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## 数据科学代写|金融统计代写Financial Statistics代考|Forecasting

In this section we make some considerations about forecasting and validation of the proposed model. Usually two data bases are used for testing tha forecasting ability of a model: one (in-sample), used for estimation, and the other (out-of-sample) used for comparing forecasts with true values. There is an extra complication in the case of volatility models: there is no unique definition of volatility. Andersen and Bollerslev (1998) show that if wrong estimates of volatility are used, evaluation of forecasting accuracy is compromised. We could use the realized volatility as a basis for comparison, or use some trading system.

We could, for example, have a model for hourly returns and use the realized volatility computed from 15 min returns for comparisons. In general, we can compute $v_{h, t}=\sum_{i=1}^{a_{h}} r_{t-i}^{2}$, where $a_{h}$ is the aggregation factor (4, in the case of 15 min returns). Then use some measure based on $s_{h}=\tilde{v}{h, t}-v{h, t}$ for example, mean squared error, where $\tilde{v}{h, t}$ is the volatility predicted by the proposed model. See Taylor and Xu (1997), for example. 32 J. Risk Financial Manag. 2020, 13,38 Now consider Model (3). The forecast of volatility at origin $t$ and horizon $\ell$ is given by \begin{aligned} \hat{\sigma}{t}^{2}(l) &=E\left(\sigma_{t+l}^{2} \mid X_{t}\right) \ &=E\left(C_{0}+C_{1}\left(r_{t+l-1}+\ldots+r_{t+l-a_{1}}\right)^{2}+\ldots+\right.\ &\left.+C_{m}\left(r_{t+l-1}+\ldots+r_{t+l-a_{m}}\right)^{2}+b_{1} \sigma_{t+l-1}^{2}+\ldots+b_{p} \sigma_{t+l-p}^{2} \mid X_{t}\right), \end{aligned}
where $X_{t}=\left(r_{t}, \sigma_{t}, r_{t-1}, \sigma_{t-1}, \ldots\right)$, for $l=1,2, \ldots$

## 数据科学代写|金融统计代写Financial Statistics代考|High Frequency Data

In this section we further elaborate on high frequency data and introduce the series that will be analyzed later. High frequency data are very important in the financial environment, mainly because there exist large movements in short intervals of time. This aspect represents an interesting opportunity for trading. Furthermore, it is well known that volatilities in different frequencies have significant cross-correlation. We can even say that coarse volatility predicts fine volatility better than the inverse, as shown in Dacorogna et al. (2001).

As an example, take the tick by tick foreign exchange (FX) time series Euro-Dollar, from January First 1999 to December 31, 2002. Returns are calculated using bid and ask prices, as
$$r_{t}=\ln \left(\left(p_{t}^{\text {bid }}+p_{t}^{a s k}\right) / 2\right)-\ln \left(\left(p_{t-1}^{\text {bid }}+p_{t-1}^{a s k}\right) / 2\right) .$$
We discard Saturdays and Sundays, and we replace holidays with the means of the last ten observations of the returns for each respective hour and day. After cleaning the data (see Dacorogna et al. (2001), for details) we will consider equally spaced returns, with sampling interval $\Delta t=15 \mathrm{~min}$. This seems to be adequate, as many studies indicate.

Figure 2 shows Euro-Dollar returns calculated as above. The length of this time series is 95,317 . The figure shows that the absolute returns present a seasonal pattern. This is due to the fact that physical time does not follow, necessarily, the same pattern as the business time. This is a typical behavior of a financial time series and we will use a seasonal adjustment procedure similar to that of Martens et al. (2002). However, we will use absolute returns instead of squared returns; that is, we will compute the seasonal pattern as
$$S_{d, s, h}=\frac{1}{s} \sum_{j=1}^{s} \mid\left(r_{d, j, h} \mid,\right.$$
where $r_{d s, h}$ is the return in the weekday $d$, week $s$ and hour $h$, and $s$ is the number of weeks from the beginning of the series. Therefore, $S_{d, N_{s}, h}$ is the rolling window mean of the absolute returns with the beginning fixed.

