## matlab代写|time series analysisEMET3007/8012 Assignment 3

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析time series analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析time series analysis代写方面经验极为丰富，各种代写时间序列分析time series analysis相关的作业也就用不着说。

• 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

## Instructions:

This assignment is worth either 20% or 25% of the final grade, and is worth a total of 75 points. All working must be shown for all questions. For questions which ask you to write a program, you must provide the code you used. If you have found code and then modified it, then the original source must be cited. The assignment is due by 5pm Friday 1st of October (Friday of Week 8), using Turnitin on Wattle. Late submissions will only be accepted with prior written approval. Good luck.

[10 marks] In this exercise we will consider four different specifications for forecasting monthly Australian total employed persons. The dataset (available on Wattle) AUSEmp 1oy 2022. csv contains three columns; the first column contains the date; the second contains the sales figures for that month (FRED data series LFEMTTTTAUM647N), and the third contains Australian GDP for that month.1] The data runs from January 1995 to January $2022 .$

Let $M_{i t}$ be a dummy variable that denotes the month of the year. Let $D_{i t}$ be a dummy variable which denotes the quarter of the year. The four specifications we consider are
\begin{aligned} &S_1: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\epsilon_t \ &S_2: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\epsilon_t \ &S_3: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\epsilon_t \ &S_4: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$.

a) For each specification, describe this specification in words.
b) For each specification, estimate the values of the parameters, and compute the MSE, $\mathrm{AIC}$, and BIC. If you make any changes to the csv file, please describe the changes you make. As always, you must include your code.
c) For each specification, compute the MSFE for the 1-step and 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$.
d) For each specification, plot the out-of-sample forecasts and comment on the results.

[10 marks] Now add to Question 1 the additional assumption that $\epsilon_t \sim \mathcal{N}\left(0, \sigma^2\right)$. One estimator ${ }^2$ for $\sigma^2$ is
$$\hat{\sigma}^2=\frac{1}{T-k} \sum_{t=1}^T\left(y_t-\hat{y}_t\right)^2$$
where $\hat{y}_t$ is the estimated value of $y_t$ in the model and $k$ is the number of regressors in the specification.
a) For each specification $\left(S_1, \ldots, S_4\right)$, compute $\hat{\sigma}^2$.
b) For each specification, make a $95 \%$ probability forecast for the sales in June $2021 .$
c) For each specification, compute the probability that the total employed persons in June 2022 will be greater than $13.5$ million. According to the FRED series LFEMTTTTAUM647N, what was the actual employment level for that month.
d) Do you think the assumption that $\epsilon_t$ is iid is a reasonable assumption for this data series.

[10 marks] Here we investigate whether adding GDP $\mathrm{Gs}^3$ as a predictor can improve our forecasts. Consider the following modified specifications:
\begin{aligned} &S_1^{\prime}: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\gamma x_{t-h}+\epsilon_t \ &S_2^{\prime}: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\gamma x_{t-h}+\epsilon_t \ &S_3^{\prime}: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\gamma x_{t-h}+\epsilon_t \ &S_4^{\prime}: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\gamma x_{t-h}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$, and $x_{t-h}$ is GDP at time $t-h$. For each specification, compute the MSFE for the 1-step ahead, and the 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$. For each specification, plot the out-of-sample forecasts and comment on the results.

[15 marks] Here we investigate whether Holt-Winters smoothing can improve our forecasts. Use a Holt-Winters smoothing method with seasonality, to produce 1-step ahead and 5-step ahead forecasts and compute the MSFE for these forecasts. You should use smoothing parameters $\alpha=\beta=\gamma=0.3$ and start the out-of-sample forecasting exercise at $T_0=50$. Plot these out-of-sample forecasts and comment on the results.
Additionally, estimate the values for $\alpha, \beta$, and $\gamma$ which minimise the MSFE. Find the MSFE for these parameter vales and compare it to the baseline $\alpha=\beta=\gamma=0.3$.

[5 marks] Questions 1, 3 and 4 each provided alternative models for forecasting Australian Total Employment. Compare the efficacy of these forecasts. Your comparison should include discussions of MSFE, but must also make qualitative observations (typically based on your graphs).

[10 marks] Develop another model, either based on material from class or otherwise, to forecast Australian Total Employment. Your new model should perform better (have a lower MSFE or MAFE) than all models from Questions 1,3, and 4. As part of your response to this question you must provide:
a) a brief written explanation of what your model is doing,
b) a brief statement on why you think your new model will perform better,
c) any relevant equations or mathematics/statistics to describe the model,
d) the code to run the model, and
e) the MSFE and/or MAFE error found by your model, and a brief discussion of how this compares to previous cases.

[15 marks] Consider the ARX(1) model
$$y_t=\mu+a t+\rho y_{t-1}+\epsilon_t$$
where the errors follow an $\mathrm{AR}(2)$ process
$$\epsilon_t=\phi_1 \epsilon_{t-1}+\phi_2 \epsilon_{t-2}+u_t, \quad \mathbf{u} \sim \mathcal{N}\left(0, \sigma^2 I\right)$$
for $t=1, \ldots, T$ and $e_{-1}=e_0=0$. Suppose $\phi_1, \phi_2$ are known. Find (analytically) the maximum likelihood estimators for $\mu, a, \rho$, and $\sigma^2$.

Hint: First write $y$ and $\epsilon$ in vector/matrix form. You may wish to use different looking forms for each. Find the distribution of $\epsilon$ and $y$. Then apply some appropriate calculus. You may want to let $H=I-\phi_1 L-\phi_2 L^2$, where $I$ is the $T \times T$ identity matrix, and $L$ is the lag matrix.

## EMET3007/8012代写

matlab代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

## 统计代写|时间序列分析代写Time-Series Analysis代考|STAT3040

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

• 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|STATIONARITY INDUCING TRANSFORMATIONS

2.9 A simple stationarity transformation is to take successive differences of a series, on defining the first-difference of $x_t$ as $\nabla x_t=x_t-x_{t-1}$. Fig. $2.6$ shows the first-differences of the wine and spirits consumption series plotted in Fig. 1.6, that is, the annual changes in consumption. The trends in both series have been eradicated by this transformation and, as will be shown in Chapter 4, ARIMA Models for Nonstationary Time Series, differencing has a lot to recommend it both practically and theoretically for transforming a nonstationary series to stationarity.

First-differencing may, on some occasions, be insufficient to induce stationarity and further differencing may be required. Fig. $2.7$ shows successive temperature readings on a chemical process, this being Series $\mathrm{C}$ of Box and Jenkins (1970). The top panel shows observed temperatures. These display a distinctive form of nonstationarity, in which there are almost random switches in trend and changes in level. Although first differencing (shown as the middle panel) mitigates these switches and changes, it by no means eliminates them; second-differences are required to achieve this, as shown in the bottom panel.
2.10 Some caution is required when taking higher-order differences. The second-differences shown in Fig. $2.7$ are defined as the first-difference of the first-difference, that is, $\nabla \nabla x_t=\nabla^2 x_t$. To provide an explicit expression for second-differences, it is convenient to introduce the lag operator $B$, defined such that $B^i x_t \equiv x_{t-j}$, so that:
$$\nabla x_t=x_t-x_{t-1}=x_t-B x_t=(1-B) x_t$$
Consequently:
$$\nabla^2 x_t=(1-B)^2 x_t=\left(1-2 B+B^2\right) x_t=x_t-2 x_{t-1}+x_{t-2}$$
which is clearly not the same as $x_t-x_{t-2}=\nabla_2 x_t$, the two-period difference, where the notation $\nabla_j=1-B^j$ for the taking of $j$-period differences has been introduced. The distinction between the two is clearly demonstrated in Fig. 2.8, where second- and two-period differences of Series $\mathrm{C}$ are displayed.
2.11 For some time series, interpretation can be made easier hy taking proportional or percentage changes rather than simple differences, that is, transforming by $\nabla x_t / x_{t-1}$ or $100 \nabla x_t / x_{t-1}$. For financial time series these are typically referred to as the return. Fig. $2.9$ plots the monthly price of gold and its percentage return from 1980 to 2017 .

