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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|Some Useful Results

This section is intended to serve as a reference for much of the rest of the book. We will essentially make a list (with occasional commentary but without proofs) of useful definitions and theorems. At the end of this we will present two sets of regularity conditions that will each have a set of desirable implications. Later, we will be able to make assumptions by which one or other of these whole sets of regularity conditions is satisfied and thereby be able to draw without further ado a wide variety of useful conclusions.

To begin with, we will concentrate on laws of large numbers and the properties that allow them to be satisfied. In all of these theorems, we consider a sequence of sums $\left{S_n\right}$ where
$$S_n \equiv \frac{1}{n} \sum_{t=1}^n y_t$$
The random variables $y_t$ will be referred to as the (random) summands. First, we present a theorem with very little in the way of moment restrictions on the random summands but very strong restrictions on their homogeneity.
Theorem 4.3. (Khinchin)
If the random variables $y_t$ of the sequence $\left{y_t\right}$ are mutually independent and all distributed according to the same distribution, which possesses a mean of $\mu$, then
$$\operatorname{Pr}\left(\lim _{n \rightarrow \infty} S_n=\mu\right)=1 .$$
Only the existence of the first moment is required, but all the summands must be identically distributed. Notice that the identical mean of the summands means that we need not bother to center the variables $y_t$.

Next, we present a theorem due to Kolmogorov, which still requires independence of the summands, and now existence of their second moments, but very little else in the way of homogeneity.

## 经济代写|计量经济学代写Econometrics代考|Asymptotic Identifiability

When we speak in econometrics of models to be estimated or tested, we refer to sets of DGPs. When we indulge in asymptotic theory, the DGPs in question must be stochastic processes, for the reasons laid out in Chapter 4 . Without further ado then, let us denote a model that is to be estimated, tested, or both, as $\mathbb{M}$ and a typical DGP belonging to $\mathbb{M}$ as $\mu$. Precisely what we mean by this notation should become clear shortly.

The simplest model in econometrics is the linear regression model, but even for it there are several different ways in which it can be specified. One possibility is to write
$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}, \quad \boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^2 \mathbf{I}_n\right)$$

where $\boldsymbol{y}$ and $\boldsymbol{u}$ are $n$-vectors and $\boldsymbol{X}$ is a nonrandom $n \times k$ matrix. Then the (possibly implicit) assumptions are made that $\boldsymbol{X}$ can be defined by some rule (see Section 4.2) for all positive integers $n$ larger than some suitable value and that, for all such $n, \boldsymbol{y}$ follows the $N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^2 \mathbf{I}_n\right)$ distribution. This distribution is unique if the parameters $\boldsymbol{\beta}$ and $\sigma^2$ are specified. We may therefore say that the DGP is completely characterized by the model parameters. In other words, knowledge of the model parameters $\boldsymbol{\beta}$ and $\sigma^2$ uniquely identify an element $\mu$ of $\mathbb{M}$.

On the other hand, the linear regression model can also be written as
$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{IID}\left(\mathbf{0}, \sigma^2 \mathbf{I}_n\right),$$
with no assumption of normality. Many aspects of the theory of linear regressions are just as applicable to (5.02) as to (5.01); for instance, the OLS estimator is unbiased, and its covariance matrix is $\sigma^2\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1}$. But the distribution of the vector $\boldsymbol{u}$, and hence also that of $\boldsymbol{y}$, is now only partially characterized even when $\boldsymbol{\beta}$ and $\sigma^2$ are known. For example, the errors $u_t$ could be skewed to the left or to the right, could have fourth moments larger or smaller than $3 \sigma^4$, or might even possess no moments of order higher than, say, the sixth. DGPs with all sorts of properties, some of them very strange, are special cases of the linear regression model if it is defined by (5.02) rather than (5.01).

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Some Useful Results

$$S_n \equiv \frac{1}{n} \sum_{t=1}^n y_t$$

$$\operatorname{Pr}\left(\lim _{n \rightarrow \infty} S_n=\mu\right)=1 .$$

## 经济代写|计量经济学代写Econometrics代考|Asymptotic Identifiability

$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}, \quad \boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^2 \mathbf{I}_n\right)$$

$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{IID}\left(\mathbf{0}, \sigma^2 \mathbf{I}_n\right)$$

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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。

assignmentutor™您的专属作业导师
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