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  • 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
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  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础

经济代写|宏观经济学代写Macroeconomics代考|The Shipbuilding Model and the Lambert Function

In his 1931 article on the “Shipbuilding cycle,” Tinbergen was interested in the increase of tonnage that followed the building of ships with a lag of about one year due to the construction period. From this connection a relation emerged between the increase of total tonnage and the volume of total tonnage two years before (Tinbergen, 1959: 2). Tinbergen proposed to model this relation as a differential equation with a delay, of the form:
\dot{f}(t)=-a f(t-\theta),
where $f(t)$ is the tonnage as a function of time, $\theta$ represents the delay between the tonnage and its increase in $t$, and $a$ is the intensity of the relation, the volume of increase above the trend (linbergen, 1959: 3). Tinbergen assumed a solution to his equation of the form $f(t)=C e^{\lambda t}$, which, inserted into the equation above yields $\lambda=-a e^{-\lambda \theta}$, once we have simplified the $C$ which only depends on the initial conditions. $\lambda$ can be a real or complex number, but because it appears both in the exponential function and alone this equation is transcendental. Now, “transcendental” means that usually the answer will only be found “experimentally” as Tinbergen put it. Indeed a transcendental equation is periodic, in the same sense that the exponential function with an imaginary argument traces a circle repeating itself as the argument increases. To find a general solution, Tinbergen (and after him Frisch and Kalecki) separated the real and the imaginary part of this equation and solved for one of the two in terms of trigonometric functions; for instance Tinbergen obtained the equation:
b \frac{\sin (y)}{y}=e^{-\frac{y}{\tan (y)}},
where $b=a \theta$ and $\lambda=x+i y, i=\sqrt{-1}$. To find the solutions for $y$ of this type of equations, they took the same approach of plotting both sides and looking for points of intersection, before improving on this solution with simple algorithms. Figure $2.4$ shows in the solid red lines the right hand side of the Eq. 2.2, while the dashed lines are the left hand side, for three different values of $b$ and both as a function of $y$.
It is readily apparent that there will be only one solution in each interval of length $2 \pi$. We can see that the leftmost solution will have the lowest frequency, that is, the largest period, and that all other solutions will have a higher frequency; thus the roots of the characteristic equation above will be ordered by their decreasing period or increasing frequency.

经济代写|宏观经济学代写Macroeconomics代考|From Natural Sciences to Economics

Hamburger claimed that it was only with new mathematical tools that economists would be able to account quantitatively and qualitatively for economic processes. Ultimately, the aim was to transform economics into a science similar to biology, ${ }^5$ a science capable of understanding the operation of social organisms beset by recurrent “pathologies.” It should be noted that Hamburger was not the only economist interested in business cycles who was showing some discontent with a mechanical analogy. Ernst Wagemann, the German head of the imperial statistical office and of the business cycle research institute of Berlin, in a book published in 1928, called as well for a biological metaphor.

Although Wagemann wrote in German, his book met enough success to warrant its translation in English only two years later under the title Economic Rhythm: A Theory of Business Cycles, with a prefatory note from Wesley C. Mitchell (Wagemann, 1930). In the preface to the English edition, Wagemann presented his contribution as a small step “toward the repayment of the debt which Europe owes to America in the field of research into economic dynamics” (Wagemann, 1930: v). However, the type of dynamics that was applied remained very empirical: although Wagemann was searching for a theory, he steered resolutely away from abstract constructions which were heavily criticized. His review of existing theories led him to propose that “while the American methods are those of engineering, and the Russian those of astronomy, the German institute represents the medical, or, better, the organicbiological point of view” (1930: 10). The “organic-biological principle” which he described (with reference to Menger) was meant to capture both the interconnection of the separate parts of an (economic) organism as well as “a peculiarity which may be defined as consisting in the power to regulate its own movement” (1930: 11), an approach which he emphasized as “anything but mechanical” (1930: 11).

Another radical opinion on the business cycle was that it was only a “myth.” This opinion was shared among American economists and statisticians, such as Carl Snyder (1930) and Irving Fisher (1925). ${ }^6$ While the former based his claim on the fact that compared to the growth of the economy, the amplitude of fluctuations remained within certain limits, the second doubted that “inherent” cyclical regularity in business could be detected. For Fisher, even if there existed a simple self-generating cycle similar to that of a pendulum swinging under the influence of the force of gravity, its tendency to materialize would be necessarily “defeated in practice” (Fisher, 1925: 192). To show this, he proposed to move away from the pendulum metaphor, toward the “physical analogue” of “the sway of the trees or of their branches.” For instance, after a tree is bended, one observes a swaying movement similar to that of the cycle: but Fisher did not think that such a movement was actually observed in the woods: “in actual experience […] twigs or tree tops seldom oscillate so regularly, even temporarily; they register instead, chiefly the variations in wind velocity” (Fisher, 1925: 192). A steady wind as well as any “outside forces”7 may thus bend the trees for weeks and annihilate completely their tendency to swing back and forth while changes in wind speed or in its direction will simply modify the angle of the tree with the ground.



