## 经济代写|计量经济学代写Econometrics代考|ECON2271

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Panel data

A panel data set consists of a time series for each cross-sectional member in the data set; as an example we could consider the sales and the number of employees for 50 firms over a five-year period. Panel data can also be collected on a geographical basis; for example, we might have GDP and money supply data for a set of 20 countries and for a 20 -year period.

Panel data are denoted by the use of both $i$ and $t$ subscripts, which we have used before for cross-sectional and time series data, respectively. This is simply because panel data have both cross-sectional and time series dimensions. So, we might denote GDP for a set of countries and for a specific time period as:
$$Y_{i t} \quad \text { for } t=1,2,3, \ldots, T \text { and } i=1,2,3, \ldots, N$$
To better understand the structure of panel data, consider a cross-sectional and a time series variable as $N \times 1$ and $T \times 1$ matrices, respectively:

$$Y_t^{\text {ARGENTINA }}=\left(\begin{array}{c} Y_{1990} \ Y_{1991} \ Y_{1992} \ \vdots \ Y_{2012} \end{array}\right), \quad Y_i^{1990}=\left(\begin{array}{c} Y_{\text {ARGENTINA }} \ Y_{\text {BRAZIL }} \ Y_{U R U G U A Y} \ \vdots \ Y_{\text {VENEZUELA }} \end{array}\right)$$
Here $Y_t^{A R G E N T I N A}$ is the GDP for Argentina from 1990 to 2012 and $Y_i^{1990}$ is the GDP for 20 different Latin American countries.

## 经济代写|计量经济学代写Econometrics代考|The classical linear regression model

The classical linear regression model is a way of examining the nature and form of the relationships between two or more variables. In this chapter we consider the case of only two variables. One important issue in the regression analysis is the direction of causation between the two variables; in other words, we want to know which variable is affecting the other. Alternatively, this can be stated as which variable depends on the other. Therefore, we refer to the variables as the dependent variable (usually denoted by $Y$ ) and the independent or explanatory variable (usually denoted by $X$ ). We want to explain/predict the value of $Y$ for different values of the explanatory variable $X$. Let us assume that $X$ and $Y$ are linked by a simple linear relationship:
$$E\left(Y_t\right)=a+\beta X_t$$
where $E\left(Y_t\right)$ denotes the average value of $Y_t$ for given $X_t$ and unknown population parameters $a$ and $\beta$ (the subscript $t$ indicates that we have time series data). Equation (3.1) is called the population regression equation. The actual value of $Y_t$ will not always equal its expected value $E\left(Y_t\right)$. There are various factors that can ‘disturb’ its actual behaviour and therefore we can write actual $Y_t$ as:
$$Y_t=E\left(Y_t\right)+u_t$$
or
$$Y_t=a+\beta X_t+u_t$$
where $u_t$ is a disturbance. There are several reasons why a disturbance exists:
1 Omission of explanatory variables. There might be other factors (other than $X_t$ ) affecting $Y_t$ that have been left out of Equation (3.2). This may be because we do not know these factors, or even if we know them we might be unable to measure them in order to use them in a regression analysis.
2 Aggregation of variables. In some cases it is desirable to avoid having too many variables and therefore we attempt to summarize in aggregate a number of relationships in only one variable. Therefore, eventually we have only a good approximation of $Y_t$, with discrepancies that are captured by the disturbance term.
3 Model specification. We might have a misspecified model in terms of its structure. For example, it might be that $Y_t$ is not affected by $X_t$, but it is affected by the value of $X$ in the previous period (that is, $X_{t-1}$ ). In this case, if $X_t$ and $X_{t-1}$ are closely related, the estimation of Equation (3.2) will lead to discrepancies that are again captured by the error term.
4 Functional misspecification. The relationship between $X$ and $Y$ might be non-linear. We shall deal with non-linearities in other chapters of this text.
5 Measurement errors. If the measurement of one or more variables is not correct then errors appear in the relationship and these contribute to the disturbance term.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Panel data

$Y_{i t} \quad$ for $t=1,2,3, \ldots, T$ and $i=1,2,3, \ldots, N$

$Y_t^{\text {ARGENTINA }}=\left(Y_{1990} Y_{1991} Y_{1992} \vdots Y_{2012}\right), \quad Y_i^{1990}=\left(Y_{\text {ARGENTINA }} Y_{\text {BRAZIL }} Y_{\text {URUGUAY }}\right.$

## 经济代写|计量经济学代写Econometrics代考|The classical linear regression model

$$E\left(Y_t\right)=a+\beta X_t$$

$$Y_t=E\left(Y_t\right)+u_t$$

$$Y_t=a+\beta X_t+u_t$$

1 解释变量的遗漏。可能还有其他因素 (除了 $X_t$ ) 影响 $Y_t$ 已被排除在等式 (3.2)之外。这可能是因为我 们不知道这些因素，或者即使我们知道它们也可能无法测量它们以便在回归分析中使用它们。
2 变量的聚合。在某些情况下，希望避免有太多变量，因此我们试图仅在一个变量中汇总总结许多关系。 因此，最终我们只有一个很好的近似值 $Y_t$ ，具有由扰动项捕获的差异。
3 型号说明。就其结构而言，我们可能有一个错误指定的模型。例如，它可能是 $Y_t$ 不受 $X_t$ ，但它受值的影 响 $X$ 在上一时期 (即 $X_{t-1}$ ). 在这种情况下，如果 $X_t$ 和 $X_{t-1}$ 密切相关，方程 (3.2) 的估计将导致误差项 再次捕获的差异。
4 功能性错误说明。之间的关系 $X$ 和 $Y$ 可能是非线性的。我们将在本文的其他章节中处理非线性问题。
5 测量误差。如果一个或多个变量的测量不正确，则关系中会出现错误，这些错误会导致干扰项。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|计量经济学代写Econometrics代考|ECON2300

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Central limit theorem

If a set of data is iid with $n$ observations, $\left(\mathrm{Y}_1, \mathrm{Y}_2, \ldots \mathrm{Y}_n\right)$, and with a finite variance then as $n$ goes to infinity the distribution of $\bar{Y}$ becomes normal. So as long as $n$ is reasonably large we can think of the distribution of the mean as being approximately normal.

This is a remarkable result; what it says is that, regardless of the form of the population distribution, the sampling distribution will be normal as long as it is based on a large enough sample. To take an extreme example, suppose we think of a lottery which pays out one winning ticket for every 100 tickets sold. If the prize for a winning ticket is $\$ 100$and the cost of each ticket is$\$1$, then, on average, we would expect to earn $\$ 1$per ticket bought. But the population distribution would look very strange; 99 out of every 100 tickets would have a return of zero and one ticket would have a return of$\$100$. If we tried to graph the distribution of returns it would have a huge spike at zero and a small spike at $\$ 100$and no observations anywhere else. But, as long as we draw a reasonably large sample, when we calculate the mean return over the sample it will be centred on$\$1$ with a normal distribution around 1 .

The importance of the central limit theorem is that it allows us to know what the sampling distribution of the mean should look like as long as the mean is based on a reasonably large sample. So we can now replace the arbitrary triangular distribution in Figure 1.1 with a much more reasonable one, the normal distribution.

A final small piece of our statistical framework is the law of large numbers. This simply states that if a sample $\left(\mathrm{Y}_1, \mathrm{Y}_2, \ldots \mathrm{Y}_n\right)$ is IID with a finite variance then $\bar{Y}$ is a consistent estimator of $\mu$, the true population mean. This can be formally stated as $\operatorname{Pr}(|\bar{Y}-\mu|<\varepsilon) \rightarrow 1$ as $n \rightarrow \infty$, meaning that the probability that the absolute difference between the mean estimate and the true population mean will be less than a small positive number tends to one as the sample size tends to infinity. This can be proved straightforwardly, since, as we have seen, the variance of the sampling distribution of the mean is inversely proportional to $n$; hence as $n$ goes to infinity the variance of the sampling distribution goes to zero and the mean is forced to the true population mean.

We can now summarize: $\bar{Y}$ is an unbiased and consistent estimate of the true population mean $\mu$; it is approximately distributed as a normal distribution with a variance which is inversely proportional to $n$; this may be expressed as $N\left(\mu, \sigma^2 / n\right)$. So if we subtract the population mean from $\bar{Y}$ and divide by its standard deviation we will create a variable which has a mean of zero and a unit variance. This is called standardizing the variable.

