## 统计代写|时间序列分析代写Time-Series Analysis代考|STAT758

2022年9月24日

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## 统计代写|时间序列分析代写Time-Series Analysis代考|THRESHOLD AND SMOOTH TRANSITION AUTOREGRESSIONS

11.18 A popular class of nonlinear model is the self-exciting threshold autoregressive (SETAR) process, which allows for asymmetry by defining a set of piecewise autoregressive models whose switch points, or “thresholds,” are generally unknown (see Tong and Lim, 1980; Tong, 1990; Teräsvirta, 2006):
$$x_t=\sum_{j=1}^r\left(\phi_{j, 1} x_{t-1}+\cdots+\phi_{j, p} x_{t-p}+a_{j, t}\right) \mathbf{1}\left(c_{j-1}<x_{t-d} \leq c_j\right)$$
Here $d$ is the (integer-valued) delay parameter and $c_1<c_2<\ldots<c_{r-1}$ are the thresholds: the model is often denoted $\operatorname{SETAR}(r: p, d) .{ }^3$ It is assumed that $a_{j, t} \sim W N\left(0, \sigma_j^2\right), j=1, \ldots, r$, so that the error variance is allowed to alter across the $r$ “regimes.” A popular version of (11.7) is the two-regime SETAR(2: $p, d)$ model:
\begin{aligned} x_t=&\left(\phi_{1,1} x_{t-1}+\cdots+\phi_{1, p} x_{t-p}+a_{1, t}\right) \mathbf{1}\left(x_{t-d} \leq c_1\right) \ &+\left(\phi_{2,1} x_{t-1}+\cdots+\phi_{2, p} x_{t-p}+a_{2, t}\right)\left(1-\mathbf{1}\left(x_{t-d} \leq c_1\right)\right) \end{aligned}
An important feature of the SETAR model is its ability to generate “limit cycles”: if (11.7) is extrapolated assuming that the error terms equal zero, then the extrapolated series displays oscillations of a given length that do not die out.

As previously stated, asymmetry may be captured by the regimes: for example, if $x_{t-d}$ measures the phase of an economic business cycle, a tworegime SETAR could describe processes whose dynamic properties differ across expansions and recessions. If the transition variable $x_{t-d}$ is replaced by its difference $\nabla x_{t-d}$, then any asymmetry lies in the growth rate of the series so that, for example, increases in growth rates may be rapid but the return to a lower level of growth may be slow.

If the transition variable $x_{t-d}$ is replaced by $t$ then the model becomes an autoregression with $r \quad 1$ breaks at times $c_1, \ldots, c_{r-1}$.

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

11.21 Yet another way of introducing asymmetry is to consider “regime switching” models. Hamilton $(1989,1990)$, Engle and Hamilton (1990), and Lam (1990) all propose variants of a switching-regime Markov model, which can be regarded as a nonlinear extension of an ARMA process that can accommodate complicated dynamics, such as asymmetry and conditional heteroskedasticity. The setup is that of the UC model of $\S 8.1$. i.e.. Eq. (8.1), where $z_t$ now evolves as a two-state Markov process:
$$z_t=\alpha_0+\alpha_1 S_t$$
where
$$\begin{gathered} P\left(S_t=1 \mid S_{t-1}=1\right)=p \ P\left(S_t=0 \mid S_{t-1}=1\right)=1-p \ P\left(S_t=1 \mid S_{t-1}=0\right)=1-q \ P\left(S_t=0 \mid S_{t-1}=0\right)=q \end{gathered}$$
The noise component $u_t$ is assumed to follow an $\operatorname{AR}(r)$ process $\phi(B) u_t=\varepsilon_t$, where the innovation sequence $\varepsilon_t$ is strict white noise but $\phi(B)$ may contain a unit root, so that, unlike the conventional UC specification, $u_t$ can be nonstationary. In fact, a special case of the conventional UC model results when $p=1-q$. The random walk component then has an innovation restricted to be a two-point random variable, taking the values 0 and 1 with probabilities $q$ and $1-q$ respectively, rather than a zero-mean random variable drawn from a continuous distribution, such as the normal.
11.22 The stochastic process for $S_t$ is strictly stationary, having the $\operatorname{AR}(1)$ representation:
$$S_t=(1-q)+\lambda S_{t-1}+V_t$$