经济代写|宏观经济学代写Macroeconomics代考|The Shipbuilding Model and the Lambert Function

在他 1931 年关于“造船周期”的文章中,Tinbergen 对船舶建造后吨位的增加感兴趣,由于建造周期的原 因,滞后大约一年。由此可见,总吨位的增加与两年前的总吨位体积之间存在一种关系 (Tinbergen, 1959:2)。Tinbergen 建议将这种关系建模为具有延迟的微分方程,形式如下:
\dot{f}(t)=-a f(t-\theta),
在哪里 $f(t)$ 吨位是时间的函数, $\theta$ 代表吨位与其增加量之间的延迟 $t ,$ 和 $a$ 是关系的强度,高于趋势的增 加量 (linbergen, 1959: 3)。 Tinbergen 假设了他的方程式的解 $f(t)=C e^{\lambda t}$ ,揷入到上面的等式中得到 $\lambda=-a e^{-\lambda \theta}$ ,一旦我们简化了 $C$ 这仅取决于初始条件。 $\lambda$ 可以是实数或复数,但因为它既出现在指数 函数中又单独出现,这个方程是超越的。现在,“先验意味着答案通常只能像丁伯根所说的那样“通过实 验”找到。事实上,超越方程是周期性的,就像具有虚参数的指数函数跟踪一个随着参数增加而重复自身 的圆圈一样。为了找到一个通解,Tinbergen(以及在他之后的 Frisch 和 Kalecki) 将这个方程的实部和 虚部分开,并根据三角函数求解其中一个;例如 Tinbergen 获得了等式:
b \frac{\sin (y)}{y}=e^{-\frac{y}{\tan (y)}},
在哪里 $b=a \theta$ 和 $\lambda=x+i y, i=\sqrt{-1}$. 寻找解决方案 $y$ 对于此类方程式,他们采用相同的方法绘制两 边并寻找交点,然后使用简单的算法改进此解决方案。数字 $2.4$ 以红色实线显示等式的右侧。2.2,而虚 线是左侧,对于三个不同的值 $b$ 两者都作为函数 $y$.
很明显,在每个长度区间内只有一个解 $2 \pi$. 我们可以看到,最左边的解将具有最低的频率,即最大的周 期,而所有其他解将具有更高的频率;因此,上述特征方程的根将按周期递减或频率递增排序。

经济代写|宏观经济学代写Macroeconomics代考|From Natural Sciences to Economics

Hamburger 声称,只有使用新的数学工具,经济学家才能对经济过程进行定量和定性分析。最终,目标 是将经济学转变为类似于生物学的科学, ${ }^5$ 一门能够理解被反复出现的”病态” 困扰的社会有机体运作的科 学。应该指出的是,汉堡并不是唯一对商业周期感兴趣并对机械类比表示不满的经济学家。德国帝国统 计局局长兼柏林商业周期研究所所长恩斯特. 瓦格曼 (Ernst Wagemann) 在 1928 年出版的一本书中也呼 吁使用生物学隐喻。
尽管 Wagemann 用德语写作,但他的书取得了足够的成功,仅在两年后就被翻译成英文,标题为”经济 节奏:商业周期理论”,并附有 Wesley C. Mitchell 的序言(Wagemann,1930 年)。在英文版的序言 中,Wagemann 将他的贡献描述为”朝着偿还欧洲在经济动态研究领域欠美国的债务”迈出的一小步 (Wagemann,1930:v) 。然而,所应用的动力学类型仍然非常经验主义:尽管 Wagemann 正在寻 找一种理论,但他坚决避开了受到严厉批评的抽象结构。他对现有理论的回顾使他提出“虽然美国的方法 是工程学的方法,而俄罗斯的方法是天文学的方法,德国研究所代表了医学,或者更确切地说,有机生 物学的观点” (1930:10) 。他描述的“有机生物学原理” (参考门格尔) 旨在捕捉 (经济) 有机体各个部 分之间的相互联系以及”可以定义为存在于调节能力中的特性”它自己的运动”(1930:11),他强调这种 方法”绝不是机械的” (1930: 11)。
另一种关于商业周期的激进观点是,它只是一个”神话”。这一观点在美国经济学家和统计学家之间得到了 认同,例如卡尔·斯奈德 (Carl Snyder) (1930) 和欧文·费雪 (Irving Fisher) (1925)。 ${ }^6$ 前者的主张基于这样 一个事实,即与经济增长相比,波动幅度保持在一定限度内,而第二个则怀疑是否可以检测到商业中“固 有的”周期性规律。对于费舍尔来说,即使存在类似于钟摆在重力作用下摆动的简单自生循环,其物化趋 势也必然会”在实践中被击败”(费舍尔,1925:192)。为了表明这一点,他提议摆脱钟摆的隐喻,转向 “树木或树枝摇摆”的“物理类比”。例如,一棵树被弯曲后,人们会观察到类似于循环的摇摆运动:但费舍 尔并不认为这种运动实际上是在树林中观察到的: “在实际经验中 [ … 树枝或树梢很少有规律地摆动,即 使是暂时的;相反,它们记录的主要是风速的变化” (Fisher,1925:192)。稳定的风以及任何“外力”7 可能会因此使树木弯曲数周并完全消除它们来回摆动的趋势,而风速或风向的变化只会改变树木与地面的角度。

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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。





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。


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 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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