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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|时间序列分析代写Time-Series Analysis代考|FIRST-ORDER AUTOREGRESSIVE PROCESSES

3.8 Although Eq. (3.2) may appear complicated, many realistic models result from specific choices for the $\psi$-weights. Taking $\mu=0$ without loss of generality, choosing $\psi_j=\phi^j$ allows (3.2) to be written as:
\begin{aligned} x_t &=a_t+\phi a_{t-1}+\phi^2 a_{t-2}+\cdots \ &=a_t+\phi\left(a_{t-1}+\phi a_{t-2}+\cdots\right) \ &=\phi x_{t-1}+a_t \end{aligned}
or
$$x_t-\phi x_{t-1}=a_t$$
This is known as a first-order autoregressive process, often given the acronym $\operatorname{AR}(1) .^4$
3.9 The lag operator $B$ introduced in $\S 2.10$ allows (possibly infinite) lag expressions to be written in a concise way. For example, by using this operator the AR(1) process can be written as:
$$(1-\phi B) x_t=a_t$$
so that
\begin{aligned} x_t &=(1-\phi B)^{-1} a_t=\left(1+\phi B+\phi^2 B^2+\cdots\right) a_t \ &=a_t+\phi a_{t-1}+\phi^2 a_{t-2}+\cdots \end{aligned}
This linear filter representation will converge if $|\phi|<1$, which is, therefore, the stationarity condition. 3.10 The ACF of an AR(1) process may now be deduced. Multiplying both sides of (3.3) by $x_{t-k}, k>0$, and taking expectations yields:
$$\gamma_k-\phi \gamma_{k-1}=E\left(a_t x_{t-k}\right) .$$
From (3.4), $a_t x_{t-k}=\sum_{i=0}^{\infty} \phi^i a_t a_{t-k-i}$. As $a_t$ is white noise, any term in $a_t a_{t-k-i}$ has zero expectation if $k+i>0$. Thus (3.5) simplifies to:
$$\gamma_k=\phi \gamma_{k-1} \quad \text { for all } k>0$$
and, consequently, $\gamma_k=\phi^k \gamma_0$. An AR(1) process, therefore, has an ACF given by $\rho_k=\phi^k$. Thus, if $\phi>0$ the ACF decays exponentially to zero, while if $\phi<0$ the ACF decays in an oscillatory pattern, both decays being slow if $\phi$ is close to the nonstationary boundaries of $+1$ and $-1$.

## 统计代写|时间序列分析代写Time-Series Analysis代考|FIRST-ORDER MOVING AVERAGE PROCESSES

3.12 Now consider the model obtained by choosing $\psi_1=-\theta$ and $\psi_j=0$, $j \geq 2$, in $(3.2)$
$$x_t=a_t-\theta a_{t-1}$$
Or
$$x_t=(1-\theta B) a_t$$
This is known as the first-order moving average (MA(1)) process and it follows immediately that: ${ }^5$
$$\gamma_0=\sigma^2\left(1+\theta^2\right) \quad \gamma_1=-\sigma^2 \theta \quad \gamma_k=0 \text { for } k>1$$
and, hence, its $\mathrm{ACF}$ is described by $$\rho_1=-\frac{\theta}{1+\theta^2} \quad \rho_k=0 \text { for } k>1$$
Thus, although observations one period apart are correlated, observations more than one period apart are not, so that the memory of the process is just one period: this “jump” to zero autocorrelation at $k=2$ may be contrasted with the smooth, exponential decay of the ACF of an AR(1) process.
3.13 The expression for $\rho_1$ can be written as the quadratic equation $\rho_1 \theta^2+\theta+\rho_1=0$. Since $\theta$ must be real, it follows that $\left|\rho_1\right|<0.5$. $^6$ However, both $\theta$ and $1 / \theta$ will satisfy this equation, and thus, two MA(1) processes can always be found that correspond to the same ACF.
3.14 Since any MA model consicts of a finite number of $\imath /$-weights, all MA models are stationary. To ohtnin a converging nutoregressive representntion, however, the restriction $\theta<1$ must be imposed. This restriction is known as the invertibility condition and implies that the process can be written in terms of an infinite autoregressive representation:
$$x_t=\pi_1 x_{t-1}+\pi_2 x_{t-2}+\cdots+a_t$$
where the $\pi$-weights converge: $\sum_{j=1}^{\infty}\left|\pi_j\right|<\infty$. In fact, the $\mathrm{MA}(1)$ model can be written as:
$$(1-\theta B)^{-1} x_t=a_t$$
and expanding $(1-\theta B)^{-1}$ yields
$$\left(1+\theta B+\theta^2 B^2+\cdots\right) x_t=a_t .$$

