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# 经济代写|计量经济学代考ECONOMETRICS代考|ECON335 Best Predictor

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## 经济代写|计量经济学代考ECONOMETRICS代考|Best Predictor

Suppose that given a random vector $X$ we want to predict or forecast $Y$. We can write any predictor as a function $g(X)$ of $X$. The (ex-post) prediction error is the realized difference $Y-g(X)$. A non-stochastic measure of the magnitude of the prediction error is the expectation of its square
$$\mathbb{E}\left[(Y-g(X))^{2}\right] .$$
We can define the best predictor as the function $g(X)$ which minimizes (2.10). What function is the best predictor? It turns out that the answer is the CEF $m(X)$. This holds regardless of the joint distribution of $(Y, X)$

To see this, note that the mean squared error of a predictor $g(X)$ is
\begin{aligned} \mathbb{E}\left[(Y-g(X))^{2}\right] &=\mathbb{E}\left[(e+m(X)-g(X))^{2}\right] \ &=\mathbb{E}\left[e^{2}\right]+2 \mathbb{E}[e(m(X)-g(X))]+\mathbb{E}\left[(m(X)-g(X))^{2}\right] \ &=\mathbb{E}\left[e^{2}\right]+\mathbb{E}\left[(m(X)-g(X))^{2}\right] \ & \geq \mathbb{E}\left[e^{2}\right] \ &=\mathbb{E}\left[(Y-m(X))^{2}\right] \end{aligned}
where the first equality makes the substitution $Y=m(X)+e$ and the third equality uses Theorem 2.4.4. The right-hand-side after the third equality is minimized by setting $g(X)=m(X)$, yielding the inequality in the fourth line. The minimum is finite under the assumption $\mathbb{E}\left[Y^{2}\right]<\infty$ as shown by Theorem 2.5.
We state this formally in the following result.

## 经济代写|计量经济学代考ECONOMETRICS代考|Conditional Variance

While the conditional mean is a good measure of the location of a conditional distribution it does not provide information about the spread of the distribution. A common measure of the dispersion is the conditional variance. We first give the general definition of the conditional variance of a random variable $W$
Definition 2.1 If $\mathbb{E}\left[W^{2}\right]<\infty$, the conditional variance of $W$ given $X=x$ is
$$\sigma^{2}(x)=\operatorname{var}[W \mid X=x]=\mathbb{E}\left[(W-\mathbb{E}[W \mid X=x])^{2} \mid X=x\right] .$$
The conditional variance treated as a random variable is $\operatorname{var}[W \mid X]=\sigma^{2}(X)$.

The conditional variance is distinct from the unconditional variance var $[W]$. The difference is that the conditional variance is a function of the conditioning variables. Notice that the conditional variance is the conditional second moment, centered around the conditional first moment.
Given this definition we define the conditional variance of the regression error.
Definition 2.2 If $\mathbb{E}\left[e^{2}\right]<\infty$, the conditional variance of the regression error $e$ given $X=x$ is
$$\sigma^{2}(x)=\operatorname{var}[e \mid X=x]=\mathbb{E}\left[e^{2} \mid X=x\right] .$$
The conditional variance of $e$ treated as a random variable is $\operatorname{var}[e \mid X]=\sigma^{2}(X)$.
Again, the conditional variance $\sigma^{2}(x)$ is distinct from the unconditional variance $\sigma^{2}$. The conditional variance is a function of the regressors, the unconditional variance is not. Generally, $\sigma^{2}(x)$ is a non-trivial function of $x$ and can take any form subject to the restriction that it is non-negative. One way to think about $\sigma^{2}(x)$ is that it is the conditional mean of $e^{2}$ given $X$. Notice as well that $\sigma^{2}(x)=\operatorname{var}[Y \mid X=x]$ so it is equivalently the conditional variance of the dependent variable.

## 经济代写|计量经济学代考ECONOMETRICS代考|Best Predictor

$$\mathbb{E}\left[(Y-g(X))^{2}\right] .$$

$$\mathbb{E}\left[(Y-g(X))^{2}\right]=\mathbb{E}\left[(e+m(X)-g(X))^{2}\right] \quad=\mathbb{E}\left[e^{2}\right]+2 \mathbb{E}[e(m(X)-g(X))]+\mathbb{E}\left[(m(X)-g(X))^{2}\right]=\mathbb{E}\left[e^{2}\right]+\mathbb{E}\left[(m(X)-g(X))^{2}\right] \quad \geq$$

## 经济代写|计量经济学代考ECONOMETRICS代考|Conditional Variance

$$\sigma^{2}(x)=\operatorname{var}[W \mid X=x]=\mathbb{E}\left[(W-\mathbb{E}[W \mid X=x])^{2} \mid X=x\right] .$$

$$\sigma^{2}(x)=\operatorname{var}[e \mid X=x]=\mathbb{E}\left[e^{2} \mid X=x\right] .$$

$\sigma^{2}(x)=\operatorname{var}[Y \mid X=x]$ 所以它等效地是因变量的条件方差。

## 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。