where $\lambda=p+q-1$ and where the innovation $V_t$ has the conditional probability distribution
\begin{aligned} &P\left(V_t=(1-p) \mid S_{t-1}=1\right)=p, \ &P\left(V_t=-p \mid S_{t-1}=1\right)=1-p, \ &P\left(V_t=-(1-q) \mid S_{t-1}=0\right)=q, \ &P\left(V_t=q \mid S_{t-1}=0\right)=1-q \end{aligned}
This innovation is uncorrelated with lagged values of $S_t$, since
$$E\left(V_t \mid S_{t-j}=1\right)=E\left(V_t \mid S_{t-j}=0\right)=0 \quad \text { for } j \geq 1$$
but it is not independent of such lagged values, as, for example,
\begin{aligned} &E\left(V_t^2 \mid S_{t-1}=1\right)=p(1-p) \ &E\left(V_t^2 \mid S_{t-1}=0\right)=q(1-q) \end{aligned}

# 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|阈值和平滑过渡自回归

$$x_t=\sum_{j=1}^r\left(\phi_{j, 1} x_{t-1}+\cdots+\phi_{j, p} x_{t-p}+a_{j, t}\right) \mathbf{1}\left(c_{j-1}<x_{t-d} \leq c_j\right)$$

\begin{aligned} x_t=&\left(\phi_{1,1} x_{t-1}+\cdots+\phi_{1, p} x_{t-p}+a_{1, t}\right) \mathbf{1}\left(x_{t-d} \leq c_1\right) \ &+\left(\phi_{2,1} x_{t-1}+\cdots+\phi_{2, p} x_{t-p}+a_{2, t}\right)\left(1-\mathbf{1}\left(x_{t-d} \leq c_1\right)\right) \end{aligned}
SETAR模型的一个重要特征是它能够产生“极限环”:如果(11.7)外推，假设误差项等于零，那么外推的系列显示给定长度的振荡不消失

## 统计代写|时间序列分析代写时间序列分析代考|马尔可夫交换模型

$$z_t=\alpha_0+\alpha_1 S_t$$

$$\begin{gathered} P\left(S_t=1 \mid S_{t-1}=1\right)=p \ P\left(S_t=0 \mid S_{t-1}=1\right)=1-p \ P\left(S_t=1 \mid S_{t-1}=0\right)=1-q \ P\left(S_t=0 \mid S_{t-1}=0\right)=q \end{gathered}$$噪声成分$u_t$被假设遵循一个$\operatorname{AR}(r)$过程$\phi(B) u_t=\varepsilon_t$，其中创新序列$\varepsilon_t$是严格的白噪声，但$\phi(B)$可能包含一个单位根，因此，与传统的UC规范不同，$u_t$可以是非平稳的。事实上，传统UC模型的一个特例是$p=1-q$。随机游走组件有一个创新限制为两点随机变量，分别取概率为$q$和$1-q$的值0和1，而不是从连续分布中提取的零均值随机变量，如正态。
11.22 $S_t$的随机过程是严格平稳的，具有$\operatorname{AR}(1)$表示:
$$S_t=(1-q)+\lambda S_{t-1}+V_t$$

，其中$\lambda=p+q-1$和创新$V_t$具有条件概率分布
\begin{aligned} &P\left(V_t=(1-p) \mid S_{t-1}=1\right)=p, \ &P\left(V_t=-p \mid S_{t-1}=1\right)=1-p, \ &P\left(V_t=-(1-q) \mid S_{t-1}=0\right)=q, \ &P\left(V_t=q \mid S_{t-1}=0\right)=1-q \end{aligned}

$$E\left(V_t \mid S_{t-j}=1\right)=E\left(V_t \mid S_{t-j}=0\right)=0 \quad \text { for } j \geq 1$$
，但它不是独立于这些滞后值，例如，
\begin{aligned} &E\left(V_t^2 \mid S_{t-1}=1\right)=p(1-p) \ &E\left(V_t^2 \mid S_{t-1}=0\right)=q(1-q) \end{aligned}

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## MATLAB代写

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