## 统计代写|时间序列分析代写Time-Series Analysis代考|DECOMPOSING A TIME SERIES AND SMOOTHING TRANSFORMATIONS

2.17 WMAs may also arise when a simple MA with an even number of terms is used but centering is required. For example, a four-term MA may be defined as:
$$\mathrm{MA}{t-1 / 2}(4)=\frac{1}{4}\left(x{t-2}+x_{t-1}+x_t+x_{t+1}\right)$$
where the notation makes clear that the “central” date to which the MA relates to is a noninteger, being halfway between $t-1$ and $t$, that is, $t-(1 / 2)$, but of course $x_{t-(1 / 2)}$ does not exist and the centering property is lost. At $t+1$, however, the MA is:
$$\operatorname{MA}{t+(1 / 2)}(4)=\frac{1}{4}\left(x{t-1}+x_t+x_{t+1}+x_{t+2}\right)$$
which has a central date of $t+(1 / 2)$. Taking the average of these two simple MAs produces a weighted MA centered on the average of $t-(1 / 2)$ and $t+(1 / 2)$, which is, of course, $t$ :
$$\mathrm{WMA}t(5)=\frac{1}{8} x{t-2}+\frac{1}{4} x_{t-1}+\frac{1}{4} x_t+\frac{1}{4} x_{t+1}+\frac{1}{8} x_{t+2}$$
This is an example of (2.9) with $n=2$ and where there are “half-weights” on the two extreme observations of the MA.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|DSC425

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

• 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|TIME SERIES HAVING NATURAL CONSTRAINTS

1.16 Some time series have natural constraints placed upon them. Fig. 1.11, for example, shows the consumption (c), investment $(i)$, government $(g)$, and “other” $(x)$ shares in the United Kingdom’s gross final expenditure for the period $1955 \mathrm{q} 1$ to $2017 \mathrm{q} 2$. Because these shares must be lie between zero and one and must also add up to one for each observation, these restrictions need to be accounted for, as to ignore them would make standard analysis of covariances and correlations invalid. Such compositional time series require distinctive treatment through special transformations before they can be analyzed, as is done in Chapter 16, Compositional and Count Time Series.
1.17 All the time series introduced so far may be regarded as being measured on a continuous scale, or at least can be assumed to be wellapproximated as being continuous. Some series, however, occur naturally as (small) integers and these are often referred to as counts. Fig. $1.12$ shows the annual number of North Atlantic storms and hurricanes (the latter being a subset of the former) between 1851 and 2017. The annual number of storms ranges from a minimum of one (in 1914) to a maximum of 28 in 2005; that year also saw the maximum number of hurricanes, 15, while there were no hurricanes in either 1907 and 1914. The figure uses spike graphs to emphasize the integer nature of these time series and this feature requires special techniques to analyze count data successfully, and will be discussed in Chapter 16, Compositional and Count Time Series.
1.18 Understanding the features exhibited by time series, both individually and in groups, is a key step in their successful analysis and clearly a great deal can be learnt by an initial plot of the data. Such plots may also suggest possible transformations of the data which may expedite formal analysis and modeling of time series, and it is to this topic that Chapter 2, Transforming Time Series, is devoted.

## 统计代写|时间序列分析代写Time-Series Analysis代考|DISTRIBUTIONAL TRANSFORMATIONS

2.2 Many statistical procedures perform more effectively on data that are normally distributed, or at least are symmetric and not excessively kurtotic (fat-tailed), and where the mean and variance are approximately constant. Observed time series frequently require some form of transformation before they exhibit these distributional properties, for in their “raw” form they are often asymmetric. For example, if a series is only able to take positive (or at least nonnegative) values, then its distribution will usually be skewed to the right, because although there is a natural lower bound to the data, often zero, no upper bound exists and the values are able to “stretch out,” possibly to infinity. In this case a simple and popular transformation is to take logarithms, usually to the base $e$ (natural logarithms).
2.3 Fig. $2.1$ displays histograms of the levels and logarithms of the monthly UK retail price index (RPI) series plotted in Fig. 1.7. Taking logarithms clearly reduces the extreme right-skewness found in the levels, but it certainly does not induce normality, for the distribution of the logarithms is distinctively bimodal.

The reason for this is clearly seen in Fig. 2.2, which shows a time series plot of the logarithms of the RPI. The central part of the distribution, which has the lower relative frequency, is transited swiftly during the 1970s, as this was a decade of high inflation characterized by the steepness of the slope of the series during this period.

Clearly, transforming to logarithms does not induce stationarity, but on comparing Fig. $2.2$ with Fig. 1.7, taking logarithms does “straighten out” the trend, at least to the extent that the periods before 1970 and after 1980 are both approximately linear with roughly the same slope. ${ }^1$ Taking logarithms also stabilizes the variance. Fig. $2.3$ plots the ratio of cumulative standard deviations, $s_i(\mathrm{RPI}) / s_i(\log \mathrm{RPI})$, defined using (1.2) and (1.3) as:
$$s_i^2(x)=i^{-1} \sum_{t=1}^i\left(x_t-\bar{x}i\right)^2 \quad \bar{x}_i=i^{-1} \sum{t=1}^i x_t$$ Since this ratio increases monotonically throughout the observation period, the logarithmic transformation clearly helps to stabilize the variance and it will, in fact, do so whenever the standard deviation of a series is proportional to its level. ${ }^2$

## 统计代写|时间序列分析代写Time-Series Analysis代考|DISTRIBUTIONAL transforms

.

$2.1$ 显示了英国零售价格指数(RPI)系列的水平和对数的直方图，如图1.7所示。取对数显然减少了在水平中发现的极端右偏性，但它肯定不能诱导正态性，因为对数的分布明显是双峰的

$$s_i^2(x)=i^{-1} \sum_{t=1}^i\left(x_t-\bar{x}i\right)^2 \quad \bar{x}_i=i^{-1} \sum{t=1}^i x_t$$由于这个比值在整个观察期间是单调增加的，对数变换显然有助于稳定方差，事实上，每当一个系列的标准差与它的水平成正比时，它就会这样做。${ }^2$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|STAT758

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

• 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|Time Series and Their Features

1.1 As stated in the Introduction, time series are indeed ubiquitous, appearing in almost every research field where data are analyzed. However, their formal study requires special statistical concepts and techniques without which erroneous inferences and conclusions may all too readily be drawn, a problem that statisticians have found necessary to confront since at least Udny Yule’s Presidential Address to the Royal Statistical Society in 1925, provocatively titled “Why do we sometimes get nonsense-correlations between time series? A study in sampling and the nature of time series.” 1
1.2 In general, a time series on some variable $x$ will be denoted as $x_t$, where the subscript $t$ represents time, with $t=1$ being the first observation available on $x$ and $t=T$ being the last. The complete set of times $t=1,2, \ldots, T$ will often be referred to as the observation period. The observations are typically measured at equally spaced intervals, say every minute, hour, or day, etc., so the order in which observations arrive is paramount. This is unlike, say, data on a cross section of a population taken at a given point in time, where the ordering of the data is usually irrelevant unless some form of spatial dependence exists between observations. ${ }^2$
1.3 Time series display a wide variety of features and an appreciation of these is essential for understanding both their properties and their evolution, including calculating future forecasts and, therefore, unknown values of $x_t$ at, say, times $T+1, T+2, \ldots, T+h$, where $h$ is referred to as the forecast horizon.