## 经济代写|计量经济学代写Econometrics代考|Cross-sectional data

A cross-sectional data set consists of a sample of individuals, households, firms, cities, countries, regions or any other type of unit at a specific point in time. In some cases, the data across all units do not correspond to exactly the same time period. Consider a survey that collects data from questionnaire surveys of different families on different days within a month. In this case, we can ignore the minor time differences in collection and the data collected will still be viewed as a cross-sectional data set.

In econometrics, cross-sectional variables are usually denoted by the subscript $i$, with $i$ taking values of $1,2,3, \ldots, N$, for $N$ number of cross-sections. So if, for example, $Y$ denotes the income data we have collected for $N$ individuals, this variable, in a cross-sectional framework, will be denoted by:
$$Y_i \quad \text { for } i=1,2,3, \ldots, N$$
Cross-sectional data are widely used in economics and other social sciences. In economics, the analysis of cross-sectional data is associated mainly with applied microeconomics. Labour economics, state and local public finance, business economics, demographic economics and health economics are some of the prominent fields in microeconomics. Data collected at a given point in time are used in these cases to test microeconomic hypotheses and evaluate economic policies.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Cross-sectional data

$$Y_i \quad \text { for } i=1,2,3, \ldots, N$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|计量经济学代写Econometrics代考|ECON2271

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Variance Reduction: Antithetic Variates

As we have seen, obtaining sufficiently accurate results from a Monte Carlo experiment can often require that a great many replications be computed. This is not always feasible. In some cases, the number of replications that is needed can be substantially reduced by using certain techniques for reducing the variance of experimental results. In the econometric literature, the variance reduction techniques which have achieved prominence are the use of antithetic variates and control variates. We discuss the former method in this section and the latter method in the next one.

The idea of antithetic variates is to calculate two different estimates of the quantity of interest in such a way that the two estimates are negatively correlated. Their average will then be substantially more accurate than either of them individually. Suppose that we wish to estimate some quantity $\theta$, and that in a single Monte Carlo experiment we can obtain two different unbiased estimators of $\theta$, say $\dot{\theta}$ and $\grave{\theta}$. These are the antithetic variates. Then the pooled estimator
$$\bar{\theta}=\frac{1}{2}(\dot{\theta}+\grave{\theta})$$
has variance
$$V(\bar{\theta})=\frac{1}{4}(V(\dot{\theta})+V(\grave{\theta})+2 \operatorname{Cov}(\dot{\theta}, \grave{\theta}))$$
where $V(\dot{\theta})$ and $V(\grave{\theta})$ denote the variances of $\dot{\theta}$ and $\grave{\theta}$. If $\operatorname{Cov}(\dot{\theta}, \grave{\theta})$ is negative, $V(\bar{\theta})$ will be smaller than $\frac{1}{4}(V(\dot{\theta})+V(\grave{\theta}))$, which is the variance that we would have obtained using the same number of replications to estimate $\theta$ from two independent experiments. The extent to which we can gain by using antithetic variates thus depends on how strong the negative correlation is between $\dot{\theta}$ and $\grave{\theta}$

One might ask why $\dot{\theta}$ and $\grave{\theta}$ should receive equal weight in computing $\bar{\theta}$. Let us therefore consider the weighted estimator
$$\ddot{\theta} \equiv w \dot{\theta}+(1-w) \dot{\theta}$$
Differentiating the variance of $\ddot{\theta}$ with respect to $w$ and setting the result to zero, we find that
$$w=\frac{V(\grave{\theta})-\operatorname{Cov}(\dot{\theta}, \grave{\theta})}{V(\dot{\theta})+V(\grave{\theta})-2 \operatorname{Cov}(\dot{\theta}, \grave{\theta})}$$

## 经济代写|计量经济学代写Econometrics代考|Variance Reduction: Control Variates

The second widely used technique for variance reduction is to employ control variates. A control variate is a random variable of which the distribution (or at least certain properties of the distribution) is known and that is correlated with the estimator(s) or test statistic(s) which are being investigated. The principal property that a control variate must have is a known population mean. The divergence between the sample mean of the control variate in the experiment and its known population mean is then used to improve the estimates from the Monte Carlo experiment. This obviously works best if the control variate is highly correlated with the estimators or test statistics with which the experiment is concerned.

Typically, control variates are statistics which could never be computed in practice but which can be calculated in the context of a Monte Carlo experiment, because the DGP is known. For example, suppose the experiment concerns the estimates of $\boldsymbol{\beta}$ from a nonlinear regression model with normal errors,
$$\boldsymbol{y}=\boldsymbol{x}(\boldsymbol{\beta})+\boldsymbol{u}, \quad \boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^2 \mathbf{I}\right),$$
where $\boldsymbol{x}(\boldsymbol{\beta})$ depends only on $\boldsymbol{\beta}$ and on regressors that are fixed or least independent of $\boldsymbol{u}$. We saw in Section 5.4 that
$$n^{1 / 2}\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\right)=\left(n^{-1} \boldsymbol{X}_0^{\top} \boldsymbol{X}_0\right)^{-1} n^{-1 / 2} \boldsymbol{X}_0^{\top} \boldsymbol{u}+o(1) .$$
Thus it is natural to consider using the vector
$$\ddot{\boldsymbol{\beta}}=\left(\boldsymbol{X}_0^{\top} \boldsymbol{X}_0\right)^{-1} \boldsymbol{X}_0^{\top} \boldsymbol{u}$$
as a source of control variates. This vector will evidently be normally distributed with mean vector zero and covariance matrix $\sigma_0^2\left(\boldsymbol{X}_0^{\top} \boldsymbol{X}_0\right)^{-1}$. It would be impossible to compute $\ddot{\boldsymbol{\beta}}$ from a real data set, but in the context of a Monte Carlo experiment, it is perfectly easy to do so. We know $\boldsymbol{\beta}_0$ and hence $\boldsymbol{X}_0 \equiv \boldsymbol{X}\left(\boldsymbol{\beta}_0\right)$. Using these and the error vector $\boldsymbol{u}^j$ that we generate at each replication, we can easily compute $\ddot{\boldsymbol{\beta}}^j$.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Variance Reduction: Antithetic Variates

$$\bar{\theta}=\frac{1}{2}(\dot{\theta}+\grave{\theta})$$

$$V(\bar{\theta})=\frac{1}{4}(V(\dot{\theta})+V(\grave{\theta})+2 \operatorname{Cov}(\dot{\theta}, \grave{\theta}))$$

$$\ddot{\theta} \equiv w \dot{\theta}+(1-w) \dot{\theta}$$

$$w=\frac{V(\grave{\theta})-\operatorname{Cov}(\dot{\theta}, \grave{\theta})}{V(\dot{\theta})+V(\grave{\theta})-2 \operatorname{Cov}(\dot{\theta}, \grave{\theta})}$$

## 经济代写|计量经济学代写Econometrics代考|Variance Reduction: Control Variates

$$\boldsymbol{y}=\boldsymbol{x}(\boldsymbol{\beta})+\boldsymbol{u}, \quad \boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^2 \mathbf{I}\right)$$

$$n^{1 / 2}\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\right)=\left(n^{-1} \boldsymbol{X}_0^{\top} \boldsymbol{X}_0\right)^{-1} n^{-1 / 2} \boldsymbol{X}_0^{\top} \boldsymbol{u}+o(1)$$

$$\ddot{\boldsymbol{\beta}}=\left(\boldsymbol{X}_0^{\top} \boldsymbol{X}_0\right)^{-1} \boldsymbol{X}_0^{\top} \boldsymbol{u}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|计量经济学代写Econometrics代考|BEA472

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Monte Carlo Experiments

Most of the methods for estimation and hypothesis testing discussed in this book have statistical properties that are known only asymptotically. This is true for nonlinear models of all types, for linear simultaneous equations models, and even for the univariate linear regression model once we dispense with the strong assumption of fixed regressors or the even stronger assumption that the error terms are normally and identically distributed. Thus, in practice, exact finite-sample theory can rarely be used to interpret estimates or test statistics. Unfortunately, unless the sample size is very large indeed, it is difficult to know whether asymptotic theory is sufficiently accurate to allow us to interpret our results with confidence.

