### 统计代写|应用时间序列分析代写applied time series analysis代考|Portmanteau Test

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Portmanteau Test

Financial applications often require to test jointly that several autocorrelations of $r_{I}$ are zero. Box and Pierce (1970) propose the Portmanteau statistic
$$Q^{}(m)=T \sum_{\ell=1}^{m} \hat{\rho}{\ell}^{2}$$ as a test statistic for the null hypothesis $H{o}: \rho_{1}=\cdots=\rho_{m}=0$ against the alternative hypothesis $H_{a}: \rho_{i} \neq 0$ for some $i \in{1, \ldots, m}$. Under the assumption that $\left{r_{t}\right}$ is an iid sequence with certain moment conditions, $Q^{}(\mathrm{~m})$ is asymptotically a chi-squared random variable with $m$ degrees of freedom.

Ljung and Box (1978) modify the $Q^{*}(\mathrm{~m})$ statistic as below to increase the power of the test in finite samples,
$$Q(m)=T(T+2) \sum_{\ell=1}^{m} \frac{\hat{\rho}_{\ell}^{2}}{T-\ell} .$$
In practice, the selection of $m$ may affect the performance of the $Q(m)$ statistic. Several values of $m$ are often used. Simulation studies suggest that the choice of $m \approx \ln (T)$ provides better power performance.

The function $\hat{\rho}{1}, \hat{\rho}{2}, \ldots$ is called the sample autocorrelation function (ACF) of $r_{t}$. It plays an important role in linear time series analysis. As a matter of fact, a linear time series model can be characterized by its ACF, and linear time series modeling makes use of the sample ACF to capture the linear dynamic of the data. Figure $2.1$ shows the sample autocorrelation functions of monthly simple and log returns of IBM stock from January 1926 to December 1997. The two sample ACFs are very close to each other, and they suggest that the serial correlations of monthly IBM stock returns are very small, if any. The sample ACFs are all within their two standard-error limits, indicating that they are not significant at the $5 \%$ level. In addition, for the simple returns, the Ljung-Box statistics give $Q(5)=5.4$ and $Q(10)=14.1$, which correspond to $p$ value of $0.37$ and $0.17$, respectively, based on chi-squared distributions with 5 and 10 degrees of freedom. For the log returns, we have $Q(5)=5.8$ and $Q(10)=13.7$ with $p$ value $0.33$ and $0.19$, respectively. The joint tests confirm that monthly IBM stock returns have no significant serial correlations. Figure $2.2$ shows the same for the monthly returns of the value-weighted index from the Center for Research in Security Prices (CRSP), University of Chicago. There are some significant serial correlations at the $5 \%$ level for both return series. The Ljung-Box statistics give $Q(5)=27.8$ and $Q(10)=36.0$ for the simple returns and $Q(5)=26.9$

## 统计代写|应用时间序列分析代写applied time series anakysis代考| WHITE NOISE AND LINEAR TIME SERIES

A time series $r_{t}$ is called a white noise if $\left{r_{t}\right}$ is a sequence of independent and identically distributed random variables with finite mean and variance. In particular,

Figure 2.2. Sample autocorrelation functions of monthly simple and log returns of the valueweighted index of U.S. Markets from January 1926 to December 1997. In each plot, the two horizontal lines denote two standard-error limits of the sample ACF.
if $r_{t}$ is normally distributed with mean zero and variance $\sigma^{2}$, the series is called a Gaussian white noise. For a white noise series, all the ACFs are zero. In practice, if all sample ACFs are close to zero, then the series is a white noise series. Based on Figures $2.1$ and $2.2$, the monthly returns of IBM stock are close to white noise, whereas those of the value-weighted index are not.

The behavior of sample autocorrelations of the value-weighted index returns indicates that for some asset returns it is necessary to model the serial dependence before further analysis can be made. In what follows, we discuss some simple time series models that are useful in modeling the dynamic structure of a time series. The concepts presented are also useful later in modeling volatility of asset returns.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Linear Time Series

A time series $r_{I}$ is said to be linear if it can be written as
$$r_{t}=\mu+\sum_{i=0}^{\infty} \psi_{i} a_{t-i}$$

where $\mu$ is the mean of $r_{t}, \psi_{0}=1$ and $\left{a_{t}\right}$ is a sequence of independent and identically distributed random variables with mean zero and a well-defined distribution (i.e., $\left{a_{t}\right}$ is a white noise series). In this book, we are mainly concerned with the case where $a_{t}$ is a continuous random variable. Not all financial time series are linear, however. We study nonlinearity and nonlinear models in Chapter $4 .$

For a linear time series in Eq. (2.4), the dynamic structure of $r_{l}$ is governed by the coefficients $\psi_{i}$, which are called the $\psi$-weights of $r_{t}$ in the time series literature. If $r_{t}$ is weakly stationary, we can obtain its mean and variance easily by using the independence of $\left{a_{t}\right}$ as
$$E\left(r_{t}\right)=\mu, \quad \operatorname{Var}\left(r_{t}\right)=\sigma_{a}^{2} \sum_{i=0}^{\infty} \psi_{i}^{2},$$
where $\sigma_{a}^{2}$ is the variance of $a_{t}$. Furthermore, the lag- $\ell$ autocovariance of $r_{t}$ is
\begin{aligned} \gamma_{\ell} &=\operatorname{Cov}\left(r_{t}, r_{t-\ell}\right)=E\left[\left(\sum_{i=0}^{\infty} \psi_{i} a_{t-i}\right)\left(\sum_{j=0}^{\infty} \psi_{j} a_{t-\ell-j}\right)\right] \ &=E\left(\sum_{i, j=0}^{\infty} \psi_{i} \psi_{j} a_{t-i} a_{t-\ell-j}\right) \ &=\sum_{j=0}^{\infty} \psi_{j+\ell} \psi_{j} E\left(a_{t-\ell-j}^{2}\right)=\sigma_{a}^{2} \sum_{j=0}^{\infty} \psi_{j} \psi_{j+\ell} \end{aligned}
Consequently, the $\psi$-weights are related to the autocorrelations of $r_{t}$ as follows:
$$\rho_{\ell}=\frac{\gamma_{\ell}}{\gamma_{0}}=\frac{\sum_{i=0}^{\infty} \psi_{i} \psi_{i+\ell}}{1+\sum_{i=1}^{\infty} \psi_{i}^{2}}, \quad \ell \geq 0,$$
where $\psi_{0}=1$. Linear time series models are econometric and statistical models used to describe the pattern of the $\psi$-weights of $r_{t}$.

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

Ljung 和 Box (1978) 修改了问∗( 米)统计如下，以增加有限样本中的检验能力，

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Linear Time Series

r吨=μ+∑一世=0∞ψ一世一种吨−一世

Cℓ=这⁡(r吨,r吨−ℓ)=和[(∑一世=0∞ψ一世一种吨−一世)(∑j=0∞ψj一种吨−ℓ−j)] =和(∑一世,j=0∞ψ一世ψj一种吨−一世一种吨−ℓ−j) =∑j=0∞ψj+ℓψj和(一种吨−ℓ−j2)=σ一种2∑j=0∞ψjψj+ℓ

ρℓ=CℓC0=∑一世=0∞ψ一世ψ一世+ℓ1+∑一世=1∞ψ一世2,ℓ≥0,

## 广义线性模型代考

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## MATLAB代写

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