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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|Asymptotic Relative Efficiency

Since all consistent tests reject with probability one as the sample size tends to infinity, it is not obvious how to compare the power of tests of which the distributions are known only asymptotically. Various approaches have been proposed in the statistical literature, of which the best known is probably the concept of asymptotic relative efficiency, or ARE. This concept, which is closely related to the idea of local alternatives, is due to Pitman (1949), and has since been developed by many other authors; see Kendall and Stuart (1979, Chapter 25). Suppose that we have two test statistics, say $\tau_1$ and $\tau_2$, both of which have the same asymptotic distribution under the null and both of which, like all the test statistics we have discussed in this chapter, are root- $n$ consistent. This means that, for the test to have a nondegenerate asymptotic distribution, the drifting DGP must approach the simple null hypothesis at a rate proportional to $n^{-1 / 2}$. In this case, the asymptotic efficiency of $\tau_2$ relative to $\tau_1$ is defined as
$$\mathrm{ARE}{21}=\lim {n \rightarrow \infty}\left(\frac{n_1}{n_2}\right),$$
where $n_1$ and $n_2$ are sample sizes such that $\tau_1$ and $\tau_2$ have the same power, and the limit is taken as both $n_1$ and $n_2$ tend to infinity. If, for example, $\mathrm{ARE}_{21}$ were $0.25, \tau_2$ would asymptotically require 4 times as large a sample as $\tau_1$ to achieve the same power.

For tests with the same number of degrees of freedom, it is easy to see that
$$\mathrm{ARE}{21}=\frac{\cos ^2 \phi_2}{\cos ^2 \phi_1} .$$ Recall from expression (12.23) that the NCP is proportional to $\cos ^2 \phi$. If the DGP did not drift, it would also be proportional to the sample size. If $\tau_1$ and $\tau_2$ are to be equally powerful in this case, they must have the same NCP. This means that $n_1 / n_2$ must be equal to $\cos ^2 \phi_2 / \cos ^2 \phi_1$. Suppose, for example, that $\cos ^2 \phi_1=1$ and $\cos ^2 \phi_2=0.5$. Then the implicit alternative hypothesis for $\tau_1$ must include the DGP, while the implicit alternative for $\tau_2$ does not. Thus the directions in which $\tau_1$ is testing explain all of the divergence between the null hypothesis and the DGP, while the directions in which $\tau_2$ is testing explain only half of it. But we can compensate for this reduced explanatory power by making $n_2$ twice as large as $n_1$, so as to make both tests equally powerful asymptotically. Hence $\mathrm{ARE}{21}$ must be $0.5$. See Davidson and MacKinnon (1987) for more on this special case.

## 经济代写|计量经济学代写Econometrics代考|Interpreting Test Statistics that Reject the Null

Suppose that we test a regression model in one or more regression directions and obtain a test statistic that is inconsistent with the null hypothesis at whatever significance level we have chosen. How are we to interpret it? We have decided that the DGP does not belong to the implicit null hypothesis of the test, since we have rejected the null and hence rejected the proposition that $\cos ^2 \phi$ is zero. Does the DGP belong to the implicit alternative, then? Possibly it does, but by no means necessarily. The NCP is the product of expression (12.24), which does not depend in any way on the alternative we are testing against, and $\cos ^2 \phi$, which does. For a given value of (12.24), the NCP will be maximized when $\cos ^2 \phi=1$. But the fact that the NCP is nonzero (which is all that a single significant test statistic tells us) merely implies that neither $\cos ^2 \phi$ nor expression (12.24) is zero. Thus all we can conclude from a single significant test statistic is that the DGP is not a special case of the model under test and that the directions represented by $\boldsymbol{Z}$ have some explanatory power for the direction $\boldsymbol{a}$ in which the model is actually false.

If we are going to make any inferences at all about the directions in which a model under test is wrong, we will evidently have to calculate more than one test statistic. Since expression (12.24) is the same for all tests in regression directions, any differences between the values of the various test statistics must be due to differences in numbers of degrees of freedom, differences in $\cos ^2 \phi$, or simple randomness (including of course differences between finitesample and asymptotic behavior of the tests). Suppose that we test a model against several sets of regression directions, represented by regressor matrices $Z_1, Z_2$, and so on. Suppose further that the $j^{\text {th }}$ regressor matrix, $Z_j$, has $r_j$ columns and generates a test statistic $T_j$, which is distributed as $\chi^2\left(r_j\right)$ asymptotically under the null. Each of the test statistics $T_j$ can be used to estimate the corresponding NCP, say $\Lambda_j$. Since the mean of a noncentral chi-squared random variable with $r$ degrees of freedom is $r$ plus the NCP, the obvious estimate of $\Lambda_j$ is $T_j-r_j$. Of course, this estimator is necessarily inconsistent, since under a drifting DGP the test statistic is a random variable no matter how large the sample size. Nevertheless, it seems reasonable that if $T_l-r_l$ is substantially larger than $T_j-r_j$ for all $j \neq l$, the logical place to look for a better model is in the directions tested by $\boldsymbol{Z}_l$.

# 计量经济学代考

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

$$\text { ARE21 }=\lim n \rightarrow \infty\left(\frac{n_1}{n_2}\right),$$

$$\text { ARE21 }=\frac{\cos ^2 \phi_2}{\cos ^2 \phi_1} .$$

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师