# 统计代写|时间序列分析代写Time-Series Analysis代考|STAT510

## 统计代写|时间序列分析代写Time-Series Analysis代考|DETERMINING THE LAG ORDER OF A VECTOR AUTOREGRESSION

13.8 To enable the VAR to become operational the lag order $p$, which will typically be unknown, needs to be determined empirically. A traditional way of selecting the lag order is to use a sequential testing procedure. Consider the model (13.3) with error covariance matrix $\Omega_p=E\left(\mathbf{u}t \mathbf{u}_t^{\prime}\right)$, where a $p$ subscript is included to emphasize that the matrix is related to a $\operatorname{VAR}(p)$. An estimate of this matrix is given by: $$\hat{\boldsymbol{\Omega}}_p=(T-p)^{-1} \hat{\mathbf{U}}_p \hat{\mathbf{U}}_p^{\prime}$$ where $\hat{\mathbf{U}}_p=\left(\hat{\mathbf{u}}{p, 1}^{\prime}, \ldots, \hat{\mathbf{u}}{p, n}^{\prime}\right)^{\prime}$ is the matrix of residuals obtained by OLS estimation of the $\operatorname{VAR}(p), \hat{\mathbf{u}}{p, r}=\left(\hat{u}{r, p+1}, \ldots, \hat{u}{r, T}\right)^{\prime}$ being the residual vector from the $r$ th equation (noting that with a sample of size $T, p$ observations will be lost through lagging). A likelihood ratio (LR) statistic for testing the order $p$ against the order $m, m<p$, is
$$L R(p, m)=(T-n p) \log \left(\frac{\left|\hat{\boldsymbol{\Omega}}m\right|}{\left|\hat{\boldsymbol{\Omega}}_p\right|}\right) \sim \chi{n^2(p-m)}^2$$
Thus, if $L R(p, m)$ exceeds the $\alpha$ critical value of the $\chi^2$ distribution with $n^2(p-m)$ degrees of freedom, then the hypothesis that the VAR order is $m$ is rejected at the $\alpha$ level of significance in favor of the higher order $p$. The statistic uses the scaling factor $T-n p$ rather than $T-p$ to account for possible small sample bias.

The statistic (13.4) may then be used sequentially beginning with a maximum value of $p, p_{\max }$ say, testing first $p_{\max }$ against $p_{\max }-1$ using $L R\left(p_{\max }, p_{\max }-1\right)$ and, if this statistic is not significant, then testing $p_{\max }-1$ against $p_{\max }-2$ using $L R\left(p_{\max }-1, p_{\max }-2\right)$, continuing until a significant test is obtained.

Alternatively, some type of information criterion can be minimized. These are essentially multivariate extensions of those initially introduced in §3.35: for example, the multivariate AIC and BIC criteria are defined as:
$\operatorname{MAIC}(p)=\log \left|\hat{\boldsymbol{\Omega}}p\right|+\left(2+n^2 p\right) T^{-1}$ $\operatorname{MBIC}(p)=\log \left|\hat{\Omega}_p\right|+n^2 p T^{-1} \ln T \quad p=0,1, \ldots, p{\max }$

## 统计代写|时间序列分析代写Time-Series Analysis代考|VARIANCE DECOMPOSITIONS AND INNOVATION ACCOUNTING

13.10 While the estimated coefficients of a VAR(1) are relatively easy to interpret, this quickly hecomes problematic for higher order VARs hecause not only do the number of coefficients increase rapidly (each additional lag introduces a further $n^2$ coefficients), but many of these coefficients will be imprecisely estimated and highly intercorrelated, so becoming statistically insignificant. This can be seen in the estimated VAR(2) of Example 13.1, where only $\hat{a}{22,2}$ in $\hat{\mathbf{A}}_2$ is significant. 13.11 This has led to the development of several techniques for examining the “information content” of a VAR that are based on the vector moving average representation (VMA) of $\mathbf{y}_t$. Suppose that the VAR is written in lag operator form as $$\mathbf{A}(B) \mathbf{y}_t=\mathbf{u}_t$$ where, as in $\S$ 13.4, $$\mathbf{A}(B)=\mathbf{I}_n-\mathbf{A}_1 B-\cdots-\mathbf{A}_p B^p$$ is a matrix polynomial in $B$. Analogous to the univariate case (recall $\S \S \mathbf{3 . 8}-\mathbf{3 . 9}$ ), the (infinite order) VMA representation is $$\mathbf{y}_t=\mathbf{A}^{-1}(B) \mathbf{u}_t=\boldsymbol{\Psi}(B) \mathbf{u}_t=\mathbf{u}_t+\sum{i=1}^{\infty} \Psi_i \mathbf{u}{t-i}$$ where $$\boldsymbol{\Psi}_i=\sum{j=1}^i \mathbf{A}j \boldsymbol{\Psi}{i-j} \quad \boldsymbol{\Psi}_0=\mathbf{I}_n \quad \boldsymbol{\Psi}_i=\mathbf{0} \quad i<0$$
this recursion being obtained by equating coefficients of $B$ in $\Psi(B) \mathbf{A}(B)=\mathbf{I}_n$.

