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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|计量经济学代写Econometrics代考|The Method of Maximum Likelihood

The estimation techniques we have discussed so far – least squares and instrumental variables – are applicable only to regression models. But not every model can be written so that the dependent variable is equal to a regression function plus an additive error term or so that a set of dependent variables, arranged as a vector, is equal to a vector of regression functions plus a vector of errors (see Chapter 9). If not, then least squares and instrumental variables are simply not appropriate. In this chapter, we therefore introduce a third estimation method, which is much more widely applicable than the techniques we have discussed so far, but also requires fairly strong assumptions. This is the method of maximum likelihood, or ML, estimation.

As an extreme example of how inappropriate least squares can be, consider the model
$$y_t^\gamma=\beta_0+\beta_1 x_t+u_t, \quad u_t \sim \operatorname{IID}\left(0, \sigma^2\right),$$
which looks almost like a regression model. This model makes sense so long as the right-hand side of $(8.01)$ is always positive, and it may even be an attractive model in certain cases. ${ }^1$ For example, suppose that the observations on $y_t$ are skewed to the right but those on $x_t$ are not. Then a conventional regression model could reconcile these two facts only if the error terms $u_t$ were right-skewed, which one would probably not want to assume and which would make the use of least squares dubious. On the other hand, the model (8.01) with $\gamma<1$ might well be able to reconcile these facts while allowing the error terms to be symmetrically distributed.

If $\gamma$ were known, (8.01) would be a regression model. But if $\gamma$ is to be estimated, (8.01) is not a regression model. As a result, it cannot sensibly be estimated by least squares. The sum-of-squares function is
$$\operatorname{SSR}(\boldsymbol{\beta}, \gamma)=\sum_{t=1}^n\left(y_t^\gamma-\beta_0-\beta_1 x_t\right)^2,$$
1 Strictly speaking, of course, it is impossible to guarantee that the right-hand side of (8.01) will always be positive, but this model may be regarded as a very good approximation if $\beta_0+\beta_1 x_{\ell}$ is always much larger than $\sigma$.

## 经济代写|计量经济学代写Econometrics代考|Fundamental Concepts and Notation

Maximum likelihood estimation depends on the notion of the likelihood of a given set of observations relative to a model, or set of DGPs. A DGP, being a stochastic process, can be characterized in a number of ways. We now develop notation in which we can readily express one such characterization that is particularly useful for present purposes. We assume that each observation in any sample of size $n$ is a realization of a random variable $y_t, t=1, \ldots, n$, taking values in $\mathbb{R}^m$. Although the notation $y_t$ ignores the possibility that the observation is in general a vector, it is more convenient to let the vector notation $\boldsymbol{y}$ (or $\boldsymbol{y}^n$ if we wish to make the sample size explicit) denote the entire sample. Thus
$$\boldsymbol{y}^n=\left[\begin{array}{l:l:l:l} y_1 & y_2 & \cdots & y_n \end{array}\right]$$
If each observation is a scalar, $\boldsymbol{y}$ is an $n$-vector, while if each observation is an $m$-vector, $\boldsymbol{y}$ is an $n \times m$ matrix. The vector or matrix $\boldsymbol{y}$ may possess a probability density, namely, the joint density of its elements under the DGP. This density, if it exists, is a map to the real line from the set of possible realizations of $\boldsymbol{y}$, a set that we will denote by $\mathrm{y}^n$ and that is in general an arbitrary subset of $\mathbb{R}^{n m}$. It will be necessary to exercise some care over the definition of the density in certain cases, but for the present it is enough to suppose that it is the ordinary density with respect to Lebesgue measure on $\mathbb{R}^{n m} \cdot 3$ When other possibilities exist, it will turn out that the choice among them is irrelevant for our purposes.

We may now define formally the likelihood function associated with a given model for a given sample $\boldsymbol{y}$. This function is a function of both the parameters of the model and the given data set $\boldsymbol{y}$; its value is just the density associated with the DGP characterized by the parameter vector $\boldsymbol{\theta} \in \Theta$, evaluated at the sample point $\boldsymbol{y}$. Here $\Theta$ denotes the parameter space in which the parameter vector $\theta$ lies; we will assume that it is a subset of $\mathbb{R}^k$. We will denote the likelihood function by $L: y^n \times \Theta \rightarrow \mathbb{R}$ and its value for $\boldsymbol{\theta}$ and $\boldsymbol{y}$ by $L(\boldsymbol{y}, \boldsymbol{\theta})$. In many practical cases, such as the one examined in the preceding section, the $y_t$ ‘s are independent and each $y_t$ has probability density $L_t\left(y_t, \boldsymbol{\theta}\right)$. The likelihood function for this special case is then
$$L(\boldsymbol{y}, \boldsymbol{\theta})=\prod_{t=1}^n L_t\left(y_t, \boldsymbol{\theta}\right) .$$
The likelihood function (8.03) of the preceding section is evidently a special case of this special case. When each of the $y_t$ ‘s is identically distributed with density $f\left(y_t, \boldsymbol{\theta}\right)$, as in that example, $L_t\left(y_t, \boldsymbol{\theta}\right)$ is equal to $f\left(y_t, \boldsymbol{\theta}\right)$ for all $t$.

# 计量经济学代考

## 经济代写|计量经济学代写econometrics代考|The Method of The Maximum Likelihood

$$y_t^\gamma=\beta_0+\beta_1 x_t+u_t, \quad u_t \sim \operatorname{IID}\left(0, \sigma^2\right),$$
，它看起来几乎像一个回归模型。只要$(8.01)$的右边总是正的，这个模型就有意义，在某些情况下，它甚至可能是一个有吸引力的模型。${ }^1$例如，假设$y_t$上的观察结果向右倾斜，而$x_t$上的观察结果没有。传统的回归模型只有在误差项$u_t$是右偏的情况下才能调和这两个事实，这可能是人们不希望假设的，并且会使最小二乘的使用令人怀疑。另一方面，带有$\gamma<1$的模型(8.01)可能很好地调和了这些事实，同时允许错误项对称分布

If $\gamma$ 如果已知，(8.01)将是一个回归模型。但是如果 $\gamma$ 是估计的，(8.01)不是一个回归模型。因此，它不能合理地用最小二乘估计。平方和函数是
$$\operatorname{SSR}(\boldsymbol{\beta}, \gamma)=\sum_{t=1}^n\left(y_t^\gamma-\beta_0-\beta_1 x_t\right)^2,$$当然，严格地说，不可能保证(8.01)的右边总是正的，但是这个模型可以被认为是一个非常好的近似，如果 $\beta_0+\beta_1 x_{\ell}$ 总是比 $\sigma$.

## 经济代写|计量经济学代写econometrics代考|基本概念和符号

.

$$\boldsymbol{y}^n=\left[\begin{array}{l:l:l:l} y_1 & y_2 & \cdots & y_n \end{array}\right]$$

$$L(\boldsymbol{y}, \boldsymbol{\theta})=\prod_{t=1}^n L_t\left(y_t, \boldsymbol{\theta}\right) .$$上一节的似然函数(8.03)显然是这个特例中的特例。当每一个 $y_t$ 的密度与密度分布相同 $f\left(y_t, \boldsymbol{\theta}\right)$，就像在那个例子中， $L_t\left(y_t, \boldsymbol{\theta}\right)$ 等于 $f\left(y_t, \boldsymbol{\theta}\right)$ 为所有人 $t$.

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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
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