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经济代写|计量经济学代写Introduction to Econometrics代考|EC2C1 Variance of Least Squares Estimator

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经济代写|计量经济学代写Introduction to Econometrics代考|Variance of Least Squares Estimator

In this section we calculate the conditional variance of the OLS estimator.
For any $r \times 1$ random vector $\boldsymbol{Z}$ define the $r \times r$ covariance matrix
\begin{aligned} \operatorname{var}[\boldsymbol{Z}] & =\mathbb{E}\left[(\boldsymbol{Z}-\mathbb{E}[\boldsymbol{Z}])(\boldsymbol{Z}-\mathbb{E}[\boldsymbol{Z}])^{\prime}\right] \ & =\mathbb{E}\left[\boldsymbol{Z} \boldsymbol{Z}^{\prime}\right]-(\mathbb{E}[\boldsymbol{Z}])(\mathbb{E}[\boldsymbol{Z}])^{\prime} \end{aligned}
and for any pair $(\boldsymbol{Z}, \boldsymbol{X})$ define the conditional covariance matrix
$$\operatorname{var}[\boldsymbol{Z} \mid \boldsymbol{X}]=\mathbb{E}\left[(\boldsymbol{Z}-\mathbb{E}[\boldsymbol{Z} \mid \boldsymbol{X}])(\boldsymbol{Z}-\mathbb{E}[\boldsymbol{Z} \mid \boldsymbol{X}])^{\prime} \mid \boldsymbol{X}\right]$$
We define
$$\boldsymbol{V}_{\widehat{\boldsymbol{\beta}}} \stackrel{\text { def }}{=} \operatorname{var}[\widehat{\boldsymbol{\beta}} \mid \boldsymbol{X}]$$
as the conditional covariance matrix of the regression coefficient estimates. We now derive its form.
The conditional covariance matrix of the $n \times 1$ regression error $e$ is the $n \times n$ matrix
$$\operatorname{var}[\boldsymbol{e} \mid \boldsymbol{X}]=\mathbb{E}\left[\boldsymbol{e} \boldsymbol{e}^{\prime} \mid \boldsymbol{X}\right] \stackrel{\text { def }}{=} \boldsymbol{D}$$

经济代写|计量经济学代写Introduction to Econometrics代考|Unconditional Moments

The previous sections derived the form of the conditional mean and variance of least-squares estimator where we conditioned on the regressor matrix $\boldsymbol{X}$. What about the unconditional mean and variance? Another goal is to state conditions under which the unconditional moments of the estimator are finite. For example, if it determined that $\mathbb{E}|\widehat{\boldsymbol{\beta}}|<\infty$ then applying the law of iterated expectations (Theorem 2.1), we find that the unconditional mean of $\widehat{\boldsymbol{\beta}}$ is $$\mathbb{E}[\widehat{\boldsymbol{\beta}}]=\mathbb{E}[\mathbb{E}[\widehat{\boldsymbol{\beta}} \mid \boldsymbol{X}]]=\boldsymbol{\beta}$$ which means that $\widehat{\boldsymbol{\beta}}$ is unconditionally unbiased. A challenge is that $\widehat{\boldsymbol{\beta}}$ may not have finite moments. Take the case of a single dummy variable regressor $d_i$ with no intercept. Assume $\mathbb{P}\left[d_i=1\right]=p<1$. Then $$\widehat{\beta}=\frac{\sum_{i=1}^n d_i y_i}{\sum_{i=1}^n d_i}$$ is well defined if $\sum_{i=1}^n d_i>0$. However, $\mathbb{P}\left[\sum_{i=1}^n d_i=0\right]=(1-p)^n>0$. This means that with positive (but small) probability $\widehat{\beta}$ does not exist. Consequently $\widehat{\beta}$ has no finite moments! We ignore this complication in practice but it does pose a conundrum for theory. This existence problem arises whenever there are discrete regressors.

经济代写|计量经济学代写Introduction to Econometrics代考|Variance of Least Squares Estimator

$$\operatorname{var}[\boldsymbol{Z}]=\mathbb{E}\left[(\boldsymbol{Z}-\mathbb{E}[\boldsymbol{Z}])(\boldsymbol{Z}-\mathbb{E}[\boldsymbol{Z}])^{\prime}\right] \quad=\mathbb{E}\left[\boldsymbol{Z} \boldsymbol{Z}^{\prime}\right]-(\mathbb{E}[\boldsymbol{Z}])(\mathbb{E}[\boldsymbol{Z}])^{\prime}$$

$$\operatorname{var}[\boldsymbol{Z} \mid \boldsymbol{X}]=\mathbb{E}\left[(\boldsymbol{Z}-\mathbb{E}[\boldsymbol{Z} \mid \boldsymbol{X}])(\boldsymbol{Z}-\mathbb{E}[\boldsymbol{Z} \mid \boldsymbol{X}])^{\prime} \mid \boldsymbol{X}\right]$$

$$\boldsymbol{V}_{\widehat{\boldsymbol{\beta}}} \stackrel{\text { def }}{=} \operatorname{var}[\widehat{\boldsymbol{\beta}} \mid \boldsymbol{X}]$$

$$\operatorname{var}[\boldsymbol{e} \mid \boldsymbol{X}]=\mathbb{E}\left[\boldsymbol{e} \boldsymbol{e}^{\prime} \mid \boldsymbol{X}\right] \stackrel{\text { def }}{=} D$$

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。