# 统计代写|时间序列分析代写Time-Series Analysis代考|STA457H1

## 统计代写|时间序列分析代写Time-Series Analysis代考|VECTOR AUTOREGRESSIONS

13.4 Suppose the model (13.2) does not contain any exogenous variables, so that all the $\mathbf{B}i$ matrices are zero, and that there are $p$ lags of the endogenous variables in every equation: $$\mathbf{y}_t=\mathbf{c}+\sum{i=1}^p \mathbf{A}i \mathbf{y}{t-i}+\mathbf{u}_t$$
Because (13.3) is now simply a $p$ th order autoregression in the vector $\mathbf{y}_t$ it is known as a vector autoregression $(\operatorname{VAR}(p))$ of dimension $n$ and, again, can be estimated by multivariate least squares. ${ }^3$ It is assumed that all the series contained in $\mathbf{y}_t$ are stationary, which requires that the roots of the characteristic equation associated with (13.3),
$$\mathbf{A}(B)=\mathbf{I}_n-\mathbf{A}_1 B-\cdots-\mathbf{A}_p B^p=\mathbf{0}$$ have moduli that are less than unity (bearing in mind that some of the $n p$ roots may appear as complex conjugates).

VARs have become extremely popular for modeling multivariate systems of time series because the absence of $\mathbf{x}_t$ terms precludes having to make any endogenous-exogenous classification of the variables, for such distinctions are often considered to be highly contentious.

## 统计代写|时间序列分析代写Time-Series Analysis代考|GRANGER CAUSALITY

13.5 In the VAR (13.3) the presence of nonzero off-diagonal elements in the $\mathbf{A}i$ matrices, $a{r s, i} \neq 0, r \neq s$, implies that there are dynamic relationships between the variables, otherwise the model would collapse to a set of $n$ univariate AR processes. The presence of such dynamic relationships is known as Granger (-Sims) causality. ${ }^4$ The variable $y_s$ does not Granger-cause the variable $y_r$ if $a_{r s, i}=0$ for all $i=1,2, \ldots, p$. If, on the other hand, there is at least one $a_{r s, i} \neq 0$ then $y_s$ is said to Granger-cause $y_r$ because if that is the case then past values of $y_s$ are useful in forecasting the current value of $y_r$ : Granger causality is, thus, a criterion of “forecastability.” If $y_r$ also Grangercauses $y_s$, the pair of variables are said to exhibit feedback.
13.6 Within a $\operatorname{VAR}(p)$, Granger causality running from $y_s$ to $y_r$, which may be depicted as $y_s \rightarrow y_r$, can be evaluated by setting up the null hypothesis of non-Granger causality $\left(y_s\right.$ does not $\left.\rightarrow y_r\right), H_0: a_{r s, 1}=\cdots=a_{r s, p}=0$, and testing this with a Wald statistic; a multivariate extension of the standard $F$-statistic for testing a set of zero restrictions in a conventional regression model: see, for example, Mills $(2014, \S 13.3)$.
13.7 The presence of nonzero off-diagonal elements in the error covariance matrix $\boldsymbol{\Omega}$ signals the presence of simultaneity. For example, $\sigma_{r s} \neq 0$ implies that $y_{r, t}$ and $y_{s, t}$ are contemporaneously correlated. It might be tempting to try and model such correlation by including $y_{r, t}$ in the equation for $y_{s, t}$ but, if this is done, then $y_{s, t}$ could equally well be included in the $y_{r, t}$ equation. As was pointed out in $\S$ 13.1, this would lead to an identification problem, since the two equations would be statistically indistinguishable and the VAR could no longer be estimated. The presence of $\sigma_{r s} \neq 0$ is sometimes referred to as instantaneous causality, although we should be careful when interpreting this phrase, as no causal direction can be inferred from $\sigma_{r s}$ being nonzero (recall the “correlation does not imply causation” argument found in any basic statistics text: see, e.g., Mills, 2014, §5.4).

## 统计代写|时间序列分析代写Time-Series Analysis代考|VECTOR AUTOREGRESSIONS

13.4假设模型(13.2)不包含任何外生变量，因此所有$\mathbf{B}i$矩阵为零，并且每个方程中内生变量有$p$个滞后:$$\mathbf{y}_t=\mathbf{c}+\sum{i=1}^p \mathbf{A}i \mathbf{y}{t-i}+\mathbf{u}_t$$

$$\mathbf{A}(B)=\mathbf{I}_n-\mathbf{A}_1 B-\cdots-\mathbf{A}_p B^p=\mathbf{0}$$相关的特征方程的根具有小于单位的模(记住，$n p$中的一些根可能作为复共轭出现)

var在建模时间序列的多元系统时已经变得非常流行，因为$\mathbf{x}_t$项的缺乏排除了对变量进行任何内生-外生分类的可能性，因为这种区别通常被认为是极具争议的

## 统计代写|时间序列分析代写时间序列分析代考|格兰杰因果关系

13.5在VAR(13.3)中存在非零非对角线元素 $\mathbf{A}i$ 矩阵， $a{r s, i} \neq 0, r \neq s$，意味着变量之间存在动态关系，否则模型将崩溃为一组 $n$ 单变量AR过程。这种动态关系的存在被称为格兰杰(-西姆斯)因果关系。 ${ }^4$ 变量 $y_s$ 不是格兰杰引起的变量吗 $y_r$ 如果 $a_{r s, i}=0$ 为所有人 $i=1,2, \ldots, p$。另一方面，如果至少有一个 $a_{r s, i} \neq 0$ 然后 $y_s$ 是格兰杰的原因吗 $y_r$ 因为如果是这种情况，那么过去的值 $y_s$ 对预测的当前价值有用吗 $y_r$ 因此，格兰杰因果关系是“可预测性”的标准。如果 $y_r$ 还有格兰杰原因 $y_s$
13.6在a $\operatorname{VAR}(p)$，格兰杰因果关系从 $y_s$ 到 $y_r$，可以描述为 $y_s \rightarrow y_r$，可以通过建立非格兰杰因果关系的零假设进行评估 $\left(y_s\right.$ 不 $\left.\rightarrow y_r\right), H_0: a_{r s, 1}=\cdots=a_{r s, p}=0$，并使用Wald统计量进行测试;标准的多元扩展 $F$-statistic用于测试传统回归模型中的一组零限制:参见Mills $(2014, \S 13.3)$
13.7误差协方差矩阵中存在非零非对角元素 $\boldsymbol{\Omega}$ 同时存在的信号。例如， $\sigma_{r s} \neq 0$ 意味着 $y_{r, t}$ 和 $y_{s, t}$ 是同时代相关的。它可能是诱人的尝试和模型这种相关性包括 $y_{r, t}$ 在等式中 $y_{s, t}$ 但是，如果这样做了，那么 $y_{s, t}$ 同样可以被纳入 $y_{r, t}$ 方程。正如在 $\S$ 13.1，这将导致识别问题，因为两个方程将在统计上不可区分，VAR将无法再估计。的存在 $\sigma_{r s} \neq 0$ 有时被称为瞬时因果关系，尽管我们在解释这个短语时应该小心，因为没有因果方向可以推断 $\sigma_{r s}$ 非零(回想一下在任何基本统计文本中发现的“相关性并不意味着因果关系”论证:参见，例如Mills, 2014，§5.4)

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