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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|STOCHASTIC PROCESSES AND STATIONARITY

3.1 The concept of a stationary time series was introduced informally in Chapter 1, Time Series and Their Features, but to proceed further it is necessary to consider the concept rather more rigorously. To this end, it is often useful to regard the observations $x_1, x_2, \ldots, x_T$ on the series $x_t$ as a realization of a stochastic process. In general, such a stochastic process may be described by a $T$-dimensional probability distribution, so that the relationship between a realization and a stochastic process is analogous. in classical statistics, to that between a sample and the population from which it has been drawn from.
Specifying the complete form of the probability distribution, however, will typically be too ambitious a task, so attention is usually concentrated on the first and second moments; the $T$ means:
$$E\left(x_1\right), E\left(x_2\right), \ldots, E\left(x_T\right)$$
$T$ variances:
$$V\left(x_1\right), V\left(x_2\right), \ldots, V\left(x_T\right)$$
and $T(T-1) / 2$ covariances:
$$\operatorname{Cov}\left(x_i, x_j\right), \quad i<j$$
If the distribution could be assumed to be (multivariate) normal, then this set of expectations would completely characterize the properties of the stochastic process. As has been seen from the examples in Chapter 2, Transforming Time Series, however, such an assumption will not always be appropriate, but if the process is taken to be linear, in the sense that the current value $x_t$ is generated by a linear combination of previous values $x_{t-1}, x_{t-2}, \ldots$ of the process itself plus current and past values of any other related processes. then again this set of expectations would capture its major properties.

In either case, however, it will be impossible to infer all the values of the first and second moments from just a single realization of the process, since there are only $T$ observations but $T+T(T+1) / 2$ unknown parameters. Consequently, further simplifying assumptions must be made to reduce the number of unknown parameters to more manageable proportions.

## 统计代写|时间序列分析代写Time-Series Analysis代考|WOLD’S DECOMPOSITION AND AUTOCORRELATION

3.6 A fundamental theorem in time series analysis, known as Wold’s decomposition, states that every weakly stationary, purely nondeterministic, stochastic process $x_t-\mu$ can be written as a linear combination (or linear filter) of a sequence of uncorrelated random variables. ” “Purely nondeterministic” means that any deterministic components have been subtracted from $x_t-\mu$. Such components are those that can be perfectly predicted from past values of themselves and examples commonly found are a (constant) mean, as is implied by writing the process as $x_t-\mu$, periodic sequences (e.g., sine and cosine functions), and polynomial or exponential sequences in $t$.
This linear filter representation is given by:
$$x_t-\mu=a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots=\sum_{j=0}^{\infty} \psi_j a_{t-j} \quad \psi_0=1$$

The $a_t, t=0, \pm 1, \pm 2, \ldots$ are a sequence of uncorrelated random variables, often known as innovations, drawn from a fixed distribution with:
$$E\left(a_t\right)=0 \quad V\left(a_t\right)=E\left(a_t^2\right)=\sigma^2<\infty$$
and
$$\operatorname{Cov}\left(a_t, a_{t-k}\right)=E\left(a_t a_{t-k}\right)=0, \text { for all } k \neq 0$$
Such a sequence is known as a white noise process, and occasionally the innovations will be denoted as $a_l \sim \mathrm{WN}\left(0, \sigma^2\right) .^3$ The coefficients (possibly infinite in number) in the linear filter (3.2) are known as $\psi$-weights.
$3.7$ It is easy to show that the model (3.2) leads to autocorrelation in $x_t$. From this equation it follows that:
$$E\left(x_t\right)=\mu$$
and
\begin{aligned} \gamma_0 &=V\left(x_t\right)=E\left(x_t-\mu\right)^2 \ &=E\left(a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots\right)^2 \ &=E\left(a_t^2\right)+\psi_1^2 E\left(a_{t-1}^2\right)+\psi_2^2 E\left(a_{t-2}^2\right)+\cdots \ &=\sigma^2+\psi_1^2 \sigma^2+\psi_2^2 \sigma^2+\cdots \ &=\sigma^2 \sum_{j=0}^{\infty} \psi_j^2 \end{aligned}
by using the white noise result that $E\left(a_{t-i} a_{t-j}\right)=0$ for $i \neq j$. Now:
\begin{aligned} \gamma_k &=E\left(x_t-\mu\right)\left(x_{t-k}-\mu\right) \ &=E\left(a_t+\psi_1 a_{t-1}+\cdots+\psi_k a_{t-k}+\cdots\right)\left(a_{t-k}+\psi_1 a_{t-k-1}+\cdots\right) \ &=\sigma^2\left(1 \cdot \psi_k+\psi_1 \psi_{k+1}+\psi_2 \psi_{k+2}+\cdots\right) \ &=\sigma^2 \sum_{j=0}^{\infty} \psi_j \psi_{j+k} \end{aligned}
and this implies
$$\rho_h=\frac{\sum_{j=0}^{\infty} \psi_j \psi_{j+k}}{\sum_{j=0}^{\infty} \psi_j^2}$$

# 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|STOCHASTIC PROCESSES AND stationary

$$E\left(x_1\right), E\left(x_2\right), \ldots, E\left(x_T\right)$$
$T$方差:
$$V\left(x_1\right), V\left(x_2\right), \ldots, V\left(x_T\right)$$

$$\operatorname{Cov}\left(x_i, x_j\right), \quad i<j$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|WOLD ‘S DECOMPOSITION – AND – AUTOCORRELATION

$$x_t-\mu=a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots=\sum_{j=0}^{\infty} \psi_j a_{t-j} \quad \psi_0=1$$

$a_t, t=0, \pm 1, \pm 2, \ldots$是一个不相关的随机变量序列，通常被称为创新，从一个固定的分布:
$$E\left(a_t\right)=0 \quad V\left(a_t\right)=E\left(a_t^2\right)=\sigma^2<\infty$$

$$\operatorname{Cov}\left(a_t, a_{t-k}\right)=E\left(a_t a_{t-k}\right)=0, \text { for all } k \neq 0$$

$3.7$很容易看出，模型(3.2)在$x_t$中导致自相关。从这个方程可以得到:
$$E\left(x_t\right)=\mu$$

\begin{aligned} \gamma_0 &=V\left(x_t\right)=E\left(x_t-\mu\right)^2 \ &=E\left(a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots\right)^2 \ &=E\left(a_t^2\right)+\psi_1^2 E\left(a_{t-1}^2\right)+\psi_2^2 E\left(a_{t-2}^2\right)+\cdots \ &=\sigma^2+\psi_1^2 \sigma^2+\psi_2^2 \sigma^2+\cdots \ &=\sigma^2 \sum_{j=0}^{\infty} \psi_j^2 \end{aligned}
，通过使用白噪声结果$E\left(a_{t-i} a_{t-j}\right)=0$ for $i \neq j$。现在:
\begin{aligned} \gamma_k &=E\left(x_t-\mu\right)\left(x_{t-k}-\mu\right) \ &=E\left(a_t+\psi_1 a_{t-1}+\cdots+\psi_k a_{t-k}+\cdots\right)\left(a_{t-k}+\psi_1 a_{t-k-1}+\cdots\right) \ &=\sigma^2\left(1 \cdot \psi_k+\psi_1 \psi_{k+1}+\psi_2 \psi_{k+2}+\cdots\right) \ &=\sigma^2 \sum_{j=0}^{\infty} \psi_j \psi_{j+k} \end{aligned}
，这意味着
$$\rho_h=\frac{\sum_{j=0}^{\infty} \psi_j \psi_{j+k}}{\sum_{j=0}^{\infty} \psi_j^2}$$

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