## 经济代写|计量经济学代写Econometrics代考|Best27

statistics-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
• (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|Asymptotic Normality and Central Limit Theorems

There is the same sort of close connection between the property of asymptotic normality and central limit theorems as there is between consistency and laws of large numbers. The easiest way to demonstrate this close connection is by means of an example. Suppose that samples are generated by random drawings from distributions with an unknown mean $\mu$ and unknown and variable variances. For example, it might be that the variance of the distribution from which the $t^{\text {th }}$ observation is drawn is
$$\sigma_t^2 \equiv \omega^2\left(1+\frac{1}{2}(t(\bmod 3))\right) .$$
Then $\sigma_t^2$ will take on the values $\omega^2, 1.5 \omega^2$, and $2 \omega^2$ with equal probability. Thus $\sigma_t^2$ varies systematically with $t$ but always remains within certain limits, in this case $\omega^2$ and $2 \omega^2$.

We will suppose that the investigator does not know the exact relation (4.26) and is prepared to assume only that the variances $\sigma_t^2$ vary between two positive bounds and average out asymptotically to some value $\sigma_0^2$, which may or not be known, defined as
$$\sigma_0^2 \equiv \lim {n \rightarrow \infty}\left(\frac{1}{n} \sum{t=1}^n \sigma_t^2\right) .$$
The sample mean may still be used as an estimator of the population mean, since our law of large numbers, Theorem 4.1, is applicable. The investigator is also prepared to assume that the distributions from which the observations are drawn have absolute third moments that are bounded, and so we too will assume that this is so. The investigator wishes to perform asymptotic statistical inference on the estimate derived from a realized sample and is therefore interested in the nondegenerate asymptotic distribution of the sample mean as an estimator. We saw in Section $4.3$ that for this purpose we should look at the distribution of $n^{1 / 2}\left(m_1-\mu\right)$, where $m_1$ is the sample mean. Specifically, we wish to study
$$n^{1 / 2}\left(m_1-\mu\right)=n^{-1 / 2} \sum_{t=1}^n\left(y_t-\mu\right),$$
where $y_t-\mu$ has variance $\sigma_t^2$.

## 经济代写|计量经济学代写Econometrics代考|Asymptotic Normality and Central Limit Theorems

Since all the $y_t$ ‘s are mutually independent and have mean zero, no term in the quadruple sum of (4.27) can be nonzero unless the indices either are all the same or fall into pairs (with, for instance, $r=t$ and $s=u$ with $r \neq s$ ). If all the indices are the same, then the value of the corresponding term is just the fourth moment of the distribution of the $y_t$ ‘s. But there can only be $n$ such terms. With the factor of $n^{-2}$ in (4.27), we see that these terms contribute to (4.27) only to order $n^{-1}$. On the other hand, the number of terms for which the indices fall in pairs is $3 n(n-1),{ }^4$ which is $O\left(n^2\right)$. Thus the latter terms contribute to (4.27) to the order of unity. But, and this is the crux of the argument, the value of each of these terms is just the square of the variance of each $y_t$, or $\sigma^4$. Thus, to leading order, the fourth moment of $S_n$ depends only on the variance of the $y_t$ ‘s; it does not depend on the fourth moment of the distribution of the $y_t$ ‘s. ${ }^5$.

A similar argument applies to all the moments of $S_n$ of order higher than 2. Thus, to leading order, all these moments depend only on the variance $\sigma^2$ and not on any other property of the distribution of the $y_t$ ‘s. This being so, if it is legitimate to characterize a distribution by its moments, then the limiting distribution of the sequence $\left{S_n\right}_{n=1}^{\infty}$ depends only on $\sigma^2$. Consequently, the limiting distribution must be the same for all possible distributions with the variance of $y_t$ equal to $\sigma^2$, regardless of other properties of that distribution. This means that we may calculate the limiting distribution making use of whatever distribution we choose, provided it has mean 0 and variance $\sigma^2$, and the answer will be independent of our choice.