Fig. $1.1$ shows monthly observations of an index of the North Atlantic Oscillation (NAO) between 1950 and 2017. The NAO is a weather phenomenon in the North Atlantic Ocean and measures fluctuations in the difference of atmospheric pressure at sea level between two stable air pressure areas, the Subpolar low and the Subtropical (Azores) high. Strong positive phases of the NAO tend to be associated with above-normal temperatures in eastern United States and across northern Europe and with below-normal temperatures in Greenland and across southern Europe and the Middle East. These positive phases are also associated with above-normal precipitation over northern Europe and Scandinavia and with below-normal precipitation over southern and central Europe. Upposite patterns of temperature and precipitation anomalies are typically observed during strong negative phases of the NAO (see Hurrell et al., 2003).

Clearly, being able to identify recurring patterns in the NAO would be very useful for medium- to long-range weather forecasting, but, as Fig. $1.1$ illustrates, no readily discernible patterns seem to exist.

## 统计代写|时间序列分析代写Time-Series Analysis代考|AUTOCORRELATION AND PERIODIC MOVEMENTS

1.4 Such a conclusion may, however, be premature for there might well be internal correlations within the index that could be useful for identifying interesting periodic movements and for forecasting future values of the index. These are typically referred to as the autocorrelations between a current value, $x_t$, and previous, or lagged, values, $x_{t-k}$, for $k=1,2, \ldots$ The lag- $k$ (sample) autocorrelation is defined as
$$r_k=\frac{\sum_{t=k+1}^T\left(x_t-\bar{x}\right)\left(x_{t-k}-\bar{x}\right)}{T s^2}$$

where
$$\bar{x}=T^{-1} \sum_{t=1}^T x_t$$
and
$$s^2=T^{-1} \sum_{t=1}^T\left(x_t-\bar{x}\right)^2$$
are the sample mean and variance of $x_t$, respectively. The set of sample autocorrelations for various values of $k$ is known as the sample autocorrelation function (SACF) and plays a key role in time series analysis. An examination of the SACF of the NAO index is provided in Example 3.1.
1.5 A second physical time series that has a much more pronounced periodic movement is the annual sunspot number from 1700 to 2017 as shown in Fig. 1.2. As has been well-documented, sunspots display a periodic cycle (the elapsed time from one minimum (maximum) to the next) of approximately 11 years; see, for example, Hathaway (2010). The SACF can be used to calculate an estimate of the length of this cycle, as is done in Example 3.3.
1.6 Fig. $1.3$ shows the temperature of a hospital ward taken every hour for several months during 2011 and 2012 (see Iddon et al., 2015, for more details and description of the data). Here there is a long cyclical movement-an annual swing through the seasons-superimposed upon which are short-term diurnal movements as well as a considerable amount of random fluctuation (known as noise), typically the consequence of windows being left open on the ward for short periods of time and more persistent movements which are related to external temperatures and solar irradiation (sunshine).

## 统计代写|时间序列分析代写时间序列分析代考|时间序列及其特征

1.1如导言所述，时间序列确实是无处不在的，几乎出现在每一个分析数据的研究领域。然而，他们的正式研究需要特殊的统计概念和技术，没有这些概念和技术，就很容易得出错误的推论和结论。至少从1925年乌德尼·尤尔(Udny Yule)在皇家统计学会(Royal statistical Society)发表主席演讲以来，统计学家就发现有必要面对这个问题，该演讲的标题颇具挑衅意味:“为什么我们有时会在时间序列之间得到无意义的相关性?”一项关于采样和时间序列性质的研究。”1
1.2一般来说，一个关于某个变量的时间序列 $x$ 将表示为 $x_t$，其中下标 $t$ 表示时间，其中 $t=1$ 作为第一个观察到的 $x$ 和 $t=T$ 最后一个。全套的时间 $t=1,2, \ldots, T$ 通常被称为观察期。这些观测通常是在等距间隔的时间内进行测量的，比如每分钟、每小时或每一天等，因此观测的到达顺序是最重要的。这与在给定时间点上采集的种群横截面数据不同，后者数据的顺序通常是不相关的，除非观察之间存在某种形式的空间依赖性。 ${ }^2$时间序列显示出各种各样的特征，对这些特征的认识对于理解它们的性质和演变是至关重要的，包括计算未来的预测，以及因此的未知值 $x_t$ 在，比如说，几次 $T+1, T+2, \ldots, T+h$，其中 $h$ 称为预测层位。

$1.1$显示了1950年至2017年期间北大西洋涛动指数(NAO)的月度观测结果。NAO是北大西洋的一种天气现象，测量两个稳定气压区(亚极地低压区和亚热带(亚速尔群岛)高压区)之间的海平面大气压力差的波动。NAO的强正相位往往与美国东部和整个北欧的高于正常温度有关，与格陵兰岛和整个南欧和中东的低于正常温度有关。这些正相也与北欧和斯堪的那维亚半岛上空高于正常的降水和南欧和中欧上空低于正常的降水有关。温度和降水异常的反向模式通常在NAO的强负相期间观察到(见Hurrell等人，2003年) 显然，能够识别NAO中反复出现的模式对中期到长期天气预报非常有用，但是，如图$1.1$所示，似乎不存在容易识别的模式

## 统计代写|时间序列分析代写Time-Series Analysis代考|AUTOCORRELATION AND PERIODIC movement

1.4然而，这样的结论可能还为时过早，因为指数内部很可能存在内在的相互关系，有助于确定有趣的周期性波动和预测指数的未来值。这些通常被称为当前值$x_t$和之前的或滞后的值$x_{t-k}$(用于$k=1,2, \ldots$)之间的自相关性。滞后- $k$(样本)自相关性定义为
$$r_k=\frac{\sum_{t=k+1}^T\left(x_t-\bar{x}\right)\left(x_{t-k}-\bar{x}\right)}{T s^2}$$

$$\bar{x}=T^{-1} \sum_{t=1}^T x_t$$

$$s^2=T^{-1} \sum_{t=1}^T\left(x_t-\bar{x}\right)^2$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|STAT435

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

• 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|Seasonal ARIMA Modeling of Beer Sales

