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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|Transformations and Reparametrizations

In this and the subsequent sections of this chapter, we develop the classical theory of maximum likelihood estimation and, in particular, demonstrate the properties that make it a desirable estimation method. We will also point out that in some circumstances these properties fail. As we discussed in Section 8.1, the major desirable features of ML estimators are invariance, consistency, asymptotic normality, asymptotic efficiency, and computability. In this section, we will discuss the first of these, the invariance of ML estimators to reparametrization of the model.

The idea of invariance is an important one in econometric analysis. Let us denote by $\mathbb{M}$ the model in which we are interested. A parametrization of the model $\mathbb{M}$ is a mapping, say $\lambda$, from a parameter space $\Theta$ to $\mathbb{M}$. For any given model $\mathbb{M}$ there may in general exist an infinite number of parametrizations. There are, after all, few constraints on the parameter space $\Theta$, other than its dimensionality. A subset of $\mathbb{R}^k$ of full dimension can be mapped in a one-to-one and differentiable manner onto virtually any other subset of $\mathbb{R}^k$ of full dimension by such devices as translation, rotation, dilation, and so on, and subsequently any of these other subsets can perfectly well serve as the parameter space for the model $\mathbb{M}$. It is because of this fact that one appeals to invariance as a desirable property of estimators. “Invariance” is understood in this context as invariance under the sort of transformation we have been discussing, which we call formally reparametrization.

As an illustration of the fact that any model may be parametrized in an infinite number of ways, consider the case of the exponential distribution, which was discussed in Section 8.1. The likelihood function for a sample of independent drawings from this distribution was seen to be (8.03). If we make the definition $\theta \equiv \delta^\alpha$, we can define a whole family of parametrizations indexed by $\alpha$. We may choose $\alpha$ to be any finite, nonzero number. The likelihood function corresponding to this family of parametrizations is
$$L(\boldsymbol{y}, \delta)=\prod_{t=1}^n \delta^\alpha e^{-\delta^\alpha y_t}$$
Evidently, $\alpha=1$ corresponds to the $\theta$ parametrization of (8.02) and $\alpha=-1$ corresponds to the $\phi$ parametrization of (8.07).

It is easy to see that ML estimators are invariant to reparametrizations of the model. Let $\eta: \Theta \rightarrow \Phi \subseteq \mathbb{R}^k$ denote a smooth mapping that transforms the vector $\theta$ uniquely into another vector $\boldsymbol{\phi} \equiv \boldsymbol{\eta} \boldsymbol{\theta})$. The likelihood function for the model $\mathbb{M}$ in terms of the new parameters $\phi$, say $L^{\prime}$, is defined by the relation
$$L^{\prime}(\boldsymbol{y}, \boldsymbol{\phi})=L(\boldsymbol{y}, \boldsymbol{\theta}) \quad \text { for } \boldsymbol{\phi}=\boldsymbol{\eta}(\boldsymbol{\theta}) .$$
Equation (8.23) follows at once from the facts that a likelihood function is the density of a stochastic process and that $\boldsymbol{\theta}$ and $\boldsymbol{\phi}=\boldsymbol{\eta}(\boldsymbol{\theta})$ describe the same stochastic process. Let us define $\hat{\boldsymbol{\phi}}$ as $\boldsymbol{\eta}(\hat{\boldsymbol{\theta}})$ and $\boldsymbol{\phi}^$ as $\boldsymbol{\eta}\left(\boldsymbol{\theta}^\right)$. Then if
$$L(\boldsymbol{y}, \hat{\boldsymbol{\theta}})>L\left(\boldsymbol{y}, \boldsymbol{\theta}^\right) \text { for all } \boldsymbol{\theta}^ \neq \hat{\boldsymbol{\theta}},$$
it follows that
$$L^{\prime}(\boldsymbol{y}, \hat{\boldsymbol{\phi}})=L^{\prime}(\boldsymbol{y}, \boldsymbol{\eta}(\hat{\boldsymbol{\theta}}))=L(\boldsymbol{y}, \hat{\boldsymbol{\theta}})>L\left(\boldsymbol{y}, \boldsymbol{\theta}^\right)=L^{\prime}\left(\boldsymbol{y}, \boldsymbol{\phi}^\right) \text { for all } \boldsymbol{\phi}^* \neq \hat{\boldsymbol{\phi}}$$

