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

#### Doug I. Jones

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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## 统计代写|时间序列分析代写Time-Series Analysis代考|TESTING FOR NONLINEARITY

11.36 As the previous sections have demonstrated, there have been a wide variety of nonlinear processes proposed for modeling time series. We have, for example, compared ARCH and bilinear models, and in so doing have discussed LM tests for each. Nevertheless, given the range of alternative nonlinear models, it is not surprising that other tests for nonlinearity have also been proposed. Since the form of the departure from linearity is often difficult to specify a priori, many tests are diagnostic in nature, i.e., a clear alternative to the null hypothesis of linearity is not specified. This, of course, leads to difficulties in discriminating between the possible causes of any “nonlinear misspecification” that might be uncovered by such tests.
11.37 The detection of nonlinearity is further complicated by the fact that it has similar symptoms to other types of time series behavior. For example, Andersson et al. (1999) have shown that long memory may lead to spurious rejection of the linearity hypothesis. As demonstrated by Granger and Teräsvirta (1999) and Diebold and Inoue (2001), the opposite may also be true, since some nonlinear processes exhibit characteristics that might justify modeling via a long memory model.
11.38 A related approach considers testing and modeling nonlinearity within a long memory process (see, for example, Baillie and Kapetanios, 2007). Koop and Potter (2001) have shown that unpredictable structural instability in a time series may also produce erroneous evidence of threshold-type nonlinearity. An alarming finding by Ghysels et al. (1996) is that nonlinear transformations, such as the X11 seasonal adjustment procedure, that are routinely applied prior to time series modeling, may also induce nonlinear behavior. Equally, seasonal adjustments may smooth out 1999). Finally, as discussed by Van Dijk et al. (1999) and De Lima (1997), neglecting outliers and nonnormalities may also lead to spurious evidence of nonlinearity.

Despite these difficulties, testing for nonlinearity is usually an effort well spent, since the burden associated with the specification and estimation of nonlinear models is often substantial and complex.
11.39 Empirical applications and simulation studies (e.g., Lee et al., 1993; Barnett et al., 1997) have shown that no nonlinearity test dominates in all situations and that power varies with sample size and the characteristics of the underlying stochastic process. This means that, in practice, it is advisable to apply a variety of nonlinearity tests to the data to guide the model specification process.
11.40 Based on Volterra expansions, Ramsey (1969), Keenan (1985), and Tsay (1986b) provide regression-type tests of linearity against unspecified alternatives. These appear to have good power against the nonlinear moving average and bilinear alternatives, but possibly low power against ARCH models. In developing these tests, we assume that an $\operatorname{AR}(p)$ process has been fitted to the observed series $x_t$ and that the residuals, $e_t$, and the fitted values, $\hat{x}_t=x_t-e_t$, have been calculated.

## 统计代写|时间序列分析代写Time-Series Analysis代考|TRANSFER FUNCTION-NOISE MODELS

12.1 The models that have been developed so far in this book have all been univariate, so that the current value of a time series depends, linearly or otherwise, only on past values of itself and, perhaps, a deterministic function of time. While univariate models are important in themselves, they also play a key role in providing a “baseline” to which multivariate models may be compared. We shall analyze several multivariate models over the next chapters, but our development begins with the simplest. This is the single-input transfer function-noise model, in which an endogenous, or output, variable $y_t$ is related to a single input, or exogenous, variable $x_t$ through the dynamic model ${ }^1$
$$y_t=v(B) x_t+n_t$$
where the lag polynomial $v(B)=v_0+v_1 B+v_2 B^2+\cdots$ allows $x$ to influence $y$ via a distributed lag: $v(B)$ is often referred to as the transfer function and the coefficients $v_i$ as the impulse response weights.
12.2 It is assumed that both input and output variables are stationary, perhaps after appropriate transformation. The relationship between the two is not, however, deterministic – rather, it will be contaminated by noise captured by the stochastic process $n_t$, which will generally be serially correlated. A crucial assumption made in (12.1) is that $x_t$ and $n_t$ are independent, so that past $x$ ‘s influence future $y$ ‘s but not vice-versa, so ruling out feedback from $y$ to $x$.
12.3 In general, $v(B)$ will be of infinite order and, hence, some restrictions must be placed on the transfer function before empirical modeling of (12.1)becomes feasible. The typical way in which restrictions are imposed is analogous to the approximation of the linear filter representation of a univariate stochastic process by a ratio of low order polynomials in $B$, which leads to the familiar ARMA model (cf. $\S \S \mathbf{3 . 2 5}-\mathbf{3 . 2 7}$ ). More precisely, $v(B)$ may be written as the rational distributed lag
$$v(B)=\frac{\omega(B) B^b}{\delta(B)}$$

# 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|测试FOR非线性

11.38一个相关的方法考虑了在长记忆过程中测试和建模非线性(例如，Baillie和Kapetanios, 2007)。库普和波特(2001)已经表明，时间序列中不可预测的结构不稳定性也可能产生阈值型非线性的错误证据。Ghysels等人(1996)的一个惊人发现是，在时间序列建模之前通常应用的非线性转换，如X11季节调整过程，也可能导致非线性行为。同样，季节性调整可能会使1999年平稳下来)。最后，正如Van Dijk等人(1999)和De Lima(1997)所讨论的，忽略异常值和非正态也可能导致非线性的虚假证据

## 统计代写|时间序列分析代写Time-Series Analysis代考|传递函数-噪声模型

12.1到目前为止，在本书中开发的模型都是单变量的，因此，时间序列的当前值，线性或其他，只依赖于自身的过去值，也许，一个时间的确定性函数。虽然单变量模型本身很重要，但它们在提供多变量模型可以比较的“基线”方面也起着关键作用。我们将在接下来的章节中分析几个多元模型，但是我们的开发从最简单的开始。这是单输入传递函数-噪声模型，其中内生或输出变量$y_t$通过动态模型${ }^1$
$$y_t=v(B) x_t+n_t$$

12.2假设输入和输出变量都是平稳的，也许经过适当的变换。然而，两者之间的关系不是确定的——相反，它会受到随机过程$n_t$捕获的噪声的污染，该过程通常是串行相关的。(12.1)中做的一个关键假设是$x_t$和$n_t$是独立的，因此过去的$x$会影响未来的$y$，反之亦然，因此排除了从$y$到$x$的反馈。
12.3一般来说，$v(B)$的顺序是无限的，因此，在(12.1)的经验建模变得可行之前，必须对传递函数施加一些限制。施加限制的典型方式类似于$B$中通过低阶多项式的比率逼近单变量随机过程的线性滤波表示，这导致了我们熟悉的ARMA模型(参见$\S \S \mathbf{3 . 2 5}-\mathbf{3 . 2 7}$)。更准确地说，$v(B)$可以写成有理分布滞后
$$v(B)=\frac{\omega(B) B^b}{\delta(B)}$$

## 有限元方法代写

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