The SACF for the $\nabla \nabla^4$ transformation of beer sales shown in Fig. $9.3$ has $r_1=-0.56, r_2=-0.03, r_3=0.44, r_4=-0.65$, and $r_5=0.30$. Since $r_2 \approx 0$ and $r_1 r_4=0.36$, these first five sample autocorrelations are, within reasonable sampling limits, consistent with the ACF of an $\operatorname{ARIMA}(0,1,1)(0,1,1)4$ airline model. Using (9.7) the standard error of the sample autocorrelations for lags greater than 5 is calculated to be $0.20$. Only $r{16}$ exceeds two-standard errors, suggesting that this airline model could represent a satisfactory representation of the beer sales data. Fitting this model obtains ${ }^4$ :
$$\nabla_1 \nabla_4 x_t=\left(1-\begin{array}{c} 0.694 \ (0.098) \end{array}\right)\left(1-\begin{array}{c} 0.604 \ (0.110) \end{array} B^4\right) a_t \quad \hat{\sigma}=271.9$$
The more general seasonal ARIMA model is estimated to be:
$$\nabla \nabla_4 x_t=\left(1-\underset{(0.072)}{0.802} B-\underset{(0.095)}{0.552} B^4+\begin{array}{c} 0.631 \ (0.098) \end{array} B^5\right) a_t \quad \hat{\sigma}=265.0$$
The multiplicative model imposes the nonlinear restriction $\theta_1 \theta_4+\theta_5=0$. The log-likelihoods of the two models are $-547.76$ and $-545.64$, leading to a likelihood ratio test statistic of $4.24$, which is distributed as chi-square with one degree of freedom and so is not quite significant at the $2.5 \%$ level, although a Wald test does prove to be significant.

Using $\theta=0.7$ and $\Theta=0.6$ for simplicity, then the $\psi$-weights of the airline model are given, in general, by:
$$\begin{gathered} \psi_{4 r+1}=\psi_{4 r+2}=\psi_{4(r+1)-1}=0.3(r+1-0.6 r)=0.3+0.12 r \ \psi_{4(r+1)}=0.3(r+1-0.6 r)+0.4=0.7+0.12 r \end{gathered}$$
Thus,
\begin{aligned} &\psi_1=\psi_2=\psi_3=0.3 \ &\psi_4=0.7 \ &\psi_5=\psi_6=\psi_7=0.42 \ &\psi_8=0.82 \ &\psi_9=\psi_{10}=\psi_{11}=0.54, \text { etc. } \end{aligned}

9.13 In $\S 2.16$ we introduced a decomposition of an observed time series into trend, seasonal, and irregular (or noise) components, focusing attention on estimating the seasonal component and then eliminating it to provide a seasonally adjusted series. Extending the notation introduced in (8.1), this implicit UC decomposition can be written as
$$x_t=z_t+s_t+u_t$$
where the additional seasonal component $s_t$ is assumed to be independent of both $z_t$ and $u_t$. On obtaining an estimate of the seasonal component, $\hat{s}_t$, the seasonally adjusted series can then be defined as $x_t^a=x_t-\hat{s}_t$.
9.14 An important question is why we would wish to remove the seasonal component, rather than modeling it as an integral part of the stochastic process generating the data, as in fitting a seasonal ARIMA model, for example. A commonly held view is that the ability to recognize, interpret, or react to important nonseasonal movements in a series, such as turning points and other cyclical events, emerging patterns, or unexpected occurrences for which potential causes might be sought, may well be hindered by the presence of seasonal movements. Consequently, seasonal adjustment is carried out to simplify data so that they may be more easily interpreted by “statistically unsophisticated” users without this simplification being accompanied by too large a loss of information.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|DSC425

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

• 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

8.8 Given a UC model of the form of (8.1) and models for $z_t$ and $u_t$, it is often useful to provide estimates of these two unobserved components, a procedure that is known as signal extraction. A MMSE estimate of $z_t$, is an estimate $\hat{z}t$ which minimizes $E\left(\zeta_t^2\right)$, where $\zeta_t=z_t-\hat{z}_t$ is the estimation error (cf. \$7.2). From, for example. Pierce (1979). given an infinite sample of observations, denoted$\left{x_t,-\infty \leq t \leq \infty\right}$, such an estimator is: $$\hat{z}_t=\nu_z(B) x_t=\sum{j=-\infty}^{\infty} \nu_{z j} x_{t-j}$$ where the filter$\nu_z(B)$is defined as: $$\nu_z(B)=\frac{\sigma_v^2 \gamma(B) \gamma\left(B^{-1}\right)}{\sigma_e^2 \theta(B) \theta\left(B^{-1}\right)}$$ in which case the noise component can be estimated as: $$\hat{u}t=x_t-\hat{z}_t=\left(1-\nu_z(B)\right) x_t=\nu_u(B) x_t$$ For example, for the Muth model of a random walk overlaid with white noise: $$\nu_z(B)=\frac{\sigma_v^2}{\sigma_e^2}(1-\theta B)^{-1}\left(1-\theta B^{-1}\right)^{-1}=\frac{\sigma_v^2}{\sigma_e^2} \frac{1}{\left(1-\theta^2\right)} \sum{j=-\infty}^{\infty} \theta^{|j|} B^j$$ so that, using$\sigma_v^2=(1-\theta)^2 \sigma_e^2$, obtained using (8.6), we have: $$\hat{z}t=\left.\frac{(1-\theta)^2}{1-\theta^2} \sum{j=-\infty}^{\infty} \theta^{\mid j}\right|{x{t-j}}$$ Thus, for values of$\theta$close to unity,$\hat{z}_t$will be given by an extremely long moving average of future and past values of$x$. If$\theta$is close to zero, however,$\hat{z}_t$will be almost equal to the most recently observed value of$x$. From (8.3), large values of$\theta$are seen to correspond to small values of the signal-tonoise variance ratio$\kappa=\sigma_v^2 / \sigma_u^2$. When the noise component dominates, a long moving average of$x$values will provide the best estimate of the trend, while if the noise component is only small then the trend is essentially given by the current position of$x$. ## 统计代写|时间序列分析代写Time-Series Analysis代考|MODELING STOCHASTIC SEASONALITY 9.3 It would, however, be imprudent to rule out the possibility of an evolving seasonal pattern: in other words, the presence of stochastic seasonality. As in the modeling of stochastic trends, ARIMA processes have been found to do an excellent job in modeling stochastic seasonality, albeit in an extended form to that developed in previous chapters. 9.4 An important consideration when attempting to model a seasonal time series with an ARIMA model is to determine what sort of process will best match the SACFs and PACFs that characterize the data. Concentrating on the beer sales series, we have already noted the seasonal pattern in the SACF for$\nabla x_t$shown in Fig. 9.1. In considering the SACF further, we note that the seasonality manifests itself in large positive autocorrelations at the seasonal lags ($4 k, k \geq 1$) being flanked by negative autocorrelations at the “satellites”$[4(k-1), 4(k+1)]$. The slow decline of these seasonal autocorrelations is indicative of seasonal nonstationarity and, analogous to the analysis of “nonseasonal nonstationarity,” this may be dealt with by seasonal differencing, that is, by using the$\nabla_4=1-B^4$operator in conjunction with the usual$\nabla$operator. Fig.$9.3$shows the SACF of$\nabla \nabla^4$transformed beer sales and this is now clearly stationary and, thus, potentially amenable to ARIMA identification. 9.5 In general, if we have a seasonal period of$m$then the seasonal differencing operator may be denoted as$\nabla_m$. The nonseasonal and seasonal differencing operators may then be applied$d$and$D$times, respectively, so that a seasonal ARIMA model may take the general form $$\nabla^d \nabla_m^D \phi(B) x_t=\theta(B) a_t$$ Appropriate forms of the$\theta(B)$and$\phi(B)$polynomials can then, at least in principle, be obtained by the usual methods of identification and/or model selection. Unfortunately, two difficulties are typically encountered. First, the PACFs of seasonal models are difficult both to derive and to interpret, so that conventional identification is usually based solely on the behavior of the appropriate SACF. Second, since the$\theta(B)$and$\phi(B)$polynomials need to account for the seasonal autocorrelation, at least one of them must be of minimum order$m$. This often means that the number of models which need to be considered in model selection procedures can become prohibitively large. ## 时间序列分析代考 ## 统计代写|时间序列分析代写时间序列分析代考|信号提取 给定(8.1)形式的UC模型和$z_t$和$u_t$的模型，提供这两个未观测成分的估计通常是有用的，这个过程被称为信号提取。对$z_t$的MMSE估计是将$E\left(\zeta_t^2\right)$最小化的估计$\hat{z}t$，其中$\zeta_t=z_t-\hat{z}t$是估计误差(cf$7.2)。例如，从。皮尔斯(1979)。给定一个无限的观察样本，记为$\left{x_t,-\infty \leq t \leq \infty\right}$，这样的估计量是:$$\hat{z}_t=\nu_z(B) x_t=\sum{j=-\infty}^{\infty} \nu{z j} x_{t-j}$$
，其中滤波器$\nu_z(B)$定义为:
$$\nu_z(B)=\frac{\sigma_v^2 \gamma(B) \gamma\left(B^{-1}\right)}{\sigma_e^2 \theta(B) \theta\left(B^{-1}\right)}$$
，在这种情况下，噪声分量可以估计为:
$$\hat{u}t=x_t-\hat{z}_t=\left(1-\nu_z(B)\right) x_t=\nu_u(B) x_t$$例如，对于覆盖着白噪声的随机漫步的Muth模型:$$\nu_z(B)=\frac{\sigma_v^2}{\sigma_e^2}(1-\theta B)^{-1}\left(1-\theta B^{-1}\right)^{-1}=\frac{\sigma_v^2}{\sigma_e^2} \frac{1}{\left(1-\theta^2\right)} \sum{j=-\infty}^{\infty} \theta^{|j|} B^j$$
，因此，使用(8.6)得到的$\sigma_v^2=(1-\theta)^2 \sigma_e^2$，我们有:
$$\hat{z}t=\left.\frac{(1-\theta)^2}{1-\theta^2} \sum{j=-\infty}^{\infty} \theta^{\mid j}\right|{x{t-j}}$$

