Chapter 60: autoregression in time series
張翔老師. 2015. “ARMA Part1.” https://www.youtube.com/watch?v=G-0dR57W-fo.
張翔老師. 2015. “ARMA Part2.” https://www.youtube.com/watch?v=fQaZzO7E6FE.
張翔老師. 2015. “ARMA Part3.” https://www.youtube.com/watch?v=Ocw4NXoO8Xo.
time series [data]
\[ \cdots,Y_{t-2},Y_{t-1},Y_{t},Y_{t+1},Y_{t+2},\cdots \]
- lag
- \(1^\text{st}\) lag = lag \(1\) \[ Y_{t-1} \]
- \(k^\text{th}\) lag = lag \(k\) \[ Y_{t-k} \]
\(1^\text{st}\) difference
\[ \Delta Y_{t}=Y_{t}-Y_{t-1} \] approximation for RoR = rate of return
\[ \begin{aligned} \Delta\ln Y_{t}=\ln Y_{t}-\ln Y_{t-1}= & \ln\dfrac{Y_{t}}{Y_{t-1}}=\ln\left(1+\dfrac{Y_{t}-Y_{t-1}}{Y_{t-1}}\right)\\ = & \dfrac{Y_{t}-Y_{t-1}}{Y_{t-1}}+\mathrm{O}\left(\left(\dfrac{Y_{t}-Y_{t-1}}{Y_{t-1}}\right)^{2}\right)\\ \approx & \dfrac{Y_{t}-Y_{t-1}}{Y_{t-1}}=\mathrm{RoR} \end{aligned} \]
\[ \begin{cases} \ln(1+x)=\sum\limits _{n=1}^{\infty}(-1)^{n+1}\dfrac{x^{n}}{n}=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots\\ \ln(1-x)=-\sum\limits _{n=1}^{\infty}\dfrac{x^{n}}{n}=-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}-\cdots \end{cases} \]
\[ \begin{aligned} \frac{1}{1+t}= & 1-t+t^{2}-t^{3}+\cdots\\ \ln(1+x)= & \int_{0}^{x}\frac{1}{1+t}dt\\ = & x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots \end{aligned} \]
Definition 37.1 autocorrelation = serial correlation
\[ \exists t_{1}\ne t_{2}\left[\mathrm{V}\left(Y_{t_{1}},Y_{t_{2}}\right)\ne0\right] \]
\[ \begin{cases} \mathrm{E}\left(Y_{t}\right)\triangleq\mu_{t}\\ \mathrm{V}\left(Y_{t}\right)\triangleq\sigma_{t}^{2} \end{cases} \]
Definition 15.1 \(k^\text{th}\) order autocovariance
\[ \mathrm{V}\left(Y_{t},Y_{t-k}\right)\triangleq\gamma\left(t,k\right) \]
\[ \sigma_{t}^{2}\triangleq\mathrm{V}\left(Y_{t}\right)=\mathrm{V}\left(Y_{t},Y_{t}\right)=\mathrm{V}\left(Y_{t},Y_{t-0}\right)=\gamma\left(t,k=0\right)=\gamma\left(t,0\right) \]
Definition 15.2 \(k^\text{th}\) order autocorrelation
\[ \rho_{t,t-k}=\dfrac{\sigma_{t,t-k}}{\sqrt{\sigma_{tt}}\sqrt{\sigma_{t-k,t-k}}}=\dfrac{\sigma_{t,t-k}}{\sqrt{\sigma_{t}^{2}}\sqrt{\sigma_{t-k}^{2}}}=\dfrac{\sigma_{t,t-k}}{\sigma_{t}\sigma_{t-k}}=\dfrac{\mathrm{V}\left(Y_{t},Y_{t-k}\right)}{\sqrt{\mathrm{V}\left(Y_{t}\right)}\sqrt{\mathrm{V}\left(Y_{t-k}\right)}}\triangleq\mathrm{R}\left(Y_{t},Y_{t-k}\right)\triangleq\rho\left(t,k\right) \]
Definition 15.3 stationary time series
\[ \begin{cases} \mathrm{E}\left(Y_{t}\right)\triangleq\mu_{t}=\mu & \left(1\right)\text{independent of }t\\ \mathrm{V}\left(Y_{t}\right)\triangleq\sigma_{t}^{2}=\sigma^{2}<\infty & \left(2\right)\text{independent of }t\\ \mathrm{V}\left(Y_{t},Y_{t-k}\right)\triangleq\gamma\left(t,k\right)=\gamma\left(k\right) & \left(3\right)\text{independent of }t \end{cases} \]