The simplest choice is the normal distribution, $N\left(0, \sigma^2\right)$. The calculation of the limiting distribution is very easy for this choice: $S_n$ is just a sum of $n$ independent normal variables, namely, the $n^{-1 / 2} y_t$ ‘s, all of which have mean 0 and variance $n^{-1} \sigma^2$. Consequently, $S_n$ itself is distributed as $N\left(0, \sigma^2\right)$ for all $n$. If the distribution is $N\left(0, \sigma^2\right)$ for all $n$ independent of $n$, then the limiting distribution is just the $N\left(0, \sigma^2\right)$ distribution as well. But if this is so for normal summands, we may conclude by our earlier argument that the limiting distribution of any sequence $S_n$ made up from independent mean-zero summands, all with variance $\sigma^2$, will be $N\left(0, \sigma^2\right)$

## 经济代写|计量经济学代写Econometrics代考|Asymptotic Normality and Central Limit Theorems

$$\sigma_t^2 \equiv \omega^2\left(1+\frac{1}{2}(t(\bmod 3))\right) .$$

$$\sigma_0^2 \equiv \lim n \rightarrow \infty\left(\frac{1}{n} \sum t=1^n \sigma_t^2\right)$$

$$n^{1 / 2}\left(m_1-\mu\right)=n^{-1 / 2} \sum_{t=1}^n\left(y_t-\mu\right),$$

## 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。

## 经济代写|计量经济学代写Econometrics代考|Best22

statistics-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
• (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 $\hat{\boldsymbol{\beta}}$.

## 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。

## 经济代写|计量经济学代写Econometrics代考|Find2022

statistics-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
• (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|Sequences, Limits, and Convergence

The concept of infinity is one of unending fascination for mathematicians. One noted twentieth-century mathematician, Stanislaw Ulam, wrote that the continuing evolution of various notions of infinity is one of the chief driving forces behind research in mathematics (Ulam, 1976). However that may be, seemingly impractical and certainly unattainable infinities are at the heart of almost all valuable and useful applications of mathematics presently in use, among which we may count econometrics.

The reason for the widespread use of infinity is that it can provide workable approximations in circumstances in which exact results are difficult or impossible to obtain. The crucial mathematical operation which yields these approximations is that of passage to the limit, the limit being where the notion of infinity comes in. The limits of interest may be zero, finite, or infinite. Zero or finite limits usually provide the approximations that are sought: Things difficult to calculate in a realistic, finite, context are replaced by their limits as an approximation.

The first and most frequently encountered mathematical construct which may possess a limit is that of a sequence. A sequence is a countably infinite collection of things, such as numbers, vectors, matrices, or more general mathematical objects, and thus by its mere definition cannot be represented in the actual physical world. But some sequences are nevertheless very familiar. Consider the most famous sequence of all: the sequence
$${1,2,3, \ldots}$$
of the natural numbers. This is a simple-minded example perhaps, but one that exhibits some of the important properties which sequences may possess.

## 经济代写|计量经济学代写Econometrics代考|Rates of Convergence

We covered a lot of ground in the last section, so much so that we have by now, even if very briefly, touched on all the important purely mathematical topics to be discussed in this chapter. What remains is to flesh out the treatment of some matters and to begin to apply our theory to statistics and econometrics. The subject of this section is rates of convergence. In treating it we will introduce some very important notation, called the $O$, o notation, which is read as “big- $O$, little-o notation.” Here $O$ and $o$ stand for order and are often referred to as order symbols. Roughly speaking, when we say that some quantity is, say, $O(x)$, we mean that is of the same order, asymptotically, as the quantity $x$, while when we say that it is $o(x)$, we mean that it is of lower order than the quantity $x$. Just what this means will be made precise below.

In the last section, we discussed the random variable $b_n$ at some length and saw from (4.05) that its variance converged to zero, because it was proportional to $n^{-1}$. This implies that the sequence converges in probability to zero, and it can be seen that the higher moments of $b_n$, the third, fourth, and so on, must also tend to zero as $n \rightarrow \infty$. A somewhat tricky calculation, which interested readers are invited to try for themselves, reveals that the fourth moment of $b_n$ is
$$E\left(b_n^4\right)=\frac{3}{16} n^{-2}-\frac{1}{8} n^{-3},$$
that is, the sum of two terms, one proportional to $n^{-2}$ and the other to $n^{-3}$. The third moment of $b_n$, like the first, is zero, simply because the random variable is symmetric about zero, a fact which implies that all its odd-numbered moments vanish. Thus the second, third, and fourth moments of $b_n$ all converge to zero, but at different rates. Again, the two terms in the fourth moment (4.11) converge at different rates, and it is the term which is proportional to $n^{-2}$ that has the greatest importance asymptotically.

1,2,3,…

## 经济代写|计量经济学代写Econometrics代考|Rates of Convergence

$$E\left(b_n^4\right)=\frac{3}{16} n^{-2}-\frac{1}{8} n^{-3},$$

## 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。

## 经济代写|计量经济学代写Econometrics代考|Best27

statistics-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
• (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|Computing OLS Estimates

(1.34) yields a seriously inaccurate answer even when double precision is employed.

