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统计代写|时间序列分析代写Time-Series Analysis代考|Are Shocks to British GDP Temporary or Permanent?

Fig. $5.5$ shows the logarithms of British real GDP per capita annually over the period of 1822-1913, i.e., from just after the end of the Napoleonic wars to the beginning of World War I, a period which covers the whole of the Victorian era. A linear trend line has been superimposed on the plot, calculated from a model of the form of (5.8):
$$
\begin{gathered}
x_t=\underset{(0.019)}{0.329}+\underset{(0.0004)}{0.0103 t} t+\varepsilon_t \
\varepsilon_t=\underset{(0.076)}{0.696} \varepsilon_{t-1}+\hat{a}_t
\end{gathered}
$$
This TS model implies that, since we are dealing with logarithms, trend growth in real GDP per capita was $1.03 \%$ per annum, with there being stationary deviations about the trend line. Consequently, all shocks that push real GDP per capita away from its long-run trend path have only short-run, “transitory” impacts, with the series always returning to this trend path. Since the error component is modeled by an AR(1) process with a parameter of around $0.7$, such shocks die away geometrically and rather quickly; being reduced by over $90 \%$ in size after 7 years $\left(0.7^7=0.082\right)$.

Note that the error component displays no evidence of a “business cycle,” for this would require $\varepsilon_t$ to follow (at least) an $\mathrm{AR}(2)$ process with complex roots, for which there is no evidence, since the inclusion of a second autoregressive term produces an insignificant coefficient.
This TS representation may be contrasted with the DS representation
$$
\nabla x_t=\underset{(0.0003)}{0.0104}+\hat{a}_t
$$
obtained by replacing the autoregressive coefficient of $0.7$ with one of unity. This model is a drifting random walk with the drift parameter estimated as $1.04 \%$ per annum. The interpretation of this model, however, is one in which all shocks are permanent; remaining in the series for all time with no dissipation.

统计代写|时间序列分析代写Time-Series Analysis代考|OTHER APPROACHES TO TESTING FOR A UNIT ROOT

5.12 An alternative unit root test to the ADF for dealing with autocorrelation in $a_t$, which also allows for heterogeneity of variance, has been proposed by Phillips and Perron (1988). Rather than including extra lags of $\nabla x_t$ to ensure that the errors of $(5.4)$ are indeed white noise, the idea here is to estimate an “unaugmented” model-(5.3), say-and to modify the test statistics so that the effects of any autocorrelation are accounted for. This will enable the same DF limiting distributions and, hence, critical values to be used.

Under a specific set of conditions placed upon $a_t$, known as weak dependency, which are described in detail by Phillips (1987), the $\tau_\mu$ statistic obtained from the estimation of $(5.3)$ is modified to
$$
Z\left(\tau_\mu\right)=\tau_\mu\left(\hat{\sigma}0 / \hat{\sigma}{\ell}\right)-\frac{1}{2}\left(\hat{\sigma}{\ell}^2-\hat{\sigma}_0^2\right) / \Sigma{\ell}
$$
in which
$$
\begin{aligned}
&\hat{\sigma}0^2=T^{-1} \sum{t=1}^T \hat{a}t^2 \ &\hat{\sigma}{\ell}^2=\hat{\sigma}0^2+2 T^{-1} \sum{j=1}^{\ell} w_j(\ell)\left(\sum_{t=j+1}^T \hat{a}t \hat{a}{t-j}\right) \
&\Sigma_{\ell}^2=T^{-2} \hat{\sigma}{\ell}^2 \sum{t=2}^T\left(x_{t-1}-\bar{x}{-1}\right)^2 \quad \bar{x}{-1}=(T-1)^{-1} \sum_{t=1}^{T-1} x_t
\end{aligned}
$$

$\hat{\sigma}{\ell}^2$ is a consistent estimator of the long-run variance and employs a window or kernel function $w_j(\ell)$ to weight the sample autocovariances appearing in the formula. This ensures that the estimator remains positive, with $\ell$ acting as a truncation lag, much like $k$ in the ADF regression. A range of kernel functions are available, such as the “triangular” set of lag weights $w_j(\ell)=\ell-j /(\ell+1) . Z\left(\tau\mu\right)$ is often referred to as the Phillips-Perron (PP) non-parametric unit root test.
$Z\left(\tau_\mu\right)$ has the same limiting distribution as $\tau_\mu$, so that the latter’s critical values may again be used. If $x_t$ has zero mean, the adjusted statistic, $Z(\tau)$, is as in (5.10) with $\bar{x}{-1}$ removed and has the same limiting distribution as $\tau$. If a time trend is included then a further adjustment is required to enable the statistic, now denoted $Z\left(\tau\tau\right)$, to have the limiting $\tau_\tau$ distribution (Mills and Markellos, 2008, page 87, for example, provide details).
5.13 Many alternative unit root tests have been developed since the initial ADF and PP tests were introduced. A recurring theme of unit root testing is the low power and severe size distortion inherent in many tests: see, especially, the review by Haldrup and Jansson (2006). For example, the PP tests suffer severe size distortions when there are moving average errors with a large negative root and, although their ADF counterparts are better behaved in this respect, the problem is not negligible even here. Moreover, many tests have low power when the largest autoregressive root is close to, but nevertheless less than, unity.

A related issue is that unlike many hypothesis testing situations, the power of tests of the unit root hypothesis against stationary alternatives depends less on the number of observations per se and more on the span of the data (i.e., the length of the observation period). For a given number of observations, power has been found to be highest when the span is longest; conversely, for a given span, additional observations obtained using data sampled more frequently lead to only a marginal increase in power, the increase becoming negligible as the sampling interval is decreased. Hence, a series containing fewer annual observations over an extended time period will often lead to unit root tests having higher power than those computed from a series containing more observations over a shorter period.

