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# 经济代写|计量经济学代写Introduction to Econometrics代考|EC2C1 Numerical Computation

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Numerical Computation

Modern econometric estimation involves large samples and many covariates. Consequently calculation of even simple statistics such as the least squares estimator requires a large number (millions) of arithmetic operations. In practice most economists don’t need to think much about this as it is done swiftly and effortlessly on our personal computers. Nevertheless it is useful to understand the underlying calculation methods as occasionally choices can make substantive differences.

While today nearly all statistical computations are made using statistical software running on personal computers, this was not always the case. In the nineteenth and early twentieth centures, “computer” was a job label for workers who made computations by hand. Computers were employed by astronomers and statistical laboratories to execute numerical calculations. This fascinating job (and the fact that most computers employed in laboratories were women) has entered popular culture. For example the lives of several computers who worked for the early U.S. space program is described in the book and popular movie Hidden Figures, and the life of computer/astronomer Henrietta Swan Leavitt is dramatized in the moving play Silent Sky.

Until programmable electronic computers became available in the 1960s, economics graduate students were routinely employed as computers. Sample sizes were considerably smaller than those seen today, but still the effort required to calculate by hand (for example) a regression with $n=100$ observations and $k=5$ variables is considerable! If you are a current graduate student, you should feel fortunate that the profession has moved on from the era of human computers! (Now research assistants do more elevated tasks such as writing Stata and Matlab code.)

To obtain the least squares estimator $\widehat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1}\left(\boldsymbol{X}^{\prime} \boldsymbol{y}\right)$ we need to either invert $\boldsymbol{X}^{\prime} \boldsymbol{X}$ or solve a system of equations. To be specific, let $\boldsymbol{A}=\boldsymbol{X}^{\prime} \boldsymbol{X}$ and $\boldsymbol{c}=\boldsymbol{X}^{\prime} \boldsymbol{y}$ so that the least squares estimator can be written as either the solution to
$$A \widehat{\boldsymbol{\beta}}=\boldsymbol{c}$$
or as
$$\widehat{\boldsymbol{\beta}}=A^{-1} \boldsymbol{c} .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Collinearity Errors

For the least squares estimator to be uniquely defined the regressors cannot be linearly dependent. However, it is quite easy to attempt to calculate a regression with linearly dependent regressors. This can occur for many reasons, including the following.

1. Including the same regressor twice.
2. Including regressors which are a linear combination of one another, such as education, experience and age in the CPS data set example (recall, experience is defined as age-education-6).
3. Including a dummy variable and its square.
4. Estimating a regression on a sub-sample for which a dummy variable is either all zeros or all ones.
5. Including a dummy variable interaction which yields all zeros.
6. Including more regressors than observations.
In any of the above cases the regressors are linearly dependent so $\boldsymbol{X}^{\prime} \boldsymbol{X}$ is singular and the least squares estimator is not defined. If you attempt to estimate the regression, you are likely to encounter an error message. (A possible exception is Matlab using “A $\mathrm{lb}$ “, as discussed below.) The message may be that “system is exactly singular”, “system is computationally singular”, a variable is “omitted because of collinearity”, or a coefficient is listed as “NA”. In some cases (such as estimation in R using explicit matrix computation or Matlab using the regress command) the program will stop execution. In other cases the program will continue to run. In Stata (and in the lm package in R), a regression will be reported but one or more variables will be omitted to achieve non-singularity.

## 经济代写|计量经济学代写Introduction to Econometrics代 考|Numerical Computation

$$A \widehat{\boldsymbol{\beta}}=\boldsymbol{c}$$

$$\widehat{\boldsymbol{\beta}}=A^{-1} \boldsymbol{c}$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Collinearity Errors

1. 包括两次相同的回归变量。
2. 包括相互线性组合的回归变量，例如 CPS 数据集示例中的教育、经验和年龄（回想一下，经验定义为年 龄-教育-6)。
3. 包括一个虚拟变量及其平方。
4. 估计虚拟变量全为零或全为 1 的子样本的回归。
5. 包括产生全䨐的虚拟变量交互。
6. 包括比观察更多的回归变量。
在上述任何一种情况下，回归变量都是线性相关的，因此 $\boldsymbol{X}^{\prime} \boldsymbol{X}$ 是奇异的，并且末定义最小二乘估计 量。如果您尝试估计回归，您可能会遇到一条错误消息。（一个可能的例外是 Matlab 使用“Alb”，如下 所述。）消息可能是“系统完全是奇异的”“系统在计算上是奇异的”、变量“由于共线性而被省略”，或者 系数被列为“NA”。在某些情况下 (例如在 R 中使用显式矩阵计算或在 Matlab 中使用回归命令进行估 计）程序将停止执行。在其他情况下，程序将继续运行。在 Stata（以及 $R$ 中的 Im 包) 中，将报告回 归，但将省略一个或多个变量以实现非奇异性。

## 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。