## 统计代写|时间序列分析代写时间序列分析代考|建模随机季节性

9.5一般来说，如果我们有一个$m$的季节性周期，那么季节差异算子可以表示为$\nabla_m$。然后可以分别应用非季节性和季节性差分算子$d$和$D$次，这样季节性ARIMA模型就可以采用一般形式
$$\nabla^d \nabla_m^D \phi(B) x_t=\theta(B) a_t$$然后，至少在原则上，可以通过通常的识别和/或模型选择方法得到$\theta(B)$和$\phi(B)$多项式的适当形式。不幸的是，通常会遇到两个困难。首先，季节模型的pacf难以推导和解释，因此传统的识别通常仅基于适当的SACF的行为。其次，由于$\theta(B)$和$\phi(B)$多项式需要考虑季节自相关，它们中至少有一个必须是最小阶$m$。这通常意味着在模型选择过程中需要考虑的模型数量会变得大得令人望而却步

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|STAT758

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

• 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|FORECASTING WITH AUTOREGRESSIVE-INTEGRATED

7.1 An important feature of the univariate models introduced in previous chapters is their ability to provide forecasts of future values of the observed series. There are two aspects to forecasting: the provision of a forecast for a future value of the series and the provision of a forecast error that can be attached to this point forecast. This forecast error may then be used to construct forecast intervals to provide an indication of the precision these forecasts are likely to possess. The setup is, thus, analogous to the classic statistical problem of estimating an unknown parameter of a model and providing a confidence interval for that parameter.

What is often not realized when forecasting is that the type of model used to construct point and interval forecasts will necessarily determine the properties of these forecasts. Consequently, forecasting from an incorrect or misspecified model may lead to forecasts that are inaccurate and which incorrectly measure the precision that may be attached to them. ${ }^1$
7.2 To formalize the forecasting problem, suppose we have a realization $\left(x_{1-d}, x_{2-d}, \ldots, x_T\right)$ from a general ARIMA $(p, d, q)$ process
$$\phi(B) \nabla^d x_t=\theta_0+\theta(B) a_t$$
and that we wish to forecast a future value $x_{T+h}, h$ being known as the lead time or forecast horizon. ${ }^2$ If we let
$$\alpha(B)=\phi(B) \nabla^d=\left(1-\alpha_1 B-\alpha_2 B^2-\cdots-\alpha_{p+d} B^{p+d}\right)$$ then (7.1) becomes, for time $T+h$,
$$\alpha(B) x_{T+h}=\theta_0+\theta(B) a_{T+h}$$
that is.
\begin{aligned} x_{T+h}=& \alpha_1 x_{T+h-1}+\alpha_2 x_{T+h-2}+\cdots+\alpha_{p+d} x_{T+h+p-d}+\theta_0+a_{T+h} \ &-\theta_1 a_{T+h-1}-\cdots-\theta_q a_{T+h-q} \end{aligned}
Clearly, observations from $T+1$ onwards are unavailable, but a minimum mean square error (MMSE) forecast of $x_{T+h}$ made at time $T$ (known as the origin), and denoted $f_{T, h}$, is given by the conditional expectation
\begin{aligned} f_{T, h}=& E\left(\alpha_1 x_{T+h-1}+\alpha_2 x_{T+h-2}+\cdots+\alpha_{p+d} x_{T+h-p-d}+\theta_0\right.\ &\left.+a_{T+h}-\theta_1 a_{T+h-1}-\cdots-\theta_q a_{T+h-q} \mid x_T, x_{T-1}, \ldots\right) . \end{aligned}

## 统计代写|时间序列分析代写Time-Series Analysis代考|UNOBSERVED COMPONENT MODELS

8.1 A difference stationary, that is, $I(1)$, time series may always be decomposed into a stochastic nonstationary trend, or signal, component and a stationary noise, or irregular, component:
$$x_t=z_t+u_t$$
Such a decomposition can be performed in several ways. For instance, Muth’s (1960) classic example assumes that the trend component $z_t$ is a random walk
$$z_t=\mu+z_{t-1}+v_t$$
while $u_t$ is white noise and independent of $v_t$, that is, $u_t \sim \mathrm{WN}\left(0, \sigma_u^2\right)$ and $v_I \sim \operatorname{WN}\left(0, \sigma_v^2\right)$, with $E\left(u_t v_{t-i}\right)=0$ for all $i$. Thus, it follows that $\nabla x_t$ is the stationary process
$$\nabla x_t=\mu+v_t+u_t-u_{t-1}$$
which has an autocorrelation function that cuts off at lag one with coefficient
$$\rho_1=-\frac{\sigma_u^2}{\sigma_u^2+2 \sigma_v^2}$$
It is clear from (8.3) that $-0.5 \leq \rho_1 \leq 0$, the exact value depending on the relative sizes of the two variances, so that $\nabla x_t$ can be written as the MA(1) process:
$$\nabla x_t=\mu+e_t-\theta e_{t-1}$$

where $e_t \sim \mathrm{WN}\left(0, \sigma_e^2\right)$. On defining $\kappa=\sigma_v^2 / \sigma_u^2$ to be the signal-to-noise variance ratio, the relationship between the parameters of (8.2) and (8.4) can be shown to be:
$$\theta=\frac{1}{2}\left((\kappa+2)-\left(\kappa^2+4 \kappa\right)^{1 / 2}\right), \quad \kappa=\frac{(1-\theta)^2}{\theta}, \quad \kappa \geq 0, \quad|\theta|<1$$
and
$$\sigma_u^2=\theta \sigma_e^2$$

