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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|GENERAL AR AND MA PROCESSES

3.16 Extensions to the AR(1) and $\mathrm{MA}(1)$ models are immediate. The general autoregressive model of order $p(\operatorname{AR}(p))$ can be written as:
$$x_t-\phi_1 x_{t-1}-\phi_2 x_{t-2}-\cdots-\phi_p x_{t-p}=a_t$$
or $$\left(1-\phi_1 B-\phi_2 B^2-\cdots-\phi_p B^p\right) x_t=\phi(B) x_t=a_t$$
The linear filter representation $x_t=\phi^{-1}(B) a_t=\psi(B) a_t$ can be obtained by equating coefficients in $\phi(B) \psi(B)=1^7$
3.17 The stationarity conditions required for convergence of the $\psi$-weights are that the roots of the characteristic equation:
$$\phi(B)=\left(1-g_1 B\right)\left(1-g_2 B\right) \cdots\left(1-g_p B\right)=0$$ are such that $\left|g_i\right|<1$ for $i=1,2, \ldots, p$. The behavior of the ACF is determined by the difference equation: $$\phi(B) \rho_k=0 \quad k>0$$
which has the solution
$$\rho_k=A_1 g_1^k+A_2 g_2^k+\cdots+A_p g_p^k$$
Since $\left|g_i\right|<1$, the ACF is thus described by a mixture of damped exponentials (for real roots) and damped sine waves (for complex roots). As an example, consider the AR(2) process:
$$\left(1-\phi_1 B-\phi_2 B^2\right) x_t=a_t$$
with characteristic equation
$$\phi(B)=\left(1-g_1 B\right)\left(1-g_2 B\right)=0$$
The roots $g_1$ and $g_2$ are given by:
$$g_1, g_2=\frac{1}{2}\left(\phi_1 \pm\left(\phi_1^2+4 \phi_2\right)^{1 / 2}\right)$$
and can both be real, or they can be a pair of complex numbers. For stationarity, it is required that the roots be such that $\left|g_1\right|<1$ and $\left|g_2\right|<1$, and it can be shown that these conditions imply this set of restrictions on $\phi_1$ and $\phi_2:{ }^8$
$$\phi_1+\phi_2<1 \quad-\phi_1+\phi_2<1 \quad-1<\phi_2<1$$
The roots will be complex if $\phi_1^2+4 \phi_2<0$, although a necessary condition for complex roots is simply that $\phi_2<0$.

## 统计代写|时间序列分析代写Time-Series Analysis代考|ARMA MODEL BUILDING AND ESTIMATION

3.29 An essential first step in fitting ARMA models to observed time series is to obtain estimates of the generally unknown parameters $\mu, \sigma_x^2$, and the $\rho_k$. With the stationarity and (implicit) ergodicity assumptions, $\mu$ and $\sigma_x^2$ can be estimated by the sample mean and sample variance, respectively, of the realization $x_1, x_2, \ldots, x_T$, that is, by Eqs. (1.2) and (1.3). An estimate of $\rho_k$ is then provided by the lag $k$ sample autocorrelation given by Eq. (1.1), which, because of its importance, is reproduced here:
$$r_k=\frac{\sum_{t=k+1}^T\left(x_t-\bar{x}\right)\left(x_{t-k}-\bar{x}\right)}{T s^2} \quad k=1,2, \ldots$$
Recall from $\S 1.2$ that the set of $r_k \mathrm{~s}$ defines the sample ACF (SACF), which is sometimes referred to as the correlogram.
3.30 Consider a time series generated as independent observations drawn from a fixed distribution with finite variance (i.e., $\rho_k=0$ for all $k \neq 0$ ). Such a series is said to be independent and identically distributed or i.i.d. For such a series the variance of $r_k$ is approximately given by $T^{-1}$. If $T$ is large as well, $\sqrt{T} r_k$ will be approximately standard normal, so that $r_k \stackrel{a}{\sim} N\left(0, T^{-1}\right)$, implying that an absolute value of $r_k$ in excess of $2 / \sqrt{T}$ may be regarded as “significantly” different from zero at the $5 \%$ significance level. More generally, if $\rho_k=0$ for $k>q$, the variance of $r_k$, for $k>q$, is:
$$V\left(r_k\right)=T^{-1}\left(1+2 \rho_1^2+\cdots+2 \rho_q^2\right) .$$
Thus, by successively increasing the value of $q$ and replacing the $\rho_k \mathrm{~s}$ by their sample estimates, the variances of the sequence $r_1, r_2, \ldots, r_k$ can be estimated as $T^{-1}, T^{-1}\left(1+2 r_1^2\right), \ldots, T^{-1}\left(1+2 r_1^2+\cdots+2 r_{k-1}^2\right)$, and, of course, these will be larger for $k>1$ than those calculated using the simple formula $T^{-1}$. Taking the square root of $V\left(r_k\right)$ gives the standard error to be attached to $r_k$ and these are often referred to as Bartlett standard errors, as (3.12) was derived in Bartlett (1946).