We hope this example makes it clear that attempts to compute estimates by evaluating standard algebraic expressions without regard for issues of machine precision and numerical stability are extremely unwise. The best rule to follow is always to use software written by experts who have taken such considerations into account. If such software is not available, one should probably always use double precision, with higher precision (if available) being used for sensitive calculations. ${ }^{10}$ As the example indicates, even numerically stable procedures may give nonsense results with 32-bit single precision if the data are ill-conditioned.

Let us now return to the principal topic of this scction, which is the computation of ordinary least squares estimates. Many good references on this subject exist – see, among others, Chambers (1977), Kennedy and Gentle (1980), Maindonald (1984), Farebrother (1988), and Golub and Van Loan (1989) – and we will therefore not go into many details.

The obvious way to obtain $\hat{\boldsymbol{\beta}}$ is first to form a matrix of sums of squares and cross-products of the regressors and the regressand or, equivalently, the matrix $\boldsymbol{X}^{\top} \boldsymbol{X}$ and the vector $\boldsymbol{X}^{\top} \boldsymbol{y}$. One would then invert the former by a general matrix inversion routine and postmultiply $\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1}$ by $\boldsymbol{X}^{\top} \boldsymbol{y}$. Unfortunately, this procedure has all the disadvantages of expression (1.34). It may work satisfactorily if double precision is used throughout, all the columns of $\boldsymbol{X}$ are similar in magnitude, and the $\boldsymbol{X}^{\top} \boldsymbol{X}$ matrix is not too close to being singular, but it cannot be recommended for general use.

The use of geometry as an aid to the understanding of linear regression has a long history; see Herr (1980). Early and important papers include Fisher (1915), Durbin and Kendall (1951), Kruskal $(1961,1968,1975)$, and Seber (1964). One valuable reference on linear models that takes the geometric approach is Seber (1980), although that book may be too terse for many readers. A recent expository paper that is quite accessible is Bryant (1984). The approach has not been used as much in econometrics as it has in statistics, but a number of econometrics texts – notably Malinvaud (1970a) and also Madansky (1976), Pollock (1979), and Wonnacott and Wonnacott (1979) use it to a greater or lesser degree. Our approach could be termed semigeometric, since we have not emphasized the coordinate-free nature of the analysis quite as much as some authors; see Kruskal’s papers, the Seber book or, in econometrics, Fisher $(1981,1983)$ and Fisher and McAleer (1984).
In this chapter, we have entirely ignored statistical models. Linear regression has been treated purely as a computational device which has a geometrical interpretation, rather than as an estimation procedure for a family of statistical models. All the results discussed have been true numerically, as a consequence of how ordinary least squares estimates are computed, and have not depended in any way on how the data were actually generated. We emphasize this, because conventional treatments of the linear regression model often fail to distinguish between the numerical and statistical properties of least squares.

## 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。

## 经济代写|计量经济学代写Econometrics代考|Best22

statistics-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
• (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
$$\boldsymbol{y}=\boldsymbol{X} 1 \boldsymbol{\beta} 1+\boldsymbol{X} 2 \boldsymbol{\beta} 2+\text { residuals, }$$

## 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。

## 经济代写|计量经济学代写Econometrics代考|Find2022

statistics-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
• (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 Geometry of Least Squares

The essential ingredients of a linear regression are a regressand $y$ and a matrix of regressors $\boldsymbol{X} \equiv\left[\boldsymbol{x}{1} \ldots \boldsymbol{x}{k}\right]$. The regressand $\boldsymbol{y}$ is an $n$-vector, and the matrix of regressors $\boldsymbol{X}$ is an $n \times k$ matrix, each column $\boldsymbol{x}{i}$ of which is an $n$-vector. The regressand $\boldsymbol{y}$ and each of the regressors $\boldsymbol{x}{1}$ through $\boldsymbol{x}_{k}$ can be thought of as points in $\boldsymbol{n}$-dimensional Euclidean space, $E^{n}$. The $k$ regressors, provided they are linearly independent, span a $k$-dimensional subspace of $E^{n}$. We will denote this subspace by $\mathcal{S}(\boldsymbol{X}) .^{1}$
‘the subspace $\mathcal{S}(\boldsymbol{X})$ consists of all points $\boldsymbol{z}$ in $E^{\text {n }}$ such that $\boldsymbol{z}=\boldsymbol{X} \boldsymbol{\gamma}$ for some $\gamma$, where $\gamma$ is a $k$-vector. Strictly speaking, we should refer to $\mathcal{S}(\boldsymbol{X})$ as the subspace spanned by the columns of $\boldsymbol{X}$, but less formally we will often refer to it simply as the span of $\boldsymbol{X}$. The dimension of $\mathcal{S}(\boldsymbol{X})$ is always equal to $\rho(\boldsymbol{X})$, the rank of $\boldsymbol{X}$ (i.e., the number of columns of $\boldsymbol{X}$ that are linearly independent). We will assume that $k$ is strictly less than $n$, something which it is reasonable to do in almost all practical cases. If $n$ were less than $k$, it would be impossible for $\boldsymbol{X}$ to have full column rank $k$.