## 统计代写|时间序列分析代写时间序列分析代考|预测与自回归-集成

$$\phi(B) \nabla^d x_t=\theta_0+\theta(B) a_t$$
，我们希望预测未来的值 $x_{T+h}, h$ 被称为提前期或预测期。 ${ }^2$ 如果我们让
$$\alpha(B)=\phi(B) \nabla^d=\left(1-\alpha_1 B-\alpha_2 B^2-\cdots-\alpha_{p+d} B^{p+d}\right)$$ 那么(7.1)变成，就时间而言 $T+h$，
$$\alpha(B) x_{T+h}=\theta_0+\theta(B) a_{T+h}$$

\begin{aligned} x_{T+h}=& \alpha_1 x_{T+h-1}+\alpha_2 x_{T+h-2}+\cdots+\alpha_{p+d} x_{T+h+p-d}+\theta_0+a_{T+h} \ &-\theta_1 a_{T+h-1}-\cdots-\theta_q a_{T+h-q} \end{aligned}显然，观察 $T+1$ 的最小均方误差(MMSE)预测 $x_{T+h}$ 制作时间 $T$ (称为原点)，并表示 $f_{T, h}$，由条件期望

## 统计代写|时间序列分析代写时间序列分析代考|未观察到的组件模型

$$x_t=z_t+u_t$$

$$z_t=\mu+z_{t-1}+v_t$$
，而$u_t$是白噪声，独立于$v_t$，即$u_t \sim \mathrm{WN}\left(0, \sigma_u^2\right)$和$v_I \sim \operatorname{WN}\left(0, \sigma_v^2\right)$, $E\left(u_t v_{t-i}\right)=0$对所有$i$。因此，可以得出$\nabla x_t$是平稳过程
$$\nabla x_t=\mu+v_t+u_t-u_{t-1}$$
，它具有一个自相关函数，在系数
$$\rho_1=-\frac{\sigma_u^2}{\sigma_u^2+2 \sigma_v^2}$$

$$\nabla x_t=\mu+e_t-\theta e_{t-1}$$

where $e_t \sim \mathrm{WN}\left(0, \sigma_e^2\right)$。将$\kappa=\sigma_v^2 / \sigma_u^2$定义为信噪比，(8.2)和(8.4)的参数之间的关系可以表示为:
$$\theta=\frac{1}{2}\left((\kappa+2)-\left(\kappa^2+4 \kappa\right)^{1 / 2}\right), \quad \kappa=\frac{(1-\theta)^2}{\theta}, \quad \kappa \geq 0, \quad|\theta|<1$$

$$\sigma_u^2=\theta \sigma_e^2$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## matlab代写|time series analysisEMET3007/8012 Assignment 2

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析time series analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析time series analysis代写方面经验极为丰富，各种代写时间序列分析time series analysis相关的作业也就用不着说。

• 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

## Instructions:

This assignment is worth either 20% or 25% of the final grade, and is worth a total of 75 points. All working must be shown for all questions. For questions which ask you to write a program, you must provide the code you used. If you have found code and then modified it, then the original source must be cited. The assignment is due by 5pm Friday 1st of October (Friday of Week 8), using Turnitin on Wattle. Late submissions will only be accepted with prior written approval. Good luck.

[10 marks] In this exercise we will consider four different specifications for forecasting monthly Australian total employed persons. The dataset (available on Wattle) AUSEmp 1oy 2022. csv contains three columns; the first column contains the date; the second contains the sales figures for that month (FRED data series LFEMTTTTAUM647N), and the third contains Australian GDP for that month.1] The data runs from January 1995 to January $2022 .$

Let $M_{i t}$ be a dummy variable that denotes the month of the year. Let $D_{i t}$ be a dummy variable which denotes the quarter of the year. The four specifications we consider are
\begin{aligned} &S_1: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\epsilon_t \ &S_2: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\epsilon_t \ &S_3: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\epsilon_t \ &S_4: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$.

a) For each specification, describe this specification in words.
b) For each specification, estimate the values of the parameters, and compute the MSE, $\mathrm{AIC}$, and BIC. If you make any changes to the csv file, please describe the changes you make. As always, you must include your code.
c) For each specification, compute the MSFE for the 1-step and 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$.
d) For each specification, plot the out-of-sample forecasts and comment on the results.

[10 marks] Now add to Question 1 the additional assumption that $\epsilon_t \sim \mathcal{N}\left(0, \sigma^2\right)$. One estimator ${ }^2$ for $\sigma^2$ is
$$\hat{\sigma}^2=\frac{1}{T-k} \sum_{t=1}^T\left(y_t-\hat{y}_t\right)^2$$
where $\hat{y}_t$ is the estimated value of $y_t$ in the model and $k$ is the number of regressors in the specification.
a) For each specification $\left(S_1, \ldots, S_4\right)$, compute $\hat{\sigma}^2$.
b) For each specification, make a $95 \%$ probability forecast for the sales in June $2021 .$
c) For each specification, compute the probability that the total employed persons in June 2022 will be greater than $13.5$ million. According to the FRED series LFEMTTTTAUM647N, what was the actual employment level for that month.
d) Do you think the assumption that $\epsilon_t$ is iid is a reasonable assumption for this data series.

[10 marks] Here we investigate whether adding GDP $\mathrm{Gs}^3$ as a predictor can improve our forecasts. Consider the following modified specifications:
\begin{aligned} &S_1^{\prime}: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\gamma x_{t-h}+\epsilon_t \ &S_2^{\prime}: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\gamma x_{t-h}+\epsilon_t \ &S_3^{\prime}: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\gamma x_{t-h}+\epsilon_t \ &S_4^{\prime}: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\gamma x_{t-h}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$, and $x_{t-h}$ is GDP at time $t-h$. For each specification, compute the MSFE for the 1-step ahead, and the 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$. For each specification, plot the out-of-sample forecasts and comment on the results.

[15 marks] Here we investigate whether Holt-Winters smoothing can improve our forecasts. Use a Holt-Winters smoothing method with seasonality, to produce 1-step ahead and 5-step ahead forecasts and compute the MSFE for these forecasts. You should use smoothing parameters $\alpha=\beta=\gamma=0.3$ and start the out-of-sample forecasting exercise at $T_0=50$. Plot these out-of-sample forecasts and comment on the results.
Additionally, estimate the values for $\alpha, \beta$, and $\gamma$ which minimise the MSFE. Find the MSFE for these parameter vales and compare it to the baseline $\alpha=\beta=\gamma=0.3$.

