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• (Generalized) Linear Models 广义线性模型
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• Foundations of Data Science 数据科学基础
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## 统计代写|金融统计代写Financial Statistics代考|Term Structure of Interest Rates

In this paper, interest rates are treated as a multidimensional variable that represents the return on investment expressed by three related quantities: spot rate, forward rate, and the discount value.
Each of these quantities depends on several economical, political, and social information, such as supply and demand of money and the expectation of its future value, risk, and trust perception, consequences of political acts, etc. The term structure of interest rates is a valuable tool not only for banks and financial firms, or governments and policy makers, but, for society itself, helping to understand the movements of markets and flows of money.

It is assumed that fixed income government bonds can be considered risk-free so we can define a special type of yield that is the spot interest rate, $s(\tau)$. This function is the return of a fixed income zero-coupon risk-free bond that expires in $\tau$ periods. Today’s price of such financial instrument whose future value is $\$ 1.00$, assuming that its interest rate is continuously compounded, is given by the discount function,$d(\tau), represented by $$d(\tau)=e^{-s(\tau) \times \tau}$$ The relationship between the discount value and the spot rate can be recovered by $$s(\tau)=-\frac{\log (d(\tau))}{\tau}$$ Based on the available bonds in the market with different maturities, it is possible to plan at an instant a financial transaction that will take place in another future instant, starting at the maturity of the shorter bond and expiring at the maturity of the longer bond. The interest rate of this future transaction is called the forward rate. ## 统计代写|金融统计代写Financial Statistics代考|Dynamic Nelson–Siegel Diebold and Li (2003) proposed the following dynamic version of Nelson-Siegel yield curve model $$N S(\tau)=\beta_{1} L_{1}(\tau)+\beta_{2} L_{2}(\tau)+\beta_{3} L_{3}(\tau)$$ where \begin{aligned} &L_{1}(\tau)=1 \ &L_{2}(\tau)=\frac{1-e^{-\lambda \tau}}{\lambda \tau} \ &L_{3}(\tau)=\frac{1-e^{-\lambda \tau}}{\lambda \tau}-e^{-\lambda \tau} \end{aligned} and the parameter\lambda$is a constant interpreted as the conductor of the curve exponential decay rate. A yield curve is fitted according to the Nelson-Siegel model to relate yields and maturities of available contracts for a specific day. We will refer to such yield curve as the Nelson-Siegel static yield curvé. Let$\theta$be$\left{\beta_{1}, \beta_{2}, \beta_{3}\right}$. The curves are fitted by constructing a simplex solver which computes appropriate values for$\theta$to minimize the distance between$N S(\tau)$and market data points. The coefficients$\beta_{1}, \beta_{2}$, and$\beta_{3}$are interpreted as three latent dynamic factors. The loading on$\beta_{1}$is constant and do not change in the limit; then,$\beta_{1}$can be viewed as a long-term factor. The loading on$\beta_{2}$starts at 1 and decays quickly to zero; then,$\beta_{2}$can be viewed as a short-term factor. Finally, the loading on$\beta_{3}$starts at zero, increases, and decays back to zero; then,$\beta_{3}$can be viewed as a medium term factor. The Dynamic Nelson-Siegel (DNS) model is defined by $$\begin{array}{r} s_{t}(\tau)=\beta_{1, t} L_{1}(\tau)+\beta_{2, t} L_{2}(\tau)+\beta_{3, t} L_{3}(\tau)+\epsilon_{t} \ t=1, \ldots, T \end{array}$$ where the coefficients$\beta_{i, t}$are AR(1) processes defined by $$\beta_{i, t}=c_{i}+\phi_{i} \beta_{i, t-1}+\eta_{i, t} \quad i=1,2,3 .$$ ## 统计代写|金融统计代写Financial Statistics代考|Arbitrage-Free Nelson–Siegel The Arbitrage-Free Nelson-Siegel (AFNS) static model for daily yield curve fitting was derived by Christensen et al. (2011) from the standard continuous-time affine Arbitrage-Free formulation of Duffie and Kan (1996). The AFNS model almost matches the NS model except by the yield-adjustment term$-\frac{C\left(\tau, \tau_{M}\right)}{\tau_{M}-\tau}$. In fact, the definition of the AFNS static model in Christensen et al. (2011) is given by $$\operatorname{AFNS}(\tau)=\beta_{1} L_{1}(\tau)+\beta_{2} L_{2}(\tau)+\beta_{3} L_{3}(\tau)-\frac{C\left(\tau, \tau_{M}\right)}{\tau_{M}-\tau}$$ The AFNS model built by Christensen et al. (2011) considers the mean levels of the state variable under the$Q$-measure at zero, i.e.,$\Theta^{Q}=0$. Thus,$-\frac{C\left(\tau, \tau_{M}\right)}{\tau_{M}-\tau}have the form \begin{aligned} -\frac{C\left(\tau, \tau_{M}\right)}{\tau_{M}-\tau}=&-\frac{1}{2} \frac{1}{\tau_{M}-\tau} \ \sum_{j=1}^{3} \int_{\tau}^{\tau_{M}}\left(\Sigma^{\prime} B\left(s, \tau_{M}\right) B\left(s, \tau_{M}\right)^{\prime} \Sigma\right){j, j} d s \end{aligned} Considering a general volatility matrix (not related to the dynamic model for forecasting the yield curve) $$\Sigma=\left(\begin{array}{lll} \sigma{11} & \sigma_{12} & \sigma_{13} \ \sigma_{21} & \sigma_{22} & \sigma_{23} \ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{array}\right)$$ Christensen et al. (2011) show that an analytical form of the yield-adjustment term can be derived as $$\frac{C\left(\tau, \tau_{M}\right)}{\tau_{M}-\tau}=\frac{1}{2} \frac{1}{\tau_{M}-\tau} \int_{\tau}^{\tau_{M}} \sum_{j=1}^{3}\left(\Sigma^{\prime} B\left(s, \tau_{M}\right) B\left(s, \tau_{M}\right)^{\prime} \Sigma\right)_{j, j} d s .$$ ## 金融统计代考 ## 统计代写|金融统计代写Financial Statistics代考|Term Structure of Interest Rates 在本文中，利率被视为一个多维变量，它表示由三个相关量表示的投资回报：即期利率、远期利率和贴现值。 这些数量中的每一个都取决于若干经济、政治和社会信息，例如货币的供求及其末来价值的预期、风险和信任感知、政治行为的后果等。 利率的期限结构是不仅对银行和金融公司，或政府和政策制定者，而且对社会本身来说，它都是一个有价值的工具，有助于了解市场的运 动和资金的流动。 假设固定收益政府债券可以被认为是无风险的，因此我们可以定义一种特殊类型的收益率，即即期利率，s(\tau)$. 这个函数是固定收益零息 无风险债券的回报$\tau$期间。末来价值为的此类金融工具的今日价格$\$1.00$ ，假设其利率是连续复利的，由贴现函数给出， $d(\tau)$ ，表示为
$$d(\tau)=e^{-s(\tau) \times \tau}$$