properties
\[ \begin{cases} \mathrm{V}\left(Y_{t}\right)\triangleq\sigma_{t}^{2}=\gamma\left(t,k=0\right)=\gamma\left(k=0\right)\overset{\left(3\right)}{=}\gamma\left(0\right)\triangleq\gamma_{0}\overset{\left(2\right)}{=}\sigma^{2} & \left(4\right)\Rightarrow\mathrm{V}\left(Y_{t-k}\right)=\gamma\left(0\right)\\ \mathrm{R}\left(Y_{t},Y_{t-k}\right)\triangleq\rho\left(t,k\right)\triangleq\dfrac{\mathrm{V}\left(Y_{t},Y_{t-k}\right)}{\sqrt{\mathrm{V}\left(Y_{t}\right)}\sqrt{\mathrm{V}\left(Y_{t-k}\right)}}\\ =\dfrac{\gamma\left(t,k\right)}{\sqrt{\gamma\left(0\right)}\sqrt{\gamma\left(0\right)}}\overset{\left(3\right)}{\underset{\left(4\right)}{=}}\dfrac{\gamma\left(k\right)}{\gamma\left(0\right)}\triangleq\dfrac{\gamma_{k}}{\gamma_{0}}\triangleq\rho\left(k\right)\triangleq\rho_{k} & \left(5\right)\\ \gamma\left(k\right)=\gamma\left(-k\right) & \left(6\right)\Rightarrow\rho\left(k\right)=\rho\left(-k\right) \end{cases} \]
\[ \gamma\left(k\right)\overset{\left(3\right)}{=}\mathrm{V}\left(Y_{t},Y_{t-k}\right)=\mathrm{V}\left(Y_{t-k},Y_{t}\right)=\mathrm{V}\left(Y_{t^{\prime}},Y_{t^{\prime}+k}\right)=\mathrm{V}\left(Y_{t^{\prime}},Y_{t^{\prime}-\left(-k\right)}\right)\overset{\left(3\right)}{=}\gamma\left(-k\right)\Rightarrow\left(6\right) \]
point estimation
\[ \begin{cases} \widehat{\mathrm{E}}\left(Y_{t}\right)\triangleq\overline{Y}\triangleq\widehat{\mu}=\dfrac{1}{T}\sum\limits _{t=1}^{T}Y_{t} & \rightarrow\mathrm{E}\left(Y_{t}\right)=\mu\\ \widehat{\mathrm{V}}\left(Y_{t}\right)\triangleq\widehat{\gamma}_{0}=\dfrac{1}{T}\sum\limits _{t=1}^{T}\left(Y_{t}-\overline{Y}\right)^{2} & \rightarrow\mathrm{V}\left(Y_{t}\right)=\sigma^{2}=\gamma_{0}\\ \widehat{\mathrm{V}}\left(Y_{t},Y_{t-k}\right)\triangleq\widehat{\gamma}_{k}\\ =\dfrac{1}{T}\sum\limits _{t=k+1}^{T}\left(Y_{t}-\overline{Y}\right)\left(Y_{t-k}-\overline{Y}\right)=\dfrac{1}{T}\sum\limits _{t=1}^{T-k}\left(Y_{t}-\overline{Y}\right)\left(Y_{t+k}-\overline{Y}\right) & \rightarrow\mathrm{V}\left(Y_{t},Y_{t-k}\right)=\gamma_{k}\\ \dfrac{\widehat{\mathrm{V}}\left(Y_{t},Y_{t-k}\right)}{\sqrt{\mathrm{\widehat{\mathrm{V}}}\left(Y_{t}\right)}\sqrt{\mathrm{\widehat{\mathrm{V}}}\left(Y_{t-k}\right)}}\triangleq\widehat{\rho}_{k}=\dfrac{\widehat{\gamma}_{k}}{\widehat{\gamma}_{0}}=\dfrac{\dfrac{1}{T}\sum\limits _{t=k+1}^{T}\left(Y_{t}-\overline{Y}\right)\left(Y_{t-k}-\overline{Y}\right)}{\dfrac{1}{T}\sum\limits _{t=1}^{T}\left(Y_{t}-\overline{Y}\right)^{2}} & \rightarrow\mathrm{R}\left(Y_{t},Y_{t-k}\right)=\rho_{k} \end{cases} \]
\[ Y_{1}Y_{1+k}+Y_{2}Y_{2+k}+\cdot\cdots+Y_{t}Y_{t+k}+\cdot\cdots+Y_{T-k}Y_{T} \]
\(1^\text{st}\)-order autocorrelation estimation