## 经济代写|计量经济学代写Econometrics代考|Asymptotic Efficiency of the ML Estimator

In this section, we will demonstrate the asymptotic efficiency of the ML estimator or, strictly speaking, of the Type $2 \mathrm{ML}$ estimator. Asymptotic efficiency means that the variance of the asymptotic distribution of any consistent estimator of the model parameters differs from that of an asymptotically efficient estimator by a positive semidefinite matrix; see Definition 5.6. One says an asymptotically efficient estimator rather than the asymptotically efficient estimator because, since the property of asymptotic efficiency is a property only of the asymptotic distribution, there can (and do) exist many estimators that differ in finite samples but have the same, efficient, asymptotic distribution. An example can be taken from the nonlinear regression model, in which, as we will see in Section 8.10, NLS is equivalent to ML estimation if we assume normality of the error terms. As we saw in Section 6.6, there are nonlinear models that are just linear models with some nonlinear restrictions imposed on them. In such cases, one-step estimation starting from the estimates of the linear model was seen to be asymptotically equivalent to NLS, and hence asymptotically efficient. One-step estimation is possible in the general maximum likelihood context as well and can often provide an efficient estimator that is easier to compute than the ML estimator itself.

We will begin our proof of the asymptotic efficiency of the ML estimator by a discussion applicable to any root- $n$ consistent and asymptotically unbiased estimator of the parameters of the model represented by the loglikelihood function $\ell(\boldsymbol{y}, \boldsymbol{\theta})$. Note that consistency by itself does not imply asymptotic unbiasedness without the imposition of various regularity conditions. Since every econometrically interesting consistent estimator that we are aware of is in fact asymptotically unbiased, we will deal only with such estimators here. Let such an estimator be denoted by $\hat{\boldsymbol{\theta}}(\boldsymbol{y})$, where the notation emphasizes the fact that the estimator is a random variable, dependent on the realized sample $\boldsymbol{y}$. Note that we have changed notation here, since $\hat{\boldsymbol{\theta}}(\boldsymbol{y})$ is in general not the ML estimator. Instead, the latter will be denoted $\tilde{\boldsymbol{\theta}}(\boldsymbol{y})$; the new notation is designed to be consistent with our treatment throughout the book of restricted and unrestricted estimators, since in an important sense the ML estimator corresponds to the former and the arbitrary consistent estimator $\hat{\boldsymbol{\theta}}(\boldsymbol{y})$ corresponds to the latter.

# 计量经济学代考

## 经济代写|计量经济学代写econometrics代考| transforms -and- reparameterizations

$$L(\boldsymbol{y}, \delta)=\prod_{t=1}^n \delta^\alpha e^{-\delta^\alpha y_t}$$

$$L^{\prime}(\boldsymbol{y}, \boldsymbol{\phi})=L(\boldsymbol{y}, \boldsymbol{\theta}) \quad \text { for } \boldsymbol{\phi}=\boldsymbol{\eta}(\boldsymbol{\theta}) .$$

$$L(\boldsymbol{y}, \hat{\boldsymbol{\theta}})>L\left(\boldsymbol{y}, \boldsymbol{\theta}^\right) \text { for all } \boldsymbol{\theta}^ \neq \hat{\boldsymbol{\theta}},$$
，则得到
$$L^{\prime}(\boldsymbol{y}, \hat{\boldsymbol{\phi}})=L^{\prime}(\boldsymbol{y}, \boldsymbol{\eta}(\hat{\boldsymbol{\theta}}))=L(\boldsymbol{y}, \hat{\boldsymbol{\theta}})>L\left(\boldsymbol{y}, \boldsymbol{\theta}^\right)=L^{\prime}\left(\boldsymbol{y}, \boldsymbol{\phi}^\right) \text { for all } \boldsymbol{\phi}^* \neq \hat{\boldsymbol{\phi}}$$

## 有限元方法代写

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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