## 经济代写|计量经济学代写Econometrics代考|The double-log functional form

2023年4月13日
couryes™为您提供可以保分的包课服务

couryes-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|计量经济学代写Econometrics代考|The double-log functional form

The double-log model is very popular in cases where we expect variables to have constant ratios. A common specification is the Cobb-Douglas type of production function of the form:
$$Y_t=A K_t^a L_t^\beta$$
where standard notation is used. Taking logarithms of both sides and adding an error term we get:
$$\ln Y_t=\gamma+a \ln K_t+\beta \ln L_t+u_t$$ and it can be shown here that $a$ and $\beta$ are the elasticities of $K_t$ and $L_t$, respectively. To demonstrate that, consider changes in $K$ while keeping $L$ constant. We have:
$$a=\frac{d(\ln Y)}{d(\ln K)}=\frac{(1 / Y) d Y}{(1 / K) d K}=\frac{K}{Y} \frac{d Y}{d K}$$
Another way to show this is by taking the derivative of $Y$ with respect to $K$; from the initial function in Equation (8.26):
$$\frac{d Y}{d K}=a A K_t^{a-1} L_t^\beta=a \frac{A K_t^a L_t^\beta}{K}=a \frac{Y}{K}$$
and therefore:
$$a=\frac{d Y}{d K} \frac{K}{Y}$$
It can be shown that the same holds for $\beta$. We leave this as an exercise for the reader. Table 8.2 provides interpretations of the marginal effects in the various logarithmic models.

## 经济代写|计量经济学代写Econometrics代考|The Box–Cox transformation

As was demonstrated above, the choice of functional form plays a very important role in the interpretation of the estimated coefficients, and therefore a formal test is needed to direct the choice of functional form where there is uncertainty about the population relationship.

For example, consider a model with two explanatory variables ( $X_2$ and $\left.X_3\right)$. We must be able to determine whether to use the linear, log-linear, linear-log or double$\log$ specification. The choice between the linear and linear-log model, or between the log-linear and double-log specification, is simple because we have the same dependent variable in each of the two models. So, we can estimate both models and choose the

functional form that yields the higher $R^2$. However, in cases where the dependent variable is not the same, as for example in the linear form:
$$Y=\beta_1+\beta_2 X$$
and the double-log form:
$$\ln Y=\beta_1+\beta_2 \ln X$$
it is not possible to compare the two by using $R^2$.
In such examples, the $Y$-variable must be scaled in such a way that the two models can be compared. The procedure is based on the work of Box and Cox (1964) and is usually known as the Box-Cox transformation. The procedure follows these steps:
Step 1 Obtain the geometric mean of the sample $Y$-values. This is:
$$\tilde{Y}=\left(Y_1 Y_2 Y_3 \ldots Y_n\right)^{1 / n}=\exp \left[(1 / n) \sum \ln Y_i\right]$$
Step 2 Transform the sample $Y$-values by dividing each of them by $\tilde{Y}$ obtained above to get:
$$Y_i^=Y_i / \tilde{Y}$$ Step 3 Estimate Equations (8.31) and (8.32), substituting $Y_i^$ as the dependent variable in both. The RSSs of the two equations are now directly comparable, and the equation with the lower RSS is preferred.

Step 4 If we need to know whether one of the equations is significantly better than the other, we have to calculate the following statistic:
$$\left(\frac{1}{2} n\right) \ln \left(\frac{R S S_2}{R S S_1}\right)$$
where $R_S$ is the higher RSS, and $R_S S_1$ is the lower. The above statistic follows a $x^2$ distribution with 1 degree of freedom. If $x^2$-statistical exceeds the $x^2$ critical value we can say with confidence that the model with the lower RSS is superior at the level of significance for which the $\chi^2$-critical is obtained.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|The double-log functional form

$$Y_t=A K_t^a L_t^\beta$$

$$\ln Y_t=\gamma+a \ln K_t+\beta \ln L_t+u_t$$

$$a=\frac{d(\ln Y)}{d(\ln K)}=\frac{(1 / Y) d Y}{(1 / K) d K}=\frac{K}{Y} \frac{d Y}{d K}$$

$$\frac{d Y}{d K}=a A K_t^{a-1} L_t^\beta=a \frac{A K_t^a L_t^\beta}{K}=a \frac{Y}{K}$$

$$a=\frac{d Y}{d K} \frac{K}{Y}$$

## 经济代写|计量经济学代写Econometrics代考|The Box–Cox transformation

$$Y=\beta_1+\beta_2 X$$

$$\ln Y=\beta_1+\beta_2 \ln X$$

$$\tilde{Y}=\left(Y_1 Y_2 Y_3 \ldots Y_n\right)^{1 / n}=\exp \left[(1 / n) \sum \ln Y_i\right]$$

$$Y_i=Y_i / \tilde{Y}$$

$$\left(\frac{1}{2} n\right) \ln \left(\frac{R S S_2}{R S S_1}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。