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# 统计代写|时间序列代写Time Series代考|STAT435 MODELING PERSISTENCE: A PREVIEW

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## 统计代写|时间序列代写Time Series代考|MODELING PERSISTENCE: A PREVIEW

It has been observed that $\mathrm{H}=\mathrm{d}+\frac{1}{2}$. Hence, the estimate of $\mathrm{d}$ is obviously given by $\hat{d}=\hat{H}-\frac{1}{2}$. Several estimates of $\mathrm{H}$ using the principle have been suggested, but they are all highly biased, and their standard error is usually large. Keeping this in view, we generally do not recommend these methods based on the Hurst exponent.
Since spectral density provides another characterization of persistence, the estimator of $\mathrm{d}$ based on spectral density function can be constructed as discussed in detail in Singh and Kumar (1995). Given a realization $Z_{1}, Z_{2}, \ldots, Z_{n}$ of series $\left{Z_{t}\right}$, it is assumed that $\left{Z_{t}\right}$, when differenced d times, gives rise to a stationary series $\left{X_{t}\right}$ with a rational spectrum $f_{X}(\omega)$. Then, the spectrum of $Z_{t}$ is given by
$$f_{Z}(\omega)=\left|1-e^{-i \omega}\right|^{-2 d} f_{X}(\omega)$$
Let
$$I_{n}(\omega, \mathrm{Z})=\frac{2}{n}\left|\sum Z_{t} e^{-i \omega t}\right|^{2}$$
be the periodogram at frequency $\omega$. Janacek (1982) suggested an estimator of d, which fits a fractional exponential model to the log of periodogram ordinates at all available frequencies using a linear regression. The estimator of $d$ is given by
102
Recent Advances in Time Series Forecasting
$$\hat{d}{M}=\left(S-\sum{k=1}^{M} \frac{\hat{c}{k}}{k} / \sum{k=1}^{M} k^{-2}\right)$$
where
\begin{aligned} &S=\pi^{-1} \int_{0}^{\pi} W(\omega) \ln \hat{f}{Z}(\omega) d \omega \ &W(\omega)=\sum{k=1}^{\infty} \cos (k \omega) / k \ &\left.\hat{c}{k}=n^{-1} \sum{p=1}^{n^{+1}} \ln I_{n}(b, Z) \cos \left(k \omega_{p}\right)+\left(2 n^{s}\right)^{-1} \ln I_{n}(0, Z)-\delta_{n} \ln I_{n}(\pi, Z)\right} \end{aligned}

## 统计代写|时间序列代写Time Series代考|APPLICATION OF ARIMA TO CRUDE OIL DATA

This section provides data application of the ARFIMA (p, d, q) time series model, using the “arfima” package in $\mathrm{R}$. We collected the adjusted crude oil price (dollars per barrel) data for the time period January 1, 2011 to January 1, 2019 (weekly series) from Yahoo! Finance. The ACF plot in Figure 7.1 shows that ACF decays slowly and not exponentially.

To check stationarity of the series, we used the Augmented Dickey Fuller (ADF) test. The ADF test examines the null hypothesis that a time series $Y_{t}$ is stationary against the alternative that it is non-stationary. The p-value turned out to be $0.066$, which is greater than $5 \%$. Thus, we can conclude that the series is not stationary.
We use the Hurst exponent $(\mathrm{H})$ (using function available in “pracma” package in R) to test the presence of long memory in the data. The value of $\mathrm{H}$ turned out to be $0.8652$, indicating that the data has a long-memory structure since $0.5<\mathrm{H}<1$.
The long memory parameter $d$ is estimated using the package “fracdiff” in $\mathrm{R}$. Estimated value of the parameter, its asymptotic deviation value and regression standard deviation values are shown in Table 7.1.

After conducting an ADF test for the fractionally differenced series, we inferred that the series is stationary and can be used for further analysis.

We used the best models function in the ARTFIMA package in $\mathrm{R}$ to identify the best ARFIMA model (Table 7.2).

## 统计代写|时间序列代写Time Series代考|MODELING PERSISTENCE: A PREVIEW

\left 的分隔符缺失或无法识别 ，，当差分 $d$ 次时，产生一个平稳序列
〈left 的分隔符竾失或无法识别 有一个合理的频谱 $f_{X}(\omega)$. 那/，频谱 $Z_{t}$ 是 (谁) 给的
$$f_{Z}(\omega)=\left|1-e^{-i \omega}\right|^{-2 d} f_{X}(\omega)$$

$$I_{n}(\omega, \mathrm{Z})=\frac{2}{n} \mid \sum Z_{t} e^{-\left.i \omega t\right|^{2}}$$

102

$$\hat{d} M=\left(S-\sum k=1^{M} \frac{\hat{c} k}{k} / \sum k=1^{M} k^{-2}\right)$$

\right 的分隔符缺失或无法识别

## 统计代写|时间序列代写Time Series代考|APPLICATION OF ARIMA TO CRUDE OIL DATA

1日期间 (每周系列) 的调整后原油价格 (美元/桶) 数据。金融。图 $7.1$ 中的 ACF 图显示 ACF 哀減缓慢而不是指数哀减。

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