There are basically two ways to deal with this situation. One is to refine asymptotic approximations such as those we have derived in this book by adding terms of lower order in the sample size $n$, typically terms that are $O\left(n^{-1 / 2}\right)$ or $O\left(n^{-1}\right)$. These more refined approximations are variously referred to as finite-sample approximations or as asymptotic expansions. The asymptotic expansions approach has been most extensively employed in studying the properties of estimators of simultaneous equations models and estimators of univariate linear dynamic models. This approach can, in some cases, yield valuable insights into the behavior of estimators and test statistics. Unfortunately, it frequently involves mathematics that are either more advanced or more tedious than most applied econometricians are comfortable with. It is often applicable only to relatively simple models, and it tends to produce results that are complicated and very difficult to interpret, in part because they often depend on unknown parameters. Moreover, these results are themselves only approximations; while they are generally better than asymptotic approximations, they may not be accurate enough. Ideally, one would like to be able to use asymptotic expansions routinely, as part of econometric software packages, in order to obtain confidence intervals and hypothesis tests more accurate than the asymptotic ones discussed in this book. Unfortunately, this ideal situation appears to be a very long way off, although recent work such as Rothenberg (1988) has perhaps moved us a little closer. Two useful surveys of methods based on asymptotic expansions are Phillips (1983) and Rothenberg (1984). A somewhat critical review of the literature is Taylor (1983).

## 经济代写|计量经济学代写Econometrics代考|Designing Monte Carlo Experiments

The hardest part of doing a set of Monte Carlo experiments is usually designing them. Limitations on computing resources, the experimenter’s time, and the amount of space that can reasonably be devoted to presenting the results mean that it is usually practical to perform only a small number of experiments. These must be designed to shed as much light as possible on the issues of interest.

The first thing to recognize is that results from Monte Carlo experiments are necessarily random. At a minimum, this means that results must be reported in a way which allows readers to appreciate the extent of experimental randomness. Moreover, it is essential to perform enough replications so that the results are sufficiently accurate for the purpose at hand. The number of replications that is needed can sometimes be substantially reduced by using variance reduction techniques, which will be discussed in the next two sections. Such techniques are by no means always readily available, however. In this section, we consider various other aspects of the design of Monte Carlo experiments.

We first consider the problem of determining how many replications to perform. As an example, suppose that the investigator is interested in calculating the size of a certain test statistic (i.e., the probability of rejecting the null hypothesis when it is true) at, say, the nominal .05 level. Let us denote this unknown quantity by $p$. Each replication will generate a test statistic that either exceeds or does not exceed the nominal critical value. These can be thought of as independent Bernoulli trials. Suppose $N$ replications are performed and $R$ rejections are obtained. Then the obvious estimator of $p$, which is also the ML estimator, is $R / N$. The variance of this estimator is $N^{-1} p(1-p)$, which can be estimated by $R(N-R) / N^3$.

Now suppose that one wants the length of a $95 \%$ confidence interval on the estimate of $p$ to be approximately .01. Using the normal approximation to the binomial, which is surely valid here since $N$ will be a large number, we see that the confidence interval must cover $2 \times 1.96=3.92$ standard errors. Hence we require that
$$3.92\left(\frac{p(1-p)}{N}\right)^{1 / 2}=.01$$

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Designing Monte Carlo Experiments

$$3.92\left(\frac{p(1-p)}{N}\right)^{1 / 2}=.01$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|计量经济学代写Econometrics代考|BEA472

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Simultaneous Equations Models

For many years, the linear simultaneous equations model was the centerpiece of econometric theory. We discussed a special case of this model, a two-equation demand-supply model, in Section 7.3. The purpose of that discussion was simply to show that simultaneity induces correlation between the regressors and error terms of each equation of the system, thus causing OLS to be inconsistent and justifying the use of instrumental variables. The inconsistency of least squares estimators of individual equations in simultaneous equations models is by no means the only econometric issue that arises in such models. In this chapter, we therefore discuss simultaneous equations models at some length.

Much of the early work on simultaneous equations models was done under the auspices of the Cowles Commission; references include Koopmans (1950) and Hood and Koopmans (1953). This work heavily influenced the direction of econometric theory for many years. For a history of the early development of econometrics, see Morgan (1990). Because the literature on simultaneous equations models is vast, we will be able to deal with only a small part of it. There are many surveys of the field and many textbook treatments at various levels. Two useful survey articles are Hausman (1983), which deals with the mainstream literature, and Phillips (1983), which deals with the rather specialized field of small-sample theory in simultaneous equations models, a subject that we will not discuss at all.

The essential feature of simultaneous equations models is that two or more endogenous variables are determined jointly within the model, as a function of exogenous variables, predetermined variables, and error terms. Up to this point, we have said very little about what we mean by exogenous and predetermined variables. Since the role of such variables in simultaneous equations models is critical, it is time to rectify that omission. In Section 18.2, we will therefore discuss the important concept of exogeneity at some length.
Most of the chapter will be concerned with the linear simultaneous equations model. Suppose there are $g$ endogenous variables, and hence $g$ equations,and $k$ exogenous or predetermined variables. Then the model can be written in matrix form as
$$\boldsymbol{Y} \boldsymbol{\Gamma}=\boldsymbol{X} \boldsymbol{B}+\boldsymbol{U}$$

## 经济代写|计量经济学代写Econometrics代考|Exogeneity and Causality

In the case of a single regression equation, we are estimating the distribution, or at least the mean and variance, of an endogenous variable conditional on the values of some explanatory variables. In the case of a simultaneous equations model, we are estimating the joint distribution of two or more endogenous variables conditional on the values of some explanatory variables. But we have not yet discussed the conditions under which one can validly treat a variable as explanatory. This includes the use of such variables as regressors in least squares estimation and as instruments in instrumental variables or GMM estimation. For conditional inference to be valid, the explanatory variables must be either predetermined or exogenous in one or other of a variety of senses to be defined below.

In a time-series context, we have seen that random variables which are predetermined can safely be used as explanatory variables in least squares estimation, at least asymptotically. In fact, lagged endogenous variables are regularly used both as explanatory variables and as instrumental variables. However, there are a great many cases, including of course models estimated with cross-section data, in which we want to use variables that are not predetermined as explanatory variables. Moreover, the concept of predeterminedness turns out to be somewhat trickier than one might expect, since it is not invariant to reparametrizations of the model. Thus it is clear that we need a much more general concept than that of predeterminedness.

It is convenient to begin with formal definitions of the concept of predeterminedness and the related concept of strict exogeneity. In this, we are following the standard exposition of these matters, given in Engle, Hendry, and Richard (1983). Readers should be warned that this paper, although a standard reference, is not easy to read. Our discussion will be greatly simplified relative to theirs and will be given in a more general context, since they restrict themselves to fully specified parametric models capable of being estimated by maximum likelihood. We will, however, make use of one of their specific examples as a concrete illustration of a number of points.

Let the $1 \times g$ vector $\boldsymbol{Y}_t$ denote the $t^{\text {th }}$ observation on a set of variables that we wish to model as a simultaneous process, and let the $1 \times k$ vector $\boldsymbol{X}_t$ be the $t^{\text {th }}$ observation on a set of explanatory variables, some or all of which may be lagged $\boldsymbol{Y}_t$ ‘s. We may write an, in general nonlinear, simultaneous equations model as
$$\boldsymbol{h}_t\left(\boldsymbol{Y}_t, \boldsymbol{X}_t, \boldsymbol{\theta}\right)=\boldsymbol{U}_t,$$
where $\boldsymbol{h}_t$ is a $1 \times g$ vector of functions, somewhat analogous to the regression function of a univariate model, $\boldsymbol{\theta}$ is a $p$-vector of parameters, and $\boldsymbol{U}_t$ is a $1 \times g$ vector of error terms. The linear model (18.01) is seen to be a special case of (18.04) if we rewrite it as
$$\boldsymbol{Y}_t \boldsymbol{\Gamma}=\boldsymbol{X}_t \boldsymbol{B}+\boldsymbol{U}_t$$
and define $\boldsymbol{\theta}$ so that it consists of all the elements of $\boldsymbol{\Gamma}$ and $\boldsymbol{B}$ which have to be estimated. Here $\boldsymbol{X}_t$ and $\boldsymbol{Y}_t$ are the $t^{\text {th }}$ rows of the matrices $\boldsymbol{X}$ and $\boldsymbol{Y}$. A set of (conditional) moment conditions could be based on (18.04), by writing
$$E\left(\boldsymbol{h}_t\left(\boldsymbol{Y}_t, \boldsymbol{X}_t, \boldsymbol{\theta}\right)\right)=\mathbf{0}$$
where the expectation could be interpreted as being conditional on some appropriate information set.

