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## 金融代写|金融建模代写Financial Modeling代考|Serial correlation in returns

Here, the null hypothesis is that the first $p$ serial correlations of returns are equal to $0, H_0: \rho_1=\cdots=\rho_p=0$, where the correlation of order $j$ is estimated by

$$\hat{\rho}j=\frac{\sum{t=j+1}^T\left(r_t-\bar{r}\right)\left(r_{t-j}-\bar{r}\right)}{\sum_{t=1}^T\left(r_t-\bar{r}\right)^2} \quad \text { for } \quad 0 \leq j<T-1$$
A simple test of $H_0$ could be based on the Ljung-Box $\mathrm{Q}$ statistic
$$Q_p=T(T+2) \sum_{j=1}^p \frac{1}{T-j} \hat{\rho}_j^2 .$$
Under the null hypothesis of no serial correlation, the $Q_p$ statistic is asymptotically distributed as $\chi^2(p)$. A common practice is to test $H_0$ repeatedly using several choices of $p$.

## 金融代写|金融建模代写Financial Modeling代考|Serial correlation in volatility

To test for time dependency in volatility, we need a time-varying measure of volatility. There are at least two possible ways to approach this; first using mean-adjusted squared returns and secondly using absolute returns. Assume that returns have the following dynamics
$$r_t=\mu+\varepsilon_t, \quad \text { with } \quad \varepsilon_t=\sigma_t z_t,$$
where $\mu$ is the constant mean, $\varepsilon_t$ the mean adjusted returns, $\sigma_t$ the timevarying volatility, and $z_t$ is an $\mathcal{N}(0,1)$ innovation. Then with information set $\mathcal{F}{t-1}$ at time $t-1$ $$E\left[\varepsilon_t^2 \mid \mathcal{F}{t-1}\right]=\sigma_t^2 E\left[z_t^2 \mid \mathcal{F}_{t-1}\right]=\sigma_t^2$$
because $z_t^2$ is distributed as $\chi^2(1)$. Therefore, $\varepsilon_t^2$ can be viewed as a proxy for the volatility at time $t$. Alternatively, omitting $\mu$ for the moment, we have $r_t \sim \mathcal{N}\left(0, \sigma_t^2\right)$ and
$$E\left[\left|r_t\right|\right]=\sigma_t \sqrt{2 / \pi} .$$
Consequently, $\left|r_t\right| / \sqrt{2 / \pi}$ is a proxy for $\sigma_t$. It should be noted however that these two measures are noisy estimates of conditional volatility. See Chapter 4 for more detail and more sophisticated measures of conditional volatility.

## 金融代写|金融建模代写Financial Modeling代考|Serial correlation in returns

$$\hat{\rho} j=\frac{\sum t=j+1^T\left(r_t-\bar{r}\right)\left(r_{t-j}-\bar{r}\right)}{\sum_{t=1}^T\left(r_t-\bar{r}\right)^2} \quad \text { for } \quad 0 \leq j<T-1$$

$$Q_p=T(T+2) \sum_{j=1}^p \frac{1}{T-j} \hat{\rho}j^2$$ 在无序列相关的原假设下， $Q_p$ 统计量渐近分布为 $\chi^2(p) .$ 个个常见的做法是测试 $H_0$ 反夏使用几种选择 $p$.

## 金融代写|金融建模代写Financial Modeling代考|Serial correlation in volatility

$$E\left[\left|r_t\right|\right]=\sigma_t \sqrt{2 / \pi} .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## avatest™帮您通过考试

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## 金融代写|金融建模代写Financial Modeling代考|Moments of a random variable

In this book, we will typically use non-Gaussian distributions, which are characterized by moments that are higher than the second order. ${ }^2$ Consider a continuous random variable $X$ (say the log-return series) with cumulative distribution function $(c d f)$

$$F_X(x)=\operatorname{Pr}[X \leq x]=\int_{-\infty}^x f_X(u) d u$$
where $f_X$ is the probability density function $(p d f)$. The uncentered moments of $X$ are defined as
$$m_k=E\left[X^k\right]=\int_{-\infty}^{\infty} x^k f_X(x) d x, \quad \text { for } k=1,2, \cdots .$$
The first non-central moment $m_1=E[X]=\mu$ is the mean of $X$.
Given the mean $m_1$, the centered moments of $X$ are defined as
$$\mu_k=E\left[\left(X-m_1\right)^k\right]=\int_{-\infty}^{\infty}\left(x-m_1\right)^k f_X(x) d x, \quad \text { for } k=1,2, \cdots \text {. }$$