## 统计代写|时间序列分析代写时间序列分析代考|确定向量自回归的滞后顺序

13.8为了使VAR成为可操作的延迟订单 $p$，这通常是未知的，需要通过经验来确定。选择滞后顺序的传统方法是使用顺序测试程序。考虑具有误差协方差矩阵的模型(13.3) $\Omega_p=E\left(\mathbf{u}t \mathbf{u}_t^{\prime}\right)$，其中 $p$ 加上下标是为了强调矩阵与a有关 $\operatorname{VAR}(p)$。这个矩阵的估计值由: $$\hat{\boldsymbol{\Omega}}_p=(T-p)^{-1} \hat{\mathbf{U}}_p \hat{\mathbf{U}}_p^{\prime}$$ 哪里 $\hat{\mathbf{U}}_p=\left(\hat{\mathbf{u}}{p, 1}^{\prime}, \ldots, \hat{\mathbf{u}}{p, n}^{\prime}\right)^{\prime}$ 的OLS估计得到的残差矩阵为 $\operatorname{VAR}(p), \hat{\mathbf{u}}{p, r}=\left(\hat{u}{r, p+1}, \ldots, \hat{u}{r, T}\right)^{\prime}$ 是残差向量 $r$ Th方程(注意在样本大小 $T, p$ 观测结果将因滞后而丢失)。用于检验顺序的似然比(LR)统计量 $p$ 违反命令 $m, m<p$，为
$$L R(p, m)=(T-n p) \log \left(\frac{\left|\hat{\boldsymbol{\Omega}}m\right|}{\left|\hat{\boldsymbol{\Omega}}_p\right|}\right) \sim \chi{n^2(p-m)}^2$$

$\operatorname{MAIC}(p)=\log \left|\hat{\boldsymbol{\Omega}}p\right|+\left(2+n^2 p\right) T^{-1}$$\operatorname{MBIC}(p)=\log \left|\hat{\Omega}_p\right|+n^2 p T^{-1} \ln T \quad p=0,1, \ldots, p{\max } ## 统计代写|时间序列分析代写时间序列分析代考|方差分解和创新会计 虽然VAR(1)的估计系数相对容易解释，但对于高阶VAR，这很快就会出现问题，因为不仅系数的数量迅速增加(每一个额外的滞后引入一个进一步的n^2系数)，而且这些系数中的许多将被不精确地估计和高度相关，因此在统计上变得不重要。这可以从例13.1的估计VAR(2)中看出，其中\hat{\mathbf{A}}_2中只有\hat{a}{22,2}是显著的。13.11这导致了基于\mathbf{y}_t的矢量移动平均表示(VMA)来检查VAR的“信息内容”的几种技术的发展。假设VAR以滞后算符形式写成$$ \mathbf{A}(B) \mathbf{y}_t=\mathbf{u}_t $$，其中如\S 13.4所示，$$ \mathbf{A}(B)=\mathbf{I}_n-\mathbf{A}_1 B-\cdots-\mathbf{A}_p B^p $$是B中的一个矩阵多项式。类似于单变量情况(回想一下\S \S \mathbf{3 . 8}-\mathbf{3 . 9})，(无限阶)VMA表示为$$ \mathbf{y}_t=\mathbf{A}^{-1}(B) \mathbf{u}_t=\boldsymbol{\Psi}(B) \mathbf{u}_t=\mathbf{u}_t+\sum{i=1}^{\infty} \Psi_i \mathbf{u}{t-i} $$，其中$$ \boldsymbol{\Psi}_i=\sum{j=1}^i \mathbf{A}j \boldsymbol{\Psi}{i-j} \quad \boldsymbol{\Psi}_0=\mathbf{I}_n \quad \boldsymbol{\Psi}_i=\mathbf{0} \quad i<0$$这个递归是通过将$\Psi(B) \mathbf{A}(B)=\mathbf{I}_n$中$B\$的系数相等得到的

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