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## 经济代写|计量经济学代写Introduction to Econometrics代考|Regression to the Mean

The term regression originated in an influential paper by Francis Galton (1886) where he examined the joint distribution of the stature (height) of parents and children. Effectively, he was estimating the conditional mean of children’s height given their parent’s height. Galton discovered that this conditional mean was approximately linear with a slope of $2 / 3$. This implies that on average a child’s height is more mediocre (average) than his or her parent’s height. Galton called this phenomenon regression to the mean, and the label regression has stuck to this day to describe most conditional relationships.

One of Galton’s fundamental insights was to recognize that if the marginal distributions of $y$ and $x$ are the same (e.g. the heights of children and parents in a stable environment) then the regression slope in a linear projection is always less than one.
To be more precise, take the simple linear projection
$$y=x \beta+\alpha+e$$
where $y$ equals the height of the child and $x$ equals the height of the parent. Assume that $y$ and $x$ have the same mean so that $\mu_y=\mu_x=\mu$. Then from (2.38)
$$\alpha=(1-\beta) \mu$$
so we can write the linear projection (2.48) as
$$\mathscr{P}(y \mid x)=(1-\beta) \mu+x \beta .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Reverse Regression

Galton noticed another interesting feature of the bivariate distribution. There is nothing special about a regression of $y$ on $x$. We can also regress $x$ on $y$. (In his heredity example this is the best linear predictor of the height of parents given the height of their children.) This regression takes the form
$$x=y \beta^+\alpha^+e^*$$
This is sometimes called the reverse regression. In this equation, the coefficients $\alpha^, \beta^$ and error $e^$ are defined by linear projection. In a stable population we find that \begin{aligned} & \beta^=\operatorname{corr}(x, y)=\beta \ & \alpha^*=(1-\beta) \mu=\alpha \end{aligned}
which are exactly the same as in the projection of $y$ on $x$ ! The intercept and slope have exactly the same values in the forward and reverse proiections!

## 经济代写|计量经济学代写Introduction to Econometrics代 考|Regression to the Mean

$$y=x \beta+\alpha+e$$

$$\alpha=(1-\beta) \mu$$

$$\mathscr{P}(y \mid x)=(1-\beta) \mu+x \beta$$

## 经济代写|计量经济学代写Introduction to Econometrics代 考|Reverse Regression

$$x=y \beta^{+} \alpha^{+} e^*$$

$$\beta^{=} \operatorname{corr}(x, y)=\beta \quad \alpha^*=(1-\beta) \mu=\alpha$$

## 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。

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Other Large Sample Tests: The Lagrange Multiplier Statistic

Once we enter the realm of asymptotic analysis, there are other test statistics that can be used for hypothesis testing. For most purposes, there is little reason to go beyond the usual $t$ and $F$ statistics: as we just saw, these statistics have large sample justification without the normality assumption. Nevertheless, sometimes it is useful to have other ways to test multiple exclusion restrictions, and we now cover the Lagrange multiplier LM statistic, which has achieved some popularity in modern econometrics.

The name “Lagrange multiplier statistic” comes from constrained optimization, a topic beyond the scope of this text. [See Davidson and MacKinnon (1993).] The name score statistic – which also comes from optimization using calculus-is used as well. Fortunately, in the linear regression framework, it is simple to motivate the $L M$ statistic without delving into complicated mathematics.

The form of the $L M$ statistic we derive here relies on the Gauss-Markov assumptions, the same assumptions that justify the $F$ statistic in large samples. We do not need the normality assumption.

To derive the $L M$ statistic, consider the usual multiple regression model with $k$ independent variables:
$$y=\beta_0+\beta_1 x_1+\ldots+\beta_k x_k+u .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|THE LAGRANGE MULTIPLIER STATISTIC FOR q EXCLUSION RESTRICTIONS:

