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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

## 商科代写|计量经济学代写Econometrics代考|AN ASIDE ON MAXIMUM LIKELIHOOD

We have already introduced the method of maximum likelihood in an earlier chapter. However, this method is of particular value when dealing with limited dependent variable models, and this is therefore, a good time to look at it in more detail. In doing so, we will introduce the idea of the Fisher Information Matrix and discuss how numerical methods can be used to calculate estimators when analytical solutions are not possible.

Maximum likelihood begins with assumption that we can write down the joint probability of observing a particular sample of data conditional on a set of parameters. For example, suppose we conduct a set of $N$ independent Bernouilli trials. We observe $k$ successes and $N-k$ failures. If the probability of a success is equal to $p$, then the joint probability of observing this outcome is given by the binomial distribution $p^{k}(1-p)^{N-k}$. Let us define the likelihood function for this case as
$$L(p)=p^{k}(1-p)^{N-k} .$$
This is a particularly easy likelihood function to work with because it depends on a single parameter $p$. It is usually easier to deal with a monotonic transformation of the likelihood function in the form of its logarithm which, in this case, we write as
$$L L(p)=k \ln (p)+(N-k) \ln (1-p) .$$
The score is defined as the derivative of the log-likelihood with respect to its paramcter(s). In this casc, we have
$$\frac{d L L(p)}{d p}=\frac{k}{p}-\frac{N-k}{(1-p)}$$

## 商科代写|计量经济学代写Econometrics代考|SOME ALTERNATIVE LIMITED DEPENDENT

So far, we have used the logit model to estimate conditional probabilities in a model with limited dependent variables. To do this, we have assumed that the probability that the right-hand side variable is equal to 1 can be written in terms of the formula $P\left(Y_{i}=1\right)=\exp \left(\alpha+\beta X_{i}\right) / 1+\exp \left(\alpha+\beta X_{i}\right)$. The problem is then on using an appropriate estimation technique to estimate the unknown parameters $\alpha$ and $\beta$. The choice of the logit function was made simply on the basis that it has the necessary properties. In particular, it is always positive, always lies between 0 and 1, approaches 0 as $X \rightarrow-\infty$ and approaches 1 as $X \rightarrow \infty$. However, the logit function is by no means the only functional form which has these properties and there are other candidate functions we might consider. We will consider two alternatives. These are the probit model and the extreme value model.

The probit model is based on the cumulative distribution function for the normal distribution. Consider the function
$$\Phi\left(a+\beta X_{i}\right)=\int_{-\infty}^{a+\beta X_{i}} \varphi(s) d s,$$ where $\varphi($ ) is the probability density function of the normal distribution, then it is easy to see that $(7.22)$ has all the necessary properties for a function that describes $P\left(Y_{i}=1\right)$. Moreover, although (7.22) looks quite forbidding, the normal distribution is such a well-known distribution that calculation of the probabilities implied by it are quite straightforward (though again will require numerical methods.) Therefore, we can again use maximumlikelihood methods to obtain estimates of the unknown parameters $\alpha$ and $\beta$. If we apply the probit model to our data set for British Airways share prices, then we obtain the following results.

# 计量经济学代考

## 商科代写|计量经济学代写Econometrics代考|AN ASIDE ON MAXIMUM LIKELIHOOD

$$L(p)=p^{k}(1-p)^{N-k} .$$

$$L L(p)=k \ln (p)+(N-k) \ln (1-p) .$$

$$\frac{d L L(p)}{d p}=\frac{k}{p}-\frac{N-k}{(1-p)}$$

## 商科代写|计量经济学代写Econometrics代考|SOME ALTERNATIVE LIMITED DEPENDENT

probit 模型基于正态分布的累积分布函数。考虑函数
$$\Phi\left(a+\beta X_{i}\right)=\int_{-\infty}^{a+\beta X_{i}} \varphi(s) d s,$$

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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。

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