# 统计代写|时间序列分析代写Time-Series Analysis代考|STA457H1

## 统计代写|时间序列分析代写Time-Series Analysis代考|TESTING FOR FRACTIONAL DIFFERENCING

5.28 The “classic” approach to detecting the presence of long memory in a time series is to use the range over standard deviation or rescaled rangc $(R / S)$ statistic. This was originally developed by Hurst (1951) when studying river discharges and a revised form was later proposed in an economic context by Mandelbrot (1972). It is defined as the range of partial sums of deviations of a time series from its mean, rescaled by its standard deviation, i.e.,
$$R_0=\hat{\sigma}0^{-1}\left[\max {1 \leq i \leq T} \sum_{t=1}^i\left(x_t-\bar{x}\right)-\min {1 \leq i \leq T} \sum{t=1}^i\left(x_t-\bar{x}\right)\right] \quad \hat{\sigma}0^2=T^{-1} \sum{t=1}^T\left(x_t-\bar{x}\right)^2$$
The first term in brackets is the maximum of the partial sums of the first $i$ deviations of $x_t$ from the sample mean. Since the sum of all $T$ deviations of the $x_t \mathrm{~s}$ from their mean is zero, this maximum is always nonnegative. The second term is the minimum of the same sequence of partial sums, and hence is always nonpositive. The difference between the two quantities, called the “range” for obvious reasons, is therefore always nonnegative, so that $R_0 \geq 0$.
5.29 Although it has long been established that the $R / S$ statistic is certainly able to detect long-range dependence, it is nevertheless sensitive to short-run influences. Consequently, any incompatibility between the data and the predicted behavior of the $R / S$ statistic under the null of no long run dependence need not come from long memory, but may merely be a symptom of shortrun autocorrelation.

The $R / S$ statistic was, thus, modified by Lo (1991), who incorporated short-run dependence into the estimator of the standard deviation, replacing (5.23) with
$$R_q=\hat{\sigma}q^{-1}\left[\max {1 \leq i \leq T} \sum_{t=1}^i\left(x_t-\bar{x}\right)-\min {1 \leq i \leq T} \sum{t=1}^i\left(x_t-\bar{x}\right)\right]$$
where
$$\hat{\sigma}q^2=\hat{\sigma}_0^2\left(1+\frac{2}{T} \sum{j=1}^q w_{q j} r_j\right) w_{q j}=1-\frac{j}{q \mid 1}, q<T$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|ESTIMATING THE FRACTIONAL DIFFERENCING PARAMETER

5.32 A drawback of the $F D-F$ procedure is that, if $d_1$ is not known a priori, as it is in the standard Dickey-Fuller case, then a consistent estimate must be provided. A variety of estimators have been suggested, many of which involve quite complex calculations. Perhaps the simplest is suggested by $R / S$ analysis and is
$$\tilde{d}=\frac{\log R_0}{\log T}-0.5$$
A popular and relatively simple estimator is the log-periodogram regression proposed by Geweke and Porter-Hudak (1983, GPH). From (5.22) the spectral density of $x_t$ can be written as
$$f_x(\omega)=\left(4 \sin ^2\left(\frac{\omega}{2}\right)\right)^{-d} f_y(\omega)$$
or, on taking logs,
$$\log f_x(\omega)=\log f_y(\omega)-d \log 4 \sin ^2\left(\frac{\omega}{2}\right)$$
This leads GPH to propose estimating $d$ as (minus) the slope estimator of the regression
$$\log I\left(\omega_j\right)=a-d \log 4 \sin ^2\left(\frac{\omega}{2}\right)$$
where
$$I\left(\omega_j\right)=2 \hat{\sigma}0^2\left(1+2 \sum{s=1}^{T-1} r_s \cos s \omega_j\right)$$ is the periodogram estimate of $f_x(\omega)$ at frequencies $\omega_j=2 \pi j / T, j=1, \ldots, K$, for a suitable choice of $K$, typically $K=\left[T^{0.5}\right]$. It has been shown that the GPH estimator $\hat{d}$ is consistent for $-0.5<d<1$ and asymptotically normal, so that the estimated standard error attached to $\hat{d}$ can be used for inference. Alternatively, the asymptotic result $\sqrt{K}(\hat{d}-d) \sim N\left(0, \pi^2 / 24\right)$ may be used.
5.33 With an estimate $\hat{d}$, the “truncated” form of (5.17) may be used to compute the fractionally differenced series
$$y_t=\nabla \hat{d} x_t=\sum_{k=0}^{t-1} \frac{\hat{d} !}{(\hat{d}-k) ! k !}(-1)^k x_{t-k}$$
where it is explicitly assumed that $y_t=0$ for $t \leq 0$, and this series can then be modeled as an ARMA process in the usual way.
5.34 Several other estimators of $d$ have been proposed, going under the name of semiparametric estimators, but these typically require numerical optimization methods, while it is also possible to estimate $d$ jointly with the ARMA parameters in the ARFIMA process, although specialized software is required for this, which is now available in EViews $10 .$

