统计代写|时间序列分析代写Time-Series Analysis代考|Local-Level Model and SARIMAX Models

统计代写|时间序列分析代写Time-Series Analysis代考|Local-Level Model

In this subsection, we consider a famous state space model, namely, the local-level model or random walk plus noise model. It takes the form
\begin{aligned} Y_t & =\mu_t+\varepsilon_t, \varepsilon_t \sim \operatorname{iidN}\left(0, \sigma_{\varepsilon}^2\right), \ \mu_{t+1} & =\mu_t+\eta_t, \eta_t \sim \operatorname{iidN}\left(0, \sigma_\eta^2\right) \end{aligned}
where $\left{\varepsilon_t\right}$ and $\left{\eta_t\right}$ are mutually uncorrelated and are independent of $\mu_1 \sim$ $\mathrm{N}\left(a_1, p_1\right)$. In some literature, $\mu_t$ is known as the trend of the series $Y_t$. Clearly, $Y_t$ is stationary and has no trend if $\sigma_\eta=0$ and $\mu_t$ is actually a stochastic trend of the series $Y_t$ (also a random walk) if $\sigma_\eta \neq 0$. Moreover, $Y_t=\mu_t$ if $\sigma_{\varepsilon}=0$. When $\sigma_{\varepsilon} \neq 0$, we first-difference the series $Y_t$ and arrive at
$$(1-B) Y_t=\eta_{t-1}+\varepsilon_t-\varepsilon_{t-1}$$
Let $\xi_t=\eta_{t-1}+\varepsilon_t-\varepsilon_{t-1}$. Obviously, $\xi_t$ is stationary and $\xi_t \sim \mathrm{N}\left(0,2 \sigma_{\varepsilon}^2+\sigma_\eta^2\right)$. Furthermore, the autocorrelations of $\xi_t$
$$\rho_k= \begin{cases}-\sigma_{\varepsilon}^2 /\left(2 \sigma_{\varepsilon}^2+\sigma_\eta^2\right) \neq 0 & \text { if } k=1 \ 0 & \text { if } k>1\end{cases}$$
This illustrates that $\xi_t$ follows an MA(1) model (see Table 3.1), and therefore the local-level model (8.14)-(8.15) has an $\operatorname{ARIMA}(0,1,1)$ representation
$$(1-B) Y_t=(1+\theta B) \omega_t$$
where $\omega_t$ is a white noise series. This representation can be viewed as a special case of the ARIMAX model. And this is also an example to translating a state space model into an equivalent ARIMAX model.

Let $\sigma_{\varepsilon}=2$ and $\sigma_\eta=1$. Then it turns out that $\rho_1=-2^2 /\left(2 \times 2^2+1\right)=-4 / 9$. At this point, we can simulate a sample of size 300 from the local-level model with $\sigma_{\varepsilon}=2$ and $\sigma_\eta=1$. Its time series plot is shown in Fig. 8.1, which displays nonstationarity of the series.

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

If we consider outer factors’ impact on a time series, then we need to import exogenous regressors, namely, input variables. SARIMAX models are such a kind of models as to describe functionality of input variables. They have a few forms and we here introduce an often used form as follows:
$$\left{\begin{array}{l} X_t=\beta_t \mathbf{Y}t+Z_t, \ \varphi(B) \Phi\left(B^s\right)(1-B)^d\left(1-B^s\right)^D Z_t=\theta(B) \Theta\left(B^s\right) \varepsilon_t, \varepsilon_t \sim \operatorname{WN}\left(0, \sigma\epsilon^2\right) \end{array}\right.$$
where $X_t$ is a univariate time series considered, $\mathbf{Y}t=\left(Y{t 1}, Y_{t 2}, \cdots, Y_{t k}\right)^{\prime}$ is the input variables (exogenous regressors), $\boldsymbol{\beta}t=\left(\beta{t 1}, \beta_{t 2}, \cdots, \beta_{t k}\right)$ is the coefficients corresponding to $Y_t$, and $Z_t$ is the regressing error. And Eq. (8.16b) is a $\operatorname{SARIMA}(p, d, q)(P, D, Q)_s$ model presented in Chap. 5. Thus Eqs. (8.16a)(8.16b) are a regression with SARIMA errors. Note that in many cases, $\boldsymbol{\beta}_t=\boldsymbol{\beta}$ is time-invariant. Now let us look at an example to implementing SARIMAX modeling with Python. The Python function used here is SARIMAX() in the following module statsmodels.tsa.statespace.sarimax.Example 8.4 (SARIMAX Model Building) The dataset “USEconomicChange.csv” in the folder Ptsadata is a time series that consists of changes (viz., growth rates) of the five US macroeconomic variables from the first quarter of 1970 to the third quarter of 2016. The five variables are consumption (cons), income (inc), production (prod), savings (sav), and unemployment (unem). First of all, we observe the correlation plots of the five variables shown in Fig. 8.4. It turns out that the five variables should be stationary as a five-dimensional vector series. In the light of economic common sense, we select the consumption variable cons as the endogenous (dependent) variable and all the others, namely, inc, prod, sav, and unem as the exogenous regressors. In the function SARIMAX(), let the parameters endog $=$ cons and exog $={$ inc, prod,sav, unem $}$, and by trial and error, we choose order $=(1,0,1)$ for the $\operatorname{ARIMA}(p, d, q)$ model. At the same time, all the other parameters are by default. Thus the ARMA model will be written as a state space Harvey representation. For more details on the Harvey representation, see Durbin and Koopman (2012) and Harvey (1989), among others. The estimated SARIMAX model is shown in the SARIMAX Results table in the following Python code. We see that the estimated coefficients of inc and prod are positive and the estimated coefficients of sav and unem negative. This is economically desirable. Furthermore, the ACF plot and $p$-value plot for Ljung-Box test of the residuals of the estimated SARIMAX model are, respectively, shown in Figs. 8.5 and 8.6.

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|Local-Level Model

$$Y_t=\mu_t+\varepsilon_t, \varepsilon_t \sim \operatorname{iidN}\left(0, \sigma_{\varepsilon}^2\right), \mu_{t+1} \quad=\mu_t+\eta_t, \eta_t \sim \operatorname{iidN}\left(0, \sigma_\eta^2\right)$$

$$(1-B) Y_t=\eta_{t-1}+\varepsilon_t-\varepsilon_{t-1}$$

$$\rho_k=\left{-\sigma_{\varepsilon}^2 /\left(2 \sigma_{\varepsilon}^2+\sigma_\eta^2\right) \neq 0 \quad \text { if } k=10 \quad \text { if } k>1\right.$$

$$(1-B) Y_t=(1+\theta B) \omega_t$$

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

SARIMAX 模型是描述输入变量功能的一类模型。它们有几种形式，我们在伩里介绍一 种常用的形栻如下:
$\$ \$$Vleft{$$
X_t=\beta_t \mathbf{Y} t+Z_t, \varphi(B) \Phi\left(B^s\right)(1-B)^d\left(1-B^s\right)^D Z_t=\theta(B) \Theta\left(B^s\right) \varepsilon_t, \varepsilon_t \sim \mathrm{WN}\left(0, \sigma \epsilon^2\right)
$$正确的。 \ \$$

(8.16b) 是一个SARIMA $(p, d, q)(P, D, Q)_s$ 模型在第 1 章介绍。5. 因此等式。(8.16a)
(8.16b) 是具有 SARIMA 误差的回归。请注意，在许多情况下， $\boldsymbol{\beta}_t=\boldsymbol{\beta}$ 是时不变的。现

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