### 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|Data-Generating Processes and Asymptotic Theory

In this section, we apply the mathematical theory developed in the preceding sections to econometric estimation and testing from an asymptotic point of view. In order to say anything about how estimators and test statistics are distributed, we have to specify how the data of which they are functions are generated. That is why we introduced the idea of a data-generating process, or DGP, in Section 2.4. But what precisely do we mean by a data-generating process in an asymptotic context? When we spoke of DGPs before, it was enough to restrict our attention to a particular given sample size and characterize a DGP by the law of probability that governs the random variables in a sample of that size. But, since when we say “asymptotic” we refer to a limiting process in which the sample size goes to infinity, it is clear that such a restricted characterization will no longer suffice. It is in order to resolve this difficulty that we make use of the notion of a stochastic process. Since this notion allows us to consider an infinite sequence of random variables, it is well adapted to our needs.

In full generality, a stochastic process is a collection of random variables indexed by some suitable index set. This index set may be finite, in which case we have no more than a vector of random variables, or it may be infinite, with either a discrete or a continuous infinity of elements. We are interested here almost exclusively in the case of a discrete infinity of random variables, in fact with sequences of random variables such as those we have already discussed at length in the preceding sections. To fix ideas, let the index set be $\mathbb{N}$, the set ${1,2, \ldots}$ of the natural numbers. Then a stochastic process is just a mapping from $\mathbb{N}$ to a set of random variables. It is in fact precisely what we previously defined as a sequence of random variables, and so we see that these sequences are special cases of stochastic processes. They are the only kind of stochastic process that we will need in this book; the more general notion of stochastic process is introduced here only so that we may use the numerous available results on stochastic processes for our own purposes.

## 经济代写|计量经济学代写Econometrics代考|Consistency and Laws of Large Numbers

We begin this section by introducing the notion of consistency, one of the most basic ideas of asymptotic theory. When one is interested in estimating parameters from data, it is desirable that the parameter estimates should have certain properties. In Chapters 2 and 3 , we saw that, under certain regularity conditions, the OLS estimator is unbiased and follows a normal distribution with a covariance matrix that is known up to a factor of the error variance, which factor can itself be estimated in an unbiased manner. We were not able in those chapters to prove any corresponding results for the NLS estimator, and it was remarked that asymptotic theory would be necessary in order to do so. Consistency is the first of the desirable asymptotic properties that an estimator may possess. In Chapter 5 we will provide conditions under which the NLS estimator is consistent. Here we will content ourselves with introducing the notion itself and illustrating the close link that exists between laws of large numbers and proofs of consistency.

An estimator $\hat{\boldsymbol{\beta}}$ of a vector of parameters $\boldsymbol{\beta}$ is said to be consistent if it converges to its true value as the sample size tends to infinity. That statement is not false or even seriously misleading, but it implicitly makes a number of assumptions and uses undefined terms. Let us try to rectify this and, in so doing, gain a better understanding of what consistency means.

First, how can an estimator converge? It can do so if we convert it to a sequence. To this end, we write $\hat{\boldsymbol{\beta}}^n$ for the estimator that results from a sample of size $n$ and then define the estimator $\hat{\boldsymbol{\beta}}$ itself as the sequence $\left{\hat{\boldsymbol{\beta}}^n\right}_{n=m}^{\infty}$. The lower limit $m$ of the sequence will usually be assumed to be the smallest sample size that allows $\hat{\boldsymbol{\beta}}^n$ to be computed. For example, if we denote the regressand and regressor matrix for a linear regression done on a sample of size $n$ by $\boldsymbol{y}^n$ and $\boldsymbol{X}^n$, respectively, and if $\boldsymbol{X}^n$ is an $n \times k$ matrix, then $m$ cannot be any smaller than $k$, the number of regressors. For $n>k$ we have as usual that $\hat{\boldsymbol{\beta}}^n=\left(\left(\boldsymbol{X}^n\right)^{\top} \boldsymbol{X}^n\right)^{-1}\left(\boldsymbol{X}^n\right)^{\top} \boldsymbol{y}^n$, and this formula embodies the rule which generates the sequence $\hat{\boldsymbol{\beta}}$.

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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。