A Euclidean space is not defined without defining an inner product. In this case, the inner product we are interested in is the so-called natural inner product. The natural inner product of any two points in $E^{n}$, say $\boldsymbol{z}{i}$ and $\boldsymbol{z}{j}$, may be denoted $\left\langle\boldsymbol{z}{i}, \boldsymbol{z}{j}\right\rangle$ and is defined by
$$\left\langle\boldsymbol{z}{i}, \boldsymbol{z}{j}\right\rangle \equiv \sum_{t=1}^{n} z_{i t} z_{j t} \equiv \boldsymbol{z}{i}^{\top} \boldsymbol{z}{j} \equiv \boldsymbol{z}{j}^{\top} \boldsymbol{z}{i} .$$

## 经济代写|计量经济学代写Econometrics代考|Restrictions and Reparametrizations

We have stressed the fact that $\mathcal{S}(\boldsymbol{X})$ is invariant to any nonsingular linear transformation of the columns of $\boldsymbol{X}$. This implies that we can always reparametrize any regression in whatever way is convenient, without in any way changing the ability of the regressors to explain the regressand. Suppose that we wished to run the regression
$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\text { residuals }$$
and compare the results of this regression with those from another regression in which $\boldsymbol{\beta}$ is subject to the $r(\leq k)$ linearly independent restrictions
$$\boldsymbol{R} \boldsymbol{\beta}=\boldsymbol{r},$$
where $\boldsymbol{R}$ is an $r \times k$ matrix of rank $r$ and $\boldsymbol{r}$ is an $r$-vector. While it is not difficult to do this by restricted least squares, it is often easier to reparametrize the regression so that the restrictions are zero restrictions. The restricted regression can then be estimated in the usual way by OLS. The reparametrization can be done as follows.

First, rearrange the columns of $\boldsymbol{X}$ so that the restrictions (1.12) can be written as
$$\boldsymbol{R}{1} \boldsymbol{\beta}{1}+\boldsymbol{R}{2} \boldsymbol{\beta}{2}=\boldsymbol{r},$$
where $\boldsymbol{R} \equiv\left[\begin{array}{ll}\boldsymbol{R}{1} & \boldsymbol{R}{2}\end{array}\right]$ and $\boldsymbol{\beta} \equiv\left[\boldsymbol{\beta}{1}: \boldsymbol{\beta}{2}\right]{ }^{5}{ }^{5} \boldsymbol{R}{1}$ being a nonsingular $r \times r$ matrix and $\boldsymbol{R}{2}$ an $r \times(k-r)$ matrix. It must be possible to do this if the restrictions are in fact distinct. Solving equations (1.13) for $\boldsymbol{\beta}{1}$ yields $$\beta{1}=R_{1}^{-1} \boldsymbol{r}-R_{1}^{-1} \boldsymbol{R}{2} \beta{2} .$$

## 经济代写|计量经济学代写Econometrics代考|Restrictions and Reparametrizations

$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\text { residuals }$$

$$\boldsymbol{R} \boldsymbol{\beta}=\boldsymbol{r},$$

$$\boldsymbol{R} 1 \boldsymbol{\beta} 1+\boldsymbol{R} 2 \boldsymbol{\beta} 2=\boldsymbol{r},$$

$$\beta 1=R_{1}^{-1} \boldsymbol{r}-R_{1}^{-1} \boldsymbol{R} 2 \beta 2 .$$

## 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。

## 经济代写|计量经济学代写Econometrics代考|BEA472

statistics-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
• (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|Cross-sectional data

A cross-sectional data set consists of a sample of individuals, households, firms, cities, countries, regions or any other type of unit at a specific point in time. In some cases, the data across all units do not correspond to exactly the same time period. Consider a survey that collects data from questionnaire surveys of different families on different days within a month. In this case, we can ignore the minor time differences in collection and the data collected will still be viewed as a cross-sectional data set.