$$s(\tau)=-\frac{\log (d(\tau))}{\tau}$$

## 统计代写|金融统计代写Financial Statistics代考|Dynamic Nelson–Siegel

Diebold 和 Li (2003) 提出了以下动态版本的 Nelson-Siegel 收益率曲线模型
$$N S(\tau)=\beta_{1} L_{1}(\tau)+\beta_{2} L_{2}(\tau)+\beta_{3} L_{3}(\tau)$$

$$L_{1}(\tau)=1 \quad L_{2}(\tau)=\frac{1-e^{-\lambda \tau}}{\lambda \tau} L_{3}(\tau)=\frac{1-e^{-\lambda \tau}}{\lambda \tau}-e^{-\lambda \tau}$$

$$s_{t}(\tau)=\beta_{1, t} L_{1}(\tau)+\beta_{2, t} L_{2}(\tau)+\beta_{3, t} L_{3}(\tau)+\epsilon_{t} t=1, \ldots, T$$

$$\beta_{i, t}=c_{i}+\phi_{i} \beta_{i, t-1}+\eta_{i, t} \quad i=1,2,3$$

## 统计代写|金融统计代写Financial Statistics代考|Arbitrage-Free Nelson–Siegel

$$\operatorname{AFNS}(\tau)=\beta_{1} L_{1}(\tau)+\beta_{2} L_{2}(\tau)+\beta_{3} L_{3}(\tau)-\frac{C\left(\tau, \tau_{M}\right)}{\tau_{M}-\tau}$$
Christensen 等人建立的 AFNS 模型。 $(2011)$ 考虑了状态变量的平均水平 $Q$ – 在零处测量，即 $\Theta^{Q}=0$. 因此， $-\frac{C(\tau, \tau M)}{\tau M-\tau}$ 有表格
$$-\frac{C\left(\tau, \tau_{M}\right)}{\tau_{M}-\tau}=-\frac{1}{2} \frac{1}{\tau_{M}-\tau} \sum_{j=1}^{3} \int_{\tau}^{\tau M}\left(\Sigma^{\prime} B\left(s, \tau_{M}\right) B\left(s, \tau_{M}\right)^{\prime} \Sigma\right) j, j d s$$

$$\frac{C\left(\tau, \tau_{M}\right)}{\tau_{M}-\tau}=\frac{1}{2} \frac{1}{\tau_{M}-\tau} \int_{\tau}^{\tau_{M}} \sum_{j=1}^{3}\left(\Sigma^{\prime} B\left(s, \tau_{M}\right) B\left(s, \tau_{M}\right)^{\prime} \Sigma\right)_{j, j} d s$$

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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。