## 经济代写|计量经济学代写Econometrics代考|Simultaneous Equations Models

$$\boldsymbol{Y} \boldsymbol{\Gamma}=\boldsymbol{X} \boldsymbol{B}+\boldsymbol{U}$$

## 经济代写|计量经济学代写Econometrics代考|Exogeneity and Causality

$$\boldsymbol{h}_t\left(\boldsymbol{Y}_t, \boldsymbol{X}_t, \boldsymbol{\theta}\right)=\boldsymbol{U}_t$$

$$\boldsymbol{Y}_t \boldsymbol{\Gamma}=\boldsymbol{X}_t \boldsymbol{B}+\boldsymbol{U}_t$$

$$E\left(\boldsymbol{h}_t\left(\boldsymbol{Y}_t, \boldsymbol{X}_t, \boldsymbol{\theta}\right)\right)=\mathbf{0}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|计量经济学代写Econometrics代考|EFN508

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Covariance Matrix Estimation

In previous sections, we mentioned the difficulties that can arise in estimating covariance matrices in the GMM context. In fact, problems occur at two distinct points: once for the choice of the weighting matrix to be used in constructing a criterion function and again for estimating the asymptotic covariance matrix of the estimates. Fortunately, similar considerations apply to both problems, and so we can consider them together.

Recall from (17.31) that the asymptotic covariance matrix of a GMM estimator computed using a weighting matrix $\boldsymbol{A}0$ is $$\left(\boldsymbol{D}^{\top} A_0 \boldsymbol{D}\right)^{-1} \boldsymbol{D}^{\top} \boldsymbol{A}_0 \boldsymbol{\Phi} \boldsymbol{A}_0 \boldsymbol{D}\left(\boldsymbol{D}^{\top} \boldsymbol{A}_0 \boldsymbol{D}\right)^{-1}$$ in the notation of Section 17.2. If the necessary condition for efficiency of $l \times l$ asymptotic covariance matrix of the empirical moments $n^{-1 / 2} \boldsymbol{F}^{\top}(\boldsymbol{\theta}) \iota$ with typical element $$n^{-1 / 2} \sum{t=1}^n f_{t i}\left(y_t, \boldsymbol{\theta}\right)$$

Thus the problem is to find a consistent estimator $\hat{\boldsymbol{\Phi}}$ of $\boldsymbol{\Phi}$. If we can do so, we can minimize the criterion function
$$\iota^{\top} \boldsymbol{F}(\boldsymbol{\theta}) \hat{\Phi}^{-1} \boldsymbol{F}^{\top}(\theta) \iota$$
If a typical element of $\hat{\boldsymbol{D}}$ is defined by (17.32), the asymptotic covariance matrix of $\hat{\boldsymbol{\theta}}$ may then he eatimated by
$$\frac{1}{n}\left(\hat{\boldsymbol{D}}^{\top} \hat{\boldsymbol{\Phi}}^{-1} \hat{\boldsymbol{D}}\right)^{-1}$$
It is clear that we must proceed in at least two steps, because $\hat{\boldsymbol{\Phi}}$ is to be an estimate of the covariance matrix of the empirical moments evaluated at the true parameter values. Thus before $\hat{\Phi}$ can be calculated, it is necessary to have a preliminary consistent estimator of the parameters $\boldsymbol{\theta}$. Since one can use an arbitrary weighting matrix $\boldsymbol{A}_0$ without losing consistency, there are many ways to find this preliminary estimate. Then $\hat{\Phi}$ can be computed, and subsequently, by minimizing (17.54), a new set of parameter estimates can be obtained. If it seems desirable, one may repeat these steps one or more times. In theory, one iteration is enough for asymptotic efficiency but, in practice, the original estimates may be bad enough for several iterations to be advisable.

Our previous definition of $\Phi,(17.29)$, was based on the assumption that the empirical moments $f_{t i}$ were serially independent. Since we wish to relax that assumption in this section, it is necessary to make a new definition of $\Phi$, in order that it should still be the asymptotic covariance matrix of the empirical moments. We therefore make the definition:
$$\boldsymbol{\Phi} \equiv \lim {n \rightarrow \infty}\left(\frac{1}{n} \sum{t=1}^n \sum_{s=1}^n E_\mu\left(\boldsymbol{F}_t^{\top}\left(y_t, \boldsymbol{\theta}_0\right) \boldsymbol{F}_s\left(y_t, \boldsymbol{\theta}_0\right)\right)\right),$$
where $\boldsymbol{F}_t$ is the $t^{\text {th }}$ row of the $n \times l$ matrix $\boldsymbol{F}$. Since the DGP $\mu$ will remain fixed for what follows, we will drop it from our notation. (17.56) differs from (17.29) in that it allows for any pattern of correlation of the contributions $\boldsymbol{F}_t$ to the empirical moments and remains valid even if no central limit theorem does. It is necessary, of course, to assume that the limit in (17.56) exists. Our task now is to find a consistent estimator of (17.56).

## 经济代写|计量经济学代写Econometrics代考|Inference with GMM Models

In this section, we undertake an investigation of how hypotheses may be tested in the context of GMM models. We begin by looking at tests of overidentifying restrictions and then move on to develop procedures akin to the classical tests studied in Chapter 13 for models estimated by maximum likelihood. The similarities to procedures we have already studied are striking. There is one important difference, however: We will not be able to make any great use of artificial linear regressions in order to implement the tests we discuss. The reason is simply that such artificial regressions have not yet been adequately developed. They exist only for some special cases, and their finite-sample properties are almost entirely unknown. However, there is every reason to hope and expect that in a few years it will be possible to perform inference on GMM models by means of artificial regressions still to be invented.

In the meantime, there are several testing procedures for GMM models that are not difficult to perform. The most important of these is a test of the overidentifying restrictions that are usually imposed. Suppose that we have estimated a vector $\theta$ of $k$ parameters by minimizing the criterion function
$$\iota^{\top} \boldsymbol{F}(\boldsymbol{\theta}) \hat{\Phi}^{-1} \boldsymbol{F}^{\top}(\boldsymbol{\theta}) \iota$$
in which the empirical moment matrix $\boldsymbol{F}(\boldsymbol{\theta})$ has $l>k$ columns. Observe that

we have used a weighting matrix $\hat{\boldsymbol{\Phi}}^{-1}$ that satisfies the necessary condition of Theorem 17.3 for the efficiency of the GMM estimator. Only $k$ moment conditions are needed to identify $k$ parameters, and so there are $l-k$ overidentifying restrictions implicit in the estimation we have performed. As we emphasized in Chapter 7, where we first encountered overidentifying restrictions, it should be a routine practice always to test these restrictions before making any use of the estimation results.

One way of doing so, suggested by Hansen (1982), is to use as a test statistic the minimized value of the criterion function. The test statistic is simply (17.67) evaluated at $\boldsymbol{\theta}=\hat{\boldsymbol{\theta}}$ and divided by the sample size $n$ :
$$\frac{1}{n} \iota^{\top} \hat{\boldsymbol{F}} \hat{\Phi}^{-1} \hat{\boldsymbol{F}}^{\top} \boldsymbol{\iota}$$
where, as usual, we write $\hat{\boldsymbol{F}}$ for $\boldsymbol{F}(\hat{\boldsymbol{\theta}})$. The factor of $n^{-1}$ is necessary to offset the factor of $n$ in $\hat{\Phi}^{-1}$, which arises from the fact that $\Phi$ is defined in (17.29) as the covariance matrix of $n^{-1 / 2} \boldsymbol{F}_0^{\top} \iota$. The definition (17.29) therefore implies that if the overidentifying restrictions are correct, the asymptotic distribution of $n^{-1 / 2} \boldsymbol{F}_0^{\top} \iota$ is $N(\mathbf{0}, \boldsymbol{\Phi})$.

