Posted on Categories:Financial Econometrics, 经济代写, 计量经济学

# 经济代写|计量经济学代写Introduction to Econometrics代考|Estimation of Error Variance

avatest™

## 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

•200+ 英语母语导师

•70分以下全额退款

## 经济代写|计量经济学代写Introduction to Econometrics代考|Estimation of Error Variance

The error variance $\sigma^2=\mathbb{E}\left[e^2\right]$ can be a parameter of interest even in a heteroskedastic regression or a projection model. $\sigma^2$ measures the variation in the “unexplained” part of the regression. Its method of moments estimator (MME) is the sample average of the squared residuals:
$$\widehat{\sigma}^2=\frac{1}{n} \sum_{i=1}^n \widehat{e}i^2$$ In the linear regression model we can calculate the mean of $\widehat{\sigma}^2$. From (3.28) and the properties of the trace operator observe that $$\widehat{\sigma}^2=\frac{1}{n} \boldsymbol{e}^{\prime} \boldsymbol{M e}=\frac{1}{n} \operatorname{tr}\left(\boldsymbol{e}^{\prime} \boldsymbol{M e}\right)=\frac{1}{n} \operatorname{tr}\left(\boldsymbol{M e} \boldsymbol{e}^{\prime}\right)$$ Then \begin{aligned} \mathbb{E}\left[\widehat{\sigma}^2 \mid \boldsymbol{X}\right] & =\frac{1}{n} \operatorname{tr}\left(\mathbb{E}\left[\boldsymbol{M e e}^{\prime} \mid \boldsymbol{X}\right]\right) \ & =\frac{1}{n} \operatorname{tr}\left(\boldsymbol{M} \mathbb{E}\left[\boldsymbol{e} \boldsymbol{e}^{\prime} \mid \boldsymbol{X}\right]\right) \ & =\frac{1}{n} \operatorname{tr}(\boldsymbol{M D}) \ & =\frac{1}{n} \sum{i=1}^n\left(1-h_{i i}\right) \sigma_i^2 . \end{aligned}
The final equality holds since the trace is the sum of the diagonal elements of $M D$, and since $\boldsymbol{D}$ is diagonal the diagonal elements of $\boldsymbol{M D}$ are the product of the diagonal elements of $\boldsymbol{M}$ and $\boldsymbol{D}$ which are $1-h_{i i}$ and $\sigma_i^2$, respectively.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Mean-Square Forecast Error

One use of an estimated regression is to predict out-of-sample. Consider an out-of-sample realization $\left(Y_{n+1}, X_{n+1}\right)$ where $X_{n+1}$ is observed but not $Y_{n+1}$. Given the coefficient estimator $\widehat{\beta}$ the standard point estimator of $\mathbb{E}\left[Y_{n+1} \mid X_{n+1}\right]=X_{n+1}^{\prime} \beta$ is $\widetilde{Y}{n+1}=X{n+1}^{\prime} \widehat{\beta}$. The forecast error is the difference between the actual value $Y_{n+1}$ and the point forecast $\widetilde{Y}{n+1}$. This is the forecast error $\widetilde{e}{n+1}=Y_{n+1}-\widetilde{Y}{n+1}$. The meansquared forecast error (MSFE) is its expected squared value $\operatorname{MSFE}_n=\mathbb{E}\left[\tilde{e}{n+1}^2\right]$. In the linear regression model $\widetilde{e}{n+1}=e{n+1}-X_{n+1}^{\prime}(\widehat{\beta}-\beta)$ so
$$\operatorname{MSFE}n=\mathbb{E}\left[e{n+1}^2\right]-2 \mathbb{E}\left[e_{n+1} X_{n+1}^{\prime}(\widehat{\beta}-\beta)\right]+\mathbb{E}\left[X_{n+1}^{\prime}(\widehat{\beta}-\beta)(\widehat{\beta}-\beta)^{\prime} X_{n+1}\right] .$$
The first term in (4.28) is $\sigma^2$. The second term in (4.28) is zero since $e_{n+1} X_{n+1}^{\prime}$ is independent of $\widehat{\beta}-\beta$ and both are mean zero. Using the properties of the trace operator the third term in (4.28) is
\begin{aligned} & \operatorname{tr}\left(\mathbb{E}\left[X_{n+1} X_{n+1}^{\prime}\right] \mathbb{E}\left[(\widehat{\beta}-\beta)(\widehat{\beta}-\beta)^{\prime}\right]\right) \ & =\operatorname{tr}\left(\mathbb{E}\left[X_{n+1} X_{n+1}^{\prime}\right] \mathbb{E}\left[\mathbb{E}\left[(\widehat{\beta}-\beta)(\widehat{\beta}-\beta)^{\prime} \mid \boldsymbol{X}\right]\right]\right) \ & =\operatorname{tr}\left[\mathbb{E}\left[X_{n+1} X_{n+1}^{\prime}\right] \mathbb{E}\left[\boldsymbol{V}{\widehat{\beta}}\right]\right) \ & =\mathbb{E}\left[\operatorname{tr}\left(\left(X{n+1} X_{n+1}^{\prime}\right) \boldsymbol{V}{\widehat{\beta}}\right)\right] \ & =\mathbb{E}\left[X{n+1}^{\prime} \boldsymbol{V}{\widehat{\beta}} X{n+1}\right] \end{aligned}
where we use the fact that $X_{n+1}$ is independent of $\widehat{\beta}$, the definition $V_{\widehat{\beta}}=\mathbb{E}\left[(\widehat{\beta}-\beta)(\widehat{\beta}-\beta)^{\prime} \mid \boldsymbol{X}\right]$, and the fact that $X_{n+1}$ is independent of $V_{\widehat{\beta}}$. Thus
$$\operatorname{MSFE}n=\sigma^2+\mathbb{E}\left[X{n+1}^{\prime} \boldsymbol{V}{\widehat{\beta}} X{n+1}\right]$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Estimation of Error Variance

