### 统计代写|应用时间序列分析代写applied time series analysis代考|Likelihood Function of Returns

statistics-lab™ 为您的留学生涯保驾护航 在代写应用时间序列分析applied time series analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写应用时间序列分析applied time series analysis方面经验极为丰富，各种代写应用时间序列分析applied time series analysis相关的作业也就用不着说。

• 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|Likelihood Function of Returns

The partition of Eq. (1.15) can be used to obtain the likelihood function of the log returns $\left{r_{1}, \ldots, r_{T}\right}$ of an asset, where for ease in notation the subscript $i$ is omitted from the log return. If the conditional distribution $f\left(r_{t} \mid r_{t-1}, \ldots, r_{1}, \theta\right)$ is normal with mean $\mu_{t}$ and variance $\sigma_{t}^{2}$, then $\boldsymbol{\theta}$ consists of the parameters in $\mu_{t}$ and $\sigma_{t}^{2}$ and the likelihood function of the data is
$$f\left(r_{1}, \ldots, r_{T} ; \boldsymbol{\theta}\right)=f\left(r_{1} ; \boldsymbol{\theta}\right) \prod_{t=2}^{T} \frac{1}{\sqrt{2 \pi} \sigma_{t}} \exp \left[\frac{-\left(r_{t}-\mu_{t}\right)^{2}}{2 \sigma_{t}^{2}}\right]$$
where $f\left(r_{1} ; \theta\right)$ is the marginal density function of the first observation $r_{1}$. The value of $\boldsymbol{\theta}$ that maximizes this likelihood function is the maximum likelihood estimate (MLE) of $\theta$. Since log function is monotone, the MLE can be obtained by maximizing the log likelihood function,
$\ln f\left(r_{1}, \ldots, r_{T} ; \theta\right)=\ln f\left(r_{1} ; \theta\right)-\frac{1}{2} \sum_{t=2}^{T}\left[\ln (2 \pi)+\ln \left(\sigma_{t}^{2}\right)+\frac{\left(r_{t}-\mu_{t}\right)^{2}}{\sigma_{t}^{2}}\right]$
which is casier to handle in practice. Log likelihood function of the data can be obtained in a similar manner if the conditional distribution $f\left(r_{t} \mid r_{t-1}, \ldots, r_{1} ; \theta\right)$ is not normal.

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Empirical Properties of Returns

The data used in this section are obtained from the Center for Research in Security Prices (CRSP) of the University of Chicago. Dividend payments, if any, are included in the returns. Figure $1.2$ shows the time plots of monthly simple returns and log returns of International Business Machines (IBM) stock from January 1926 to December 1997. A time plot shows the data against the time index. The upper plot is for the simple returns. Figure $1.3$ shows the same plots for the monthly returns of value-weighted market index. As expected, the plots show that the basic patterns of simple and log returns are similar.

Table $1.2$ provides some descriptive statistics of simple and log returns for selected U.S. market indexes and individual stocks. The returns are for daily and monthly sample intervals and are in percentages. The data spans and sample sizes are also given in the table. From the table, we make the following observations. (a) Daily returns of the market indexes and individual stocks tend to have high excess kurtoses. For monthly series, the returns of market indexes have higher excess kurtoses than individual stocks. (b) The mean of a daily return series is close to zero, whereas thatof a monthly return series is slightly larger. (c) Monthly returns have higher standard deviations than daily returns. (d) Among the daily returns, market indexes have smaller standard deviations than individual stocks. This is in agreement with common sense. (e) The skewness is not a serious problem for both daily and monthly

returns. (f) The descriptive statistics show that the difference between simple and log returns is not substantial.

Figure $1.4$ shows the empirical density functions of monthly simple and log returns of IBM stock. Also shown, by a dashed line, in each graph is the normal probability density function evaluated by using the sample mean and standard deviation of IBM returns given in Table 1.2. The plots indicate that the normality assumption is questionable for monthly IBM stock returns. The empirical density function has a higher peak around its mean, but fatter tails than that of the corresponding normal distribution. In other words, the empirical density function is taller, skinnier, but with a wider support than the corresponding normal density.

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

Besides the return series, we also consider the volatility process and the behavior of extreme returns of an asset. The volatility process is concerned with the evolution of conditional variance of the return over time. This is a topic of interest because, as shown in Figures $1.2$ and $1.3$, the variabilities of returns vary over time and appear in

clusters. In application, volatility plays an important role in pricing stock options. By extremes of a return series, we mean the large positive or negative returns. Table $1.2$ shows that the minimum and maximum of a return series can be substantial. The negative extreme returns are important in risk management, whereas positive extreme returns are critical to holding a short position. We study properties and applications of extreme returns, such as the frequency of occurrence, the size of an extreme, and the impacts of economic variables on the extremes, in Chapter $7 .$

Other financial time series considered in the book include interest rates, exchange rates, bond yields, and quarterly earning per share of a company. Figure $1.5$ shows the time plots of two U.S. monthly interest rates. They are the 10-year and 1-year Treasury constant maturity rates from April 1954 to January 2001. As expected, the two interest rates moved in unison, but the l-year rates appear to be more volatile. Table $1.3$ provides some descriptive statistics for selected U.S. financial time series. The monthly bond returns obtained from CRSP are from January 1942 to December 1999. The interest rates are obtained from the Federal Reserve Bank of St Louis. The weekly 3 -month Treasury Bill rate started on January 8, 1954, and the 6-month rate started on December 12, 1958. Both series ended on February 16, 2001. For the interest rate series, the sample means are proportional to the time to maturity, but the sample standard deviations are inversely proportional to the time to maturity. For

the bond returns, the sample standard deviations are positively related to the time to maturity, whereas the sample means remain stable for all maturities. Most of the series considered have positive excess kurtoses.

With respect to the empirical characteristics of returns shown in Table $1.2$, Chapters 2 to 4 focus on the first four moments of a return series and Chapter 7 on the behavior of minimum and maximum returns. Chapters 8 and 9 are concerned with moments of and the relationships between multiple asset returns, and Chapter 5 addresses properties of asset returns when the time interval is small. An introduction to mathematical finance is given in Chapter 6 .

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

F(r1,…,r吨;θ)=F(r1;θ)∏吨=2吨12圆周率σ吨经验⁡[−(r吨−μ吨)22σ吨2]

ln⁡F(r1,…,r吨;θ)=ln⁡F(r1;θ)−12∑吨=2吨[ln⁡(2圆周率)+ln⁡(σ吨2)+(r吨−μ吨)2σ吨2]

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。