# 时间序列分析代考

## 统计代写|时间序列分析代写Time-Series Analysis代考|FIRST-ORDER AUTOREGRESSIVE PROCESSES

$3.8$ 虽然方程式。(3.2) 可能看起来很复杂，许多现实的模型来自于特定的选择 $\psi$-权重。服用 $\mu=0$ 不失一般性，选择 $\psi_j=\phi^j$ 允许 (3.2) 写成:
$$x_t=a_t+\phi a_{t-1}+\phi^2 a_{t-2}+\cdots \quad=a_t+\phi\left(a_{t-1}+\phi a_{t-2}+\cdots\right)=\phi x_{t-1}+a_t$$

$$x_t-\phi x_{t-1}=a_t$$

$3.9$ 滞后算子 $B$ 介绍于 $\S 2.10$ 允许以简洁的方式编写 (可能是无限的) 滞后表达式。例如，通过使用这个算子，AR(1) 过程可以写成:
$$(1-\phi B) x_t=a_t$$

$$x_t=(1-\phi B)^{-1} a_t=\left(1+\phi B+\phi^2 B^2+\cdots\right) a_t \quad=a_t+\phi a_{t-1}+\phi^2 a_{t-2}+\cdots$$

$$\gamma_k-\phi \gamma_{k-1}=E\left(a_t x_{t-k}\right) .$$

$$\gamma_k=\phi \gamma_{k-1} \quad \text { for all } k>0$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|FIRST-ORDER MOVING AVERAGE PROCESSES

$3.12$ 现在考虑通过选择得到的模型 $\psi_1=-\theta$ 和 $\psi_j=0, j \geq 2$ ，在 $(3.2)$
$$x_t=a_t-\theta a_{t-1}$$

$$x_t=(1-\theta B) a_t$$

$$\gamma_0=\sigma^2\left(1+\theta^2\right) \quad \gamma_1=-\sigma^2 \theta \quad \gamma_k=0 \text { for } k>1$$

$$\rho_1=-\frac{\theta}{1+\theta^2} \quad \rho_k=0 \text { for } k>1$$

$3.13$ 的表达式 $\rho_1$ 可以写成二次方程 $\rho_1 \theta^2+\theta+\rho_1=0$. 自从 $\theta$ 必须是真实的，因此 $\left|\rho_1\right|<0.5 .{ }^6$ 然而，两者 $\theta$ 和 $1 / \theta$ 将满足这个方程，因此，总是可以找到对应于相 同 ACF 的两个 $M A(1)$ 过程。
$3.14$ 由于任何 MA 模型都包含有限数量的 $\imath$-权重，所有 MA 模型都是静止的。然而，对于一个收敛的 nuregressive 表示，限制 $\theta<1$ 必须施加。这种限制被称为 可逆性条件，并暗示该过程可以用无限自回归表示来编写:
$$x_t=\pi_1 x_{t-1}+\pi_2 x_{t-2}+\cdots+a_t$$

$$(1-\theta B)^{-1} x_t=a_t$$

$$\left(1+\theta B+\theta^2 B^2+\cdots\right) x_t=a_t .$$

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