$$\widehat{\sigma}^2=\frac{1}{n} \sum_{i=1}^n \hat{e} i^2$$

$$\widehat{\sigma}^2=\frac{1}{n} e^{\prime} M e=\frac{1}{n} \operatorname{tr}\left(e^{\prime} M e\right)=\frac{1}{n} \operatorname{tr}\left(M e e^{\prime}\right)$$

$$\mathbb{E}\left[\widehat{\sigma}^2 \mid \boldsymbol{X}\right]=\frac{1}{n} \operatorname{tr}\left(\mathbb{E}\left[\boldsymbol{M e} \boldsymbol{e}^{\prime} \mid \boldsymbol{X}\right]\right) \quad=\frac{1}{n} \operatorname{tr}\left(\boldsymbol{M}\left[\boldsymbol{e} \boldsymbol{e}^{\prime} \mid \boldsymbol{X}\right]\right)=\frac{1}{n} \operatorname{tr}(\boldsymbol{M D}) \quad=\frac{1}{n} \sum i=1^n\left(1-h_{i i}\right) \sigma_i^2 .$$
$1-h_{i i}$ 和 $\sigma_i^2$ ，分别。

## 经济代写|计量经济学代写Introduction to Econometrics代考|Mean-Square Forecast Error

$$\operatorname{MSFE} n=\mathbb{E}\left[e n+1^2\right]-2 \mathbb{E}\left[e_{n+1} X_{n+1}^{\prime}(\widehat{\beta}-\beta)\right]+\mathbb{E}\left[X_{n+1}^{\prime}(\widehat{\beta}-\beta)(\widehat{\beta}-\beta)^{\prime} X_{n+1}\right] .$$
(4.28) 中的第一项是 $\sigma^2 .(4.28)$ 中的第二项为零，因为 $e_{n+1} X_{n+1}^{\prime}$ 独立于 $\widehat{\beta}-\beta$ 两者的均值为零。使用迹运算符的性质，(4.28) 中的第三项是
$$\operatorname{tr}\left(\mathbb{E}\left[X_{n+1} X_{n+1}^{\prime}\right] \mathbb{E}\left[(\widehat{\beta}-\beta)(\widehat{\beta}-\beta)^{\prime}\right]\right) \quad=\operatorname{tr}\left(\mathbb{E}\left[X_{n+1} X_{n+1}^{\prime}\right] \mathbb{E}\left[\mathbb{E}\left[(\widehat{\beta}-\beta)(\widehat{\beta}-\beta)^{\prime} \mid \boldsymbol{X}\right]\right]\right)=\operatorname{tr}\left[\mathbb{E}\left[X_{n+1} X_{n+1}^{\prime}\right] \mathbb{E}[\boldsymbol{V} \widehat{\beta}]\right)$$

$$\operatorname{MSFE} n=\sigma^2+\mathbb{E}\left[X n+1^{\prime} \boldsymbol{V} \widehat{\beta} X n+1\right]$$

## 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。