# 统计代写|时间序列分析代写Time-Series Analysis代考|STAT435

## 统计代写|时间序列分析代写Time-Series Analysis代考|BREAKING TRENDS AND UNIT ROOT TESTS

6.3 How can we distinguish between TS breaking trends and breaking DS processes? Clearly unit root tests should be applicable, but what is the influence of breaking trends upon such tests? Perron $(1989,1990$ : see also Perron and Vogelsang, 1993) was the first to consider the impact of breaking trends and shifting levels on unit root tests, showing that standard tests of the type discussed in Chapter 5, Unit Roots, Difference and Trend Stationarity, and Fractional Differencing, are not consistent against TS alternatives when the trend function contains a shift in slope. Here the estimate of the largest autoregressive root is biased toward unity and, in fact, the unit root null becomes impossible to reject, even asymptotically. Although the tests are consistent against a shift in the intercept of the trend function, their power is nevertheless reduced considerably because the limiting value of the estimated autoregressive root is inflated above its true value.

6.4 Perron (1989) consequently extended the Dickey-Fuller unit root testing strategy to ensure consistency against shifting trend functions by developing two asymptotically equivalent procedures. The first uses initial regressions in which $x_t$ is detrended according to either model (A), the level shift (6.1); model (B), the segmented trend (6.2); or model (C), the combined model (6.3). Thus, let $\tilde{x}t^i, i=A, B, C$, be the residuals from a regression of $x_t$ on (1) $i=A$ : a constant, $t$, and $\mathrm{DU}_t^c$; (2) $i=B$ : a constant, $t$, and $\mathrm{DT}_t^c$; and (3) $i=C$ : a constant, $t, \mathrm{DU}_t^c$, and $\mathrm{DT}_t^c$. For models (A) and (C) a modified ADF regression (cf. (5.5)) is then estimated: $$\tilde{x}_t^i=\tilde{\phi}^i \tilde{x}_t^i+\sum{j=0}^k \gamma_j \mathrm{D}\left(\mathrm{TB}^c\right){t-j} \sum{j=1}^k \delta_j \nabla \tilde{x}t^i \quad i=A, C$$ and a $t$-test of $\tilde{\phi}^i=1$ is performed $\left(t^i, i=A, C\right)$. The inclusion of the $k+1$ dummy variables $\mathrm{D}\left(\mathrm{TB}^c\right)_t, \ldots, \mathrm{D}\left(\mathrm{TB}^c\right){t-k}$ is required to ensure that the limiting distributions of $t^A$ and $t^C$ are invariant to the correlation structure of the errors (see Perron and Vogelsang, 1993). For model (B) the “unmodified” ADF regression
$$\tilde{x}t^B=\tilde{\phi}^i \tilde{x}{t-1}^B+\sum_{j=1}^k \delta_j \nabla \tilde{x}_{t-j}^B+a_t$$
may be estimated to obtain $t^B$.

## 统计代写|时间序列分析代写Time-Series Analysis代考|UNIT ROOTS TESTS WHEN THE BREAK DATE IS UNKNOWN

6.10 The procedure set out in $\S \S 6.3-6.5$ is only valid when the break date is known independently of the data, for if a systematic search for a break is carried out then the limiting distributions of the tests are no longer appropriate. Problems also occur if an incorrect break date is selected exogenously, with the tests then suffering size distortions and loss of power.

Consequently, several approaches have been developed that treat the occurrence of the break date as unknown and needing to be estimated: see, for example, Zivot and Andrews (1992), Perron (1997), and Vogelsang and Perron (1998). Thus, suppose now that the correct break date $T_b^c$ is unknown. Clearly, if this is the case then the models of $\$ \$6.3-6.5$ are not able to be used until some break date, say $\hat{T}_b$, is selected, since none of the dummy variables that these models require can be defined until this selection has been made.
6.11 Two data-dependent methods for choosing $\hat{T}_b$ have been considered, both of which involve estimating the appropriate detrended AO regression, (6.7) or (6.8), or IO regression (6.9-6.11), for all possible break dates. The first method chooses $\hat{T}_b$ as the break date that is most likely to reject the unit root hypothesis, which is the date for which the $t$-statistic for testing $\phi=1$ is minimized (i.e., is most negative).

