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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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 Global Temperatures Using Exponential

In Example 4.3, an ARIMA( $0,1,3)$ process without drift was fitted to monthly global temperatures, and in Example $7.2$ this model was used to provide forecasts out to 2020. As $\hat{\theta}2$ and $\hat{\theta}_3$, although significant, are both small when compared to $\hat{\theta}_1$, an $\operatorname{ARIMA}(0,1,1)$ process should provide a decent fit to the series, and indeed it does, with $\hat{\theta}=0.55$ and a root mean square error (RMSE) of $0.1255$ [compared with $0.1236$ for $\operatorname{ARIMA}(0,1,3)$ ]. From the equivalence of simple exponential smoothing and the ARIMA(0,1,1), we would expect the former model to produce a similar fit and forecasts for a smoothing parameter of $\alpha=0.45$. Fitting the series by simple exponential smoothing and estimating $\alpha$ does indeed lead to this value for the smoothing parameter, an RMSE of $0.1257$, and forecasts given by $f{T, h}=z_T=0.581$. These should be compared to the $\operatorname{ARIMA}(0,1,3)$ forecasts obtained in Example 7.2, which, for $h>2$, are equal to $0.621$.

Acknowledging the possibility of a linear trend in global temperatures would require the use of either double exponential smoothing or Holt-Winters. The former estimates the single smoothing parameter to be $\gamma=0.196$, accompanied by an RMSE of $0.1319$. Interestingly, double exponential smoothing gives $z_T=0.569$ and $\tau_T=-0.014$, so that, using (9.14), forecasts will contain a negatively sloped, albeit small, linear trend. Holt-Winters estimates the smoothing parameters as $\alpha=0.45$ and $\beta=0$, which implies that the trend component is a constant, so that $\tau_t=\tau_{t-1}=\cdots=\tau$, a value that is estimated by $x_2-x_1=0.0005$. The Holt-Winters forecasts thus include a small positive linear trend which increases the forecasts from $0.582$ to $0.599$ by the end of the forecast period, December 2020. In either case, there is an absence of a significant positive drift in the forecasts, consistent with our earlier findings. Given that the RMSE of Holt-Winters was $0.1256$, this implies that simple exponential smoothing is the most appropriate of these three techniques for forecasting monthly global temperatures.

## 统计代写|时间序列分析代写Time-Series Analysis代考|AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTIC PROCESSES

10.5 Up until this point we have said nothing about how the conditional variances $\sigma_t^2$ might be generated. We now consider the case where they are a function of past values of $x_t$ :
$$\sigma_t^2=f\left(x_{t-1}, x_{t-2}, \ldots\right)$$
A simple example is:
$$\sigma_t^2=f\left(x_{t-1}\right)=\alpha_0+\alpha_1\left(x_{t-1}-\mu\right)^2$$
where $\alpha_0$ and $\alpha_1$ are both positive to ensure that $\sigma_t^2>0$. With $U_t \sim \operatorname{NID}(0,1)$ and independent of $\sigma_t, x_t=\mu+\sigma_t U_t$ is then conditionally normal,
$$x_t \mid x_{t-1}, x_{t-2}, \ldots \sim \mathrm{N}\left(\mu, \sigma_t^2\right)$$
so that
$$V\left(x_t \mid x_{t-1}\right)=\alpha_0+\alpha_1\left(x_{t-1}-\mu\right)^2$$
If $0<\alpha_1<1$ then the unconditional variance is $V\left(x_t\right)=\alpha_0 /\left(1-\alpha_1\right)$ and $x_t$ is weakly stationary. It may be shown that the fourth moment of $x_t$ is finite if $3 \alpha_1^2<1$ and, if so, the kurtosis of $x_t$ is given by $3\left(1-\alpha_1^2\right) /\left(1-3 \alpha_1^2\right)$. Since this must exceed 3, the unconditional distribution of $x_t$ is fatter tailed than the normal. If this moment condition is not satisfied, then the variance of $x_t$ will be infinite and $x_t$ will not be weakly stationary.
10.6 This model is known as the first-order autoregressive conditional het eroskedastic [ARCH(1)] process and was originally introduced by Engle (1982, 1983). ARCH processes have proven to be extremely popular for modeling volatility in time series. A more convenient notation is to define $\varepsilon_t=x_t-\mu=U_t \sigma_t$, so that the $\mathrm{ARCH}(1)$ model can be written as:
$$\begin{gathered} \varepsilon_t \mid x_{t-1}, x_{t-2}, \ldots \sim \operatorname{NID}\left(0, \sigma_t^2\right) \ \sigma_t^2=\alpha_0+\alpha_1 \varepsilon_{t-1}^2 \end{gathered}$$

# 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|自回归条件异方差过程

$$\sigma_t^2=f\left(x_{t-1}, x_{t-2}, \ldots\right)$$一个简单的例子是:
$$\sigma_t^2=f\left(x_{t-1}\right)=\alpha_0+\alpha_1\left(x_{t-1}-\mu\right)^2$$
where $\alpha_0$ 和 $\alpha_1$ 两者都是积极的吗 $\sigma_t^2>0$。用 $U_t \sim \operatorname{NID}(0,1)$ 独立于 $\sigma_t, x_t=\mu+\sigma_t U_t$ 那么是否有条件正常，
$$x_t \mid x_{t-1}, x_{t-2}, \ldots \sim \mathrm{N}\left(\mu, \sigma_t^2\right)$$

$$V\left(x_t \mid x_{t-1}\right)=\alpha_0+\alpha_1\left(x_{t-1}-\mu\right)^2$$

$$\begin{gathered} \varepsilon_t \mid x_{t-1}, x_{t-2}, \ldots \sim \operatorname{NID}\left(0, \sigma_t^2\right) \ \sigma_t^2=\alpha_0+\alpha_1 \varepsilon_{t-1}^2 \end{gathered}$$

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