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

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

The Markov switching model is widely used in econometrics and other disciplines. It involves multiple structures that characterize the time series variable behaviors in different regimes and permits switching between these structures. The Markov switching model provides such a kind of switching mechanism so that it is controlled by an unobservable state variable which follows a first-order Markov chain. It is sometimes also referred to as the (Markov) regime switching model. FrühwirthSchnatter (2006) gives a detailed Bayesian analysis of this model. Kim and Nelson (1999) elaborate on a class of state space models with Markov switching. For a time series $X_t$, a general Markov switching model is of the form
\begin{aligned} X_t= & \mu_{S_t}+\mathbf{Y}t \boldsymbol{\beta}{S_t}+\varphi_{1, S_t}\left(X_{t-1}-\mu_{S_{t-1}}-\mathbf{Y}{t-1} \boldsymbol{\beta}{S_{t-1}}\right)+\cdots \ & +\varphi_{p, S_t}\left(X_{t-p}-\mu_{S_{t-p}}-\mathbf{Y}{t-p} \boldsymbol{\beta}{S_{t-p}}\right)+\varepsilon_t, \varepsilon_t \sim \mathrm{WN}\left(0, \sigma_{S_t}^2\right) \end{aligned}
where $\mathbf{Y}t=\left(Y{t 1}, Y_{t 2}, \cdots, Y_{t k}\right)$ are the input variables (exogenous regressors), $\boldsymbol{\beta}{S_t}$ are the corresponding regression coefficient vectors, and $S_t$ is a Markov chain for regimes. Here a regime is viewed as a state that the Markov chain may take. Assume that all the possible values of state variable $S_t$ are $S={1: K}$. Then $S_t$ satisfies the Markov property $$P\left(S_t=j \mid S{t-1}=i, S_{t-2}=s_{t-2}, \cdots, S_1=s_1\right)=P\left(S_t=j \mid S_{t-1}=i\right)=p_{i j}$$
where $j, i, s_{t-2}, \cdots, s_1 \in S$ and $p_{i j}$ is called the one-step transition probability from state $i$ to state $j$. And the state transition is governed by the (state) transition (probability) matrix $\mathbf{P}=\left[p_{i j}\right]$. Besides, the error $\varepsilon_t$ is also written as $\varepsilon_t=\sigma_{S_t} \eta_t$ where $\eta_t \sim \mathrm{WN}(0,1)$. And it is often assumed that the error $\varepsilon_t$ is normally distributed, namely, $\varepsilon_t \sim \operatorname{iidN}\left(0, \sigma_{S_t}^2\right)$.
If there are no exogenous regressors in Eq. (8.17), it is reduced to
$$X_t=\mu_{S_t}+\varphi_{1, S_t}\left(X_{t-1}-\mu_{S_{t-1}}\right)+\cdots+\varphi_{p, S_t}\left(X_{t-p}-\mu_{S_{l-p}}\right)+\varepsilon_t, \varepsilon_t \sim \mathrm{WN}\left(0, \sigma_{S_t}^2\right)$$
and called the Markov switching autoregressive model.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Deterministic Trend and Stochastic Trend

What is a stochastic trend in a time series? Let us have a look at the following example before answering that question.

Example 9.1 (A Time Series with Trend Components) Consider the time series $X_t=\alpha+\beta t+\eta_t$ where $\beta \neq 0$ and $\eta_t=\eta_{t-1}+\varepsilon_t, \varepsilon_t \sim \mathrm{WN}(0,1)$ is a random walk. Its time series plot is shown in Fig. 9.1 when $\alpha=0.3$ and $\beta=0.2$. Clearly there is a deterministic trend in the time series, and thus it is nonstationary. At the same time, the series has no seasonality. Furthermore, if we detrend the deterministic trend from the time series, we arrive at the series $\eta_t$ that is a random walk and also nonstationary, seeing Fig. 9.2. In other words, there are two components in the time series, which of both make it nonstationary. One component is well known to us, that is, the deterministic trend. Another is new and will be defined below. In addition, if we difference the deterministic-trend-removed time series $\eta_t$, then the differenced series is the white noise $\nabla \eta_t=\varepsilon_t$ and so stationary, seeing Figs. 9.3 and 9.4.

Definition 9.1 (1) If a time series is still nonstationary after its deterministic components are removed from it, then it is said to have a stochastic (random) trend as long as the ordinarily differenced series of the deterministic-componentremoved series is stationary. (2) If a nonstationary time series has no deterministic components, then it is said to possess a stochastic (random) trend as long as its ordinarily differenced series is stationary.
Remarks on Definition 9.1:

• A time series with a stochastic trend should satisfy two conditions: (1) it is nonstationary, and if it has deterministic components (deterministic trend and/or seasonality), then the deterministic-component-removed series is still nonstationary, and (2) the ordinarily differenced series is stationary, which also guarantees that the stochastic component is the stochastic trend, not stochastic seasonality.
• The time series $X_t=\alpha+\beta t+\eta_t$ in Example 9.1 belongs to the case (1) of Definition 9.1, and note that it has both a deterministic trend and a stochastic trend. If we consider the random walk $Y_t=Y_{t-1}+\varepsilon_t$, it obviously belongs to the case (2) of Definition 9.1.
• It is sometimes not easy to distinguish between a stochastic trend and a deterministic trend in a time series. The background of data often helps distinguish them. For example, logarithm price series of financial products tend to possess a stochastic trend, while macroeconomic yearly or seasonally adjusted time series usually have a deterministic trend (and may sometimes also have a stochastic trend). Two real examples are given in Example 9.2 and Problem 9.13.

# 时间序列分析代考

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

$$X_t=\mu_{S_t}+\mathbf{Y} t \boldsymbol{\beta} S_t+\varphi_{1, S_t}\left(X_{t-1}-\mu_{S_{t-1}}-\mathbf{Y} t-1 \boldsymbol{\beta} S_{t-1}\right)+\cdots \quad+\varphi_{p, S_t}\left(X_{t-p}-\right.$$

$$P\left(S_t=j \mid S t-1=i, S_{t-2}=s_{t-2}, \cdots, S_1=s_1\right)=P\left(S_t=j \mid S_{t-1}=i\right)=p_{i j}$$

$$X_t=\mu_{S_t}+\varphi_{1, S_t}\left(X_{t-1}-\mu_{S_{t-1}}\right)+\cdots+\varphi_{p, S_t}\left(X_{t-p}-\mu_{S_{l-p}}\right)+\varepsilon_t, \varepsilon_t \sim \mathrm{WN}\left(0, \sigma_{S_t}^2\right)$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Deterministic Trend and Stochastic Trend

• 具有随机趋势的时间序列应满足两个条件: (1) 它是非平稳的，如果它具有确定性 成分（确定性趋势和/或季节性)，则去除确定性成分的序列仍然是非平稳的，以及
（2）常差序列是平稳的，这也保证了随机成分是随机趋势，而不是随机季节性。
• 时间序列 $X_t=\alpha+\beta t+\eta_t$ 例9.1中的属于定义9.1的情呪（1），注意它既有确定性 趋势又有随机性趋势。如果我们考虑随机游走 $Y_t=Y_{t-1}+\varepsilon_t$ ，显然属于定义9.1的 情况 (2)。
• 有时不容易区分时间序列中的随机趋势和确定性趋势。数据背景通常有助于区分它 们。例如，金融产品的对数价格序列往往具有随机趋势，而宏观经济年度或季节性 调整时间序列通常具有确定性趋势 (有时也可能具有随机趋势) 。例 9.2 和习题 9.13 给出了两个真实的例子。

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