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金融代写|金融数学FINANCIAL MATHEMATICS代写|MAP3170 Forecasting for ARIMA Processes and Properties of Forecast Errors

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金融代写|金融数学FINANCIAL MATHEMATICS代写|Forecasting for ARIMA Processes and Properties of Forecast Errors

For forecasting in the time series setting, we assume that the process $\left{Y_{t}\right}$ follows an $\operatorname{ARIMA}(p, d, q)$ model, $\phi(B)(1-B)^{d} Y_{t}=\theta(B) \varepsilon_{t}$, and we assume the white noise series $\varepsilon_{t}$ are mutually independent random variables. We are interested in forecasting the future value $Y_{t+l}$ based on observations $Y_{t}, Y_{t-1}, \cdots$. From the result in the previous section, the minimum MSE forecast of $Y_{t+l}$ based on $Y_{t}, Y_{t-1}, \ldots$, which we will denote as $\hat{Y}{t}(l)$, is such that $\hat{Y}{t}(l)=\mathrm{E}\left(Y_{t+l} \mid Y_{t}, Y_{t-1}, \ldots\right)$. The prediction $\hat{Y}{t}(l)$ is called the lead $l$ or $l$-step ahead forecast of $Y{t+l}, l$ is the lead time, and $t$ is the forecast origin.
To obtain a representation for $\hat{Y}{t}(l)$, recall the ARIMA process has the “infinite” MA form $Y{t}=\psi(B) \varepsilon_{t}=\sum_{i=0}^{\infty} \psi_{i} \varepsilon_{t-i}$, and hence a future value $Y_{t+l}$ at time $t+l$, relative to the current time or “forecast origin” $t$, can be expressed as
$$Y_{t+l}=\sum_{i=0}^{\infty} \psi_{i} \varepsilon_{t+l-i}=\sum_{i=0}^{l-1} \psi_{i} \varepsilon_{t+l-i}+\sum_{i=l}^{\infty} \psi_{i} \varepsilon_{t+l-i}$$
The information contained in the past history of the $Y_{t}^{\prime} s,\left{Y_{s}, s \leq t\right}$, is the same as that contained in the past random shocks $\varepsilon_{t}$ ‘s (because the $Y_{t}$ ‘s are generated by the $\varepsilon_{t}$ ‘s). Also, $\varepsilon_{t+h}$, for $h>0$ is independent of present and past values $Y_{t}, Y_{t-1}, \ldots$, so that $\mathrm{E}\left(\varepsilon_{t+h} \mid Y_{t}, Y_{t-1}, \ldots\right)=0, h>0$. Thus
$$\hat{Y}{t}(l)=\mathrm{E}\left(Y{t+l} \mid Y_{t}, Y_{t-1}, \ldots\right)=\mathrm{E}\left(Y_{t+l} \mid \varepsilon_{t}, \varepsilon_{t-1}, \ldots\right)=\sum_{i=l}^{\infty} \psi_{i} \varepsilon_{t+l-i}$$
using the additional property that $\mathrm{E}\left(\varepsilon_{t+l-i} \mid \varepsilon_{t}, \varepsilon_{t-1}, \ldots\right)=\varepsilon_{t+l-i}$ if $i \geq l$. The $l$-step ahead prediction error is given by $e_{t}(l)=Y_{t+l}-\hat{Y}{t}(l)=\sum{i=0}^{l-1} \psi_{i} \varepsilon_{t+l-i}$. So we have $\mathrm{E}\left[e_{t}(l)\right]=0$ and the mean squared error or variance of the $l$-step prediction error is
$$\sigma^{2}(l)=\operatorname{Var}\left(e_{t}(l)\right)=\mathrm{E}\left[e_{t}^{2}(l)\right]=\operatorname{Var}\left(\sum_{i=0}^{l-1} \psi_{i} \varepsilon_{t+l-i}\right)=\sigma^{2} \sum_{i=0}^{l-1} \psi_{i}^{2}$$

金融代写|金融数学FINANCIAL MATHEMATICS代写|Stylized Models for Asset Returns

It is generally known that the equity market is extremely efficient in quickly absorbing information about stocks: When new information arrives, it gets incorporated in the price without delay. Thus the efficient market hypothesis is widely accepted by financial economists. It would imply that neither technical analysts that study the past prices in an attempt to predict future prices nor fundamental analysts of financial information related to company earnings and asset values would carry any advantage over the returns obtained from a randomly selected portfolio of individual stocks.

The efficient market hypothesis is usually associated with the idea of the “random walk,” which implies that all future price changes represent random departures from past prices. As stated in Malkiel (2012) [258]: “The logic of the random walk is that if the flow of information is unimpeded and information is immediately reflected in stock prices, then tomorrow’s price change will reflect only tomorrow’s news and will be independent of the price changes today. But news by definition is unpredictable, and, thus resulting price changes must be unpredictable and random.”

The random walk (RW) model (Example $2.3$ ) without the drift term can be stated as:
$$p_{t}=p_{t-1}+\varepsilon_{t},$$
where $\varepsilon_{t} \sim N\left(0, \sigma^{2}\right)$ i.i.d. and $p_{t}=\ln \left(P_{t}\right)$. Note as observed earlier, this model is a particular case of $\mathrm{AR}(1)$ model if the constant term is assumed to be zero and the slope, $\phi$, is assumed to be one. Thus, the RW model is a non-stationary model and considering $\varepsilon_{t}=p_{t}-p_{t-1}=r_{t}$ is the differencing of the series, $p_{t}$, makes the series $\varepsilon_{t}$ stationary. Note $r_{t} \approx \frac{P_{t}-P_{t-1}}{P_{t-1}}$, the returns are purely random and are unpredictable. For any chronological data the decision concerning the need for differencing is based, informally, on the features of time series plot of $p_{t}$, its sample autocorrelation function; that is, its failure to dampen out sufficiently quickly.

金融代写|金融数学FINANCIAL MATHEMATICS代写|Forecasting for ARIMA Processes and Properties of Forecast Errors

$$Y_{t+l}=\sum_{i=0}^{\infty} \psi_{i} \varepsilon_{t+l-i}=\sum_{i=0}^{l-1} \psi_{i} \varepsilon_{t+l-i}+\sum_{i=l}^{\infty} \psi_{i} \varepsilon_{t+l-i}$$

$$\hat{Y} t(l)=\mathrm{E}\left(Y t+l \mid Y_{t}, Y_{t-1}, \ldots\right)=\mathrm{E}\left(Y_{t+l} \mid \varepsilon_{t}, \varepsilon_{t-1}, \ldots\right)=\sum_{i=l}^{\infty} \psi_{i} \varepsilon_{t+l-i}$$

$$\sigma^{2}(l)=\operatorname{Var}\left(e_{t}(l)\right)=\mathrm{E}\left[e_{t}^{2}(l)\right]=\operatorname{Var}\left(\sum_{i=0}^{l-1} \psi_{i} \varepsilon_{t+l-i}\right)=\sigma^{2} \sum_{i=0}^{l-1} \psi_{i}^{2}$$

金融代写|金融数学FINANCIAL MATHEMATICS代写|Stylized Models for Asset Returns

$$p_{t}=p_{t-1}+\varepsilon_{t,}$$

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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。