\ z_{2,t-p}\end{array}\right)+\left(\begin{array}{c}{ϵ}_{1t}\\ {ϵ}_{2t}\end{array}\right)\end{array}$$
• If $z_2$ fails to Granger-cause $z_1$ and $z_1$ fails to Granger-cause $z_2$, then the VAR coefficient matrices ${\Phi }_1,\cdots ,{\Phi }_p$ are diagonal. Then we can model $z_1$ and $z_2$ separately.

In the bivariate model, testing $H_0:z_2$ does not Granger-cause $z_1$ reduces to a testing $H_0:{\varphi }_{12}^1={\varphi }_{12}^2=\cdots ={\varphi }_{12}^p=0$ from the linear regression
$$\begin{array}{rl}{z}_{1t}& ={\varphi }_{10}+{\varphi }_{11}^1{z}_{1,t-1}+\cdots +{\varphi }_{11}^p{z}_{1,t-p}\\ & +{\varphi }_{12}^1{z}_{2,t-1}+\cdots +{\varphi }_{12}^p{z}_{2,t-p}+{ϵ}_{1t}\end{array}$$
The test statistic is a simple F-statistic or Wald statistic.

Block Exogeneity

The block-exogeneity test is the multivariate generalization of the Granger causality test. For example, in a trivariate model with $y_t$, $x_t$ and $w_t$ :

• the test is whether lags of $w_t$ Granger cause either $y_t$ or $x_t$;
• the test is whether lags of $w_t$ and $x_t$ Granger cause $y_t$.
• Eviews commands : Estimate VAR $\to$ View/Lag Structure/ Granger Causality/Block Exogeneity Tests

The block-exogeneity tests could be done using Wald test or the likelihood ratio test.

Lag Exclusion Tests

The lag exclusion test carries out lag exclusion tests for each lag in the VAR. For example, in a bivariate $\mathrm{VAR}(5)$ model with $y_t$ and $x_t$ :

• the test is whether the first lag of $y_t$ and $x_t$ should be excluded from each equation.
• the test is whether the second lag of $y_t$ and $x_t$ should be excluded from each equation.
• $\cdots$
• Eviews commands : Estimate VAR $\to$ View/Lag Structure/ Lag Exclusion Tests

The lag exclusion tests are done using Wald test in Eviews.

Testing for Serial Dependence?

The $Q_k(m)$ statistic can be applied to the residual series to check the assumption that there are no serial or cross correlations in the residuals.残差是否为白噪音，与其他序列是否相关

For a fitted $VAR(p)$ model, $Q_k(m)$ statistic of the residuals is asymptotically a ${\chi }^2\left(k^{2}m-g\right)$, where $g$ is the number of estimated parameters in the VAR coefficient matrices.

Multivariate Portmanteau Tests/Ljung-Box Statistics $Q(m)$ :

• $H_0:{\rho }_1=\cdots ={\rho }_m=0$ v.s. $H_a:{\rho }_i\ne 0$ for some $i$.
• Under the null hypothesis and some regularity conditions
  $$Q_k(m)=T^2\sum _{l=1}^m\frac{1}{T-I}\mathrm{tr}\left({\Gamma }_l^{\prime }{\Gamma }_0^{-1}{\Gamma }_{,}{\Gamma }_0^{-1}\right)\stackrel{\mathcal{D}}{⟶}{\chi }^2\left(k^{2}m\right)$$
  where $k$ is the dimension of $z_t$ and $tr$ is the sum of diagonal elements.

A Revisit of the Motivating Example

1. 估计VAR，选择lag的长度 model1
2. Granger 因果检验/Block 外生性检验 ： 二者确实有关系，需要用VAR
3. lag exclusion test
4. 得到Model 2,VAR restriction 继续做lag exclusion 直到所有的系数都显著
   • 发现系数不显著时，还要做一个系数都为0的联合检验，使用wald coefficient test
5. model checking: residual test
6. 看一下脉冲响应函数