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# 经济代写|计量经济学代考ECONOMETRICS代考|ECON3120 Homoskedasticity and Heteroskedasticity

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## 经济代写|计量经济学代考ECONOMETRICS代考|Homoskedasticity and Heteroskedasticity

An important special case obtains when the conditional variance $\sigma^{2}(x)$ is a constant and independent of $x$. This is called homoskedasticity.
Definition 2.3 The error is homoskedastic if $\sigma^{2}(x)=\sigma^{2}$ does not depend on $x$.

In the general case where $\sigma^{2}(x)$ depends on $x$ we say that the error $e$ is heteroskedastic.
Definition 2.4 The error is heteroskedastic if $\sigma^{2}(x)$ depends on $x$.

It is helpful to understand that the concepts homoskedasticity and heteroskedasticity concern the conditional variance, not the unconditional variance. By definition, the unconditional variance $\sigma^{2}$ is a constant and independent of the regressors $X$. So when we talk about the variance as a function of the regressors we are talking about the conditional variance $\sigma^{2}(x)$.

Some older or introductory textbooks describe heteroskedasticity as the case where “the variance of $e$ varies across observations”. This is a poor and confusing definition. It is more constructive to understand that heteroskedasticity means that the conditional variance $\sigma^{2}(x)$ depends on observables.

Older textbooks also tend to describe homoskedasticity as a component of a correct regression specification and describe heteroskedasticity as an exception or deviance. This description has influenced many generations of economists but it is unfortunately backwards. The correct view is that heteroskedasticity is generic and “standard”, while homoskedasticity is unusual and exceptional. The default in empirical work should be to assume that the errors are heteroskedastic, not the converse.

In apparent contradiction to the above statement we will still frequently impose the homoskedasticity assumption when making theoretical investigations into the properties of estimation and inference methods. The reason is that in many cases homoskedasticity greatly simplifies the theoretical calculations and it is therefore quite advantageous for teaching and learning. It should always be remembered, however, that homoskedasticity is never imposed because it is believed to be a correct feature of an empirical model but rather because of its simplicity.

## 经济代写|计量经济学代考ECONOMETRICS代考|Regression Derivative

One way to interpret the CEF $m(x)=\mathbb{E}[Y \mid X=x]$ is in terms of how marginal changes in the regressors $x$ imply changes in the conditional mean of the response variable $Y$. It is typical to consider marginal changes in a single regressor, say $X_{1}$, holding the remainder fixed. When a regressor $X_{1}$ is continuously distributed, we define the marginal effect of a change in $X_{1}$, holding the variables $X_{2}, \ldots, X_{k}$ fixed, as the partial derivative of the CEF
$$\frac{\partial}{\partial x_{1}} m\left(x_{1}, \ldots, x_{k}\right)$$
When $X_{1}$ is discrete we define the marginal effect as a discrete difference. For example, if $x_{1}$ is binary, then the marginal effect of $X_{1}$ on the CEF is
$$m\left(1, x_{2}, \ldots, x_{k}\right)-m\left(0, x_{2}, \ldots, x_{k}\right)$$

We can unify the continuous and discrete cases with the notation
$$\nabla_{1} m(x)=\left{\begin{array}{cc} \frac{\partial}{\partial x_{1}} m\left(x_{1}, \ldots, x_{k}\right), & \text { if } X_{1} \text { is continuous } \ m\left(1, x_{2}, \ldots, x_{k}\right)-m\left(0, x_{2}, \ldots, x_{k}\right), & \text { if } X_{1} \text { is binary. } \end{array}\right.$$
Collecting the $k$ effects into one $k \times 1$ vector, we define the regression derivative with respect to $X$ :
$$\nabla m(x)=\left[\begin{array}{c} \nabla_{1} m(x) \ \nabla_{2} m(x) \ \vdots \ \nabla_{k} m(x) \end{array}\right] .$$

## 经济代写|计量经济学代考ECONOMETRICS代考|Regression Derivative

$$\frac{\partial}{\partial x_{1}} m\left(x_{1}, \ldots, x_{k}\right)$$

$$m\left(1, x_{2}, \ldots, x_{k}\right)-m\left(0, x_{2}, \ldots, x_{k}\right)$$

$\frac{\partial}{\partial x_{1}} m\left(x_{1}, \ldots, x_{k}\right), \quad$ if $X_{1}$ is continuous $m\left(1, x_{2}, \ldots, x_{k}\right)-m\left(0, x_{2}, \ldots, x_{k}\right)$, if $X_{1}$ is binary.
\正确的。
Collectingthe $\$ \$$ffectsintoone \ k \times 1 \ v e c t o r, wedefinetheregressionderivativewithrespectto \ X \$$ :
|nabla $\mathrm{m}(\mathrm{x})=\backslash$ 左 [
$$\nabla_{1} m(x) \nabla_{2} m(x) \vdots \nabla_{k} m(x)$$
\正确的]。

## 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。