弱平穩(Weak Stationary)

$$\begin{aligned} \operatorname{E}(y_t)&=C \\ \operatorname{Cov}(y_t,y_s)&=R(t,s) \\ \operatorname{Var}(y_t)&=\operatorname{Cov}(y_t,y_t)=\sigma_y^2 \\\\ \rho_k&=\operatorname{Corr}(y_t,y_{t-k})=\frac{\operatorname{Cov}(y_t,y_{t-k})}{\sigma_y^2} \\ r_k&:=\hat{\rho_k}= \frac{\sum_{t=k+1}^T(y_{t-k}-\bar{y})(y_{t}-\bar{y})} {\sum_{t=1}^T(y_{t}-\bar{y})} \end{aligned}$$

白噪音

$$\begin{aligned} \operatorname{E}(y_t)&=0 \\ \operatorname{Cov}(y_t,y_s)&=0 \\ \operatorname{Var}(y_t)&=\sigma_y^2 \\\\ r_k&\sim t_{T-1} \\ \operatorname{SE}(r_k)&=\frac{1}{\sqrt{T}} \\ \operatorname{Var}(y_{T+l}-\hat{y_{T+l}})&=\operatorname{Var}(y_{T+l}-\bar{y})=\sigma_y^2(1+\frac{1}{T})\approx \sigma_y^2 \end{aligned}$$

隨機漫步

$$\begin{aligned} y_t &= y_0 + \sum_{i=1}^{t} c_i, \quad\quad c_i \sim W(\mu_c, \sigma_c^2) \\ \operatorname{E}(y_t) &= y_0 + t\mu_c \\ \operatorname{Var}(y_t) &= t\sigma_c^2 \\ \operatorname{Cov}(y_s, y_t) &= s\sigma_c^2, \quad \text{for } s \le t \end{aligned}$$

$$\begin{aligned} \hat{y_{T+l}}&=y_T+l\bar{c} \\\\ \operatorname{Var}(y_{T+l}-\hat{y_{T+l}})&=\operatorname{Var}(\sum_{i=1}^lc_{t+i})+l^2\operatorname{Var}(\bar{c}) \\&=l\sigma_c^2+\frac{l^2\sigma_c^2}{T}\approx l\sigma_c^2 \\\\ \bar{c}&=\frac{c_1+c_2+...+c_T}{T}=\frac{y_T-y_0}{T} \\ s^2&=\frac{\sum_{i=1}^T(c_i-\bar{c})^2}{T} \end{aligned}$$

平滑

$$\hat{y_{t+1}}=\hat{s_t}$$

移動平均

$$\begin{aligned} \hat{s_t}&=\frac{y_t-ks_{t-1}-y_{t-k}}{k} \\ \hat{s_t}^{(2)}&=\frac{\hat{s_t}+\hat{s_{t-1}}+...+\hat{s_{t-k+1}}}{k} \end{aligned}$$

指數平均

$$\begin{aligned} \hat{s_t}&=\frac{y_t+\omega y_{y-1}+...+\omega^t y_{0}}{1/(1-\omega)} \\ &=\omega \hat{s_{t-1}}+(1-\omega)y_t \\ \\ \hat{s_0}&=\frac{y_0}{1/{1-\omega}}=(1-\omega)y_0 \end{aligned}$$

雙指數平滑

$$\begin{aligned} \hat{y_{T+l}}&=\hat{\beta_{0,T}}+\hat{\beta_{1,T}}l\\ \\ \hat{\beta_{0,T}}&=2\hat{s}_T^{(1)}-\hat{s}T^{(2)} \\ \hat{\beta{1,T}}&=\frac{\omega}{1-\omega}(\hat{s}_T^{(1)}-\hat{s}_T^{(2)} ) \end{aligned}$$

AR(1)

$$\begin{aligned} y_t=\beta_0+\beta_1y_{t-1}+\varepsilon_t& \quad\quad \varepsilon_t\sim W(0, \sigma^2) \\ \text{stationary} \Leftrightarrow &\lvert \beta_1 \rvert<1 \end{aligned}$$

$$\begin{aligned} \operatorname{E}(y_t) &= \frac{\beta_0}{1 - \beta_1} \\\\ \operatorname{Var}(y_t) &= \frac{\operatorname{Var}(\varepsilon_t)}{1 - \beta_1^2} = \frac{\sigma^2}{1 - \beta_1^2} = \sigma_y^2 \\\\ \operatorname{Cov}(y_t, y_{t-k}) &= \beta_1^k \operatorname{Cov}(y_{t-k}, y_{t-k}) = \beta_1^k \sigma_y^2 \\\\ \rho_k &= \frac{\operatorname{Cov}(y_t, y_{t-k})}{\operatorname{Var}(y_t)} = \beta_1^k \end{aligned}$$

$$\begin{aligned} \hat{\beta_1} &\approx r_1 = \frac{\sum_{t=2}^T(y_{t-1} - \bar{y})(y_{t} - \bar{y})}{\sum_{t=1}^T(y_{t} - \bar{y})^2} \\ \hat{\beta_0} &\approx \bar{y} - r_1\bar{y} = (1 - r_1)\bar{y} \\ s^2 &= \frac{\sum_{t=2}^T(e_{t} - \bar{e})^2}{(T - 1) - 2} \end{aligned}$$

$$\begin{aligned} \hat{y}_{T+k} &= \hat{\beta}_0 (1 + \hat{\beta}_1^1 + \hat{\beta}_1^2 + \cdots + \hat{\beta}_1^{k-1}) + \hat{\beta}_1^k y_T \\ &= \frac{\hat{\beta}_0 (1 - \hat{\beta}_1^k)}{1 - \hat{\beta}_1} + \hat{\beta}1^k y_T \\ &= (1 - \hat{\beta}1^k)\hat{\mu} + \hat{\beta}1^k y_T \\ &\sim t{(T-1)-2} \\\\ \operatorname{Var}(y{T+k} - \hat{y}{T+k}) &\overset{\text{estimate}}{\approx} \sigma^2 \left( 1 + \beta_1^2 + \beta_1^4 + \cdots + \beta_1^{2(k-1)} \right) \end{aligned}$$

GARCH(p,q)

$$\varepsilon_t|\Omega_t-1\sim N(0,\sigma_t^2)$$

$$\sigma_t^2=\omega+(\gamma_1\varepsilon_{t-1}^2+\cdots+\gamma_p\varepsilon_{t-p}^2)+ (\delta_1\sigma_{t-1}^2+\cdots+\delta_p\sigma_{t-q}^2)$$

$$\text{stationary} \Leftrightarrow (\sum_{j=1}^p\gamma_j+\sum_{j=1}^q\delta_j)<1$$

$$\begin{aligned} \operatorname{Var}(\varepsilon_t)&=\frac{\omega}{1-(\sum_{j=1}^p\gamma_j+\sum_{j=1}^q\delta_j)} \\ \operatorname{E}(\varepsilon_{t+1}^2|\Omega_{T})&=\sigma_{T+1}^2 \end{aligned}$$