\[ \widehat{\rho}_{1}=\dfrac{\widehat{\gamma}_{1}}{\widehat{\gamma}_{0}}=\dfrac{\dfrac{1}{T}\sum\limits _{t=1+1=2}^{T}\left(Y_{t}-\overline{Y}\right)\left(Y_{t-1}-\overline{Y}\right)}{\dfrac{1}{T}\sum\limits _{t=1}^{T}\left(Y_{t}-\overline{Y}\right)^{2}} \]
60.1 AR(1) = \(1^\text{st}\)-order autoregressive model = first-order autoregressive model
張翔老師. 2015. “ARMA Part3.” https://www.youtube.com/watch?v=Ocw4NXoO8Xo.
Definition 15.4 AR(1) = \(1^\text{st}\)-order autoregressive model$
\[ Y_{t}=\beta_{0}+\beta_{1}Y_{t-1}+\mathcal{E}_{t} \]
\[ \mathcal{E}_{t}\sim\mathcal{N}\left(0,\sigma_{\mathcal{E}}^{2}\right) \] \[ \begin{aligned} Y_{t}= & \beta_{0}+\beta_{1}Y_{t-1}+\mathcal{E}_{t}\\ = & \beta_{0}+\beta_{1}\left(\beta_{0}+\beta_{1}Y_{t-2}+\mathcal{E}_{t-1}\right)+\mathcal{E}_{t}\\ = & \beta_{0}\left(1+\beta_{1}\right)+\beta_{1}^{2}Y_{t-2}+\mathcal{E}_{t}+\beta_{1}\mathcal{E}_{t-1}\\ = & \beta_{0}\left(1+\beta_{1}\right)+\beta_{1}^{2}\left(\beta_{0}+\beta_{1}Y_{t-3}+\mathcal{E}_{t-2}\right)+\mathcal{E}_{t}+\beta_{1}\mathcal{E}_{t-1}\\ = & \beta_{0}\left(1+\beta_{1}+\beta_{1}^{2}\right)+\beta_{1}^{3}Y_{t-3}+\mathcal{E}_{t}+\beta_{1}\mathcal{E}_{t-1}+\beta_{1}^{2}\mathcal{E}_{t-2}\\ \vdots\\ = & \beta_{0}\left(1+\beta_{1}+\beta_{1}^{2}+\cdots\right)+\mathcal{E}_{t}+\beta_{1}\mathcal{E}_{t-1}+\beta_{1}^{2}\mathcal{E}_{t-2}+\cdots\\ \overset{\left|\beta_{1}\right|<1}{=} & \dfrac{\beta_{0}}{1-\beta_{1}}+\sum_{k=0}^{\infty}\beta_{1}^{k}\mathcal{E}_{t-k} \end{aligned} \]
\[ \begin{cases} \mu=\mathrm{E}\left(Y_{t}\right)=\mathrm{E}\left(\dfrac{\beta_{0}}{1-\beta_{1}}+\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-i}\right)\\ =\dfrac{\beta_{0}}{1-\beta_{1}}+\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathrm{E}\left(\mathcal{E}_{t-i}\right)\overset{\mathcal{E}_{t}\sim\mathcal{N}\left(0,\sigma_{\mathcal{E}}^{2}\right)}{=}\dfrac{\beta_{0}}{1-\beta_{1}} & \mu=\dfrac{\beta_{0}}{1-\beta_{1}}\\ \gamma_{0}=\sigma^{2}=\mathrm{V}\left(Y_{t}\right)=\mathrm{V}\left(\mu+\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-i}\right)=\mathrm{V}\left(\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-i}\right)\\ =\sum_{i=0}^{\infty}\left(\beta_{1}^{i}\right)^{2}\mathrm{V}\left(\mathcal{E}_{t-i}\right)\overset{\mathcal{E}_{t}\sim\mathcal{N}\left(0,\sigma_{\mathcal{E}}^{2}\right)}{=}\sum\limits _{i=0}^{\infty}\left(\beta_{1}^{2}\right)^{i}\sigma_{\mathcal{E}}^{2}=\dfrac{\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}} & \gamma_{0}=\sigma^{2}=\dfrac{\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}}\\ \gamma_{k}=\mathrm{V}\left(Y_{t},Y_{t-k}\right)=\mathrm{V}\left(\mu+\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-i},\mu+\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-k-i}\right)\\ =\mathrm{V}\left(\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-i},\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-k-i}\right)=\cdots & \gamma_{k}=\dfrac{\beta_{1}^{k}\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}}\\ \rho_{k}=\mathrm{R}\left(Y_{t},Y_{t-k}\right)=\dfrac{\gamma_{k}}{\gamma_{0}}=\dfrac{\dfrac{\beta_{1}^{k}\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}}}{\dfrac{\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}}}=\beta_{1}^{k} & \rho_{k}=\dfrac{\gamma_{k}}{\gamma_{0}}=\beta_{1}^{k} \end{cases} \]
\[ \rho_{1}=\dfrac{\gamma_{1}}{\gamma_{0}}=\beta_{1}^{1}=\beta_{1} \]
\[ \begin{aligned} \gamma_{k}=\mathrm{V}\left(Y_{t},Y_{t-k}\right)= & \mathrm{V}\left(\mu+\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-i},\mu+\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-k-i}\right)\\ = & \mathrm{V}\left(\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-i},\sum\limits _{i=0}^{\infty}\beta_{1}^{i}\mathcal{E}_{t-k-i}\right)=\cdots \end{aligned} \]
\[ \boldsymbol{Y}=\left\langle Y_{1},Y_{2},\cdots,Y_{T}\right\rangle =\begin{bmatrix}Y_{1} & Y_{2} & \cdots & Y_{T}\end{bmatrix}^{\mathrm{T}}=\begin{bmatrix}Y_{1}\\ Y_{2}\\ \vdots\\ Y_{T} \end{bmatrix} \]
AR(1) covariance matrix
by (56.3)
\[ \begin{aligned} \mathrm{V}\left(\boldsymbol{Y}\right)= & \mathrm{E}\left[\left[\boldsymbol{Y}-\mathrm{E}\left(\boldsymbol{Y}\right)\right]\left[\boldsymbol{Y}-\mathrm{E}\left(\boldsymbol{Y}\right)\right]^{\mathrm{T}}\right]\\ = & \begin{bmatrix}\sigma_{1}^{2} & \sigma_{12} & \cdots & \sigma_{1T}\\ \sigma_{21} & \sigma_{2}^{2} & \cdots & \sigma_{2T}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{T1} & \sigma_{T2} & \cdots & \sigma_{T}^{2} \end{bmatrix}=\begin{bmatrix}\sigma_{11} & \sigma_{12} & \cdots & \sigma_{1T}\\ \sigma_{21} & \sigma_{22} & \cdots & \sigma_{2T}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{T1} & \sigma_{T2} & \cdots & \sigma_{TT} \end{bmatrix}=\left[\sigma_{ij}\right]_{T\times T}=\mathit{\Sigma}\\ = & \begin{bmatrix}\gamma_{0} & \gamma_{1} & \cdots & \gamma_{T-1}\\ \gamma_{1} & \gamma_{0} & \cdots & \gamma_{T-2}\\ \vdots & \vdots & \ddots & \vdots\\ \gamma_{T-1} & \gamma_{T-2} & \cdots & \gamma_{0} \end{bmatrix}=\begin{bmatrix}\dfrac{\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}} & \dfrac{\beta_{1}^{1}\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}} & \cdots & \dfrac{\beta_{1}^{T-1}\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}}\\ \dfrac{\beta_{1}^{1}\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}} & \dfrac{\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}} & \cdots & \dfrac{\beta_{1}^{T-2}\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}}\\ \vdots & \vdots & \ddots & \vdots\\ \dfrac{\beta_{1}^{T-1}\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}} & \dfrac{\beta_{1}^{T-2}\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}} & \cdots & \dfrac{\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}} \end{bmatrix}\\ = & \dfrac{\sigma_{\mathcal{E}}^{2}}{1-\beta_{1}^{2}}\begin{bmatrix}1 & \beta_{1} & \cdots & \beta_{1}^{T-1}\\ \beta_{1} & 1 & \cdots & \beta_{1}^{T-2}\\ \vdots & \vdots & \ddots & \vdots\\ \beta_{1}^{T-1} & \beta_{1}^{T-2} & \cdots & 1 \end{bmatrix}=\sigma^{2}\begin{bmatrix}1 & \beta_{1} & \beta_{1}^{2} & \cdots & \beta_{1}^{T-1}\\ \beta_{1} & 1 & \beta_{1} & \cdots & \beta_{1}^{T-2}\\ \beta_{1}^{2} & \beta_{1} & 1 & \cdots & \beta_{1}^{T-3}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \beta_{1}^{T-1} & \beta_{1}^{T-2} & \beta_{1}^{T-3} & \cdots & 1 \end{bmatrix}\\ = & \sigma^{2}\begin{bmatrix}1 & \rho_{1} & \rho_{2} & \cdots & \rho_{T-1}\\ \rho_{1} & 1 & \rho_{1} & \cdots & \rho_{T-2}\\ \rho_{2} & \rho_{1} & 1 & \cdots & \rho_{T-3}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \rho_{T-1} & \rho_{T-2} & \rho_{T-3} & \cdots & 1 \end{bmatrix}=\sigma^{2}\begin{bmatrix}1 & \rho & \rho^{2} & \cdots & \rho^{T-1}\\ \rho & 1 & \rho & \cdots & \rho^{T-2}\\ \rho^{2} & \rho & 1 & \cdots & \rho^{T-3}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \rho^{T-1} & \rho^{T-2} & \rho^{T-3} & \cdots & 1 \end{bmatrix} \end{aligned} \]
\[ \boldsymbol{Y}\sim\mathcal{D}\left(\boldsymbol{\mu},\mathit{\Sigma}\right)=\mathrm{d}\left(\boldsymbol{\mu}_{\boldsymbol{Y}},\mathit{\Sigma}_{\boldsymbol{Y}}\right)=\mathrm{d}\left(\mathrm{E}\left[\boldsymbol{Y}\right],\mathrm{V}\left[\boldsymbol{Y}\right]\right) \]
where
\[ \mathit{\Sigma}=\begin{bmatrix}\gamma_{0} & \gamma_{1} & \cdots & \gamma_{T-1}\\ \gamma_{1} & \gamma_{0} & \cdots & \gamma_{T-2}\\ \vdots & \vdots & \ddots & \vdots\\ \gamma_{T-1} & \gamma_{T-2} & \cdots & \gamma_{0} \end{bmatrix}=\sigma^{2}\begin{bmatrix}1 & \rho & \rho^{2} & \cdots & \rho^{T-1}\\ \rho & 1 & \rho & \cdots & \rho^{T-2}\\ \rho^{2} & \rho & 1 & \cdots & \rho^{T-3}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \rho^{T-1} & \rho^{T-2} & \rho^{T-3} & \cdots & 1 \end{bmatrix} \]
60.2 AR(2) = \(2^\text{nd}\)-order autoregressive model = second-order autoregressive model
Definition 60.1 AR(2) = \(2^\text{nd}\)-order autoregressive model
\[ Y_{t}=\beta_{0}+\beta_{1}Y_{t-1}+\beta_{2}Y_{t-2}+\mathcal{E}_{t} \]
\[ \mathcal{E}_{t}\sim\mathcal{N}\left(0,\sigma_{\mathcal{E}}^{2}\right) \]
60.3 AR(\(p\)) = \(p^\text{th}\)-order autoregressive model
\[ Y_{t}=\beta_{0}+\sum_{i=1}^{p}\beta_{i}Y_{t-i}+\mathcal{E}_{t} \]