# 统计代写|时间序列分析代写Time-Series Analysis代考|STAT510

## 统计代写|时间序列分析代写Time-Series Analysis代考|ROBUST TESTS FOR A BREAKING TREND

6.13 Of course, the broken trend will typically be of interest in itself, and so it is natural for the robust trend analysis of $\$ \$5.17-5.22$ to have been extended to cover such specifications, most notably by Harvey, Leybourne, and Taylor (HLT, 2009). If the break date is known to be at $T_b^c$ with break fraction $\tau_c$ then, focusing on the segmented trend model (B), the HLT method is extended by focusing on autocorrelation corrected $t$-tests of $\beta=0$ in (6.2) and (6.5), which we denote as $t_0\left(\tau^c\right)$ and $t_1\left(\tau^c\right)$. A weighted average of these two statistics is again considered,
$$t_\lambda=\lambda\left(S_0\left(\tau^c\right), S_1\left(\tau^c\right)\right) \times\left|t_0\left(\tau^c\right)\right|+\left(1-\lambda\left(S_0\left(\tau^c\right), S_1\left(\tau^c\right)\right)\right) \times\left|t_1\left(\tau^c\right)\right|$$
with the weight function now being defined as
$$\lambda\left(S_0\left(\tau^c\right), S_1\left(\tau^c\right)\right)=\exp \left(-\left(500 S_0\left(\tau^c\right) S_1\left(\tau^c\right)\right)^2\right)$$
Here $S_0\left(\tau^c\right)$ and $S_1\left(\tau^c\right)$ are KPSS $\eta_\tau$ statistics (cf. $\S \mathbf{5 . 1 6}$ ) computed from the residuals of (6.2) and (6.5) respectively. Under $H_0: \beta=0, t_\lambda$ will be asymptotically standard normal.
6.14 When $\tau^c$ is unknown but is assumed to lie between $0<\tau_{\min }, \tau_{\max }<1$ then (6.13) can be replaced by
$$t_\lambda=\lambda\left(S_0(\hat{\tau}), S_1(\tilde{\tau})\right) \times\left|t_0(\hat{\tau})\right|+m_{\xi}\left(1-\lambda\left(S_0(\hat{\tau}), S_1(\tilde{\tau})\right)\right) \times\left|t_1(\tilde{\tau})\right|$$
Here $\hat{\tau}$ and $\tilde{\tau}$ are the break fractions that maximize $\left|t_0(\tau)\right|$ and $\left|t_1(\tau)\right|$ across all break fractions in the range $\tau_{\min }<\tau<\tau_{\max }$. In these circumstances $t_\lambda$ is no longcr asymptotically standard normal. The constant $m_{\xi}$ is chosen so that, for a given significance level $\xi$, the asymptotic null critical value of $t_\lambda$ is the same irrespective of whether the errors are $I(0)$ or $I(1)$.

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

6.16 When the break date is estimated it is often useful to be able to provide a confidence interval for the unknown $T_b^c$. Perron and Zhu (2005) show that for the segmented trend model (B) and $I(1)$ errors
$$\sqrt{T}\left(\hat{\tau}-\tau^c\right) \stackrel{d}{\sim} N\left(0,2 \sigma^2 / 15 \beta^2\right)$$
while for $I(0)$ errors
$$T^{3 / 2}\left(\tilde{\tau}-\tau^c\right) \stackrel{d}{\sim} N\left(0,4 \sigma^2 /\left(\tau^c\left(1-\tau^c\right) \beta^2\right)\right)$$
so that, for example, a 95\% confidence interval for $\tau^c$ when the errors are $I(1)$ is given by
$$\hat{\tau} \pm 1.96 \sqrt{\frac{2 \hat{\sigma}^2}{15 T \hat{\beta}^2}}$$
The limiting distributions for the break date do not depend on the autocorrelation structure of the errors, only requiring an estimate of the error variance $\sigma^2$. When the errors are $I(1)$ the limiting distribution is invariant to the location of the break, whereas for $I(0)$ errors, the limiting distribution depends on the location of the break in such a way that the variance is smaller the closer the break is to the middle of the sample. In both cases the variance decreases as the shift in slope increases.

For model (C) the limiting distributions for the break date are no longer normal but are complicated functions of nuisance parameters and, thus, can only be simulated, so that no simple results are available.
6.17 In theory, all the procedures available when there is only one break, may be extended to the case of multiple breaks, but, in practice, when there are multiple breaks at unknown times, only the sequential procedure of Kejriwal and Perron (2010), which requires specialized programming, is currently available.

## 统计代写|时间序列分析代写Time-Series Analysis代考|ROBUST TESTS FOR A BREAKING TREND

$6.13$ 当然，破碎的趋势本身通常会引起人们的兴趣，因此对于稳健的趋势分析是很自然的 $\$ \$5.17-5.22$ 已扩展到涵盖此类规范，最著名的是 Harvey、Leybourne 和 Taylor (HLT，2009)。如果已知休息日期在 $T_b^c$ 帯断裂分数 $\tau_c$ 然后，关注分段趋势模型 (B)，通过关注自相关校正来扩展 HLT 方法 $t$ – 测试 $\beta=0$ 在 (6.2) 和 (6.5) 中，我们将其表示为 $t_0\left(\tau^c\right)$ 和 $t_1\left(\tau^c\right)$. 再次考虑这两个统计数据的加权平均值，
$$t_\lambda=\lambda\left(S_0\left(\tau^c\right), S_1\left(\tau^c\right)\right) \times\left|t_0\left(\tau^c\right)\right|+\left(1-\lambda\left(S_0\left(\tau^c\right), S_1\left(\tau^c\right)\right)\right) \times\left|t_1\left(\tau^c\right)\right|$$

$$\lambda\left(S_0\left(\tau^c\right), S_1\left(\tau^c\right)\right)=\exp \left(-\left(500 S_0\left(\tau^c\right) S_1\left(\tau^c\right)\right)^2\right)$$

$6.14$ 何时 $\tau^c$ 末知，但假定介于 $0<\tau_{\min }, \tau_{\max }<1$ 那么 $(6.13)$ 可以被替换为
$$t_\lambda=\lambda\left(S_0(\hat{\tau}), S_1(\tilde{\tau})\right) \times\left|t_0(\hat{\tau})\right|+m_{\xi}\left(1-\lambda\left(S_0(\hat{\tau}), S_1(\tilde{\tau})\right)\right) \times\left|t_1(\tilde{\tau})\right|$$

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

$6.16$ 在估计中断日期时，能够为末知数提供置信区间通常很有用 $T_b^c$. Perron 和 Zhu (2005) 表明，对于分段 趋势模型 (B) 和 $I(1)$ 错误
$$\sqrt{T}\left(\hat{\tau}-\tau^c\right) \stackrel{d}{\sim} N\left(0,2 \sigma^2 / 15 \beta^2\right)$$

$$T^{3 / 2}\left(\tilde{\tau}-\tau^c\right) \stackrel{d}{\sim} N\left(0,4 \sigma^2 /\left(\tau^c\left(1-\tau^c\right) \beta^2\right)\right)$$

$$\hat{\tau} \pm 1.96 \sqrt{\frac{2 \hat{\sigma}^2}{15 T \hat{\beta}^2}}$$

$6.17$ 理论上，只有一次休息时可用的所有程序，可以扩展到多次休息的情况，但实际上，当在末知时间有 多次休息时，只有 Kejriwal 和 Perron (2010) 的顺序程序，这需要专门的编程，目前可用。

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