## 统计代写|时间序列分析代写Time-Series Analysis代考|TESTING FOR FRACTIONAL DIFFERENCING

$5.28$ 检测时间序列中是否存在长记忆的“经典“方法是使用标准差上的范围或重新缩放的 $\operatorname{rangc}(R / S)$ 统 计。这最初是由 Hurst (1951) 在研究河流排放时开发的，后来 Mandelbrot (1972) 在经济背景下提出了一 种修订形式。它被定义为时间序列偏离其平均值的部分和的范围，按其标准偏差重新调整，即
$$R_0=\hat{\sigma} 0^{-1}\left[\max 1 \leq i \leq T \sum_{t=1}^i\left(x_t-\bar{x}\right)-\min 1 \leq i \leq T \sum t=1^i\left(x_t-\bar{x}\right)\right] \quad \hat{\sigma} 0^2=T^{-1} \sum t=1^T$$

$5.29$ 虽然早已确立 $R / S$ 统计当然能够检则到长期依赖，但它对短期影响很敏感。因此，数据与预测行为之 间的任何不兼容 $R / S$ 无长期依赖零下的统计量不一定来自长期记忆，而可能仅仅是短期自相关的症状。

$$R_q=\hat{\sigma} q^{-1}\left[\max 1 \leq i \leq T \sum_{t=1}^i\left(x_t-\bar{x}\right)-\min 1 \leq i \leq T \sum t=1^i\left(x_t-\bar{x}\right)\right]$$

$$\hat{\sigma} q^2=\hat{\sigma}0^2\left(1+\frac{2}{T} \sum j=1^q w{q j} r_j\right) w_{q j}=1-\frac{j}{q \mid 1}, q<T$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|ESTIMATING THE FRACTIONAL DIFFERENCING PARAMETER

$5.32$ 一个缺点 $F D-F$ 程序是，如果 $d_1$ 不知道先验，因为它是在标准的迪基-富勒案例中，那么必须提供 一致的估计。已经提出了多种估计器，其中许多涉及相当复杂的计算。也许最简单的建议是 $R / S$ 分析并且 是
$$\tilde{d}=\frac{\log R_0}{\log T}-0.5$$

$$f_x(\omega)=\left(4 \sin ^2\left(\frac{\omega}{2}\right)\right)^{-d} f_y(\omega)$$

$$\log f_x(\omega)=\log f_y(\omega)-d \log 4 \sin ^2\left(\frac{\omega}{2}\right)$$

$$\log I\left(\omega_j\right)=a-d \log 4 \sin ^2\left(\frac{\omega}{2}\right)$$

$$I\left(\omega_j\right)=2 \hat{\sigma} 0^2\left(1+2 \sum s=1^{T-1} r_s \cos s \omega_j\right)$$

$5.33$ 有估计 $\hat{d} ，(5.17)$ 的“截断””形式可用于计算分数差分序列
$$y_t=\nabla \hat{d} x_t=\sum_{k=0}^{t-1} \frac{\hat{d} !}{(\hat{d}-k) ! k !}(-1)^k x_{t-k}$$

$5.34$ 其他几个估计量 $d$ 已经提出，以半参数估计器的名义，但这些通常需要数值优化方法，同时也可以估 计 $d$ 与 ARFIMA 过程中的 ARMA 参数一起，尽管为此需要专门的软件，现在 EViews 中提供了该软件 10 .

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