The second approach involves choosing $\hat{T}b$ as the break date for which some statistic that tests the significance of the break parameters is maximized. This is equivalent to minimizing the residual sum of squares across all possible regressions, albeit after some preliminary trimming has been performed, that is, if only break fractions $\tau=T_b / T$ between $0<\tau{\min }, \tau_{\max }<1$ are considered.
6.12 Having selected $\hat{T}_b$ by one of these methods, the procedure set out in $\S \$ 6.3-6.5$may then be applied conditional upon this choice. Critical values may be found in Vogelsang and Perron (1998) for a variety of cases: typically, they are more negative than the critical values that hold when the break date is known to be at$T_b^c$. ## 统计代写|时间序列分析代写Time-Series Analysis代考|BREAKING TRENDS AND UNIT ROOT TESTS$6.3$我们如何区分 TS 破坏趋势和破坏 DS 过程? 显然，单位根测试应该适用，但突破趋势对此类测试有何 影响? 佩龙 (1989, 1990: 另见 Perron 和 Vogelsang, 1993) 是第一个考虑突破赸势和移动水平对单位根检 验的影响的人，表明第 5 章“单位根、差值和趋势平稳性以及分数差分”中讨论的类型的标准检验，当趋势 函数包含斜率变化时，与$\mathrm{TS}$替代方案不一致。这里最大自回归根的估计偏向于统一，事实上，单位根空 值变得不可能拒绝，即使是渐近的。尽管测试与趋势函数截距的变化一致，但它们的功效仍然大大降低， 因为估计的自回归根的极限值被夸大到其真实值之上。 6.4 Perron (1989) 因此扩展了 Dickey-Fuller 单位根检验策略，以通过开发两个渐近等效程序来确保与变化 趋势函数的一致性。第一个使用初始回归，其中$x_t$根据任一模型 (A)、水平位移 (6.1) 去趋势；模型 (B)，分段趋势 (6.2) ；或模型 (C)，组合模型 (6.3) 。因此，让$\tilde{x} t^i, i=A, B, C$，是回归的残差$x_t$在 (1)i$=A:$一个常数，$t$，和$\mathrm{DU}t^c ;(2) i=B:$一个常数，$t$，和$\mathrm{DT}_t^c ;(3) i=C:$一个常数，$t, \mathrm{DU}_t^c$，和$\mathrm{DT}_t^c$. 对于模型 (A) 和 (C)，然后估计修改后的 ADF 回归（参见 (5.5)) : $$\tilde{x}_t^i=\tilde{\phi}^i \tilde{x}_t^i+\sum j=0^k \gamma_j \mathrm{D}\left(\mathrm{TB}^c\right) t-j \sum j=1^k \delta_j \nabla \tilde{x} t^i \quad i=A, C$$ 和一个$t$– 测试$\tilde{\phi}^i=1$被执行$\left(t^i, i=A, C\right)$. 列入$k+1$虚拟变量$\mathrm{D}\left(\mathrm{TB}^c\right)_t, \ldots, \mathrm{D}\left(\mathrm{TB}^c\right) t-k$需要确保 的限制分布$t^A$和$t^C$对于误差的相关结构是不变的（参见 Perron 和 Vogelsang，1993)。对于模型 (B)， “末修改”的 ADF 回归 $$\tilde{x} t^B=\tilde{\phi}^i \tilde{x} t-1^B+\sum{j=1}^k \delta_j \nabla \tilde{x}_{t-j}^B+a_t$$ 估计可以得到$t^B$. ## 统计代写|时间序列分析代写Time-Series Analysis代考|UNIT ROOTS TESTS WHEN THE BREAK DATE IS UNKNOWN 6.10 中规定的程序§§§§6.3−6.5仅当独立于数据而知道中断日期时才有效，因为如果对中断进行系统搜索，则测试的限制分布不再合适。如果外部选择了错误的休息日期，也会出现问题，然后测试会遭受尺寸扭曲和功率损失。 因此，已经开发了几种方法，将中断日期的发生视为未知且需要估计：例如，参见 Zivot 和 Andrews (1992)、Perron (1997) 以及 Vogelsang 和 Perron (1998)。因此，现在假设正确的休息日期吨bC是未知的。显然，如果是这种情况，那么$$6.3−6.5直到某个休息日期才能使用，比如说吨^b, 被选中，因为这些模型所需的虚拟变量都无法定义，直到做出此选择。 6.11 两种依赖数据的选择方法吨^b已经考虑过，这两者都涉及为所有可能的中断日期估计适当的去趋势 AO 回归（6.7）或（6.8）或 IO 回归（6.9-6.11）。第一种方法选择吨^b作为最有可能拒绝单位根假设的中断日期，即吨- 测试统计φ=1被最小化（即最负）。 第二种方法涉及选择吨^b作为休息日期，一些测试休息参数重要性的统计数据被最大化。这相当于在所有可能的回归中最小化残差平方和，尽管在执行了一些初步修整之后，也就是说，如果只打破分数吨=吨b/吨之间0<吨分钟,吨最大限度<1被考虑。 6.12 已选择吨^b通过其中一种方法，程序在§§$6.3−6.5然后可以根据该选择应用。在 Vogelsang 和 Perron (1998) 中可以找到各种情况下的临界值：通常，它们比已知中断日期时保持的临界值更负吨bC.

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