In econometrics, cross-sectional variables are usually denoted by the subscript $i$, with $i$ taking values of $1,2,3, \ldots, N$, for $N$ number of cross-sections. So if, for example, $Y$ denotes the income data we have collected for $N$ individuals, this variable, in a cross-sectional framework, will be denoted by:
$$Y_{i} \quad \text { for } i=1,2,3, \ldots, N$$
Cross-sectional data are widely used in economics and other social sciences. In economics, the analysis of cross-sectional data is associated mainly with applied microeconomics. Labour economics, state and local public finance, business economics, demographic economics and health economics are some of the prominent fields in microeconomics. Data collected at a given point in time are used in these cases to test microeconomic hypotheses and evaluate economic policies.

## 经济代写|计量经济学代写Econometrics代考|Time series data

A time series data set consists of observations of one or more variables over time. Time series data are arranged in chronological order and can have different time frequencies, such as biannual, annual, quarterly, monthly, weekly, daily and hourly. Examples of time series data include stock prices, gross domestic product (GDP), money supply and ice cream sales figures, among many others.

Because past events can influence those in the future, and lags in behaviour are prevalent in the social sciences, time is a very important dimension in time series data sets. A variable that is lagged one period will be denoted as $Y_{t-1}$, and when it is lagged $s$ periods will be denoted as $Y_{t-s} .$ Similarly, if it is leading $k$ periods it will be denoted as $Y_{t+k}$

A key feature of time series data, which makes them more difficult to analyse than cross-sectional data, is that economic observations are commonly dependent across time; that is, most economic time series are closely related to their recent histories. So, while most econometric procedures can be applied to both cross-sectional and time series data sets, in the case of time series more things need to be done to specify the appropriate econometric model. Additionally, the fact that economic time series display clear trends over time has led to new econometric techniques that attempt to address these features.

Another important feature is that time series data that follow certain frequencies might exhibit a strong seasonal pattern. This feature is encountered mainly with weekly, monthly and quarterly time series. Finally, it is important to note that time series data are mainly associated with macroeconomic applications.

## 经济代写|计量经济学代写Econometrics代考|Cross-sectional data

$$Y_{i} \quad \text { for } i=1,2,3, \ldots, N$$

## 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。

## 经济代写|计量经济学代写Econometrics代考|ECON2300

statistics-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
• (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|Hypothesis testing and the central limit theorem

It would seem that the mean fulfils our two criteria for being a good estimate of the population as a whole: it is unbiased and its efficiency increases with the sample size. However, before we can begin to test a hypothesis about this mean, we need some idea of the shape of the whole sampling distribution. Unfortunately, while we have derived a simple expression for the mean and the variance, it is not in general possible to derive the shape of the complete sampling distribution. A hypothesis test proceeds by making an assumption about the truth; we call this the null hypothesis, often referred to as $H$. We then set up a specific alternative hypothesis, typically called $H_{a}$. The test consists of calculating the probability that the observed value of the statistic could have arisen purely by chance, assuming that the null hypothesis is true. Suppose that our null hypothesis is that the true population mean for age at death for men is $70, H: E\left(\bar{Y}_{m}\right)=$ 70. Having observed a mean of 75.1, we might then test the alternative that the mean is greater than 70 . We would do this by calculating the probability that $75.1$ could arise purely by chance when the true value of the population mean is 70 .

With a continuous distribution the probability of any exact point coming up is zero, so strictly what we are calculating is the probability of drawing any value for the mean that is greater than 75.1. We can then compare this probability with a predetermined value, which we call the significance level of the test. If the probability is less than the significance level, we reject the null hypothesis in favour of the alternative. In traditional statistics the significance level is usually set at $1 \%, 5 \%$ or $10 \%$. If we were using a $5 \%$ significance level and we found that the probability of observing a mean greater than $75.1$ was $0.01$, as $0.01<0.05$ we would reject the hypothesis that the true value of the population mean is 70 against the alternative that it is greater than 70 .

The alternative hypothesis can typically be specified in two ways, which give rise to either a one-sided test or a two-sided test. The example above is a one-sided test, as the alternative was that the age at death was greater than 70 , but we could equally have tested the possibility that the true mean was either greater or less than 70 , in which case we would have been conducting a two-sided test. In the case of a two-sided test we would be calculating the probability that a value either greater than $75.1$ or less than $70-(75.1-70)=64.9$ could occur by chance. Clearly this probability would be higher than in the one-sided test.

