### 统计代写|应用时间序列分析代写applied time series analysis代考|Linear Time Series Analysis and Its Applications

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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|应用时间序列分析代写applied time series anakysis代考|STATIONARITY

The foundation of time series analysis is stationarity. A time series $\left{r_{t}\right}$ is said to be strictly stationary if the joint distribution of $\left(r_{l_{1}}, \ldots, r_{t_{k}}\right)$ is identical to that of $\left(r_{t_{1}+1}, \ldots, r_{t_{k}+t}\right)$ for all $t$, where $k$ is an arbitrary positive integer and $\left(t_{1}, \ldots, t_{k}\right)$ is a collection of $k$ positive integers. In other words, strict stationarity requires that the joint distribution of $\left(r_{t_{1}}, \ldots, r_{t_{k}}\right)$ is invariant under time shift. This is a very strong condition that is hard to verify empirically. A weaker version of stationarity is often assumed. A time series $\left{r_{t}\right}$ is weakly stationary if both the mean of $r_{t}$ and the covariance between $r_{t}$ and $r_{t-\ell}$ are time-invariant, where $\ell$ is an arbitrary integer. More specifically, $\left{r_{t}\right}$ is weakly stationary if (a) $E\left(r_{l}\right)=\mu$, which is a constant, and (b) $\operatorname{Cov}\left(r_{t}, r_{t-\ell}\right)=\gamma_{\ell}$, which only depends on $\ell$. In practice, suppose that we have observed $T$ data points $\left{r_{t} \mid t=1, \ldots, T\right}$. The weak stationarity implies that the time plot of the data would show that the $T$ values fluctuate with constant variation around a constant level.

Implicitly in the condition of weak stationarity, we assume that the first two moments of $r_{t}$ are finite. From the definitions, if $r_{t}$ is strictly stationary and its first two moments are finite, then $r_{t}$ is also weakly stationary. The converse is not true in general. However, if the time series $r_{t}$ is normally distributed, then weak stationarity is equivalent to strict stationarity. In this book, we are mainly concerned with weakly stationary series.

The covariance $\gamma_{\ell}=\operatorname{Cov}\left(r_{t}, r_{t-\ell}\right)$ is called the lag- $\ell$ autocovariance of $r_{t}$. It has two important properties: (a) $\gamma_{0}=\operatorname{Var}\left(r_{t}\right)$ and (b) $\gamma_{-\ell}=\gamma_{\ell}$. The second property holds because $\operatorname{Cov}\left(r_{t}, r_{t-(-\ell)}\right)=\operatorname{Cov}\left(r_{t-(-\ell)}, r_{t}\right)=\operatorname{Cov}\left(r_{t+\ell}, r_{t}\right)=$ $\operatorname{Cov}\left(r_{t_{1}}, r_{l_{1}-\ell}\right)$, where $t_{1}=t+\ell$.

In the finance literature, it is common to assume that an asset return series is weakly stationary. This assumption can be checked empirically provided that a sufficient number of historical returns are available. For example, one can divide the data into subsamples and check the consistency of the results obtained.

## 统计代写|应用时间序列分析代写applied time series anakysis代考| CORRELATION AND AUTOCORRELATION FUNCTION

The correlation coefficient between two random variables $X$ and $Y$ is defined as
$$\rho_{x, y}=\frac{\operatorname{Cov}(X, Y)}{\sqrt{\operatorname{Var}(X) \operatorname{Var}(Y)}}=\frac{E\left[\left(X-\mu_{x}\right)\left(Y-\mu_{y}\right)\right]}{\sqrt{E\left(X-\mu_{x}\right)^{2} E\left(Y-\mu_{y}\right)^{2}}},$$
where $\mu_{x}$ and $\mu_{y}$ are the mean of $X$ and $Y$, respectively, and it is assumed that the variances exist. This coefficient measures the strength of linear dependence between $X$ and $Y$, and it can be shown that $-1 \leq \rho_{x, y} \leq 1$ and $\rho_{x, y}=\rho_{y, x}$. The two random variables are uncorrelated if $\rho_{x, y}=0$. In addition, if both $X$ and $Y$ are normal random variables, then $\rho_{x, y}=0$ if and only if $X$ and $Y$ are independent. When the sample $\left{\left(x_{l}, y_{t}\right)\right}_{t=1}^{T}$ is available, the correlation can be consistently estimated by its

sample counterpart
$$\hat{\rho}{x, y}=\frac{\sum{t=1}^{T}\left(x_{t}-\bar{x}\right)\left(y_{t}-\bar{y}\right)}{\sqrt{\sum_{t=1}^{T}\left(x_{t}-\bar{x}\right)^{2} \sum_{t=1}^{T}\left(y_{t}-\bar{y}\right)^{2}}},$$
where $\bar{x}=\sum_{t=1}^{T} x_{t} / T$ and $\bar{y}=\sum_{t=1}^{T} y_{t} / T$ are the sample mean of $X$ and $Y$, respectively.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Autocorrelation Function

Consider a weakly stationary return series $r_{t}$. When the linear dependence between $r_{t}$ and its past values $r_{t-i}$ is of interest, the concept of correlation is generalized to autocorrelation. The correlation coefficient between $r_{t}$ and $r_{t-\ell}$ is called the lag- $\ell$ autocorrelation of $r_{t}$ and is commonly denoted by $\rho_{\ell}$, which under the weak stationarity assumption is a function of $\ell$ only. Specifically, we define
$$\rho_{\ell}=\frac{\operatorname{Cov}\left(r_{t}, r_{t-\ell}\right)}{\sqrt{\operatorname{Var}\left(r_{t}\right) \operatorname{Var}\left(r_{t-\ell}\right)}}=\frac{\operatorname{Cov}\left(r_{t}, r_{t-\ell}\right)}{\operatorname{Var}\left(r_{t}\right)}=\frac{\gamma_{\ell}}{\gamma_{0}},$$
where the property $\operatorname{Var}\left(r_{t}\right)=\operatorname{Var}\left(r_{t-\ell}\right)$ for a weakly stationary series is used. From the definition, we have $\rho_{0}=1, \rho_{\ell}=\rho_{-\ell}$, and $-1 \leq \rho_{\ell} \leq 1$. In addition, a weakly stationary series $r_{t}$ is not serially correlated if and only if $\rho_{\ell}=0$ for all $\ell>0$.
For a given sample of returns $\left{r_{t}\right}_{t=1}^{T}$, let $\bar{r}$ be the sample mean (i.e., $\bar{r}=$ $\sum_{t=1}^{T} r_{t} / T$ ). Then the lag-1 sample autocorrelation of $r_{t}$ is
$$\hat{\rho}{1}=\frac{\sum{t=2}^{T}\left(r_{t}-\bar{r}\right)\left(r_{t-1}-\bar{r}\right)}{\sum_{t=1}^{T}\left(r_{t}-\bar{r}\right)^{2}} .$$

## MATLAB代写

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