# 统计代写|时间序列分析代写Time-Series Analysis代考|STAT3040

## 统计代写|时间序列分析代写Time-Series Analysis代考|AUTOREGRESSIVE DISTRIBUTED LAG MODELS

12.8 Nevertheless, it would clearly be useful if an automatic model selection procedure could be developed. This has not been done for the multiple input model (12.4), but if a restricted form of it is specified then such a procedure becomes feasible. This restricted form is known as the autoregressive distributed lag, or ARDL, model and is obtained by placing the following restrictions on (12.4):
$$\delta_1(B)=\cdots=\delta_M(B)=\phi(B) \quad \theta(B)=1$$
so that the model is, on defining $\beta_j(B)=\omega_j(B) B^{b_j}$ and including an intercept,
$$\phi(B) y_t=\beta_0+\sum_{j=1}^M \beta_j(B) x_{j, t}+a_t$$
This is known as the $\operatorname{ARDL}\left(p, s_1, \ldots, s_M\right)$ model and restricts all the autoregressive lag polynomials to be the same and excludes a moving average noise component, although this exclusion is not essential. These restrictions reduce the noise component to white noise through constraining the dynamics and enables (12.5) to be estimated by OLS, so that on selecting a maximum lag order of, say, $m$, goodness-of-fit statistics, such as information criteria, can be used to select the appropriate specification.
12.9 The ARDL representation (12.5) may be recast in a potentially useful way. Recalling the development of $\$ \mathbf{8 . 4}$, each input polynomial may be decomposed as $$\beta_j(B)=\beta_j(1)+\nabla \tilde{\beta}j(B)$$ where $$\tilde{\beta}_j(B)=\tilde{\beta}{j, 0}+\tilde{\beta}{j, 1} B+\tilde{\beta}{j, 2} B^2+\cdots+\tilde{\beta}{j, j{-1}} B^{s,-1}$$ with $$\bar{\beta}{j, i}=-\sum{l=i+1}^{s_j} \beta_{j, l}$$ ## 统计代写|时间序列分析代写Time-Series Analysis代考|MULTIVARIATE DYNAMIC REGRESSION MODELS 13.1 In a natural extension to the ARDL model of the previous chapter, suppose that there are now two endogenous variables,$y_{1, t}$and$y_{2, t}$, that may both be related to an exogenous variable$x_tand its lags as well as to lags of each other. In the simplest case, such a model would be: \begin{aligned} &y_{1, t}=c_1+a_{11} y_{1, t-1}+a_{12} y_{2, t-1}+b_{10} x_t+h_{11} x_{t-1}+u_{1, t} \ &y_{2, t}=c_2+a_{21} y_{1, t-1}+a_{22} y_{2, t-1}+b_{20} x_t+b_{21} x_{t-1}+u_{2, t} \end{aligned} The “system” contained in Eq. (13.1) is known as a multivariate dynamic regression, a model treated in some detail in Spanos (1986, Chapter 24). Note that the “contemporaneous” variables,y_{1, t}$and$y_{2, t}$, are not included as regressors in the equations for$y_{2, t}$and$y_{1, t}$, respectively, as this would lead to simultaneity and an identification problem, in the sense that the two equations making up (13.1) would then be statistically indistinguishable, there being the same variables in both. Of course,$y_{1, t}$and$y_{2, t}$may well be contemporaneously correlated, and any such correlation can be modeled by allowing the covariance between the innovations to be nonzero, so that$E\left(u_{1, t} u_{2, t}\right)=\sigma_{12}$say, the variances of the two innovations then being$E\left(u_1^2\right)=\sigma_1^2$and$E\left(u_2^2\right)=\sigma_2^2$. 13.2 The pair of equations in (13.1) may be generalized to a model containing$n$endogenous variables and$k$exogenous variables.${ }^1$Gathering these together in the vectors$\mathbf{y}t^{\prime}=\left(y{1, t}, y_{2, t}, \ldots, y_{n, t}\right)$and$\mathbf{x}t^{\prime}=\left(x{1, t}, x_{2, t}, \ldots, x_{k, t}\right)$,the general form of the multivariate dynamic regression model may be written as: $$\mathbf{y}t=\mathbf{c}+\sum{i=1}^p \mathbf{A}i \mathbf{y}{t-i}+\sum_{i=0}^q \mathbf{B}i \mathbf{x}{t-i}+\mathbf{u}_t$$ where there is a maximum of$p$lags on the endogenous variables and a maximum of$q$lags on the exogenous variables. ## 统计代写|时间序列分析代写时间序列分析代考|自回归分布滞后模型 尽管如此，如果能够开发出一种自动模型选择程序，显然是有用的。对于多重输入模型(12.4)还没有这样做，但是如果指定了它的受限形式，那么这样的过程就变得可行了。这种受限形式被称为自回归分布滞后(ARDL)模型，通过对(12.4)施加以下限制获得: $$\delta_1(B)=\cdots=\delta_M(B)=\phi(B) \quad \theta(B)=1$$ ，因此在定义$\beta_j(B)=\omega_j(B) B^{b_j}$并包含一个截距时， $$\phi(B) y_t=\beta_0+\sum_{j=1}^M \beta_j(B) x_{j, t}+a_t$$ 这被称为$\operatorname{ARDL}\left(p, s_1, \ldots, s_M\right)$模型，它限制所有的自回归滞后多项式是相同的，并排除了移动平均噪声成分，尽管这种排除不是必要的。这些限制通过约束动态将噪声成分降低为白噪声，并使OLS能够估计(12.5)，因此在选择最大滞后阶数时，例如$m$，拟合优度统计数据，如信息标准，可用于选择适当的规范。 12.9 ARDL表示(12.5)可以以一种潜在有用的方式重制。回顾$\$\mathbf{8 . 4}$的发展，每个输入多项式可以分解为
$$\beta_j(B)=\beta_j(1)+\nabla \tilde{\beta}j(B)$$，其中$$\tilde{\beta}j(B)=\tilde{\beta}{j, 0}+\tilde{\beta}{j, 1} B+\tilde{\beta}{j, 2} B^2+\cdots+\tilde{\beta}{j, j{-1}} B^{s,-1}$$

