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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

## 统计代写|时间序列分析代写Time-Series Analysis代考|GARCH Models for the Exchange Rate

Table $10.1$ presents the results of fitting various AR-GARCH models to the first differences of the $\$-£$exchange rate,$\nabla x_t$. The choice of an AR(1) model for the conditional mean equation is based on our findings from Example 4.2. Assuming homoskedastic (GARCH$(0,0)$) errors produces the estimates in the first column of Table 10.1. The$\mathrm{ARCH}(1)$statistic; the$\mathrm{LM}$test for first-order$\mathrm{ARCH}$, shows that there is strong evidence of conditional heteroskedasticity. A GARCH$(1,1)$conditional variance is fitted in the second column, using the estimation technique of quasi-maximum likelihood, which enables standard errors to be adjusted for the presence of time-varying variances: see Bollerslev and Wooldridge (1992). Both GARCH parameters are significant, and the LM test for any neglected$\mathrm{ARCH}$is insignificant. The GARCH parameters sum to just under unity, suggesting that shocks to the conditional variance are persistent. The autoregressive coefficient, although remaining significantly positive, is now even smaller in magnitude, confirming that the deviation from a “pure” random walk for the exchange rate has little economic content. The estimated “pure” GARCH$(1,1)$model is shown in the third column, with the omission of the autoregressive term being seen to have no effect on the remaining estimates of the model.${ }^2$If the lagged level of the exchange rate is added to the mean equation then this will provide a test of a unit root under$\operatorname{GARCH}(1,1)$errors: doing so yields a coefficient estimate of$-0.00001$with a$t$-statistic of just$-0.26$. The paradox found in Example$4.2$thus disappears: once the error is correctly specified as a GARCH process, there is no longer any tangible evidence against the hypothesis that the exchange rate is a random walk. The conditional standard deviations from this model are shown in Fig. 10.1. Large values of$\hat{\sigma}_t$are seen to match up with periods of high volatility in the exchange rate, most notably around the United Kingdom’s departure from the Exchange Rate Mechanism in September 1992; during the financial crisis of$2008-2009$, in which the$\$-£$ rate dropped by over a quarter in just a few months (recall Figs. $1.5$ and 1.9); and in the aftermath of the Brexit referendum of June 2016. Note also the “asymmetric” nature of $\hat{\sigma}_t$ : rapid increases are followed by much slower declines, thus, reflecting the persistence implied by the fitted models.

## 统计代写|时间序列分析代写Time-Series Analysis代考|MARTINGALES, RANDOM WALKS, AND NONLINEARITY

11.1 In $\S \mathbf{1 0 . 2}$ a distinction was drawn between serial uncorrelatedness and independence. Although this distinction lies at the heart of GARCH modeling, it is also of more general importance, manifesting itself in the concept of a martingale; a stochastic process that is a mathematical model of “fair play.” A martingale may be defined as a stochastic process $x_t$ having the following properties: ${ }^2$

1. $E\left(\left|x_t\right|\right)<\infty$ for each $t$;
2. $E\left(x_t \mid x_s, x_{s-1}, \ldots\right)=x_s$.
Written as
$$E\left(x_t-x_s \mid x_s, x_{s-1}, \ldots\right)=0, \quad s<t,$$
the martingale property implies that the MMSE forecast of a future increment of a martingale is zero. This property can be generalized to situations where:
$$E\left(x_t-x_s \mid x_s, x_{s-1}, \ldots\right) \geq 0, \quad s<t,$$
in which we have a sub-martingale, and to the case where this inequality is reversed, giving us a super-martingale.
11.2 The martingale given by (11.1) can be written equivalently as:
$$x_t=x_{t-1}+a_t,$$
where $a_t$ is known as the martingale increment or martingale difference. When written in this form, $x$, looks superficially identical to a random walk.

where $a_t$ is defined to be a stationary and uncorrelated sequence drawn from a fixed distribution, i.e., to be white noise (cf. \$4.6). Alternative definitions are, however, possible. For example,$a_t$could be defined to be strict while noise, so that it is both a stationary and independent sequence, rather than just being uncorrelated. Moreover, it is possible for$a_t$to be uncorrelated but not necessarily stationary. While the white noise assumptions rule this out, such behavior is allowed for martingale differences. This implies that there could be dependence between higher conditional moments-most notably, as we have seen in Chapter 10, Volatility and Generalized Autoregressive Conditional Heteroskedastic Processes, between conditional variances through time. # 时间序列分析代考 ## 统计代写|时间序列分析代写Time-Series Analysis代考|GARCH汇率模型 . 表$10.1$给出了各种AR-GARCH模型拟合$\$-£$汇率的第一个差异$\nabla x_t$的结果。条件均值方程AR(1)模型的选择基于例4.2中的发现。假设同方差(GARCH $(0,0)$)误差产生表10.1第一列中的估计值。$\mathrm{ARCH}(1)$统计;一阶$\mathrm{ARCH}$的$\mathrm{LM}$检验表明，有强有力的证据证明条件异方差

然而，也有可能有其他的定义。例如，$a_t$可以定义为严格而噪声，这样它既是一个平稳的和独立的序列，而不仅仅是不相关的。此外，$a_t$可能是不相关的，但不一定是平稳的。虽然白噪声假设排除了这一点，但这种行为是允许的鞅差异。这意味着更高的条件矩之间可能存在依赖性——最值得注意的是，正如我们在第10章“波动率和广义自回归条件异方差过程”中看到的那样，随着时间的推移，条件方差之间可能存在依赖性

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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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