## 经济代写|计量经济学代写Econometrics代考|Covariance Matrix Estimation

$$\left(\boldsymbol{D}^{\top} A_0 \boldsymbol{D}\right)^{-1} \boldsymbol{D}^{\top} \boldsymbol{A}0 \boldsymbol{\Phi} \boldsymbol{A}_0 \boldsymbol{D}\left(\boldsymbol{D}^{\top} \boldsymbol{A}_0 \boldsymbol{D}\right)^{-1}$$ 在第 17.2 节的符号中。如果效率的必要条件 $l \times l$ 经验矩的渐近协方差矩阵 $n^{-1 / 2} \boldsymbol{F}^{\top}(\boldsymbol{\theta}) \iota$ 具有典型元素 $$n^{-1 / 2} \sum t=1^n f{t i}\left(y_t, \boldsymbol{\theta}\right)$$

$$\iota^{\top} \boldsymbol{F}(\boldsymbol{\theta}) \hat{\Phi}^{-1} \boldsymbol{F}^{\top}(\theta) \iota$$

$$\frac{1}{n}\left(\hat{\boldsymbol{D}}^{\top} \hat{\boldsymbol{\Phi}}^{-1} \hat{\boldsymbol{D}}\right)^{-1}$$

$$\boldsymbol{\Phi} \equiv \lim n \rightarrow \infty\left(\frac{1}{n} \sum t=1^n \sum_{s=1}^n E_\mu\left(\boldsymbol{F}_t^{\top}\left(y_t, \boldsymbol{\theta}_0\right) \boldsymbol{F}_s\left(y_t, \boldsymbol{\theta}_0\right)\right)\right)$$

## 经济代写|计量经济学代写Econometrics代考|Inference with GMM Models

$$\iota^{\top} \boldsymbol{F}(\boldsymbol{\theta}) \hat{\Phi}^{-1} \boldsymbol{F}^{\top}(\boldsymbol{\theta}) \iota$$

Hansen (1982) 建议的一种方法是使用标准函数的最小值作为检验统计量。测试统计量只是 (17.67) 在 $\boldsymbol{\theta}=\hat{\boldsymbol{\theta}}$ 并除以样本量 $n:$
$$\frac{1}{n} \iota \iota^{\top} \hat{\boldsymbol{F}} \hat{\Phi}^{-1} \hat{\boldsymbol{F}}^{\top} \iota$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|计量经济学代写Econometrics代考|ECON2271

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Confidence Intervals

Estimation methods considered in Sect. $2.2$ give us a point estimate of a parameter, say $\mu$, and that is the best bet, given the data and the estimation method, of what $\mu$ might be. But it is always good policy to give the client an interval, rather than a point estimate, where with some degree of confidence, usually $95 \%$ confidence, we expect $\mu$ to lie. We have seen in Fig. $2.5$ that for a $N(0,1)$ random variable $z$, we have
$$\operatorname{Pr}\left[-z_{\alpha / 2} \leq z \leq z_{\alpha / 2}\right]=1-\alpha$$
and for $\alpha=5 \%$, this probability is $0.95$, giving the required $95 \%$ confidence. In fact, $z_{\alpha / 2}=1.96$ and
$$\operatorname{Pr}[-1.96 \leq z \leq 1.96]=0.95$$
This says that if we draw 100 random numbers from a $N(0,1)$ density, (using a normal random number generator) we expect 95 out of these 100 numbers to lie in the $[-1.96,1.96]$ interval. Now, let us get back to the problem of estimating $\mu$ from a random sample $x_1, \ldots, x_n$ drawn from a $N\left(\mu, \sigma^2\right)$ distribution. We found out that $\widehat{\mu}_{M L E}=\bar{x}$ and $\bar{x} \sim N\left(\mu, \sigma^2 / n\right)$. Hence, $z=(\bar{x}-\mu) /(\sigma / \sqrt{n})$ is $N(0,1)$. The point estimate for $\mu$ is $\bar{x}$ observed from the sample, and the $95 \%$ confidence interval for $\mu$ is obtained by replacing $z$ by its value in the above probability statement:
$$\operatorname{Pr}\left[-z_{\alpha / 2} \leq \frac{\bar{x}-\mu}{\sigma / \sqrt{n}} \leq z_{\alpha / 2}\right]=1-\alpha$$
Assuming $\sigma$ is known for the moment, one can rewrite this probability statement after some simple algebraic manipulations as
$$\operatorname{Pr}\left[\bar{x}-z_{\alpha / 2}(\sigma / \sqrt{n}) \leq \mu \leq \bar{x}+z_{\alpha / 2}(\sigma / \sqrt{n})\right]=1-\alpha$$
Note that this probability statement has random variables on both ends and the probability that these random variables sandwich the unknown parameter $\mu$ is $1-\alpha$. With the same confidence of drawing 100 random $N(0,1)$ numbers and finding 95 of them falling in the $(-1.96,1.96)$ range we are confident that if we drew a 100 samples and computed a $100 \bar{x}$ ‘s, and a 100 intervals $(\bar{x} \pm 1.96 \sigma / \sqrt{n}), \mu$ will lie in these intervals in 95 out of 100 times.

If $\sigma$ is not known, and is replaced by $s$, then Problem 12 shows that this is equivalent to dividing a $N(0,1)$ random variable by an independent $\chi_{n-1}^2$ random variable divided by its degrees of freedom, leading to a $t$-distribution with $(n-1)$ degrees of freedom. Hence, using the $t$-tables for $(n-1)$ degrees of freedom
$$\operatorname{Pr}\left[-t_{\alpha / 2 ; n-1} \leq t_{n-1} \leq t_{\alpha / 2 ; n-1}\right]=1-\alpha$$
and replacing $t_{n-1}$ by $(\bar{x}-\mu) /(s / \sqrt{n})$ one gets
$$\operatorname{Pr}\left[\bar{x}-t_{\alpha / 2 ; n-1}(s / \sqrt{n}) \leq \mu \leq \bar{x}+t_{\alpha / 2 ; n-1}(s / \sqrt{n})\right]=1-\alpha$$

## 经济代写|计量经济学代写Econometrics代考|Simple Linear Regression

In this chapter, we study extensively the estimation of a linear relationship between two variables, $Y_i$ and $X_i$, of the form:
$$Y_i=\alpha+\beta X_i+u_i \quad i=1,2, \ldots, n$$
where $Y_i$ denotes the $i$-th observation on the dependent variable $Y$ which could be consumption, investment, or output, and $X_i$ denotes the $i$-th observation on the independent variable $X$ which could be disposable income, the interest rate, or an input. These observations could be collected on firms or households at a given point in time, in which case we call the data a cross-section. Alternatively, these observations may be collected over time for a specific industry or country in which case we call the data a time-series. $n$ is the number of observations, which could be the number of firms or households in a cross-section, or the number of years if the observations are collected annually. $\alpha$ and $\beta$ are the intercept and slope of this simple linear relationship between $Y$ and $X$. They are assumed to be unknown parameters to be estimated from the data. A plot of the data, i.e., $Y$ versus $X$ would be very illustrative showing what type of relationship exists empirically between these two variables. For example, if $Y$ is consumption and $X$ is disposable income, then we would expect a positive relationship between these variables and the data may look like Fig. $3.1$ when plotted for a random sample of households. If $\alpha$ and $\beta$ were known, one could draw the straight line $(\alpha+\beta X)$ as shown in Fig. 3.1. It is clear that not all the observations $\left(X_i, Y_i\right)$ lie on the straight line $(\alpha+\beta X)$. In fact, Eq. (3.1) states that the difference between each $Y_i$ and the corresponding $\left(\alpha+\beta X_i\right)$ is due to a random error $u_i$. This error may be due to (i) the omission of relevant factors that could influence consumption, other than disposable income, like real wealth or varying tastes, or unforeseen events that induce households to consume more or less, (ii) measurement error, which could be the result of households not reporting their consumption or income accurately, or (iii) wrong choice of a linear relationship between consumption and income, when the true relationship may be nonlinear. These different causes of the error term will have different effects on the distribution of this error. In what follows, we consider only disturbances that satisfy some restrictive assumptions. In later chapters, we relax these assumptions to account for more general kinds of error terms.