$$\bar{\beta}{j, i}=-\sum{l=i+1}^{s_j} \beta {j, l}$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|多元动态回归模型

13.1在对上一章的ARDL模型的自然扩展中，假设现在有两个内生变量$y_{1, t}$和$y_{2, t}$，它们都可能与一个外生变量$x_t$及其滞后有关，也可能与彼此的滞后有关。在最简单的情况下，这样的模型将是:
\begin{aligned} &y_{1, t}=c_1+a_{11} y_{1, t-1}+a_{12} y_{2, t-1}+b_{10} x_t+h_{11} x_{t-1}+u_{1, t} \ &y_{2, t}=c_2+a_{21} y_{1, t-1}+a_{22} y_{2, t-1}+b_{20} x_t+b_{21} x_{t-1}+u_{2, t} \end{aligned}

13.2(13.1)中的方程对可以推广到一个包含$n$内生变量和$k$外生变量的模型。${ }^1$将这些集合在向量$\mathbf{y}t^{\prime}=\left(y{1, t}, y_{2, t}, \ldots, y_{n, t}\right)$和$\mathbf{x}t^{\prime}=\left(x{1, t}, x_{2, t}, \ldots, x_{k, t}\right)$中，多元动态回归模型的一般形式可以写成:
$$\mathbf{y}t=\mathbf{c}+\sum{i=1}^p \mathbf{A}i \mathbf{y}{t-i}+\sum_{i=0}^q \mathbf{B}i \mathbf{x}{t-i}+\mathbf{u}_t$$
，其中内生变量上存在最大$p$滞后，外生变量上存在最大$q$滞后。

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