(i) Regress $y$ on the restricted set of independent variables and save the residuals, $\tilde{u}$.
(ii) Regress $\tilde{u}$ on all of the independent variables and obtain the $R$-squared, say $R_u^2$ (to distinguish it from the $R$-squareds obtained with $y$ as the dependent variable).
(iii) Compute $L M=n R_u^2$ [the sample size times the $R$-squared obtained from step (ii)].
(iv) Compare $L M$ to the appropriate critical value, $c$, in a $\chi_q^2$ distribution; if $L M>$ $c$, the null hypothesis is rejected. Even better, obtain the $p$-value as the probability that a $\chi_q^2$ random variable exceeds the value of the test statistic. If the $p$-value is less than the desired significance level, then $\mathrm{H}_0$ is rejected. If not, we fail to reject $\mathrm{H}_0$. The rejection rule is essentially the same as for $F$ testing.
Because of its form, the $L M$ statistic is sometimes referred to as the $\mathbf{n}-\boldsymbol{R}$-squared statistic. Unlike with the $F$ statistic, the degrees of freedom in the unrestricted model plays no role in carrying out the $L M$ test. All that matters is the number of restrictions being tested $(q)$, the size of the auxiliary $R$-squared $\left(R_u^2\right)$, and the sample size $(n)$. The $d f$ in the unrestricted model plays no role because of the asymptotic nature of the $L M$ statistic. But we must be sure to multiply $R_u^2$ by the sample size to obtain $L M$; a seemingly low value of the $R$-squared can still lead to joint significance if $n$ is large.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Other Large Sample Tests: The Lagrange Multiplier Statistic

“Lagrange multiplier statistic”这个名称来自约束优化，这个主题超出了本文的范围。[参见 Davidson 和 MacKinnon (1993)]。也使用名称得分统计一一它也来自使用微积分的优化。幸运的是，在线性回归框架中， 很容易激发 $L M$ 无需深入研究筫杂的数学即可进行㧤计。

$$y=\beta_0+\beta_1 x_1+\ldots+\beta_k x_k+u .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|THE LAGRANGE MULTIPLIER STATISTIC FOR q EXCLUSION RESTRICTIONS:

(i) 回归 $y$ 在受限制的自变量集上并保存残差， $\tilde{u}$.
(ii) 回归 $\tilde{u}$ 在所有自变量上并获得 $R$-平方，说 $R_u^2$ (为了区别于 $R$-平方获得 $y$ 作为因变量)。
(iii) 计算 $L M=n R_u^2$ [样本量乘以 $R$-从步骤 (ii)] 获得的平方。
(iv) 比较 $L M$ 到适当的临界值， $c$ ，在一个 $\chi_q^2$ 分配; 如果 $L M>c$ ，原假设被拒绝。更好的是，获得 $p$-value 作为 $\mathrm{a}$ 的概率 $\chi_q^2$ 随机变量超过了检验统计量的值。如果 $p$ – 值小于所需的显着性水平，则 $\mathrm{H}_0$ 被拒绝了。如果没 有，我们拒绝 $\mathrm{H}_0$. 拒绝规则与 for 基本相同 $F$ 测试。

## 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。

Posted on Categories:Financial Econometrics, 经济代写, 计量经济学

## avatest™帮您通过考试

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Properties of Random Sets Related to Their Capacity Functionals

As we have seen, $\mathrm{T}$ determines uniquely the distribution of $\boldsymbol{X}$, and therefore in principle properties of $\boldsymbol{X}$ can be expressed in terms of $\mathrm{T}$. Here we give such examples (but we also caution the reader that, in other situations, deriving properties of $\boldsymbol{X}$ in terms of $\mathrm{T}$ may be extremely difficult). Specifically, we consider random sets of points, i.e., simple point processes (these are point processes such that, with probability one, no two points of the process are coincident).

Example 1.35 (Binomial process) A random closed set $\boldsymbol{X}$ is a sample of i.i.d. points $\left{\boldsymbol{x}_1, \ldots, \boldsymbol{x}_n\right}$ with the common nonatomic distribution $\mu$, and is called a binomial process, if and only if its capacity functional is