## 金融代写|金融建模代写Financial Modeling代考|Empirical moments

Now, consider a time series of realized returns $\left{r_t\right}_{t=1}^T$. The most commonly used measures of location are sample mean $\bar{r}$ and median $m$. The sample mean is calculated as ${ }^4$
$$\bar{r}=\hat{\mu}=\frac{1}{T} \sum_{t=1}^T r_t .$$
If $\left{r_t\right}_{t=1}^T$ has a symmetric distribution, then $\bar{r}$ is the optimal measure of location. Median is defined as the 50 th percentile of the sample. In other words, $50 \%$ of the sample has a value lower than $m=m e d[r]$, i.e.,
$$\operatorname{Pr}\left[r_t \leq m\right]=\operatorname{Pr}\left[r_t \geq m\right]=\frac{1}{2} .$$
Mean, as a measure for location, is sensitive to outliers. One erroneously recorded value could potentially move the mean away from the central part of the distribution. In contrast, the median is more robust against outliers, because it does not rely on the precise value of the realizations other than the median itself.

## 金融代写|金融建模代写Financial Modeling代考|Moments of a random variable

$$F_X(x)=\operatorname{Pr}[X \leq x]=\int_{-\infty}^x f_X(u) d u$$

$$m_k=E\left[X^k\right]=\int_{-\infty}^{\infty} x^k f_X(x) d x, \quad \text { for } k=1,2, \cdots .$$

$$\mu_k=E\left[\left(X-m_1\right)^k\right]=\int_{-\infty}^{\infty}\left(x-m_1\right)^k f_X(x) d x, \quad \text { for } k=1,2, \cdots .$$

## 金融代写|金融建模代写Financial Modeling代考|Empirical moments

$$\bar{r}=\hat{\mu}=\frac{1}{T} \sum_{t=1}^T r_t .$$

$$\operatorname{Pr}\left[r_t \leq m\right]=\operatorname{Pr}\left[r_t \geq m\right]=\frac{1}{2} .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Financial Modeling, 金融代写, 金融建模

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

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## 金融代写|金融建模代写Financial Modeling代考|Financial markets and financial time series

For more than four decades, distributions of financial asset returns have been known to be non-Gaussian (see Mandelbrot, 1963, and Fama, 1965). The assumption of normality is stacked against two hard facts: First, the empirical distributions of asset returns have tails thicker than those from a normal distribution and appear to be negatively skewed. This means more extreme negative values, which has a very serious implication for risk management and portfolio selection. Second, returns are time dependent. Squared returns, absolute returns, and all measures and proxies of volatility exhibit strong serial correlation. This is now known as volatility clustering or conditional heteroskedasticity (Engle, 1982).

Financial modeling is all about capturing and exploiting patterns in the data including the two phenomena mentioned above. Chapter 2 discusses the unique statistical properties of financial market data and several so-called stylized facts. These stylized facts will be the basis for Part II where each chapter will tackle some specific features of financial market returns.

Chapter 3 describes the actual functioning and the microstructure of financial markets. Here, we present some theoretical models that may help explaining why asset returns are non-normal and time dependent. The foundation is built on Clark (1973) who postulates that non-normality and volatility clustering could be due to intermittent information arrivals.

## 金融代写|金融建模代写Financial Modeling代考|Econometric modeling of asset returns

Part II is concerned with the time series aspects of asset returns. Chapters 4,5 , and 7 cover models for the second, third, and fourth moments and the tails of return distributions. These higher moments and tail measures are the hallmarks of non-Gaussian distributions. Chapter 6 deals with the dependence structure when the higher moments display significant departure from normality and returns appear to be time dependent. The dependence among the tail observations, described in Chapter 7 , is very different from the dependence, described in Chapter 6 , of the central and main part of the distributions, because of the differences in the underpinning statistical theories and the important fact that financial markets do behave very differently between normal and crisis periods.

Specifically, Chapter 4 covers models for volatility that include the better known GARCH (Generalized Autoregressive and Conditional Heteroskedasticity) class of models and some new extensions such as GARCH models with jumps and realized volatility models. With high-frequency data becoming more common these days, realized volatility is expected to remain an area of active research. This chapter also describes the lesser known or lesser discussed issues on GARCH aggregation and the relationship between stochastic volatility model in continuous time and the discrete time GARCH model.
Although time-varying volatility and volatility asymmetry may produce thick-tail and asymmetric distributions in asset return, volatility alone cannot explain away all the non-normality. To fully capture return distributions, we also need models for skewness and kurtosis. Chapter 5 does exactly that by fitting time-varying higher-moment conditional models to returns. It also describes tests for the adequacy of these conditional high moment models.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。