### 数学代写|计量经济学原理代写Principles of Econometrics代考|Covariance Decomposition

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Principles of Econometrics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。我们的专家在代写计量经济学Principles of Econometrics代写方面经验极为丰富，各种代写计量经济学Principles of Econometrics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

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Variance Decomposition

Just as we can break up the expected value using the Law of Iterated Expectations we can decompose the variance of a random variable into two parts.
Variance Decomposition: $\operatorname{var}(Y)=\operatorname{var}{X}[E(Y \mid X)]+E{X}[\operatorname{var}(Y \mid X)]$
This “beautiful” result ${ }^{9}$ says that the variance of the random variable $Y$ equals the sum of the variance of the conditional mean of $Y$ given $X$ and the mean of the conditional variance of $Y$ given $X$. In this section we will discuss this result. ${ }^{10}$

Suppose that we are interested in the wages of the population consisting of working adults. How much variation do wages display in the population? If WAGE is the wage of a randomly drawn population member, then we are asking about the variance of WAGE, that is, $\operatorname{var}(W A G E)$. The variance decomposition says
$$\operatorname{var}(W A G E)=\operatorname{var}{E D U C}[E(W A G E \mid E D U C)]+E{E D U C}[\operatorname{var}(W A G E \mid E D U C)]$$
$E(W A G E \mid E D U C)$ is the expected value of $W A G E$ given a specific value of education, such as $E D U C=12$ or $E D U C=16 . E(W A G E \mid E D U C=12)$ is the average WAGE in the population, given that we only consider workers who have 12 years of education. If $E D U C$ changes then the conditional mean $E(W A G E \mid E D U C)$ changes, so that $E(W A G E \mid E D U C=16)$ is not the same as $E(W A G E \mid E D U C=12)$, and in fact we expect $E(W A G E \mid E D U C=16)>E(W A G E \mid E D U C=12)$; more education means more “human capital” and thus the average wage should be higher. The first component in the variance decomposition $\operatorname{var}_{E D U C}[E(W A G E \mid E D U C)]$ measures the variation in $E(W A G E \mid E D U C$ ) due to variation in education.

The second part of the variance decomposition is $E_{E D U C}[\operatorname{var}(W A G E \mid E D U C)]$. If we restrict our attention to population members who have 12 years of education, the mean wage is $E(W A G E \mid E D U C=12)$. Within the group of workers who have 12 years of education we will observe wide ranges of wages. For example, using one sample of CPS data from $2013,{ }^{11}$ wages for those with 12 years of education varied from $\$ 3.11$/hour to$\$100.00 /$ hour; for those with 16 years of education wages varied from $\$ 2.75 /$hour to$\$221.10 /$ hour. For workers with 12 and 16 years of education that variation is measured by $\operatorname{var}(W A G E \mid E D U C=12)$ and

$\operatorname{var}(W A G E \mid E D U C=16)$. The term $E_{E D U C}[\operatorname{var}(W A G E \mid E D U C)]$ measures the average of $\operatorname{var}(W A G E \mid E D U C)$ as education changes.

To summarize, the variation of WAGE in the population can be attributed to two sources: variation in the conditional mean $E(W A G E \mid E D U C)$ and variation due to changes in education in the conditional variance of WAGE given education.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|The Bivariate Normal Distribution