[5 marks] Questions 1, 3 and 4 each provided alternative models for forecasting Australian Total Employment. Compare the efficacy of these forecasts. Your comparison should include discussions of MSFE, but must also make qualitative observations (typically based on your graphs).

[10 marks] Develop another model, either based on material from class or otherwise, to forecast Australian Total Employment. Your new model should perform better (have a lower MSFE or MAFE) than all models from Questions 1,3, and 4. As part of your response to this question you must provide:
a) a brief written explanation of what your model is doing,
b) a brief statement on why you think your new model will perform better,
c) any relevant equations or mathematics/statistics to describe the model,
d) the code to run the model, and
e) the MSFE and/or MAFE error found by your model, and a brief discussion of how this compares to previous cases.

[15 marks] Consider the ARX(1) model
$$y_t=\mu+a t+\rho y_{t-1}+\epsilon_t$$
where the errors follow an $\mathrm{AR}(2)$ process
$$\epsilon_t=\phi_1 \epsilon_{t-1}+\phi_2 \epsilon_{t-2}+u_t, \quad \mathbf{u} \sim \mathcal{N}\left(0, \sigma^2 I\right)$$
for $t=1, \ldots, T$ and $e_{-1}=e_0=0$. Suppose $\phi_1, \phi_2$ are known. Find (analytically) the maximum likelihood estimators for $\mu, a, \rho$, and $\sigma^2$.

Hint: First write $y$ and $\epsilon$ in vector/matrix form. You may wish to use different looking forms for each. Find the distribution of $\epsilon$ and $y$. Then apply some appropriate calculus. You may want to let $H=I-\phi_1 L-\phi_2 L^2$, where $I$ is the $T \times T$ identity matrix, and $L$ is the lag matrix.

## EMET3007/8012代写

matlab代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

## 统计代写|时间序列分析代写Time-Series Analysis代考|STAT3040

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

• 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|STOCHASTIC PROCESSES AND STATIONARITY

3.1 The concept of a stationary time series was introduced informally in Chapter 1, Time Series and Their Features, but to proceed further it is necessary to consider the concept rather more rigorously. To this end, it is often useful to regard the observations $x_1, x_2, \ldots, x_T$ on the series $x_t$ as a realization of a stochastic process. In general, such a stochastic process may be described by a $T$-dimensional probability distribution, so that the relationship between a realization and a stochastic process is analogous, in classical statistics, to that between a sample and the population from which it has been drawn from.
Specifying the complete form of the probability distribution, however, will typically be too ambitious a task, so attention is usually concentrated on the first and second moments; the $T$ means:
$$E\left(x_1\right), E\left(x_2\right), \ldots, E\left(x_T\right)$$
$T$ variances:
$$V\left(x_1\right), V\left(x_2\right), \ldots, V\left(x_T\right)$$
and $T(T-1) / 2$ covariances:
$$\operatorname{Cov}\left(x_i, x_j\right), \quad i<j$$
If the distribution could be assumed to be (multivariate) normal, then this set of expectations would completely characterize the properties of the stochastic process. As has been seen from the examples in Chapter 2, Transforming Time Series, however, such an assumption will not always be appropriate, but if the process is taken to be linear, in the sense that the current value $x_t$ is generatē by a linear combination of previous valuess $x_{t-1}, x_{t-2}, \ldots$ of the process itself plus current and past values of any other related processes, then again this set of expectations would capture its major properties.

## 统计代写|时间序列分析代写Time-Series Analysis代考|WOLD’S DECOMPOSITION AND AUTOCORRELATION

3.6 A fundamental theorem in time series analysis, known as Wold’s decomposition, states that every weakly stationary, purely nondeterministic, stochastic process $x_t-\mu$ can be written as a linear combination (or linear filter) of a sequence of uncorrelated random variables. ${ }^2$ “Purely nondeterministic” means that any deterministic components have been subtracted from $x_t-\mu$. Such components are those that can be perfectly predicted from past values of themselves and examples commonly found are a (constant) mean, as is implied by writing the process as $x_t-\mu$, periodic sequences (e.g., sine and cosine functions), and polynomial or exponential sequences in $t$.
This linear filter representation is given by:
$$x_t-\mu=a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots=\sum_{j=0}^{\infty} \psi_j a_{t-j} \quad \psi_0=1$$

The $a_t, t=0, \pm 1, \pm 2, \ldots$ are a sequence of uncorrelated random variables, often known as innovations, drawn from a fixed distribution with:
$$E\left(a_t\right)=0 \quad V\left(a_t\right)=E\left(a_t^2\right)=\sigma^2<\infty$$
and
$$\operatorname{Cov}\left(a_t, a_{t-k}\right)=E\left(a_t a_{t-k}\right)=0, \text { for all } k \neq 0$$
Such a sequence is known as a white noise process, and occasionally the innovations will be denoted as $a_t \sim \mathrm{WN}\left(0, \sigma^2\right) .^3$ The coefficients (possibly infinite in number) in the linear filter (3.2) are known as $\psi$-weights.
3.7 It is easy to show that the model (3.2) leads to autocorrelation in $x_t$. From this equation it follows that:
$$E\left(x_t\right)=\mu$$
and
\begin{aligned} \gamma_0 &=V\left(x_t\right)=E\left(x_t-\mu\right)^2 \ &=E\left(a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots\right)^2 \ &=E\left(a_t^2\right)+\psi_1^2 E\left(a_{t-1}^2\right)+\psi_2^2 E\left(a_{t-2}^2\right)+\cdots \ &=\sigma^2+\psi_1^2 \sigma^2+\psi_2^2 \sigma^2+\cdots \ &=\sigma^2 \sum_{j=0}^{\infty} \psi_j^2 \end{aligned}
by using the white noise result that $E\left(a_{t-i} a_{t-j}\right)=0$ for $i \neq j$.

## 统计代写|时间序列分析代写Time-Series Analysis代考|STOCHASTIC PROCESSES AND STATIONARITY

$3.1$ 平稳时间序列的概念在第 1 章“时间序列及其特征”中非正式地介绍过，但为了进一步深入，有必要更严格地考 虑这个概念。为此，考虑观察结果通常是有用的 $x_1, x_2, \ldots, x_T$ 在系列上 $x_t$ 作为随机过程的实现。一般来说，这 样的随机过程可以用 $T$ 维概率分布，因此在经典统计中，实现与随机过程之间的关系类似于样本与从中抽取样本的 总体之间的关系。

$$E\left(x_1\right), E\left(x_2\right), \ldots, E\left(x_T\right)$$
$T$ 差异:
$$V\left(x_1\right), V\left(x_2\right), \ldots, V\left(x_T\right)$$

$$\operatorname{Cov}\left(x_i, x_j\right), \quad i<j$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|WOLD’S DECOMPOSITION AND AUTOCORRELATION

$3.6$ 时间序列分析中的一个基本定理，称为 Wold 分解，指出每个弱平稳、纯非确定性、随机过程 $x_t-\mu$ 可以写成 一系列不相关的随机变量的线性组合 (或线性滤波器) 。 ${ }^2$ 纯非确定性”意味看已从其中减去任何确定性组件 $x_t-\mu$. 此类组件是可以从其过去的值中完美预测的组件，并且常见的示例是 (恒定的) 均值，正如将过程编写为 所暗示的那样 $x_t-\mu$ ，周期序列（例如，正弦和余弦函数），以及多项式或指数序列 $t$. 该线性滤波器表示由下式给出：
$$x_t-\mu=a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots=\sum_{j=0}^{\infty} \psi_j a_{t-j} \quad \psi_0=1$$

$$E\left(a_t\right)=0 \quad V\left(a_t\right)=E\left(a_t^2\right)=\sigma^2<\infty$$

$$\operatorname{Cov}\left(a_t, a_{t-k}\right)=E\left(a_t a_{t-k}\right)=0, \text { for all } k \neq 0$$

