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# 经济代写|计量经济学代写Introduction to Econometrics代考|At What Level to Cluster?

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## 经济代写|计量经济学代写Introduction to Econometrics代考|At What Level to Cluster?

A practical question which arises in the context of cluster-robust inference is “At what level should we cluster?” In some examples you could cluster at a very fine level, such as families or classrooms, or at higher levels of aggregation, such as neighborhoods, schools, towns, counties, or states. What is the correct level at which to cluster? Rules of thumb have been advocated by practitioners but at present there is little formal analysis to provide useful guidance. What do we know?

First, suppose cluster dependence is ignored or imposed at too fine a level (e.g. clustering by households instead of villages). Then variance estimators will be biased as they will omit covariance terms. As correlation is typically positive, this suggests that standard errors will be too small giving rise to spurious indications of significance and precision.

Second, suppose cluster dependence is imposed at too aggregate a measure (e.g. clustering by states rather than villages). This does not cause bias. But the variance estimators will contain many extra components so the precision of the covariance matrix estimator will be poor. This means that reported standard errors will be imprecise – more random – than if clustering had been less aggregate.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Technical Proofs

Proof of Theorem 4.6 The proof technique is to calculate the Cramér-Rao bound from a carefully crafted parametric model. (For the Cramér-Rao Theorem, see, for example, Chapter 10 of Introduction to Econometrics.) We use a conditional version of the Cramér-Rao Theorem: If $f(y \mid \boldsymbol{x}, \boldsymbol{\theta})$ is a correctly specified probability model which depends on a finite dimensional parameter $\boldsymbol{\theta} \in \boldsymbol{\Theta}$, the support of $y$ does not depend on $\boldsymbol{\theta}, \boldsymbol{\theta}$ lies in the interior of $\boldsymbol{\theta}$, and if $\tilde{\boldsymbol{\theta}}$ is an unbiased estimator of $\boldsymbol{\theta}$ based on a sample of size $n$, then $\operatorname{var}[\tilde{\boldsymbol{\theta}} \mid \boldsymbol{X}] \geq\left(\sum_{i=1}^n \mathscr{I}{\boldsymbol{\theta}}\left(\boldsymbol{x}_i\right)\right)^{-1}$ where $\mathscr{I}{\boldsymbol{\theta}}(\boldsymbol{x})$ is the information matrix for model $f(y \mid \boldsymbol{x}, \boldsymbol{\theta})$.

For ease of exposition we focus on the case where $e_i$ has a conditional density $f(e \mid x)$. (The same argument applies to the discrete case using instead the probability mass function.)

The idea is as follows. The Cramér-Rao Theorem shows that within a parametric model an unbiased estimator cannot have lower variance than the inverse information matrix. This is true for any correctly-specified parametric model – which means any parametric model which includes the true distribution as a special case. Thus any correctly-specified parametric model produces a valid variance lower bound. The best bound is the supremum across these variance lower bounds. Rather than computing that directly we recognize that our goal is to produce a model with the specific variance lower bound $\left(\boldsymbol{X}^{\prime} \boldsymbol{D}^{-1} \boldsymbol{X}\right)^{-1}$. This is achieved if the information matrix equals $\boldsymbol{X}^{\prime} \boldsymbol{D}^{-1} \boldsymbol{X}$, which is achieved if the model has the likelihood score $x_i e_i \sigma_i^{-2}$. This suggests the parametric model for the error $e_i$
$$f(e \mid \boldsymbol{x}, \boldsymbol{\theta})=f(e \mid \boldsymbol{x})\left(1+\frac{\boldsymbol{\theta}^{\prime} \boldsymbol{x} e}{\sigma^2(\boldsymbol{x})}\right)$$
where $f(e \mid \boldsymbol{x})$ is the true conditional density. This model does not quite work, however, since this density is not necessarily non-negative. Consequently we use a technically more detailed argument using trimming to ensure a non-negative density.
For some $0<c<\infty$ define
$$\bar{\sigma}^2(\boldsymbol{x})=\mathbb{E}\left[e_i^2 \mathbb{1}\left(\left|e_i\right| \leq c / 2\right) \mid \boldsymbol{x}i=\boldsymbol{x}\right]$$ and $\bar{\sigma}_i^2=\bar{\sigma}^2\left(\boldsymbol{x}_i\right)$. Notice that as $c \rightarrow \infty, \bar{\sigma}_i^2 \rightarrow \sigma_i^2$ for each $i$. Set $\delta>0$. Pick $c$ sufficiently large so that $\bar{\sigma}_i^2 \geq \delta$ for all $i$. Let $M=\max {i \leq n}\left|\boldsymbol{x}_i\right|$.
Define the trimmed error
$$u_i=e_i \mathbb{1}\left(\left|e_i\right| \leq c / 2\right)-\mathbb{E}\left[e_i \mathbb{1}\left(\left|e_i\right| \leq c / 2\right) \mid \boldsymbol{x}_i\right]$$
Notice that $u_i$ satisfies $\left|u_i\right| \leq c, \mathbb{E}\left[u_i \mid \boldsymbol{x}_i\right]=0$, and $\bar{\sigma}_i^2=\mathbb{E}\left[e_i u_i \mid \boldsymbol{x}_i=\boldsymbol{x}\right]$.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Technical Proofs

$$f(e \mid \boldsymbol{x}, \boldsymbol{\theta})=f(e \mid \boldsymbol{x})\left(1+\frac{\boldsymbol{\theta}^{\prime} \boldsymbol{x} e}{\sigma^2(\boldsymbol{x})}\right)$$

$$u_i=e_i \mathbb{1}\left(\left|e_i\right| \leq c / 2\right)-\mathbb{E}\left[e_i \mathbb{1}\left(\left|e_i\right| \leq c / 2\right) \mid \boldsymbol{x}_i\right]$$

## 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。