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经济代写|计量经济学代考ECONOMETRICS代考|ECO380 Linear CEF

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经济代写|计量经济学代考ECONOMETRICS代考|Linear CEF

An important special case is when the CEF $m(x)=\mathbb{E}[Y \mid X=x]$ is linear in $x$. In this case we can write the mean equation as
$$m(x)=x_{1} \beta_{1}+x_{2} \beta_{2}+\cdots+x_{k} \beta_{k}+\beta_{k+1} .$$
Notationally it is convenient to write this as a simple function of the vector $x$. An easy way to do so is to augment the regressor vector $X$ by listing the number ” 1 ” as an element. We call this the “constant” and the corresponding coefficient is called the “intercept”. Equivalently, specify that the final element ${ }^{9}$ of the vector $x$ is $x_{k}=1$. Thus (2.4) has been redefined as the $k \times 1$ vector
$$X=\left(\begin{array}{c} X_{1} \ X_{2} \ \vdots \ X_{k-1} \ 1 \end{array}\right) .$$

With this redefinition, the CEF is
$$m(x)=x_{1} \beta_{1}+x_{2} \beta_{2}+\cdots+\beta_{k}=x^{\prime} \beta$$
where
$$\beta=\left(\begin{array}{c} \beta_{1} \ \vdots \ \beta_{k} \end{array}\right)$$
is a $k \times 1$ coefficient vector. This is the linear CEF model. It is also often called the linear regression model, or the regression of $Y$ on $X$.

In the linear CEF model the regression derivative is simply the coefficient vector. That is $\nabla m(x)=\beta$. This is one of the appealing features of the linear CEF model. The coefficients have simple and natural interpretations as the marginal effects of changing one variable, holding the others constant.
If in addition the error is homoskedastic we call this the homoskedastic linear CEF model.

经济代写|计量经济学代考ECONOMETRICS代考|Linear CEF with Nonlinear Effects

The linear CEF model of the previous section is less restrictive than it might appear, as we can include as regressors nonlinear transformations of the original variables. In this sense, the linear CEF framework is flexible and can capture many nonlinear effects.

For example, suppose we have two scalar variables $X_{1}$ and $X_{2}$. The CEF could take the quadratic form
$$m\left(x_{1}, x_{2}\right)=x_{1} \beta_{1}+x_{2} \beta_{2}+x_{1}^{2} \beta_{3}+x_{2}^{2} \beta_{4}+x_{1} x_{2} \beta_{5}+\beta_{6} .$$
This equation is quadratic in the regressors $\left(x_{1}, x_{2}\right)$ yet linear in the coefficients $\beta=\left(\beta_{1}, \ldots, \beta_{6}\right)^{\prime}$. We still call (2.14) a linear CEF because it is a linear function of the coefficients. At the same time, it has nonlinear effects because it is nonlinear in the underlying variables $x_{1}$ and $x_{2}$. The key is to understand that (2.14) is quadratic in the variables $\left(x_{1}, x_{2}\right)$ yet linear in the coefficients $\beta$.

To simplify the expression we define the transformations $x_{3}=x_{1}^{2}, x_{4}=x_{2}^{2}, x_{5}=x_{1} x_{2}$, and $x_{6}=1$, and redefine the regressor vector as $x=\left(x_{1}, \ldots, x_{6}\right)^{\prime}$. With this redefinition, $m\left(x_{1}, x_{2}\right)=x^{\prime} \beta$ which is linear in $\beta$. For most econometric purposes (estimation and inference on $\beta$ ) the linearity in $\beta$ is all that is important.
An exception is in the analysis of regression derivatives. In nonlinear equations such as (2.14) the regression derivative should be defined with respect to the original variables not with respect to the transformed variables. Thus
\begin{aligned} &\frac{\partial}{\partial x_{1}} m\left(x_{1}, x_{2}\right)=\beta_{1}+2 x_{1} \beta_{3}+x_{2} \beta_{5} \ &\frac{\partial}{\partial x_{2}} m\left(x_{1}, x_{2}\right)=\beta_{2}+2 x_{2} \beta_{4}+x_{1} \beta_{5} . \end{aligned}
We see that in the model (2.14), the regression derivatives are not a simple coefficient, but are functions of several coefficients plus the levels of $\left(x_{1}, x_{2}\right)$. Consequently it is difficult to interpret the coefficients individually. It is more useful to interpret them as a group.

We typically call $\beta_{5}$ the interaction effect. Notice that it appears in both regression derivative equations and has a symmetric interpretation in each. If $\beta_{5}>0$ then the regression derivative with respect to $x_{1}$ is increasing in the level of $x_{2}$ (and the regression derivative with respect to $x_{2}$ is increasing in the level of $x_{1}$ ), while if $\beta_{5}<0$ the reverse is true.

经济代写|计量经济学代考ECONOMETRICS代考|Linear CEF

$$m(x)=x_{1} \beta_{1}+x_{2} \beta_{2}+\cdots+x_{k} \beta_{k}+\beta_{k+1} .$$

$$X=\left(X_{1} X_{2} \vdots X_{k-1} 1\right)$$

$$m(x)=x_{1} \beta_{1}+x_{2} \beta_{2}+\cdots+\beta_{k}=x^{\prime} \beta$$

$$\beta=\left(\beta_{1} \vdots \beta_{k}\right)$$

经济代写|计量经济学代考ECONOMETRICS代考|Linear CEF with Nonlinear Effects

$$m\left(x_{1}, x_{2}\right)=x_{1} \beta_{1}+x_{2} \beta_{2}+x_{1}^{2} \beta_{3}+x_{2}^{2} \beta_{4}+x_{1} x_{2} \beta_{5}+\beta_{6} .$$

$$\frac{\partial}{\partial x_{1}} m\left(x_{1}, x_{2}\right)=\beta_{1}+2 x_{1} \beta_{3}+x_{2} \beta_{5} \quad \frac{\partial}{\partial x_{2}} m\left(x_{1}, x_{2}\right)=\beta_{2}+2 x_{2} \beta_{4}+x_{1} \beta_{5} .$$

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。