## 经济代写|计量经济学代写Econometrics代考|Central limit theorem

If a set of data is iid with $n$ observations, $\left(\mathrm{Y}{1}, \mathrm{Y}{2}, \ldots \mathrm{Y}_{n}\right)$, and with a finite variance then as $n$ goes to infinity the distribution of $\bar{Y}$ becomes normal. So as long as $n$ is reasonably large we can think of the distribution of the mean as being approximately normal.

This is a remarkable result; what it says is that, regardless of the form of the population distribution, the sampling distribution will be normal as long as it is based on a large enough sample. To take an extreme example, suppose we think of a lottery which pays out one winning ticket for every 100 tickets sold. If the prize for a winning ticket is $\$ 100$and the cost of each ticket is$\$1$, then, on average, we would expect to earn $\$ 1$per ticket bought. But the population distribution would look very strange; 99 out of every 100 tickets would have a return of zero and one ticket would have a return of$\$100$. If we tried to graph the distribution of returns it would have a huge spike at zero and a small spike at $\$ 100$and no observations anywhere else. But, as long as we draw a reasonably large sample, when we calculate the mean return over the sample it will be centred on$\$1$ with a normal distribution around 1 .

The importance of the central limit theorem is that it allows us to know what the sampling distribution of the mean should look like as long as the mean is based on a reasonably large sample. So we can now replace the arbitrary triangular distribution in Figure $1.1$ with a much more reasonable one, the normal distribution.

A final small piece of our statistical framework is the law of large numbers. This simply states that if a sample $\left(\mathrm{Y}{1}, \mathrm{Y}{2}, \ldots \mathrm{Y}_{n}\right)$ is IID with a finite variance then $\bar{Y}$ is a consistent estimator of $\mu$, the true population mean. This can be formally stated as $\operatorname{Pr}(|\bar{Y}-\mu|<\varepsilon) \rightarrow 1$ as $n \rightarrow \infty$, meaning that the probability that the absolute difference between the mean estimate and the true population mean will be less than a small positive number tends to one as the sample size tends to infinity. This can be proved straightforwardly, since, as we have seen, the variance of the sampling distribution of the mean is inversely proportional to $n$; hence as $n$ goes to infinity the variance of the sampling distribution goes to zero and the mean is forced to the true population mean.

## 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。

## 经济代写|计量经济学代写Econometrics代考|Find 2022

statistics-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
• (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|A statistical framework

The statistical framework that underlies the approach above rests on a number of key concepts, the first of which is the population. We assume that there is a population of events or entities that we are interested in. This population is assumed to be infinitely large and comprises all the outcomes that concern us. The data in Table $1.1$ are for the EU15 countries for the year 2002 . If we were interested only in this one year for this one set of countries, then there would be no statistical question to be asked. According to the data, women lived longer than men in that year in that area. That is simply a fact. But the population is much larger; it comprises all men and women in all periods, and to make an inference about this population we need some statistical framework. It might, for example, just be chance that women lived longer than men in that one year. How can we determine this?

The next important concepts are random variables and the population distribution. A random variable is simply a measurement of any event that occurs in an uncertain way. So, for example, the age at which a person dies is uncertain, and therefore the age of an individual at death is a random variable. Once a person dies, the age at death ceases to be a random variable and simply becomes an observation or a number. The population distribution defines the probability of a certain event happening; for example, the population distribution would define the probability of a man dying before he is $60\left(\operatorname{Pr}\left(Y_{m}<60\right)\right)$. The population distribution has various moments that define its shape. The first two moments are the mean (sometimes called the expected value, $E\left(Y_{m}\right)=\mu_{Y_{m}}$, or the average) and the variance $\left(E\left(Y_{m}-\mu_{Y_{m}}\right)^{2}\right.$, which is the square of the standard deviation and is often defined as $\sigma_{Y_{m}}^{2}$ ).

