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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|The Three Classical Test Statistics

One of the attractive features of ML estimation is that test statistics based on the three principles we first discussed in Chapter 3 – the likelihood ratio, Lagrange multiplier, and Wald principles – are always available and are often easy to compute. These three principles of hypothesis testing were first enunciated in the context of ML estimation, and many authors still use the terms “likelihood ratio,” “Lagrange multiplier,” and “Wald” only in the context of tests based on ML estimates. In this section, we provide an introduction to what are often referred to as the three classical tests. All three of these test statistics have the same distribution asymptotically under the null hypothesis; if there are $r$ equality restrictions, they are distributed as $\chi^2(r)$. In fact, they actually tend to the same random variable asymptotically, both under the null and under all sequences of DGPs that are close to the null in a certain sense. An adequate treatment of these important results requires more space than we have available in this section. We will therefore defer it until Chapter 13 , which provides a much more detailed discussion of the three classical test statistics.

Conceptually the simplest of the three classical tests is the likelihood ratio, or LR, test. The test statistic is simply twice the difference between the restricted and unrestricted values of the loglikelihood function,
$$2(\ell(\hat{\boldsymbol{\theta}})-\ell(\tilde{\boldsymbol{\theta}}))$$
where $\hat{\boldsymbol{\theta}}$ denotes the unrestricted ML estimate of $\boldsymbol{\theta}, \tilde{\boldsymbol{\theta}}$ denotes the ML estimate subject to $r$ distinct restrictions, and the dependence of $\ell$ on $\boldsymbol{y}$ has been suppressed for notational simplicity. The LR statistic gets its name from the fact that (8.68) is equal to
$$2 \log \left(\frac{L(\hat{\boldsymbol{\theta}})}{L(\tilde{\boldsymbol{\theta}})}\right)$$
or twice the logarithm of the ratio of the likelihood functions. It is trivially easy to compute when both restricted and unrestricted estimates are available, and that is one its attractive features.

To derive the asymptotic distribution of the LR statistic one begins by taking a second-order Taylor-series approximation to $\ell(\tilde{\boldsymbol{\theta}})$ around $\hat{\boldsymbol{\theta}}$. Although we will not complete the derivation in this section, it is illuminating to go through the first few steps. The result of the Taylor-series approximation is
$$\ell(\tilde{\boldsymbol{\theta}}) \cong \ell(\hat{\boldsymbol{\theta}})+\frac{1}{2}(\tilde{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}})^{\top} \boldsymbol{H}(\hat{\boldsymbol{\theta}})(\tilde{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}})$$

## 经济代写|计量经济学代写Econometrics代考|Nonlinear Regression Models

In this section, we discuss how the method of maximum likelihood may be used to estimate univariate nonlinear regression models. When the error terms are assumed to be normally and independently distributed with constant variance, ML estimation of these models is, at least as regards the estimation of the parameters of the regression function, numerically identical to NLS estimation. The exercise is nevertheless a useful one. First of all, it provides a concrete illustration of how to use the method of maximum likelihood. Secondly, it provides an asymptotic covariance matrix for the estimates of $\boldsymbol{\beta}$ and $\sigma$ jointly, whereas NLS provides one for the estimates of $\boldsymbol{\beta}$ alone. Finally, by considering some extensions of the normal regression model, we are able to demonstrate the power of ML estimation.
The class of models that we will consider is
$$\boldsymbol{y}=\boldsymbol{x}(\boldsymbol{\beta})+\boldsymbol{u}, \quad \boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^2 \mathbf{I}\right),$$
where the regression function $\boldsymbol{x}(\boldsymbol{\beta})$ satisfies the conditions for Theorems $5.1$ and $5.2$, and the data are assumed to have been generated by a special case of (8.79). The parameter vector $\boldsymbol{\beta}$ is assumed to be of length $k$, which implies that there are $k+1$ parameters to be estimated. The notation ” $\boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^2 \mathbf{I}\right)$ ” means that the vector of error terms $\boldsymbol{u}$ is assumed to be distributed as multivariate normal with mean vector zero and covariance matrix $\sigma^2 \mathbf{I}$. Thus the individual error terms $u_t$ are independent, each distributed as $N\left(0, \sigma^2\right)$. The density of $u_t$ is
$$f\left(u_t\right)=\frac{1}{\sqrt{2 \pi}} \frac{1}{\sigma} \exp \left(-\frac{u_t^2}{2 \sigma^2}\right) .$$
In order to construct the likelihood function, we need the density of $y_t$ rather than the density of $u_t$. This requires us to use a standard result in statistics which is discussed in Appendix B.

