### 经济代写|计量经济学代写Econometrics代考|Best22

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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 Frisch-Waugh-Lovell Theorem

We now discuss an extremely important and useful property of least squares estimates, which, although widely known, is not as widely appreciated as it should be. We will refer to it as the Frisch-Waugh-Lovell Theorem, or FWL Theorem, after Frisch and Waugh (1933) and Lovell (1963), since those papers seem to have introduced, and then reintroduced, it to econometricians. The theorem is much more general, and much more generally useful, than a casual reading of those papers might suggest, however. Among other things, it almost totally eliminates the need to invert partitioned matrices when one is deriving many standard results about ordinary (and nonlinear) least squares.

The FWL Theorem applies to any regression where there are two or more regressors, and these can logically be broken up into two groups. The regression can thus be written as
$$\boldsymbol{y}=\boldsymbol{X}{1} \boldsymbol{\beta}{1}+\boldsymbol{X}{2} \boldsymbol{\beta}{2}+\text { residuals, }$$
where $\boldsymbol{X}{1}$ is $n \times k{1}$ and $\boldsymbol{X}{2}$ is $n \times k{2}$, with $\boldsymbol{X} \equiv\left[\begin{array}{ll}\boldsymbol{X}{1} & \boldsymbol{X}{2}\end{array}\right]$ and $k=k_{1}+k_{2}$. For example, $\boldsymbol{X}{1}$ might be seasonal dummy variables or trend variables and $\boldsymbol{X}{2}$ genuine economic variables. This was in fact the type of situation dealt with by Frisch and Waugh (1933) and Lovell (1963). Another possibility is that $\boldsymbol{X}{1}$ might be regressors, the joint significance of which we desire to test, and $\boldsymbol{X}{2}$ might be other regressors that are not being tested. Or $\boldsymbol{X}{1}$ might be regressors that are known to be orthogonal to the regressand, and $\boldsymbol{X}{2}$ might be regressors that are not orthogonal to it, a situation which arises very frequently when we wish to test nonlinear regression models; see Chapter 6 .

## 经济代写|计量经济学代写Econometrics代考|Computing OLS Estimates

In this section, we will briefly discuss how OLS estimates are actually calculated using digital computers. This is a subject that most students of econometrics, and not a few econometricians, are largely unfamiliar with. The vast majority of the time, well-written regression programs will yield reliable results, and applied econometricians therefore do not need to worry about how those results are actually obtained. But not all programs for OLS regression are written well, and even the best programs can run into difficulties if the data are sufficiently ill-conditioned. We therefore believe that every user of software for least squares regression should have some idea of what the software is actually doing. Moreover, the particular method for OLS regression on which we will focus is interesting from a purely theoretical perspective.
Before we discuss algorithms for least squares regression, we must say something about how digital computers represent real numbers and how this affects the accuracy of calculations carried out on such computers. With rare exceptions, the quantities of interest in regression problems $-\boldsymbol{y}, \boldsymbol{X}, \hat{\boldsymbol{\beta}}$, and so on-are real numbers rather than integers or rational numbers. In general, it requires an infinite number of digits to represent a real number exactly, and this is clearly infeasible. Trying to represent each number by as many digits as are necessary to approximate it with “sufficient” accuracy would mean using a different number of digits to represent different numbers; this would be difficult to do and would greatly slow down calculations. Computers therefore normally deal with real numbers by approximating them using a fixed number of digits (or, more accurately, bits, which correspond to digits in base 2). But in order to handle numbers that may be very large or very small, the computer has to represent real numbers as floating-point numbers. ${ }^{6}$

## 经济代写|计量经济学代写Econometrics代考|The Frisch-Waugh-Lovell Theorem

FWL 定理适用于有两个或多个回归量的任何回归，并且这些回归量在逻辑上可以分为两组。因此回归可以写成
$$\boldsymbol{y}=\boldsymbol{X} 1 \boldsymbol{\beta} 1+\boldsymbol{X} 2 \boldsymbol{\beta} 2+\text { residuals, }$$

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## 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。