5.17 Consider again the linear trend model (5.8): $x_t=\beta_0+\beta_1 t+\varepsilon_t$. As we have seen, correct specification of the trend is crucially important for unit root and stationarity testing. As was pointed out in $\$ \mathbf{5 . 9}$, incorrectly excluding a linear trend renders the$\tau_\mu$statistic inconsistent, while it is also the case that unnecessarily including a trend vastly reduces the power of the$\tau_\tau$test, with similar problems affecting the KPSS stationarity statistics$\eta_\mu$and$\eta_\tau$Often, however, the trend parameter$\beta_1$is of direct interest, especially when ascertaining whether a trend is present$\left(\beta_1 \neq 0\right)$or not$\left(\beta_1=0\right)$. This may be assessed by either constructing a direct test of the no trend hypothesis$\beta_1=0$or by forming a confidence interval for$\beta_1$. Such tests rely on whether$\varepsilon_t$, and hence,$x_t$, is either$I(0)$or$I(1)$, but this can only be established after a unit root or stationarity test has been performed-yet the properties of these latter tests rely, in turn, on whether a trend has been correctly included or not! This circularity of reasoning has prompted the development of trend function testing procedures that are robust, in the sense that, at least asymptotically, inference on the trend function is unaffected as to whether$\varepsilon_t$is$I(0)$or$I(1)$. 5.18 To develop robust tests of trend, we start with the simplest case in which$\varepsilon_t=\rho \varepsilon_{t-1}+a_t$, where$\varepsilon_t$is$I(0)$if$|\rho|<1$and$I(1)$if$\rho=1$. We then wish to test$H_0: \beta_1=\beta_1^0$against the alternative$H_1: \beta_1 \neq \beta_1^0$. If$\varepsilon_t$is known to be$I(0)$then an optimal test of$H_0$against$H_1$is given by the “slope”$t$-ratio $$z_0=\frac{\hat{\beta}1-\beta_1^0}{s_0} \quad s_0=\sqrt{\frac{\hat{\sigma}{\varepsilon}^2}{\sum_{t=1}^T(t-\bar{t})^2}}$$ where$\hat{\sigma}{\varepsilon}^2=(T-2)^{-1} \sum{t=1}^T\left(x_t-\hat{\beta}_0-\hat{\beta}_1 t\right)^2$is the error variance from OLS estimation of (5.8). Under$H_0, z_0$will be asymptotically standard normal. 统计代写|时间序列分析代写Time-Series Analysis代考|Estimating the Trend in Central England Temperatures Fig.$5.7$shows the annual Central England temperature (CET) series. This is the longest available recorded instrumental temperature series in existence, beginning in 1659, and establishing its trend is clearly of great interest for debates concerning global warming. If it is assumed that deviations from a linear trend are$l(0)$, then estimating (5.8) with an autocorrelation correction obtains$\hat{\beta}_1=0.002749$with$s_0=0.000745$, thus yielding$z_0=3.689$, which, when compared to a standard normal distribution, implies that the trend$\beta_1$is significantly positive at$0.27^{\circ} \mathrm{C}$per century. On the other hand, assuming that the deviations are$l(1)$obtains, from estimating (5.13) with an autocorrelation correction,$\tilde{\beta}_1=0.004777, s_1=0.007769$, and$z_1=0.615$, thus implying that although the estimate of the trend is nearly$50 \%$higher than the$I(0)$deviations estimate, it is nevertheless insignificantly different from zero, a consequence of$s_1$being over ten times as large as$s_0$. Determining which of the two estimates to use is not resolved by computing unit root and stationarity tests, for$D F-G L S=-9.212$and$\eta_T=0.189$, the former rejecting the unit root null at the$1 \%$level, the latter rejecting the TS null at the$5 \%$level! This is clearly a situation when estimating the trend rohustly is called for. With the information presented, we obtain$\lambda=0.554, z_{0.554}=1.987$,$\hat{\beta}_{1,0.554}=0.002965$, and a$95 \%$confidence interval for$\beta_1$of$0.002965 \pm 0.002925$, i.e., approximately$0-0.6^{\circ} \mathrm{C}$per century. The CET, therefore, has a trend that, with a$p$-value of$.023$, is a significantly positive$0.3^{\circ} \mathrm{C}$per century. 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|稳健估计趋势 再次考虑线性趋势模型(5.8):$x_t=\beta_0+\beta_1 t+\varepsilon_t$。正如我们所看到的，正确的趋势规范对于单位根和平稳性检验是至关重要的。正如在$\$\mathbf{5 . 9}$中所指出的，不正确地排除线性趋势会使$\tau_\mu$统计数据不一致，而不必要地包含趋势也会大大降低$\tau_\tau$测试的能力，类似的问题会影响KPSS平稳性统计数据$\eta_\mu$和$\eta_\tau$

5.18为了发展趋势的稳健检验，我们从最简单的情况开始 $\varepsilon_t=\rho \varepsilon_{t-1}+a_t$，其中 $\varepsilon_t$ 是 $I(0)$ 如果 $|\rho|<1$ 和 $I(1)$ 如果 $\rho=1$。然后我们希望进行测试 $H_0: \beta_1=\beta_1^0$ 反对另一种选择 $H_1: \beta_1 \neq \beta_1^0$。如果 $\varepsilon_t$ 是已知的 $I(0)$ 然后进行最优检验 $H_0$ 反对 $H_1$ 由斜率给出 $t$-ratio
$$z_0=\frac{\hat{\beta}1-\beta_1^0}{s_0} \quad s_0=\sqrt{\frac{\hat{\sigma}{\varepsilon}^2}{\sum_{t=1}^T(t-\bar{t})^2}}$$
where $\hat{\sigma}{\varepsilon}^2=(T-2)^{-1} \sum{t=1}^T\left(x_t-\hat{\beta}_0-\hat{\beta}_1 t\right)^2$ 为OLS估计的误差方差(5.8)。下面 $H_0, z_0$ 将是渐近标准法线。

统计代写|时间序列分析代写Time-Series Analysis代考|Estimating the Trend in Central England Temperatures

$5.7$显示每年英格兰中部的温度(CET)系列。从1659年开始，这是现存最长的有记录的仪器温度序列，确定其趋势显然对有关全球变暖的辩论有很大的兴趣。如果假设偏离线性趋势的偏差是$l(0)$，那么用自相关校正估计(5.8)得到$\hat{\beta}_1=0.002749$和$s_0=0.000745$，从而得到$z_0=3.689$，与标准正态分布相比，这意味着趋势$\beta_1$在每世纪$0.27^{\circ} \mathrm{C}$处显著为正

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