The result in question says that if a random variable $x_1$ has density $f_1\left(x_1\right)$ and another random variable $x_2$ is related to it by
$$x_1=h\left(x_2\right),$$
where the function $h(\cdot)$ is continuously differentiable and monotonic, then the density of $x_2$ is given by
$$f_2\left(x_2\right)=f_1\left(h\left(x_2\right)\right)\left|\frac{\partial h\left(x_2\right)}{\partial x_2}\right| .$$

# 计量经济学代考

## 经济代写|计量经济学代写econometrics代考|The Three Classical – Test – Statistics

ML估计的一个吸引人的特性是，基于我们在第三章中首先讨论的三个原则——似然比、拉格朗日乘子和瓦尔德原则——的检验统计量总是可用的，而且通常很容易计算。假设检验的这三个原则最初是在ML估计的情况下提出的，许多作者仍然使用术语“似然比”、“拉格朗日乘子”和“瓦尔德”，只在基于ML估计的检验的情况下使用。在本节中，我们将介绍通常被称为三种经典测试的内容。这三个检验统计量在零假设下具有相同的渐近分布;如果存在$r$相等限制，则以$\chi^2(r)$的形式分发。事实上，它们实际上渐近地趋向于相同的随机变量，无论是在空值下，还是在某种意义上接近于空值的所有DGPs序列下。要充分讨论这些重要的结果，需要比本节更多的篇幅。因此，我们将把它推迟到第13章，这一章提供了对三种经典测试统计数据更详细的讨论

$$2(\ell(\hat{\boldsymbol{\theta}})-\ell(\tilde{\boldsymbol{\theta}}))$$

$$2 \log \left(\frac{L(\hat{\boldsymbol{\theta}})}{L(\tilde{\boldsymbol{\theta}})}\right)$$

$$\ell(\tilde{\boldsymbol{\theta}}) \cong \ell(\hat{\boldsymbol{\theta}})+\frac{1}{2}(\tilde{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}})^{\top} \boldsymbol{H}(\hat{\boldsymbol{\theta}})(\tilde{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}})$$

## 经济代写|计量经济学代写econometrics代考|非线性回归模型

$$\boldsymbol{y}=\boldsymbol{x}(\boldsymbol{\beta})+\boldsymbol{u}, \quad \boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^2 \mathbf{I}\right),$$
，其中回归函数$\boldsymbol{x}(\boldsymbol{\beta})$满足定理$5.1$和$5.2$的条件，并且假设数据是由(8.79)的特殊情况生成的。假设参数向量$\boldsymbol{\beta}$的长度为$k$，这意味着需要估计$k+1$参数。符号“$\boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^2 \mathbf{I}\right)$”意味着误差向量$\boldsymbol{u}$假设为多元正态分布，其平均向量为零，协方差矩阵为$\sigma^2 \mathbf{I}$。因此，各个误差项$u_t$是独立的，每个分布为$N\left(0, \sigma^2\right)$。$u_t$的密度
$$f\left(u_t\right)=\frac{1}{\sqrt{2 \pi}} \frac{1}{\sigma} \exp \left(-\frac{u_t^2}{2 \sigma^2}\right) .$$

$$x_1=h\left(x_2\right),$$
，其中函数$h(\cdot)$是连续可微的单调的，那么$x_2$的密度是
$$f_2\left(x_2\right)=f_1\left(h\left(x_2\right)\right)\left|\frac{\partial h\left(x_2\right)}{\partial x_2}\right| .$$

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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