## 经济代写|计量经济学代写Econometrics代考|Confidence Intervals

$$\operatorname{Pr}\left[-z_{\alpha / 2} \leq z \leq z_{\alpha / 2}\right]=1-\alpha$$

$$\operatorname{Pr}[-1.96 \leq z \leq 1.96]=0.95$$

$$\operatorname{Pr}\left[\bar{x}-z_{\alpha / 2}(\sigma / \sqrt{n}) \leq \mu \leq \bar{x}+z_{\alpha / 2}(\sigma / \sqrt{n})\right]=1-\alpha$$

$$\operatorname{Pr}\left[-t_{\alpha / 2 ; n-1} \leq t_{n-1} \leq t_{\alpha / 2 ; n-1}\right]=1-\alpha$$

$$\operatorname{Pr}\left[\bar{x}-t_{\alpha / 2 ; n-1}(s / \sqrt{n}) \leq \mu \leq \bar{x}+t_{\alpha / 2 ; n-1}(s / \sqrt{n})\right]=1-\alpha$$

## 经济代写|计量经济学代写Econometrics代考|Simple Linear Regression

$$Y_i=\alpha+\beta X_i+u_i \quad i=1,2, \ldots, n$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|计量经济学代写Econometrics代考|BEA472

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Comparing Biased and Unbiased Estimators

Suppose we are given two estimators $\widehat{\theta}_1$ and $\widehat{\theta}_2$ of $\theta$ where the first is unbiased and has a large variance and the second is biased but with a small variance. The question is which one of these two estimators is preferable? $\widehat{\theta}_1$ is unbiased whereas $\widehat{\theta}_2$ is biased. This means that if we repeat the sampling procedure many times, then we expect $\widehat{\theta}_1$ to be on the average correct, whereas $\widehat{\theta}_2$ would be on the average different from $\theta$. However, in real life, we observe only one sample. With a large variance for $\widehat{\theta}_1$, there is a great likelihood that the sample drawn could result in a $\widehat{\theta}_1$ far away from $\theta$. However, with a small variance for $\widehat{\theta}_2$, there is a better chance of getting a $\widehat{\theta}_2$ close to $\theta$. If our loss function is quadratic so that we are penalized when $\widehat{\theta}$ is different from $\theta$ by $L(\widehat{\theta}, \theta)=(\widehat{\theta}-\theta)^2$, then our risk is
\begin{aligned} R(\widehat{\theta}, \theta) &=E[L(\widehat{\theta}, \theta)]=E(\widehat{\theta}-\theta)^2=M S E(\widehat{\theta}) \ &=E[\widehat{\theta}-E(\widehat{\theta})+E(\widehat{\theta})-\theta]^2=\operatorname{var}(\widehat{\theta})+(\operatorname{Bias}(\widehat{\theta}))^2 . \end{aligned}
Minimizing the risk when the loss function is quadratic is equivalent to minimizing the Mean Square Error (MSE). From its definition the MSE shows the trade-off between bias and variance. MVU theory sets the bias equal to zero and minimizes $\operatorname{var}(\widehat{\theta})$. In other words, it minimizes the above risk function but only over $\widehat{\theta}$ ‘s that are unbiased. If we do not restrict ourselves to unbiased estimators of $\theta$, minimizing MSE may result in a biased estimator such as $\widehat{\theta}_2$ which beats $\widehat{\theta}_1$ because the gain from its smaller variance outweighs the loss from its small bias, see Fig. 2.2.

## 经济代写|计量经济学代写Econometrics代考|Hypothesis Testing

The best way to proceed is with an example.
Example 2.10. The Economics Departments instituted a new program to teach micro-principles. We would like to test the null hypothesis that $80 \%$ of economics undergraduate students will pass the micro-principles course versus the alternative hypothesis that only $50 \%$ will pass. We draw a random sample of size 20 from the large undergraduate micro-principles class, and as a simple rule we accept the null if $x$, the number of passing students is larger or equal to 13 , otherwise the alternative hypothesis will be accepted. Note that the distribution we are drawing from is Bernoulli with the probability of success $\theta$, and we have chosen only two states of the world $H_0 ; \theta_0=0.80$ and $H_1 ; \theta_1=0.5$. This situation is known as testing a simple hypothesis versus another simple hypothesis because the distribution is completely specified under the null $H_0$ or the alternative hypothesis $H_1$. One would expect $\left(E(x)=n \theta_0\right) 16$ students under $H_0$ and $\left(n \theta_1\right) 10$ students under $H_1$ to pass the micro-principles exams. It seems then logical to take $x \geq 13$ as the cutoff point distinguishing $H_0$ from $H_1$. No theoretical justification is given at this stage to this arbitrary choice except to say that it is the mid-point of $\lfloor 10,16]$. Figure $2.3$ shows that one can make two types of errors. The first is rejecting $H_0$ when in fact it is true; this is known as type I error and the probability of committing this error is denoted by $\alpha$. The second is accepting $H_0$ when it is false. This is known as type II error, and the corresponding probability is denoted by $\beta$. For this example
\begin{aligned} \alpha &=\operatorname{Pr}\left[\text { rejecting } H_0 / H_0 \text { is true }\right]=\operatorname{Pr}[x<13 / \theta=0.8] \ &=b(n=20 ; x=0 ; \theta=0.8)+. .+b(n=20 ; x=12 ; \theta=0.8) \ &=b(n=20 ; x=20 ; \theta=0.2)+. .+b(n=20 ; x=8 ; \theta=0.2) \ &=0+. .+0+0.0001+0.0005+0.0020+0.0074+0.0222=0.0322 \end{aligned}

where we have used the fact that $b(n ; x ; \theta)=b(n ; n-x ; 1-\theta)$ and $b(n ; x ; \theta)=$ $\left(\begin{array}{l}n \ x\end{array}\right) \theta^x(1-\theta)^{n-x}$ denotes the binomial distribution for $x=0,1, \ldots, n$, see Problem 4.
\begin{aligned} \beta &=\operatorname{Pr}\left[\text { accepting } H_0 / H_0 \text { is false }\right]=\operatorname{Pr}[x \geq 13 / \theta=0.5] \ &=b(n=20 ; x=13 ; \theta=0.5)+. .+b(n=20 ; x=20 ; \theta=0.5) \ &=0.0739+0.0370+0.0148+0.0046+0.0011+0.0002+0+0=0.1316 \end{aligned}

## 经济代写|计量经济学代写Econometrics代考|Comparing Biased and Unbiased Estimators

$$R(\hat{\theta}, \theta)=E[L(\hat{\theta}, \theta)]=E(\hat{\theta}-\theta)^2=M S E(\hat{\theta}) \quad=E[\hat{\theta}-E(\hat{\theta})+E(\hat{\theta})-\theta]^2=\operatorname{var}(\hat{\theta})+(\operatorname{Bias}$$

## 经济代写|计量经济学代写Econometrics代考|Hypothesis Testing

$\alpha=\operatorname{Pr}\left[\right.$ rejecting $H_0 / H_0$ is true $]=\operatorname{Pr}[x<13 / \theta=0.8] \quad=b(n=20 ; x=0 ; \theta=0.8)+\ldots+b(n$

$\beta=\operatorname{Pr}\left[\right.$ accepting $H_0 / H_0$ is false $]=\operatorname{Pr}[x \geq 13 / \theta=0.5] \quad=b(n=20 ; x=13 ; \theta=0.5)+\ldots+b$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|计量经济学代写Econometrics代考|ECON2300

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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|Methods of Estimation

Consider a Normal distribution with mean $\mu$ and variance $\sigma^2$. This is the important “Gaussian” distribution which is symmetric and bell-shaped and completely determined by its measure of centrality, its mean $\mu$ and its measure of dispersion, its variance $\sigma^2 . \mu$ and $\sigma^2$ are called the population parameters. Draw a random sample $X_1, \ldots, X_n$ independent and identically distributed (IID) from this population. We usually estimate $\mu$ by $\widehat{\mu}=\bar{X}$ and $\sigma^2$ by
$$s^2=\sum_{i=1}^n\left(X_i-\bar{X}\right)^2 /(n-1)$$
For example, $\mu=$ mean income of a household in New York city. $\bar{X}=$ sample average of incomes of 1000 households randomly interviewed in New York city.
This estimator of $\mu$ could have been obtained by either of the following two methods of estimation:

