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# 金融代写|利率理论代写Portfolio Theory代考|FINC6009 Beta-Pricing Models and Two-Pass Cross-Sectional Regressions

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## 金融代写|利率理论代写Portfolio Theory代考|Beta-Pricing Models and Two-Pass Cross-Sectional Regressions

From Equation 3.13, the expected-return errors of the $N$ assets are given by
$$e=E[R]-B \gamma .$$

A popular goodness-of-fit measure used in many empirical studies is the crosssectional $R^{2}$. Following Kandel and Stambaugh (1995), this is defined as
$$R^{2}=1-\frac{Q}{Q_{0}},$$
where $Q=e^{\prime} W e, Q_{0}=e_{0}^{\prime} W e_{0}, e_{0}=\left[I_{N}-1_{N}\left(1_{N}^{\prime} W 1_{N}\right)^{-1} 1_{N}^{\prime} W\right] E[R]$ represents the deviations of mean returns from their cross-sectional average and $W$ is a positive-definite weighting matrix. Popular choices of $W$ in the literature are $W=$ $I_{N}$ (ordinary least squares $[\mathrm{OLS}]$ ), $W=\operatorname{Var}[R]^{-1}$ (generalized least squares $[\mathrm{GLS}]$ ), and $W=\Sigma_{d}^{-1}$ (weighted least squares $[\mathrm{WLS}]$ ), where $\Sigma_{d}$ is a diagonal matrix containing the diagonal elements of $\Sigma$, the variance-covariance matrix of the residuals from the first-pass time series regression. In order for $R^{2}$ to be well defined requires assuming that $E[R]$ is not proportional to $1_{N}$ (the expected returns are not all equal) so that $Q_{0}>0$. Note that $0<R^{2}<1$ and it is a decreasing function of the aggregate pricing-error measure $Q$. Thus, $R^{2}$ is a natural measure of goodness of fit.

## 金融代写|利率理论代写Portfolio Theory代考|Conditional Asset Pricing Models and Return Predictability

Recall that the fundamental pricing equation (Equation 3.2) is defined in terms of conditional expectations. Although the law of iterated expectations permits the estimation of the model in terms of unconditional moments, some relevant information may get lost in the process. This section explains how to incorporate conditioning information in a linear asset pricing model, describes the underlying assumptions, and provides an interpretation of the zero-beta rate and risk premia in the cross-sectional regression.

Let $z_{t}$ be an $L$-vector of observed conditioning variables (instruments) that belongs to the information set at time $t$ and define $F_{t+1}=\left[z_{t}^{\prime}, f_{t+1}^{\prime}, z_{t}^{\prime} \otimes f_{t+1}^{\prime}\right]^{\prime}$ as a $\tilde{K}=(K+1)(L+1)-1$ vector of scaled factors. Recently, many empirical studies (see, for example, Shanken, 1990; Lettau and Ludvigson, 2001; Lustig and Van Nieuwerburgh, 2005; Santos and Veronesi, 2006) have considered a cross-sectional regression of unconditional expected returns on their unconditional betas with respect to $F_{t+1}$ :
$$E\left[R_{t+1}\right]=1_{N} \gamma_{0}+\beta \gamma_{1},$$
where
$$\beta=\operatorname{Cov}\left[R_{t+1}, F_{t+1}\right] \operatorname{Var}\left[F_{t+1}\right]^{-1} .$$

# 利率理论代写

## 金融代写|利率理论代写Portfolio Theory代考|Beta-Pricing Models and TwoPass Cross-Sectional Regressions

$$e=E[R]-B \gamma .$$

$$R^{2}=1-\frac{Q}{Q_{0}},$$

## 金融代写|利率理论代写Portfolio Theory代考|Conditional Asset Pricing Models and Return Predictability

$$E\left[R_{t+1}\right]=1_{N} \gamma_{0}+\beta \gamma_{1},$$

$$\beta=\operatorname{Cov}\left[R_{t+1}, F_{t+1}\right] \operatorname{Var}\left[F_{t+1}\right]^{-1}$$

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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。