$3.7$ 很容易证明模型 (3.2) 导致自相关 $x_t$. 从这个等式可以得出:
$$E\left(x_t\right)=\mu$$

$$\gamma_0=V\left(x_t\right)=E\left(x_t-\mu\right)^2 \quad=E\left(a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots\right)^2=E\left(a_t^2\right)+\psi_1^2 E\left(a_{t-1}^2\right)+\psi_2^2 E$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|DSC425

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

• 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|DISTRIBUTIONAL TRANSFORMATIONS

2.2 Many statistical procedures perform more effectively on data that are normally distributed, or at least are symmetric and not excessively kurtotic (fat-tailed), and where the mean and variance are approximately constant. Observed time series frequently require some form of transformation before they exhibit these distributional properties, for in their “raw” form they are often asymmetric. For example, if a series is only able to take positive (or at least nonnegative) values, then its distribution will usually be skewed to the right, because although there is a natural lower bound to the data, often zero, no upper bound exists and the values are able to “stretch out,” possibly to infinity. In this case a simple and popular transformation is to take logarithms, usually to the base $e$ (natural logarithms).
2.3 Fig. $2.1$ displays histograms of the levels and logarithms of the monthly UK retail price index (RPI) series plotted in Fig. 1.7. Taking logarithms clearly reduces the extreme right-skewness found in the levels, but it certainly does not induce normality, for the distribution of the logarithms is distinctively bimodal.

The reason for this is clearly seen in Fig. 2.2, which shows a time series plot of the logarithms of the RPI. The central part of the distribution, which has the lower relative frequency, is transited swiftly during the 1970 s, as this was a decade of high inflation characterized by the steepness of the slope of the series during this period.

Clearly, transforming to logarithms does not induce stationarity, but on comparing Fig. $2.2$ with Fig. 1.7, taking logarithms does “straighten out” the trend, at least to the extent that the periods before 1970 and after 1980 are both approximately linear with roughly the same slope. ${ }^1$ Taking logarithms also stabilizes the variance. Fig. $2.3$ plots the ratio of cumulative standard deviations, $s_i(\mathrm{RPI}) / s_i(\log \mathrm{RPI})$, defined using (1.2) and (1.3) as:
$$s_i^2(x)=i^{-1} \sum_{t=1}^i\left(x_t-\bar{x}i\right)^2 \quad \bar{x}_i=i^{-1} \sum{t=1}^i x_t$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|STATIONARITY INDUCING TRANSFORMATIONS

2.9 A simple stationarity transformation is to take successive differences of a series, on defining the first-difference of $x_t$ as $\nabla x_t=x_t-x_{t-1}$. Fig. $2.6$ shows the first-differences of the wine and spirits consumption series plotted in Fig. 1.6, that is, the annual changes in consumption. The trends in both series have been eradicated by this transformation and, as will be shown in Chapter 4, ARIMA Models for Nonstationary Time Series, differencing has a lot to recommend it both practically and theoretically for transforming a nonstationary series to stationarity.

First-differencing may, on some occasions, be insufficient to induce stationarity and further differencing may be required. Fig. $2.7$ shows successive temperature readings on a chemical process, this being Series $\mathrm{C}$ of Box and Jenkins (1970). The top panel shows observed temperatures. These display a distinctive form of nonstationarity, in which there are almost random switches in trend and changes in level. Although first differencing (shown as the middle panel) mitigates these switches and changes, it by no means eliminates them; second-differences are required to achieve this, as shown in the bottom panel.
2.10 Some caution is required when taking higher-order differences. The second-differences shown in Fig. $2.7$ are defined as the first-difference of the first-difference, that is, $\nabla \nabla x_t=\nabla^2 x_t$. To provide an explicit expression for second-differences, it is convenient to introduce the lag operator $B$, defined such that $B^j x_t \equiv x_{t-j}$, so that:
$$\nabla x_t=x_t-x_{t-1}=x_t-B x_t=(1-B) x_t$$
Consequently:
$$\nabla^2 x_t=(1-B)^2 x_t=\left(1-2 B+B^2\right) x_t=x_t-2 x_{t-1}+x_{t-2}$$
which is clearly not the same as $x_t-x_{t-2}=\nabla_2 x_t$, the two-period difference, where the notation $\nabla_j=1-B^j$ for the taking of $j$-period differences has been introduced. The distinction between the two is clearly demonstrated in Fig. $2.8$, where second- and two-period differences of Series $\mathrm{C}$ are displayed.

## 统计代写|时间序列分析代写Time-Series Analysis代考|DISTRIBUTIONAL TRANSFORMATIONS

$2.2$ 许多统计程序对正态分布的数据执行更有效，或者至少是对称的且没有过度峰态 (肥尾)，并且均值和方差近 似为常数。观察到的时间序列在表现出这些分布特性之前经常需要某种形式的转换，因为它们的“原始”形式通常是 不对称的。例如，如果一个序列只能取正值 (或至少是非负值)，那么它的分布通常会向右倾斜，因为虽然数据有 一个自然的下限，通常为零，但不存在上限并且这些值能够“延伸”到无穷大。在这种情况下，一个简单而流行的变 换是取对数，通常是底数e（自然对数）。
$2.3$ 图。2.1显示图 $1.7$ 中绘制的每月英国零售价格指数 (RPI) 系列水平和对数的直方图。取对数明显减少了水平中 发现的极端右偏度，但它肯定不会导致正态性，因为对数的分布明显是双峰的。

$$s_i^2(x)=i^{-1} \sum_{t=1}^i\left(x_t-\bar{x} i\right)^2 \quad \bar{x}_i=i^{-1} \sum t=1^i x_t$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|STATIONARITY INDUCING TRANSFORMATIONS

$2.9$ 一个简单的平稳性变换是对一个系列的连续差分，定义一阶差分 $x_t$ 作为 $\nabla x_t=x_t-x_{t-1}$. 如图。 $2.6$ 显示了 图 $1.6$ 中绘制的葡萄酒和烈酒消费系列的一阶差分，即消费的年度变化。这两个序列的趋势已经被这种转换消除 了，正如将在第 4 章，非平稳时间序列的 ARIMA 模型中展示的那样，差分对于将非平稳序列转换为平稳有很多实 际和理论上的建议。

$2.10$ 取高阶差分时需要小心。二次差分如图所示。 $2.7$ 被定义为一阶差分的一阶差分，即 $\nabla \nabla x_t=\nabla^2 x_t$. 为了提 供二阶差分的显式表达式，引入滞后算子很方便 $B$ ，定义为 $B^j x_t \equiv x_{t-j}$ ，以便:
$$\nabla x_t=x_t-x_{t-1}=x_t-B x_t=(1-B) x_t$$

$$\nabla^2 x_t=(1-B)^2 x_t=\left(1-2 B+B^2\right) x_t=x_t-2 x_{t-1}+x_{t-2}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。