(i) Method of Moments
Simply stated, this method of estimation uses the following rule: Keep equating population moments to their sample counterpart until you have estimated all the population parameters.
\begin{tabular}{l|l}
Population & Sample \
\hline & \
$E(X)=\mu$ & $\sum_{i=1}^n X_i / n=\bar{X}$ \
$E\left(X^2\right)=\mu^2+\sigma^2$ & $\sum_{i=1}^n X_i^2 / n$ \
$\vdots$ & $\vdots$ \
$E\left(X^r\right)$ & $\sum_{i=1}^n X_i^r / n$
\end{tabular}
The normal density is completely identified by $\mu$ and $\sigma^2$, hence only the first 2 equations are needed
$$\widehat{\mu}=\bar{X} \quad \text { and } \quad \widehat{\mu}^2+\widehat{\sigma}^2=\sum_{i=1}^n X_i^2 / n$$
Substituting the first equation in the second one obtains
$$\widehat{\sigma}^2=\sum_{i=1}^n X_i^2 / n-\bar{X}^2=\sum_{i=1}^n\left(X_i-\bar{X}\right)^2 / n$$

## 经济代写|计量经济学代写Econometrics代考|Properties of Estimators

(i) Unbiasedness
$\widehat{\mu}$ is said to be unbiased for $\mu$ if and only if $E(\widehat{\mu})=\mu$ For $\widehat{\mu}=\bar{X}$, we have $E(\bar{X})=\sum_{i-1}^n E\left(X_i\right) / n=\mu$ and $\bar{X}$ is unbiased for $\mu$. No distributional assumption is needed as long as the $X_i$ ‘s are distributed with the same mean $\mu$. Unbiasedness means that “on the average” our estimator is on target. Let us explain this last statement. If we repeat our drawing of a random sample of 1000 households, say 200 times, then we get $200 \bar{X}$ ‘s. Some of these $\bar{X}$ ‘s will be above $\mu$ some below $\mu$, but their average should be very close to $\mu$. Since in real life situations, we observe only one random sample, there is little consolation if our observed $\bar{X}$ is far from $\mu$. But the larger $n$ is, the smaller is the dispersion of this $\bar{X}$, since $\operatorname{var}(\bar{X})=\sigma^2 / n$ and the lesser is the likelihood of this $\bar{X}$ to be very far from $\mu$. This leads us to the concept of efficiency.
(ii) Efficiency
For two unbiased estimators, we compare their efficiencies by the ratio of their variances. We say that the one with lower variance is more efficient. For example, taking $\widehat{\mu}_1=X_1$ versus $\widehat{\mu}_2=\bar{X}$, both estimators are unbiased but $\operatorname{var}\left(\widehat{\mu}_1\right)=\sigma^2$ whereas, $\operatorname{var}\left(\widehat{\mu}_2\right)=\sigma^2 / n$ and $\left{\right.$ the relative efficiency of $\widehat{\mu}_1$ with respect to $\left.\widehat{\mu}_2\right}=$ $\operatorname{var}\left(\widehat{\mu}_2\right) / \operatorname{var}\left(\widehat{\mu}_1\right)=1 / n$, see Fig. 2.1. To compare all unbiased estimators, we find the one with minimum variance. Such an estimator if it exists is called the $M V U$ (minimum variance unbiased estimator). This is also called an efficient estimator. It is centered on the right target $\mu$ (because it is unbiased), and it has the tightest distribution around $\mu$ (because it has the smallest variance among all unbiased estimators). A lower bound for the variance of any unbiased estimator $\widehat{\mu}$ of $\mu$ is known in the statistical literature as the Cramér-Rao lower bound and is given by
$$\operatorname{var}(\widehat{\mu}) \geq 1 / n{E(\partial \log f(X ; \mu) / \partial \mu)}^2=-1 /\left{n E\left(\partial^2 \log f(X ; \mu) / \partial \mu^2\right)\right}$$
where we use either representation of the bound on the right hand side of (2.2) depending on which one is the simplest to derive.