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

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|对英国GDP的冲击是暂时的还是永久性的?

$5.5$显示了1822-1913年期间英国人均实际GDP的年对数,即从拿破仑战争刚刚结束后到第一次世界大战开始,这一时期涵盖了整个维多利亚时代。在图上叠加了一条线性趋势线,根据(5.8)形式的模型计算:
$$
\begin{gathered}
x_t=\underset{(0.019)}{0.329}+\underset{(0.0004)}{0.0103 t} t+\varepsilon_t \
\varepsilon_t=\underset{(0.076)}{0.696} \varepsilon_{t-1}+\hat{a}_t
\end{gathered}
$$
这个TS模型意味着,由于我们在处理对数,实际人均GDP的趋势增长是每年$1.03 \%$,在趋势线上存在平稳偏差。因此,所有将实际人均GDP推离其长期趋势路径的冲击都只有短期的“暂时”影响,该系列总是回到这一趋势路径。由于误差分量是由一个参数为$0.7$左右的AR(1)过程建模的,这样的冲击以几何形式迅速消失;经过7年的努力,我们的规模缩小了$90 \%$以上$\left(0.7^7=0.082\right)$ .


请注意,错误组件没有显示“商业周期”的证据,因为这将要求$\varepsilon_t$(至少)遵循一个$\mathrm{AR}(2)$过程的复根,没有证据,因为包含第二个自回归项产生一个不重要的系数。TS表示可以与DS表示(
$$
\nabla x_t=\underset{(0.0003)}{0.0104}+\hat{a}_t
$$
)相比,DS表示是通过将$0.7$的自回归系数替换为单位系数得到的。该模型是一个漂移随机游走,其漂移参数估计为每年$1.04 \%$。然而,对这个模型的解释是,所有的冲击都是永久性的;

统计代写|时间序列分析代写时间序列分析代考|测试单位根的其他方法


5.12 Phillips和Perron(1988)提出了一种替代ADF的单位根检验来处理$a_t$中的自相关,该单位根检验也考虑到了方差的异质性。这里的想法不是包括$\nabla x_t$的额外滞后以确保$(5.4)$的错误确实是白噪声,而是估计一个“未增强的”模型(5.3),并修改测试统计数据,以便考虑到任何自相关的影响。这将启用相同的DF限制分布,因此,使用临界值


在置于$a_t$上的一组特定条件下,称为弱依赖关系,Phillips(1987)详细描述了这种条件,从对$(5.3)$的估计中得到的$\tau_\mu$统计量被修改为
$$
Z\left(\tau_\mu\right)=\tau_\mu\left(\hat{\sigma}0 / \hat{\sigma}{\ell}\right)-\frac{1}{2}\left(\hat{\sigma}{\ell}^2-\hat{\sigma}0^2\right) / \Sigma{\ell}
$$
,其中
$$
\begin{aligned}
&\hat{\sigma}0^2=T^{-1} \sum{t=1}^T \hat{a}t^2 \ &\hat{\sigma}{\ell}^2=\hat{\sigma}0^2+2 T^{-1} \sum{j=1}^{\ell} w_j(\ell)\left(\sum{t=j+1}^T \hat{a}t \hat{a}{t-j}\right) \
&\Sigma_{\ell}^2=T^{-2} \hat{\sigma}{\ell}^2 \sum{t=2}^T\left(x_{t-1}-\bar{x}{-1}\right)^2 \quad \bar{x}{-1}=(T-1)^{-1} \sum_{t=1}^{T-1} x_t
\end{aligned}
$$

$\hat{\sigma}{\ell}^2$是长期方差的一致估计量,并使用窗口或核函数$w_j(\ell)$对公式中出现的样本自协方差进行加权。这确保了估计器保持为正,$\ell$充当截断延迟,很像ADF回归中的$k$。可以使用一系列的核函数,例如滞后权重的“三角”集$w_j(\ell)=\ell-j /(\ell+1) . Z\left(\tau\mu\right)$通常被称为Phillips-Perron (PP)非参数单位根检验。
$Z\left(\tau_\mu\right)$与$\tau_\mu$具有相同的极限分布,因此后者的临界值可能再次被使用。如果$x_t$的均值为零,则调整后的统计量$Z(\tau)$如(5.10)中删除了$\bar{x}{-1}$,具有与$\tau$相同的极限分布。如果包含时间趋势,则需要进一步调整以使统计量(现在标记为$Z\left(\tau\tau\right)$)具有有限的$\tau_\tau$分布(例如,Mills和Markellos, 2008年,第87页提供详细信息)。
5.13自最初的ADF和PP检验引入以来,已经开发了许多替代单位根检验。单位根测试的一个反复出现的主题是在许多测试中固有的低功率和严重的尺寸失真:特别是,参见Haldrup和Jansson(2006)的评论。例如,当有一个大的负根的移动平均误差时,PP检验会遭受严重的尺寸扭曲,尽管它们的ADF在这方面表现得更好,但即使在这里问题也不能忽略。此外,当最大自回归根接近但小于单位时,许多检验的幂次较低。


一个相关的问题是,与许多假设检验情况不同,单位根假设对平稳替代方法的检验能力不太取决于观察次数本身,而更多地取决于数据的跨度(即观察周期的长度)。对于一定数量的观测,已经发现,当跨度最长时,功率最高;相反,对于给定的跨度,使用更频繁采样的数据获得的额外观测结果只会导致功率的边际增加,随着采样间隔的减小,这种增加变得可以忽略不计。因此,在较长时间内包含较少年度观测值的序列,其单位根检验的幂往往高于由包含较多观测值的较短时间内的序列计算出的幂

统计代写|时间序列分析代写Time-Series Analysis代考

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