## 经济代写|计量经济学代写Econometrics代考|Properties of the sampling distribution of the mean

In the example above, based on Table 1.1, we calculated the mean life expectancy of men and women. Why is this a good idea? The answer lies in the sampling distribution of the mean as an estimate of the population mean. The mean of the sampling distribution of the mean is given by:

$$E\left(\frac{1}{n} \sum_{i=1}^{n} Y_{i}\right)=\frac{1}{n} \sum_{i=1}^{n} E\left(Y_{i}\right)=\frac{1}{n} \sum_{i=1}^{n} \mu_{Y}=\mu_{Y}$$
So the expected value of the mean of a sample is equal to the population mean, and hence the mean of a sample is an unbiased estimate of the mean of the population distribution. The mean thus fulfils our first criterion for being a good estimator. But what about the variance of the mean?
$$\begin{array}{r} \operatorname{var}(\bar{Y})=E\left(\bar{Y}-\mu_{Y}\right)^{2}=E\left(\frac{1}{n^{2}} \sum_{i=1}^{n} \sum_{j=1}^{n}\left(Y_{i}-\mu_{Y}\right)\left(Y_{j}-\mu_{Y}\right)\right) \ \text { beginlwmath28pt] }=\frac{1}{n^{2}}\left(\sum_{i=1}^{n} \operatorname{var}\left(Y_{i}\right)+\sum_{i=1}^{n} \sum_{j=1, j \neq i}^{n} \operatorname{cov}\left(Y_{i}, Y_{j}\right)\right)=\frac{\sigma_{Y}^{2}}{n} \end{array}$$
So the variance of the mean around the true population mean is related to the sample size that is used to construct the mean and the variance of the population distribution. As the sample size increases, the variance in the population shrinks, which is quite intuitive, as a large sample gives rise to a better estimate of the population mean. If the true population distribution has a smaller mean the sampling distribution will also have a smaller mean. Again, this is very intuitive; if everyone died at exactly the same age the population variance would be zero, and any sample we drew from the population would have a mean exactly the same as the true population mean.

## 经济代写|计量经济学代写Econometrics代考|Properties of the sampling distribution of the mean

$$E\left(\frac{1}{n} \sum_{i=1}^{n} Y_{i}\right)=\frac{1}{n} \sum_{i=1}^{n} E\left(Y_{i}\right)=\frac{1}{n} \sum_{i=1}^{n} \mu_{Y}=\mu_{Y}$$

$$\left.\operatorname{var}(\bar{Y})=E\left(\bar{Y}-\mu_{Y}\right)^{2}=E\left(\frac{1}{n^{2}} \sum_{i=1}^{n} \sum_{j=1}^{n}\left(Y_{i}-\mu_{Y}\right)\left(Y_{j}-\mu_{Y}\right)\right) \text { beginlwmath28pt }\right]=\frac{1}{n^{2}}\left(\sum_{i=}^{n}\right.$$

## 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。

## 商科代写|计量经济学代写Econometrics代考|Best 27

statistics-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
• (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

## 商科代写|计量经济学代写Econometrics代考|Formal Tests for Serial Correlation

The Durbin-Watson (DW) test provides a formal test in which the null hypothesis is that the equation errors are serially uncorrelated and the alternative is that they follow a first-order autocorrelation process. This test was first introduced by Durbin and Watson in two papers published in Biometrika in 1950 and 1951 [Durbin1950] and [Durbin1951]. It is a standard part of the regression output for most econometrics packages. The DW test builds on a previous test developed by Von Neumann [VonNeumann1941] who developed a test for autocorrelation in a series of random variables with the null that the variables are independent random numbers. Unfortunately, this is not suitable when the series under examination comprises regression residuals, which are not independent by construction. Although Von Neumann’s statistic has a relatively simple distribution, that is, the standard normal distribution, Durbin and Watson showed that the distribution of their test statistic was necessarily more complex. The nature of the test statistic means that it is not possible to derive unique critical values for a test of the null of no autocorrelation against the alternative of first-order autocorrelation. However, they did demonstrate that the critical values for their test were bounded and were able to tabulate the bounds for small sample sizes.
The DW test is concerned with a specific form of serial correlation, that is, first-order autocorrelation but is arguably sensitive to other forms. Consider the following regression model with an error that follows an AR process of order one:
\begin{aligned} &Y_{t}=\beta X_{t}+u_{t} \ &u_{t}=\rho u_{t-1}+\varepsilon_{t} . \end{aligned}
Taking the residuals from an OLS regression of $Y$ on $X$, we can construct the test statistic
$$D W=\sum_{t=2}^{T}\left(\hat{u}{t}-\hat{u}{t-1}\right)^{2} / \sum_{t=1}^{T} \hat{u}_{t}^{2} .$$

## 商科代写|计量经济学代写Econometrics代考|DEALING WITH SERIAL CORRELATION

If serial correlation is present, then there are several ways to deal with the issue. Of course, the priority is to identify the nature, and hopefully the cause, of the serial correlation. If the root cause of the problem is the omission of a relevant variable from the model, then the natural solution is to include that variable. If it is determined that modeling of the serial correlation process is appropriate, then we have several different methods available for the estimation of such models by adjusting for the presence of serially correlation errors. It should be noted that mechanical adjustments, of the type we will describe in this section, are potentially dangerous. This process has been much criticized on the grounds that there is a risk that these methods disguise an underlying problem rather than dealing with it. McGuirk and Spanos [McGuirk2009] are particularly critical of mechanical adjustments to deal with autocorrelated arguments. In this paper, they show that unless we can assume that the regress and does not Granger-cause the regressors, adjusting for autocorrelation means that least squares yield biased and inconsistent estimates. However, these methods are still used and reported in applied work and it is therefore important that we consider how they work.