## 经济代写|计量经济学代写Econometrics代考|Methods of Estimation

$$s^2=\sum_{i=1}^n\left(X_i-\bar{X}\right)^2 /(n-1)$$

(i) 矩量法

$\backslash$ begin ${$ tabular $}|| \mid}$ 人口和样本 $\backslash \backslash$ hline \& $\backslash \$ E(X)=\backslash m u \$\& \$ \backslash s u m_{-}{i=1} \wedge n X_{-} i / n=\backslash b a r{X} \$\backslash \$ E \backslash l$eft$\left(X^{\wedge} 2 \backslash r i g h t\right)=\backslash m^{\wedge} 2+1 S$正常密度完全由下式确定$\mu$和$\sigma^2$，因此只需要前两个方程 $$\widehat{\mu}=\bar{X} \quad \text { and } \quad \widehat{\mu}^2+\widehat{\sigma}^2=\sum_{i=1}^n X_i^2 / n$$ 将第一个方程代入第二个方程得到 $$\widehat{\sigma}^2=\sum_{i=1}^n X_i^2 / n-\bar{X}^2=\sum_{i=1}^n\left(X_i-\bar{X}\right)^2 / n$$ ## 经济代写|计量经济学代写Econometrics代考|Properties of Estimators (i) 公正性$\widehat{\mu}$据说是无偏见的$\mu$当且仅当$E(\widehat{\mu})=\mu$为了$\widehat{\mu}=\bar{X}$，我们有$E(\bar{X})=\sum_{i-1}^n E\left(X_i\right) / n=\mu$和$\bar{X}$不偏不倚$\mu$. 不需要分布假设，只要$X_i$的分布具有相同的均值$\mu$. 无偏意味着我们的估计“平均”是在目标上的。让我们解释 一下这最后一句话。如果我们重复抽取 1000 个家庭的随机样本，比如说 200 次，那么我们得到$200 \bar{X}$的。其中 一些$\bar{X}$会在上面$\mu$下面一些$\mu$，但他们的平均值应该非常接近$\mu$. 由于在现实生活中，我们只观察到一个随机样 本，如果我们观察到$\bar{X}$远离$\mu$. 但较大的$n$就是，这个的分散度越小$\bar{X}$，自从$\operatorname{var}(\bar{X})=\sigma^2 / n$并且这种可能性 越小$\bar{X}$离得很远$\mu$. 这使我们想到了效率的概念。 (ii) 效率 对于两个无偏估计量，我们通过方差比来比较它们的效率。我们说方差越低的越有效率。例如，取$\hat{\mu}_1=X_1$相 对$\widehat{\mu}_2=\bar{X}$，两个估计量都是无偏的，但是$\operatorname{var}\left(\widehat{\mu}_1\right)=\sigma^2$然而，$\operatorname{var}\left(\widehat{\mu}_2\right)=\sigma^2 / n$和 见图 2.1。为了比较所有无偏估计量，我们找到方差最小的估计量。如果存在这样的估计量，则称为$M V U$(最 小方差无偏估计量) 。这也称为有效估计器。它以正确的目标为中心$\mu$(因为它是无偏的)，并且它有最紧密的 分布$\mu$(因为它在所有无偏估计量中方差最小) 。任何无偏估计量方差的下限$\mu$的$\mu$在统计文献中称为 CramérRao 下界，由下式给出 loperatorname${v a r}(\backslash$widehat${\backslash \mathrm{mu}}) \backslash \operatorname{lgeq} 1 / \mathrm{n}{\mathrm{E}(\backslash \text { partial } \backslash \log \mathrm{f}(X ; \backslash \mathrm{Xu}) / \backslash \text { partial } \backslash \mathrm{mu})}^{\wedge} 2=-1 / \backslash \mathrm{eft}\left{\mathrm{n} \mathrm{E} \backslash \mathrm{left}\left(\backslash \mathrm{partia} \wedge^{\wedge} 2 \backslash\right.\right.$我们在 (2.2) 的右侧使用任一边界表示，具体取决于哪一个最容易推导。 统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。 ## 金融工程代写 金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。 ## 非参数统计代写 非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。 ## 广义线性模型代考 广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。 术语 广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。 ## 有限元方法代写 有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。 有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。 tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。 ## 随机分析代写 随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。 ## 时间序列分析代写 随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。 ## 回归分析代写 多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。 ## 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。 ## 经济代写|计量经济学代写Econometrics代考|BEA472 如果你也在 怎样代写计量经济学Econometrics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。 计量经济学，对经济关系的统计和数学分析，通常作为经济预测的基础。这种信息有时被政府用来制定经济政策，也被私人企业用来帮助价格、库存和生产方面的决策。 statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。 我们提供的计量经济学Econometrics及其相关学科的代写，服务范围广, 其中包括但不限于: • Statistical Inference 统计推断 • Statistical Computing 统计计算 • Advanced Probability Theory 高等概率论 • Advanced Mathematical Statistics 高等数理统计学 • (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|Binary Response Models In a binary response model, the value of the dependent variable$y_t$can take on only two values, 1 and 0 , which indicate whether or not some event occurs. We can think of$y_t=1$as indicating that the event occurred for observation$t$and$y_t=0$as indicating that it did not. Let$P_t$denote the (conditional) probability that the event occurred. Thus a binary response model is really trying to model$P_t$conditional on a certain information set, say$\Omega_t$, that consists of exogenous and predetermined variables. Specifying$y_t$so that it is either 0 or 1 is very convenient, because$P_t$is then simply the expectation of$y_t$conditional on$\Omega_t$: $$P_t \equiv \operatorname{Pr}\left(y_t=1 \mid \Omega_t\right)=E\left(y_t \mid \Omega_t\right)$$ The objective of a binary response model is to model this conditional expectation. From this perspective, it is clear that the linear regression model makes no sense as a binary response model. Suppose that$\boldsymbol{X}_t$denotes a row vector of length$k$of variables that belong to the information set$\Omega_t$, including a constant term or the equivalent. Then a linear regression model would specify$E\left(y_t \mid \Omega_t\right)$as$\boldsymbol{X}_t \boldsymbol{\beta}$. But$E\left(y_t \mid \Omega_t\right)$is a probability, and probabilities must lie between 0 and 1. The quantity$\boldsymbol{X}_t \boldsymbol{\beta}$is not constrained to do so and therefore cannot be interpreted as a probability. Nevertheless, a good deal of (mostly older) empirical work simply uses OLS to estimate what is (rather inappropriately) called the linear probability model,${ }^1$that is, the model $$y_t=\boldsymbol{X}_t \boldsymbol{\beta}+u_t$$ 1 See, for example, Bowen and Finegan (1969). ## 经济代写|计量经济学代写Econometrics代考|Binary Response Models In view of the much better models that are available, and the ease of estimating them using modern computer technology, this model has almost nothing to recommend it. Even if$\boldsymbol{X}_t \boldsymbol{\beta}$happens to lie between 0 and 1 for some$\boldsymbol{\beta}$and all observations in a particular sample, it is impossible to constrain$\boldsymbol{X}_t \boldsymbol{\beta}$to lie in that interval for all possible values of$\boldsymbol{X}_t$, unless the values that the independent variables can take are limited in some way (for example, they might all be dummy variables). Thus the linear probability model is not a sensible way to model conditional probabilities. Several binary response models that do make sense are available and are quite easy to deal with. The key is to make use of a transformation function$F(x)that has the properties \begin{aligned} &F(-\infty)=0, \quad F(\infty)=1, \text { and } \ &f(x) \equiv \frac{\partial F(x)}{\partial x}>0 . \end{aligned} ThusF(x)$is a monotonically increasing function that maps from the real line to the 0-1 interval. Many cumulative distribution functions have these properties, and we will shortly discuss some specific examples. Using various specifications for the transformation function, we can model the conditional expectation of$y_t$in a variety of ways. The binary response models that we will discuss consist of a transformation function$F(x)$applied to an index function that depends on the independent variables and the parameters of the model. An index function is simply a function that has the properties of a regression function, whether linear or nonlinear. Thus a very general specification of a binary response model is $$E\left(y_t \mid \Omega_t\right)=F\left(h\left(\boldsymbol{X}_t, \boldsymbol{\beta}\right)\right),$$ where$h\left(\boldsymbol{X}_t, \boldsymbol{\beta}\right)$is the index function. A more restrictive, but much more commonly encountered, specification, is $$E\left(y_t \mid \Omega_t\right)=F\left(\boldsymbol{X}_t \boldsymbol{\beta}\right)$$ ## 计量经济学代考 ## 经济代写|计量经济学代写Econometrics代考|Binary Response Models 在二元响应模型中，因变量的值$y_t$只能取两个值， 1 和 0 ，表示某个事件是否发生。我们可以想到$y_t=1$表明 事件的发生是为了观察$t$和$y_t=0$表明它没有。让$P_t$表示事件发生的（条件）概率。因此，二元响应模型实际上 是在尝试建模$P_t$以特定信息集为条件，比如说$\Omega_t$，它由外生变量和预定变量组成。指定$y_t$所以它是 0 或 1 是非 常方便的，因为$P_t$那么就是对的期望$y_t$有条件的$\Omega_t$: $$P_t \equiv \operatorname{Pr}\left(y_t=1 \mid \Omega_t\right)=E\left(y_t \mid \Omega_t\right)$$ 二元响应模型的目标是模拟这种条件期望。 从这个角度来看，很明显线性回归模型作为二元响应模型没有意义。假设$\boldsymbol{X}_t$表示长度的行向量$k$属于信息集的 变量$\Omega_t$，包括常数项或等价物。然后线性回归模型将指定$E\left(y_t \mid \Omega_t\right)$作为$\boldsymbol{X}_t \boldsymbol{\beta}$. 但$E\left(y_t \mid \Omega_t\right)$是概率，概率 必须介于 0 和 1 之间。数量$\boldsymbol{X}_t \boldsymbol{\beta}$不受限于这样做，因此不能解释为概率。尽管如此，大量 (大部分是较旧的) 实证工作只是使用 OLS 来估计 (相当不恰当地) 称为线性概率模型的东西，${ }^1$也就是说，模型 $$y_t=\boldsymbol{X}_t \boldsymbol{\beta}+u_t$$ 1 例如，参见 Bowen 和 Finegan (1969)。 ## 经济代写|计量经济学代写Econometrics代考|Binary Response Models 鉴于可用的更好的模型，以及使用现代计算机技术对其进行估算的简便性，该模型几乎没有什么可推荐的。即使$\boldsymbol{X}_t \boldsymbol{\beta}$对于某些人来说恰好位于 0 和 1 之间$\boldsymbol{\beta}$以及特定样本中的所有观察结果，不可能约束$\boldsymbol{X}_t \boldsymbol{\beta}$位于所有可能值 的区间内$\boldsymbol{X}_t$，除非自变量可以取的值以某种方式受到限制（例如，它们可能都是虚拟变量）。因此，线性概率 模型不是模拟条件概率的明智方法。 有几个确实有意义的二元响应模型可用并且很容易处理。关键是利用转换功能$F(x)$具有属性 $$F(-\infty)=0, \quad F(\infty)=1, \text { and } \quad f(x) \equiv \frac{\partial F(x)}{\partial x}>0 .$$ 因此$F(x)$是从实线映射到0-1区间的单调递增函数。许多㽧积分布函数都具有这些性质，我们将很快讨论一些具 体的例子。使用变换函数的各种规范，我们可以对条件期望进行建模$y_t$以多种方式。 我们将讨论的二元响应模型包含一个转换函数$F(x)$应用于依赖于自变量和模型参数的索引函数。指数函数只是 具有回归函数属性的函数，无论是线性的还是非线性的。因此，二元响应模型的一个非常通用的规范是 $$E\left(y_t \mid \Omega_t\right)=F\left(h\left(\boldsymbol{X}_t, \boldsymbol{\beta}\right)\right),$$ 在哪里$h\left(\boldsymbol{X}_t, \boldsymbol{\beta}\right)\$ 是指标函数。一个更严格但更常见的规范是
$$E\left(y_t \mid \Omega_t\right)=F\left(\boldsymbol{X}_t \boldsymbol{\beta}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。