# 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|GENERAL AR AND MA PROCESSES

3.16 AR(1) 的扩展和MA(1)模型是即时的。序的一般自回归模型 $p(\mathrm{AR}(p))$ 可以写成:
$$x_t-\phi_1 x_{t-1}-\phi_2 x_{t-2}-\cdots-\phi_p x_{t-p}=a_t$$

$$\left(1-\phi_1 B-\phi_2 B^2-\cdots-\phi_p B^p\right) x_t=\phi(B) x_t=a_t$$

$3.17$ 收敛所需的平稳性条件 $\psi$-权重是特征方程的根:
$$\phi(B)=\left(1-g_1 B\right)\left(1-g_2 B\right) \cdots\left(1-g_p B\right)=0$$

$$\rho_k=A_1 g_1^k+A_2 g_2^k+\cdots+A_p g_p^k$$

$$\left(1-\phi_1 B-\phi_2 B^2\right) x_t=a_t$$

$$\phi(B)=\left(1-g_1 B\right)\left(1-g_2 B\right)=0$$

$$g_1, g_2=\frac{1}{2}\left(\phi_1 \pm\left(\phi_1^2+4 \phi_2\right)^{1 / 2}\right)$$

$$\phi_1+\phi_2<1 \quad-\phi_1+\phi_2<1 \quad-1<\phi_2<1$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|ARMA MODEL BUILDING AND ESTIMATION

$3.29$ 将 ARMA 模型拟合到观察到的时间序列的一个重要的第一步是获得对一般末知参数的估计 $\mu, \sigma_{x^{\prime}}^2$ 和 $\rho_k$. 在平稳性和（隐式）遍历性假设下， $\mu$ 和 $\sigma_x^2$ 可以分别通 过实现的样本均值和样本方差来估计 $x_1, x_2, \ldots, x_T$ ，也就是说，由方程。(1.2) 和 (1.3)。估计 $\rho_k$ 然后由滞后提供 $k$ 由方程给出的样本自相关。(1.1)，由于其重要性，在此复制:
$$r_k=\frac{\sum_{t=k+1}^T\left(x_t-\bar{x}\right)\left(x_{t-k}-\bar{x}\right)}{T s^2} \quad k=1,2, \ldots$$

$3.30$ 考虑从具有有限方差的固定分布（即， $\rho_k=0$ 对所有人 $k \neq 0$ )。这样的序列被称为独立同分布或独立同分布。对于这样的序列，方差 $r_k$ 大约由 $T^{-1}$. 如果 $T$ 也 很大， $\sqrt{T} r_k$ 将近似于标准正常，因此 $r_k \stackrel{a}{\sim} N\left(0, T^{-1}\right)$ ，意味着绝对值 $r_k$ 在过量的 $2 / \sqrt{T}$ 可被视为“显着“不同于零 $5 \%$ 显着性水平。更一般地说，如果 $\rho_k=0$ 为了 $k>q$ ，的方差 $r_k$ ，为了 $k>q$ ， 是:
$$V\left(r_k\right)=T^{-1}\left(1+2 \rho_1^2+\cdots+2 \rho_q^2\right) .$$

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