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• Statistical Inference 统计推断
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• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
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## 经济代写|宏观经济学代写Macroeconomics代考|Precautionary savings

Let’s ask ourselves how savings and consumption react when uncertainty increases. Our intuition sugdubbed precautionary savings. To illustrate how this works we go back to our Euler equation:
$$u^{\prime}\left(c_t\right)=\frac{1}{1+\rho} E_t\left[\left(1+r_{t+1}^i\right) u^{\prime}\left(c_{t+1}\right)\right] .$$
Assume again that $r_{t+1}^i=\rho=0$, to simplify matters. Thus, the condition reduces to (we’ve seen this before!):
$$u^{\prime}\left(c_t\right)=E_t\left[u^{\prime}\left(c_{t+1}\right)\right] .$$
Now assume, in addition to the typical $u^{\prime}>0$ and $u^{\prime \prime}<0$, that $u^{\prime \prime \prime}>0$. This last condition is new and says that marginal utility is convex. This seems to be a very realistic assumption. It means that the marginal utility of consumption grows very fast as consumption approaches very low levels. Roughly speaking, people with convex marginal utility will be very concerned with very low levels of consumption. Figure $12.1$ shows how marginal utility behaves if this condition is met.
Notice that for a quadratic utility
$$E\left[u^{\prime}(c)\right]=u^{\prime}(E[c]) .$$
But the graph shows clearly that if marginal utility is convex then
$$E\left[u^{\prime}(c)\right]>u^{\prime}(E[c]),$$
and that the stronger the convexity, the larger the difference. The bigger $E\left[u^{\prime}(c)\right]$ is, the bigger $c_{t+1}$ needs to be to keep the expected future utility equal to $u^{\prime}(c)$, the marginal utility of consumption today. Imagine, for example that you expect one of your consumption possibilities for next period to be zero. If marginal utility at zero is $\infty$ then $E\left[u^{\prime}(c)\right]$ will also be $\infty$, and therefore you want to increase future consumption as much as possible to bring this expected marginal utility down as much as possible. In the extreme you may choose not to consume anything today! This means that you keep some extra assets, a buffer stock, to get you through the possibility of really lean times. This is what is called precautionary savings. Precautionary savings represents a departure from the permanent income hypothesis, in that it will lead individuals to save more than would be predicted by the latter, because of uncertainty.

## 经济代写|宏观经济学代写Macroeconomics代考|The Caballero model

Caballero (1990) provides a nice example that allows for a simple solution. Consider the case of a constant absolute risk aversion function.
$$u\left(c_t\right)=-\frac{1}{\theta} e^{-\theta c_t} .$$
Assuming that the interest rate is equal to the discount rate for simplification, this problem has a traditional Euler equation of the form
$$e^{-\theta c_l}=E_t\left[e^{-\Delta c_{l+1}}\right] .$$
Caballero proposes a solution of the form
$$c_{t+1}=\Gamma_t+c_t+v_{t+1},$$
were $v$ is related to the shock to income, the source of uncertainty in the model. Replacing in the Euler equation gives
$$e^{-\theta c_t}=E_t\left[e^{-\theta\left[\Gamma_t+c_t+v_{t+1}\right]}\right],$$

which, taking logs, simplifies to
$$\theta \Gamma_t=\log E_t\left[e^{-\theta v_{t+1}}\right] .$$
If $v$ is distributed $N\left(0, \sigma^2\right)$, then we can use the fact that $E e^x=e^{E x+\frac{\sigma_x^2}{2}}$ to find the value of $\Gamma$ (as the value is constant, we can do away with the subscript) in (12.33):
$$\theta \Gamma=\log \left[e^{\frac{\theta^2 \sigma_1^2}{2}}\right],$$
or, simply,
$$\Gamma=\frac{\theta \pi_v^2}{2} .$$
This is a very simple expression. It says that even when the interest rate equals the discount rate the consumption profile is upward sloping. The higher the variance, the higher the slope.

The precautionary savings hypothesis is also useful to capture other stylised facts: families tend to show an upward-sloping consumption path while the uncertainties of their labour life get sorted out. Eventually, they reach a point were consumption stabilises and they accumulate assets. Gourinchas and Parker (2002) describe these dynamics. Roughly the pattern that emerges is that families have an increasing consumption pattern until sometime in the early 30s, after which consumption starts to flatten.

# 宏观经济学代考

## 经济代写|宏观经济学代写宏观经济代考|预防性储蓄

.

$$u^{\prime}\left(c_t\right)=\frac{1}{1+\rho} E_t\left[\left(1+r_{t+1}^i\right) u^{\prime}\left(c_{t+1}\right)\right] .$$

$$u^{\prime}\left(c_t\right)=E_t\left[u^{\prime}\left(c_{t+1}\right)\right] .$$

$$E\left[u^{\prime}(c)\right]=u^{\prime}(E[c]) .$$
，但图表清楚地表明，如果边际效用是凸的，那么
$$E\left[u^{\prime}(c)\right]>u^{\prime}(E[c]),$$
，凸性越强，差异越大。$E\left[u^{\prime}(c)\right]$越大，$c_{t+1}$就需要越大，以保证未来预期效用等于$u^{\prime}(c)$，即今天消费的边际效用。想象一下，例如，你期望下一段时间的消费可能性为零。如果边际效用在0处是$\infty$那么$E\left[u^{\prime}(c)\right]$也将是$\infty$，因此你希望尽可能增加未来消费尽可能降低期望边际效用。在极端情况下，你可以选择今天不消费任何东西!这意味着你要保留一些额外的资产，一个缓冲库存，以帮助你度过真正困难的时期。这就是所谓的预防性储蓄。预防性储蓄背离了永久收入假设，因为由于不确定性，它将导致个人的储蓄超过后者的预测

## 经济代写|宏观经济学代写宏观经济学代考|卡巴列罗模型

Caballero(1990)提供了一个很好的例子，允许一个简单的解决方案。考虑绝对风险规避函数恒定的情况。
$$u\left(c_t\right)=-\frac{1}{\theta} e^{-\theta c_t} .$$

$$e^{-\theta c_l}=E_t\left[e^{-\Delta c_{l+1}}\right] .$$
Caballero提出了一个形式
$$c_{t+1}=\Gamma_t+c_t+v_{t+1},$$

$$e^{-\theta c_t}=E_t\left[e^{-\theta\left[\Gamma_t+c_t+v_{t+1}\right]}\right],$$

，在记录日志时，简化为
$$\theta \Gamma_t=\log E_t\left[e^{-\theta v_{t+1}}\right] .$$

$$\theta \Gamma=\log \left[e^{\frac{\theta^2 \sigma_1^2}{2}}\right],$$

$$\Gamma=\frac{\theta \pi_v^2}{2} .$$这是一个非常简单的表达式。它表示，即使利率等于贴现率，消费曲线也是向上倾斜的。方差越高，斜率越高

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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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