The first method we will consider is that of Cochrane-Orcutt estimation. This uses an iterative algorithm proposed by Cochrane and Orcutt [Cochrane1949] in which we use the structure of the problem to separate out the estimation of the behavioral parameters of the main equation from those of the AR process that describes the errors. Let us consider the case of an $\mathrm{AR}(1)$ error process as an example. Suppose we wish to estimate a model of the form (5.6). The two equations can be combined to give a single equation of the form
$$Y_{t}-\rho Y_{t-1}=\beta\left(X_{t}-\rho X_{t-1}\right)+\varepsilon_{t},$$
that is, an equation in “quasi-differences” of the data. If $\rho$ was known, then it would be straightforward to construct these quasi-differences and estimate the behavioral parameter $\beta$ by least squares. In the absence of such knowledge, we make a guess at $\rho$ and construct an estimate of $\beta$ on this basis. We then generate the residuals $\hat{u}{t}=Y{t}-\beta X_{t}$ on this basis and calculate an estimate of $\rho$ of the form $\hat{\rho}=\sum_{t=2}^{T} \hat{u}{t} \hat{u}{t-1} / \sum_{t=1}^{T} \hat{u}_{t}^{2}$. If, by some lucky chance, this estimate coincides with our assumption, then we stop. Otherwise, we use our estimate to recalculate the quasi-differences, reestimate $\beta$, and continue until our estimates of $\beta$ and $p$ converge. If a solution exists, then this provides a robust algorithm for estimation.

## 商科代写|计量经济学代写Econometrics代考|Formal Tests for Serial Correlation

Durbin-Watson (DW) 检验提供了一种形式检验，其中零假设是方程误差是序列不相关的，而另一种方法是它们 遵循一阶自相关过程。该测试由 Durbin 和 Watson 在 1950 年和 1951 年在 Biometrika 发表的两篇论文 [Durbin1950] 和 [Durbin1951] 中首次引入。它是大多数计量经济学软件包回归输出的标准部分。DW 测试建立 在 Von Neumann [VonNeumann1941] 开发的先前测试的基础上，该测试开发了一系列随机变量的自相关测 试，其中变量为独立随机数。不幸的是，当检查的序列包含回归残差时，这不适合，这些回归残差在构造上不是 独立的。尽管冯诺依曼的统计量具有相对简单的分布，即标准正态分布，但 Durbin 和 Watson 表明，他们的检 验统计量的分布必然更复杂。检验统计量的性质意味着不可能为无自相关的零点与一阶自相关的备选方案的检验 推导出唯一的临界值。然而，他们确实证明了他们测试的临界值是有界的，并且能够将小样本的界限制表。检验 统计量的性质意味着不可能为无自相关的零点与一阶自相关的备选方案的检验推导出唯一的临界值。然而，他们 确实证明了他们测试的临界值是有界的，并且能够将小样本的界限制表。检验统计量的性质意味着不可能为无自 相关的零点与一阶自相关的备选方案的检验推导出唯一的临界值。然而，他们确实证明了他们测试的临界值是有 界的，并且能够将小样本的界限制表。

DW 检验关注特定形式的序列相关，即一阶自相关，但可以说对其他形式敏感。考虑以下回归模型，其误差遵循 一阶 AR 过程:
$$Y_{t}=\beta X_{t}+u_{t} \quad u_{t}=\rho u_{t-1}+\varepsilon_{t} .$$

$$D W=\sum_{t=2}^{T}(\hat{u} t-\hat{u} t-1)^{2} / \sum_{t=1}^{T} \hat{u}_{t}^{2} .$$

## 商科代写|计量经济学代写Econometrics代考|DEALING WITH SERIAL CORRELATION

$$Y_{t}-\rho Y_{t-1}=\beta\left(X_{t}-\rho X_{t-1}\right)+\varepsilon_{t},$$

## 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。