$$\mathrm{T}(K)=1-(1-\mu(K))^n$$
for all compact $K \subset \mathbb{R}^d$.
Example 1.36 (Poisson point process) Define
$$\mathrm{T}(K)=1-e^{-\Lambda(K)}, \quad K \in \mathcal{K},$$
with $\Lambda$ being a locally finite measure on $\mathbb{R}^d$ (that is, each point admits a neighborhood of finite measure) and such that $\Lambda$ attaches zero mass to any single point. The corresponding random closed set $\boldsymbol{X}$ is the Poisson process in $\mathbb{R}^d$ with intensity measure $\Lambda$. Indeed, if $\boldsymbol{X}$ is a point set such that the number of points in any set $K$ is Poisson distributed with mean $\Lambda(K)$, then $X$ hits $K$ if and only if the Poisson random variable with mean $\Lambda(K)$ does not vanish. The probability of this latter event is exactly the right-hand side of (1.9). The random set $\boldsymbol{X}$ is stationary (see Definition 1.38) and then called a homogeneous Poisson process if and only if $\Lambda$ is proportional to the Lebesgue measure on $\mathbb{R}^d$; the coefficient of proportionality is called the intensity of the process.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Weak Convergence

The weak convergence of random closed sets is defined by specializing the general weak convergence definition for random elements in $\mathcal{F}$ or, equivalently, the weak convergence of probability measures on the space $\mathcal{F}$ of closed sets. For a set $K$, denote Int $K$ its interior, that is, the largest open subset of $K$.
Theorem 1.40 A sequence $\left{\boldsymbol{X}n, n \geq 1\right}$ of random closed sets in $\mathbb{R}^d$ converges weakly to $\boldsymbol{X}$ if and only if $\mathrm{T}{X_n}(K) \rightarrow \mathrm{T}_X(K)$ as $n \rightarrow \infty$ for all compact sets $K$ such that $\mathrm{T}_X(K)=\mathrm{T}_X(\operatorname{Int} K)$.

Therefore, the weak convergence of random closed sets is equivalent to the pointwise convergence of their capacity functionals on continuity sets of the limiting capacity functional $\mathrm{T}X$. The condition on $K$ in Theorem $1.40$ is akin to the conventional requirement for the weak convergence of random variables, which asks for convergence of their cumulative distribution functions at all points of continuity of the limit. The condition $\mathrm{T}_X(K)=\mathrm{T}_X($ Int $K)$ means that $X$ “touches” $K$ with probability zero, i.e., $$\mathbf{P}{\boldsymbol{X} \cap K \neq \emptyset, \boldsymbol{X} \cap \operatorname{Int} K=\emptyset}=0 .$$ Recall that $\mathrm{T}_X$ (Int $\left.K\right)$ is defined using the extension of the capacity functional to open sets. If $\boldsymbol{X}_n=\left{\boldsymbol{x}_n\right}$ are random singletons, then $\mathrm{T}{\boldsymbol{X}_n}(K)$ is the probability distribution of $\boldsymbol{x}_n$ and $\mathrm{T}_X(K)=\mathrm{T}_X(\operatorname{Int} K)$ for the weak limit $\boldsymbol{X}={\boldsymbol{x}}$ means that $\boldsymbol{x}$ belongs to the boundary of the set $K$ with probability zero. Therefore, the weak convergence of random singletons corresponds to the classical definition of weak convergence for random elements. The weak convergence is denoted by $\Rightarrow$.

As usual, the weak convergence of $\boldsymbol{X}_n$ to $\boldsymbol{X}$ implies the weak convergence of $f\left(\boldsymbol{X}_n\right)$ to $f(\boldsymbol{X})$ for any continuous map applied to sets. However, many important maps are not continuous, e.g., the Lebesgue measure is not continuous. This can be seen by noticing that a finite set of points dense in the ball converges to the ball, while the Lebesgue measure of any finite set vanishes and the ball has a positive measure. However, the Lebesgue measure becomes continuous if restricted to the family of convex sets.

Since the space of closed sets $\mathcal{F}$ in $\mathbb{R}^d$ is compact, a variant of Helly’s theorem for random closed sets establishes that each family of random closed sets has a subsequence that converges in distribution. Therefore, quite differently to the studies of random functions and stochastic processes, there is no need to check the tightness conditions when proving the weak convergence of random closed sets.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Properties of Random Sets Related to Their Capacity Functionals

$$\mathrm{T}(K)=1-(1-\mu(K))^n$$

$$\mathrm{T}(K)=1-e^{-\Lambda(K)}, \quad K \in \mathcal{K},$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Weak Convergence

$$\mathbf{P} \boldsymbol{X} \cap K \neq \emptyset, \boldsymbol{X} \cap \operatorname{Int} K=\emptyset=0 .$$

## 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。