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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 Efficiency of Nonlinear Least Squares

Up to this point, we have said nothing at all about how well either the OLS or the NLS estimator compares with other estimators. One estimator is said to be more efficient than another if, on average, the first estimator yields more accurate estimates than the second. The reason for the terminology is that an estimator which yields more accurate estimates can be said to utilize the information available in the sample more efficiently. We could define efficiency in as many different ways as we could think of to evaluate the relative accuracy of two estimators, and there are thus many definitions of efficiency in the literature. We will deal with only two of the most widely used ones here.
Suppose that $\hat{\boldsymbol{\theta}}$ and $\check{\boldsymbol{\theta}}$ are two unbiased estimators of a $k$-vector of parameters $\boldsymbol{\theta}$, with true value $\boldsymbol{\theta}_0$, and that these two estimators have covariance matrices
\begin{aligned} &\boldsymbol{V}(\hat{\boldsymbol{\theta}}) \equiv E\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)^{\top} \text { and } \ &\boldsymbol{V}(\check{\boldsymbol{\theta}}) \equiv E\left(\ddot{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)\left(\ddot{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)^{\top} \end{aligned}
respectively. Then we have:
Definition $5.5 .$
The unbiased estimator $\hat{\boldsymbol{\theta}}$, with covariance matrix $\boldsymbol{V}(\hat{\boldsymbol{\theta}})$, is said to be more efficient than the unbiased estimator $\ddot{\theta}$, with covariance matrix $\boldsymbol{V}(\check{\boldsymbol{\theta}})$, if and only if $\boldsymbol{V}(\check{\boldsymbol{\theta}})-\boldsymbol{V}(\hat{\boldsymbol{\theta}})$, the difference of the two covariance matrices, is a positive semidefinite matrix.
If $\hat{\boldsymbol{\theta}}$ is more efficient than $\check{\boldsymbol{\theta}}$ in the sense of this definition, then every individual parameter in the vector $\boldsymbol{\theta}$, and every linear combination of those parameters, is estimated at least as efficiently by $\hat{\boldsymbol{\theta}}$ as by $\ddot{\boldsymbol{\theta}}$, by which we mean that the variance of the estimator based on $\hat{\boldsymbol{\theta}}$ is never greater than that of the estimator based on $\ddot{\boldsymbol{\theta}}$. To see this, consider an arbitrary linear combination of the parameters in $\boldsymbol{\theta}$, say $\boldsymbol{w}^{\top} \boldsymbol{\theta}$, where $\boldsymbol{w}$ is a $k$-vector. Then the variances of the two estimates of this quantity are $\boldsymbol{w}^{\top} \boldsymbol{V}(\breve{\boldsymbol{\theta}}) \boldsymbol{w}$ and $\boldsymbol{w}^{\top} \boldsymbol{V}(\hat{\boldsymbol{\theta}}) \boldsymbol{w}$, and so the difference between them is
$$\boldsymbol{w}^{\top} \boldsymbol{V}(\breve{\boldsymbol{\theta}}) \boldsymbol{w}-\boldsymbol{w}^{\top} \boldsymbol{V}(\hat{\boldsymbol{\theta}}) \boldsymbol{w}=\boldsymbol{w}^{\top}(\boldsymbol{V}(\check{\boldsymbol{\theta}})-\boldsymbol{V}(\hat{\boldsymbol{\theta}})) \boldsymbol{w}$$

## 经济代写|计量经济学代写Econometrics代考|Properties of Nonlinear Least Squares Residuals

We have by now discussed most of the points of interest concerning the asymptotic properties of the nonlinear least squares estimator. In this section, we wish to discuss the properties of the NLS residuals, that is, the sequence $\left{y_t-\hat{x}_t\right}$. These properties are important for a variety of reasons, not least because the residuals will be used to estimate the error variance $\sigma^2$.

In order to obtain the asymptotic properties of the NLS residuals, we begin by making a Taylor expansion of a typical residual around $\boldsymbol{\beta}=\boldsymbol{\beta}_0$.

This expansion is
\begin{aligned} \hat{u}t \equiv y_t-x_t(\hat{\boldsymbol{\beta}}) &=y_t-x{0 t}-\boldsymbol{X}t^\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\right) \ &=u_t-\boldsymbol{X}_t^\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\right), \end{aligned}
where, as usual, $\boldsymbol{X}_t^* \equiv \boldsymbol{X}_t\left(\boldsymbol{\beta}^\right)$ for some convex combination $\boldsymbol{\beta}^$ of $\hat{\boldsymbol{\beta}}$ and $\boldsymbol{\beta}_0$. Under the conditions of Theorem 5.2, $\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0=O\left(n^{-1 / 2}\right)$. Thus we can conclude immediately that
$$\hat{u}_t=u_t+O\left(n^{-1 / 2}\right),$$
which implies that the residuals consistently estimate the actual disturbances.
The simple result (5.52) is immensely valuable, but it is not detailed enough for all purposes. To see why not, consider the expression
$$n^{-1 / 2} \boldsymbol{a}^{\top} \hat{\boldsymbol{u}}=n^{-1 / 2} \sum{t=1}^n a_t \hat{u}t$$ for some vector $\boldsymbol{a}$ with elements forming a nonstochastic sequence $\left{a_t\right}$. If each $a_t$ is of order unity, then substituting (5.52) into (5.53) shows that the latter is equal to $$n^{-1 / 2} \sum{t=1}^n a_t u_t+n^{-1 / 2} \sum_{t=1}^n O\left(n^{-1 / 2}\right) .$$

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Asymptotic Efficiency of Nonlinear Least Squares

$$\boldsymbol{V}(\hat{\boldsymbol{\theta}}) \equiv E\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)^{\top} \text { and } \quad \boldsymbol{V}(\check{\boldsymbol{\theta}}) \equiv E\left(\ddot{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)\left(\ddot{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)^{\top}$$

$$\boldsymbol{w}^{\top} \boldsymbol{V}(\breve{\boldsymbol{\theta}}) \boldsymbol{w}-\boldsymbol{w}^{\top} \boldsymbol{V}(\hat{\boldsymbol{\theta}}) \boldsymbol{w}=\boldsymbol{w}^{\top}(\boldsymbol{V}(\check{\boldsymbol{\theta}})-\boldsymbol{V}(\hat{\boldsymbol{\theta}})) \boldsymbol{w}$$

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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