## 数据科学代写|金融统计代写Financial Statistics代考|Forecasting

$$\hat{\sigma} t^{2}(l)=E\left(\sigma_{t+l}^{2} \mid X_{t}\right) \quad=E\left(C_{0}+C_{1}\left(r_{t+l-1}+\ldots+r_{t+l-a_{1}}\right)^{2}+\ldots++C_{m}\left(r_{t+l-1}+\ldots+r_{t+l-a_{m}}\right)^{2}+b_{1} \sigma_{t+l-1}^{2}+\ldots+b_{p} \sigma_{t+l-p}^{2} \mid X_{t}\right)$$

## 数据科学代写|金融统计代写Financial Statistics代考|High Frequency Data

$$r_{t}=\ln \left(\left(p_{t}^{\text {bid }}+p_{t}^{a s k}\right) / 2\right)-\ln \left(\left(p_{t-1}^{\text {bid }}+p_{t-1}^{a s k}\right) / 2\right) .$$

(2001)），我们将考䖍等距回报，采样间隔 $\Delta t=15 \mathrm{~min}$. 正如许多研究表明的那样，这似乎是足够的。

(2002 年) 。但是，我们将使用绝对收益而不是平方收益; 也就是涚，我们将计算痵节性模式为
$$S_{d, s, h}=\frac{1}{s} \sum_{j=1}^{s} \mid\left(r_{d, j, h} \mid,\right.$$

## 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。

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## avatest™帮您通过考试

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## 数据科学代写|金融统计代写Financial Statistics代考|Inference on the Gold Price Data (In US Dollars) (1980–2013)

Gold price data, say $x_{t}$, were collected per ounce in US dollars over the years $1980-2013 .$ These were transformed as $z_{t}=100\left(\ln \left(x_{f}\right)-\ln \left(x_{t-1}\right)\right)$, which were then “wrapped” to obtain $\theta_{t}=z_{t} \bmod 2 \pi$ and finally transformed to $\hat{\theta}=\left(\theta_{t}-\bar{\theta}\right) \bmod 2 \pi$, where $\bar{\theta}$ denotes the mean direction of $\theta_{t}$ and $\hat{\theta}$ denotes the variable thetamod as used in the graphs. The Durbin-Watson test performed on the log ratio transformed data shows that the autocorrelation is zero. The test statistic of Watson’s goodness of fit Jammalamadaka and SenGupta (2001) for wrapped stable distribution was obtained as $0.01632691$ and the corresponding p-value was obtained as $0.9970284$, which is greater than $0.05$, indicating that the wrapped stable distribution fits the transformed gold price data (in US dollars). The modified truncated estimate $\hat{\alpha_{1}^{*}}$ is $0.3752206$ while the estimate by characteristic function method is $0.401409$. The value of the objective function using the characteristic function estimate is $2.218941$ while that using our modified truncated estimate is 2.411018.

## 数据科学代写|金融统计代写Financial Statistics代考|Inference on the Silver Price Data (In US Dollars) (1980–2013)

Data on the price of silver in US dollars collected per ounce over the same time period also underwent the same transformation. The Durbin-Watson test performed on the log ratio transformed data shows that the autocorrelation is zero. Here, the Watson’s goodness of fit test for wrapped stable distribution was also performed and the value of the statistic was obtained as $0.02530653$ and the corresponding $p$-value is $0.9639666$, which is greater than $0.05$, indicating that the wrapped stable distribution also fits the transformed silver price data (in US dollars). The modified truncated estimate of the index parameter $\alpha$ is $0.4112475$ while the estimate by characteristic function method is $0.644846$. The value of the objective function using the characteristic function estimate is $2.234203$ while that using our modified truncated estimate is $2.234432$.