Two continuous random variables, $X$ and $Y$, have a joint normal, or bivariate normal, distribution if their joint $p d f$ takes the form
\begin{aligned} f(x, y)=\frac{1}{2 \pi \sigma_{X} \sigma_{Y} \sqrt{1-\rho^{2}}} \exp {-& {\left[\left(\frac{x-\mu_{X}}{\sigma_{X}}\right)^{2}-2 \rho\left(\frac{x-\mu_{X}}{\sigma_{X}}\right)\left(\frac{y-\mu_{Y}}{\sigma_{Y}}\right)\right.} \ &\left.\left.+\left(\frac{y-\mu_{X}}{\sigma_{Y}}\right)^{2}\right] / 2\left(1-\rho^{2}\right)\right} \end{aligned}
where $-\infty<x<\infty,-\infty<y<\infty$. The parameters $\mu_{X}$ and $\mu_{Y}$ are the means of $X$ and $Y$, $\sigma_{X}^{2}$ and $\sigma_{Y}^{2}$ are the variances of $X$ and $Y$, so that $\sigma_{X}$ and $\sigma_{Y}$ are the standard deviations. The parameter $\rho$ is the correlation between $X$ and $Y$. If $\operatorname{cov}(X, Y)=\sigma_{X Y}$ then
$$\rho=\frac{\operatorname{cov}(X, Y)}{\sqrt{\operatorname{var}(X)} \sqrt{\operatorname{var}(Y)}}=\frac{\sigma_{X Y}}{\sigma_{X} \sigma_{Y}}$$
The complex equation for $f(x, y)$ defines a surface in three-dimensional space. In Figure P.6a ${ }^{13}$ we depict the surface if $\mu_{X}=\mu_{Y}=0, \sigma_{X}=\sigma_{Y}=1$, and $\rho=0.7$. The positive correlation means there is a positive linear association between the values of $X$ and $Y$, as described in Figure P.4. Figure P.6b depicts the contours of the density, the result of slicing the density horizontally, at a given height. The contours are more “cigar-shaped” the larger the absolute value of the correlation $\rho$. In Figure P.7a the correlation is $\rho=0$. In this case the joint density is symmetrical and the contours in Figure P.7b are circles. If $X$ and $Y$ are jointly normal then they are statistically independent if, and only if, $\rho=0$.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|An Economic Model

In order to develop the ideas of regression models, we are going to use a simple, but important, economic example. Suppose that we are interested in studying the relationship between household income and expenditure on food. Consider the “experiment” of randomly selecting households from a particular population. The population might consist of households within a particular city, state, province, or country. For the present, suppose that we are interested only in households with an income of $\$ 1000$per week. In this experiment, we randomly select a number of households from this population and interview them. We ask the question “How much did you spend per person on food last week?” Weekly food expenditure, which we denote as$y$. is a random variable since the value is unknown to us until a household is selected and the question is asked and answered. The continuous random variable$y$has a probability density function (which we will abbreviate as$p d f$) that describes the probabilities of obtaining various food expenditure values. If you are rusty or uncertain about probability concepts, see the Probability Primer and Appendix$B$at the end of this book for a comprehensive review. The amount spent on food per person will vary from one household to another for a variety of reasons: some households will be devoted to gourmet food, some will contain teenagers, some will contain senior citizens, some will be vegetarian, and some will eat at restaurants more frequently. All of these factors and many others, including random, impulsive buying, will cause weekly expenditures on food to vary from one household to another, even if they all have the same income. The$p d f f(y)$describes how expenditures are “distributed” over the population and might look like Figure 2.1. The$p d f$in Figure 2.1a is actually a conditional pdf since it is “conditional” upon household income. If$x=$weekly household income$=\$1000$, then the conditional $p d f$ is $f(y \mid x=\$ 1000)$. The conditional mean, or expected value, of$y$is$E(y \mid x=\$1000)=\mu_{y \mid x}$ and is our population’s mean weekly food expenditure per person.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|The Bivariate Normal Distribution

\begin{对齐} f(x, y)=\frac{1}{2 \pi \sigma_{X} \sigma_{Y} \sqrt{1-\rho^{2}}} \exp {-& { \left[\left(\frac{x-\mu_{X}}{\sigma_{X}}\right)^{2}-2 \rho\left(\frac{x-\mu_{X}}{ \sigma_{X}}\right)\left(\frac{y-\mu_{Y}}{\sigma_{Y}}\right)\right.} \ &\left.\left.+\left(\ frac{y-\mu_{X}}{\sigma_{Y}}\right)^{2}\right] / 2\left(1-\rho^{2}\right)\right} \end{对齐}\begin{对齐} f(x, y)=\frac{1}{2 \pi \sigma_{X} \sigma_{Y} \sqrt{1-\rho^{2}}} \exp {-& { \left[\left(\frac{x-\mu_{X}}{\sigma_{X}}\right)^{2}-2 \rho\left(\frac{x-\mu_{X}}{ \sigma_{X}}\right)\left(\frac{y-\mu_{Y}}{\sigma_{Y}}\right)\right.} \ &\left.\left.+\left(\ frac{y-\mu_{X}}{\sigma_{Y}}\right)^{2}\right] / 2\left(1-\rho^{2}\right)\right} \end{对齐}

ρ=这⁡(X,是)曾是⁡(X)曾是⁡(是)=σX是σXσ是

## 有限元方法代写

tatistics-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。