## 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。

Posted on Categories:Financial Statistics, 数据科学代写, 统计代写, 统计代考, 金融统计

## avatest™帮您通过考试

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## 数据科学代写|金融统计代写Financial Statistics代考|The Trigonometric Moment Estimator

The regular symmetric stable distribution is defined through its characteristic function given by
$$\varphi(t)=\exp \left(i t \mu-|\sigma t|^{a}\right)$$
where $\mu$ is the location parameter; $\sigma$ is the scale parameter, which we take as 1 ; and $\alpha$ is the index or shape parameter of the distribution. Here, without loss of generality, we take $\mu=0$.

From the stable distribution, we can obtain the wrapped stable distribution (the process of wrapping explained in Jammalamadaka and SenGupta (2001)). Suppose $\theta_{1}, \theta_{2}, \ldots, \theta_{m}$ is a random sample of size $m$ drawn from the wrapped stable (given in Jammalamadaka and SenGupta (2001)) distribution whose probability density function is given by
$$f(\theta, \rho, \alpha, \mu)=\frac{1}{2 \pi}\left[1+2 \sum_{p=1}^{\infty} \rho^{p^{\alpha}} \cos p(\theta-\mu)\right] \quad 0<\rho \leq 1,0<\alpha \leq 2,0<\mu \leq 2 \pi$$
It is known in general from Jammalamadaka and SenGupta (2001) that the characteristic function of $\theta$ at the integer $p$ is defined as,
$$\psi_{\theta}(p)=E[\exp (i p(\theta-\mu))]=\alpha_{p}+i \beta_{p}$$
where $\quad \alpha_{p}=E \cos p(\theta-\mu) \quad$ and $\quad \beta_{p}=E \sin p(\theta-\mu)$

## 数据科学代写|金融统计代写Financial Statistics代考|Derivation of the Asymptotic Distribution of the Moment Estimator

Lemma $1 .$
$$\sqrt{m}\left(T_{m}-\mu\right) \stackrel{L}{\rightarrow} N_{4}(0, \Sigma)$$
where $T_{m}=\left(\bar{C}{1}, \overline{C{2}}, \overline{S_{1}}, \overline{S_{2}}\right)^{\prime}$,
$\mu$ is the mean vector given by
$$\mu=\left(\rho \cos \mu_{0}, \rho^{2^{\alpha}} \cos 2 \mu_{0}, \rho \sin \mu_{0}, \rho^{2^{a}} \sin 2 \mu_{0}\right)^{\prime}$$
and $\Sigma$ is the dispersion matrix given by

$$\Sigma=\left(\begin{array}{cccc} A & B & C & D \ B & E & F & G \ C & F & H & I \ D & G & I & J \end{array}\right)$$
where
$A=\frac{\rho^{2 \pi} \cos 2 \mu_{0}+1-2 \rho^{2} \cos ^{2} \mu_{0}}{2}$
$B=\frac{\rho \cos \mu_{0}+\rho^{3 \pi} \cos 3 \mu_{0}-22^{22^{\alpha}+1} \cos \mu_{0} \cos 2 \mu_{0}}{2}$
$C=\frac{\rho^{2 \pi} \sin 2 \mu_{0}-2 \rho^{2} \cos \mu_{0} \sin \mu_{0}}{2}$
$D=\frac{\rho^{3 \pi} \sin 3 \mu_{0}+\rho \sin \mu_{0}-22^{22^{\alpha}+1} \cos \mu_{0} \sin 2 \mu_{0}}{\alpha^{2}}$
$E=\frac{\rho^{4^{a}} \cos 4 \mu_{0}+1-2\left(\rho^{2 \alpha^{2}}\right)^{2} \cos ^{2} 2 \mu_{0}}{2}$
$F=\frac{\rho^{3^{\pi}} \sin 3 \mu_{0}-\rho \sin \mu_{0}-2 \rho^{2^{\alpha}}+1 \cos 2 \mu_{0} \sin \mu_{0}}{2}$
$G=\frac{\rho^{\rho^{a x}} \sin 4 \mu_{0}-2\left(\rho^{2 x}\right)^{2} \cos 2 \mu_{0} \sin 2 \mu_{0}}{2}$
$H=\frac{1-\rho^{2 \alpha} \cos 2 \mu_{0}-2 \rho^{2} \sin ^{2} \mu_{0}}{2}$
$I=\frac{\rho \cos \mu_{0}-\rho^{3^{\alpha}} \cos 3 \mu_{0}-2 \rho^{2^{\alpha}}+1 \sin \mu_{0} \sin 2 \mu_{0}}{2}$
$J=\frac{1-\rho^{4^{\alpha}} \cos 4 \mu_{0}-2\left(\rho^{2^{2}}\right)^{2} \sin ^{2} 2 \mu_{0}}{2}$
Proof. The derivations for the proof are given in Appendix A.

## 数据科学代写|金融统计代写Financial Statistics代考|The Trigonometric Moment Estimator

$$\varphi(t)=\exp \left(i t \mu-|\sigma t|^{a}\right)$$

$$f(\theta, \rho, \alpha, \mu)=\frac{1}{2 \pi}\left[1+2 \sum_{p=1}^{\infty} \rho^{p^{\alpha}} \cos p(\theta-\mu)\right] \quad 0<\rho \leq 1,0<\alpha \leq 2,0<\mu \leq 2 \pi$$

$$\psi_{\theta}(p)=E[\exp (i p(\theta-\mu))]=\alpha_{p}+i \beta_{p}$$

## 数据科学代写|金融统计代写Financial Statistics代考|Derivation of the Asymptotic Distribution of the Moment Estimator

$$\sqrt{m}\left(T_{m}-\mu\right) \stackrel{L}{\rightarrow} N_{4}(0, \Sigma)$$

$\mu$ 是由下式给出的平均向量
$$\mu=\left(\rho \cos \mu_{0}, \rho^{2^{\alpha}} \cos 2 \mu_{0}, \rho \sin \mu_{0}, \rho^{2^{a}} \sin 2 \mu_{0}\right)^{\prime}$$

$A=\frac{\rho^{2 \pi} \cos 2 \mu_{0}+1-2 \rho^{2} \cos ^{2} \mu_{0}}{2}$
$B=\frac{\rho \cos \mu_{0}+\rho^{3 \pi} \cos 3 \mu_{0}-22^{2 \alpha^{2}+1} \cos \mu_{0} \cos 2 \mu_{0}}{2}$
\begin{aligned} &C=\frac{\rho^{2 \pi} \sin 2 \mu_{0}-2 \rho^{2} \cos \mu_{0} \sin \mu_{0}}{2} \ &D=\frac{\rho^{3 \pi} \sin 3 \mu_{0}+\rho \sin \mu_{0}-22^{22^{\alpha}+1} \cos \mu_{0} \sin 2 \mu_{0}}{\alpha^{2}} \ &E=\frac{\rho^{4 a} \cos 4 \mu_{0}+1-2\left(\rho^{22^{2}}\right)^{2} \cos ^{2} 2 \mu_{0}}{2} \ &F=\frac{\rho^{3 \pi} \sin 3 \mu_{0}-\rho \sin \mu_{0}-2 \rho^{22^{\alpha}}+1 \cos 2 \mu 0 \sin \mu_{0}}{2} \ &G=\frac{\rho^{\rho^{\alpha x}} \sin 4 \mu_{0}-2\left(\rho^{2 x}\right)^{2} \cos 2 \mu_{0} \sin 2 \mu_{0}}{2} \ &H=\frac{1-\rho^{2 \alpha} \cos 2 \mu_{0}-2 \rho^{2} \sin ^{2} \mu_{0}}{2} \ &I=\frac{\rho \cos \mu_{0}-\rho^{3 \alpha} \cos 3 \mu_{0}-2 \rho^{2 \alpha}+1 \sin \mu_{0} \sin 2 \mu_{0}}{2} \ &J=\frac{1-\rho^{\rho^{\alpha}} \cos 4 \mu_{0}-2\left(\rho^{2}\right)^{2} \sin ^{2} 2 \mu_{0}}{2} \end{aligned}

## 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。