Review

def 1: weak stationary/ covariance stationary time series:

1. $\begin{cases}E(y_{t})=\mu,\mu\in(-\infty,\infty)\\V(y_{t})=\sigma^{2}<\infty,\forall t\\Cov(y_{t},y_{t-k})=\gamma(k),\forall t\end{cases}$. 有: $\mu$ 未知且有限, variance 和cov都不和时间有关.

def 2: strictly stationary time series:$f(y_{1},\ldots,y_{r})=f(y_{1+k},\ldots,y_{r+k})$, 即他们有一样的distribution.

我们关注weak stationary是因为只要知道方差和cov就可以判断是不是weak stationary了, 很方便.

def 3: white noise: $\begin{cases}E[\varepsilon_{t}]=0,\forall t\\V[\varepsilon_{t}]=\sigma_{e}^{2},\forall t\\Cov[\varepsilon_{t},\varepsilon_{t-k}]=0,\forall t,k\neq0\end{cases}$. 白噪音满足weak stationary的条件. 然而weakly stationary的序列不一定是白噪音. 注意白噪音不需要identical distribution.

def 4: Gaussian white noise: 指正态分布的白噪音.

Example 1: Linear Trending Series. 考虑$y_{t}=\alpha+\beta t+\epsilon_{t},t=1\ldots T$, 因为t不是random的, 所以可得$E(t)=t$. 检查这个序列的moment: $E[y_{t}]=\alpha+\beta t+E[\epsilon_{t}]=\alpha+\beta t$​, 不是平稳序列, 因为随t的变化而变化.

def 5: Integration Order: d阶integration意味着一个序列做d次difference之后得到平稳且不可逆序列: $\Delta^{d}y_{t}=(1-L)^{d}y_{t}$平稳且invertible (指AR是否可以转化为MA).

def 6: ARMA(p,q) Model: 指$y_{t}=\phi_{1}y_{t-1}+\cdots+\phi_{p}y_{t-p}+\epsilon_{t}+\theta_{1}\epsilon_{t-1}+\cdots+\theta_{q}\epsilon_{t-q}\leftrightarrow\phi(L)y_{t}=\theta(L)\epsilon_{t}$.

• 证: $y_{t}=\phi_{1}y_{t-1}+\cdots+\phi_{p}y_{t-p}+\epsilon_{t}+\theta_{1}\epsilon_{t-1}+\cdots+\theta_{q}\epsilon_{t-q}\to y-\phi_{1}y_{t-1}-\ldots\phi_{p}y_{t-p}=\varepsilon_{t}+\theta_{1}\varepsilon_{t-1}+\ldots+\theta_{q}\varepsilon_{t-q}$

  $\to \underbrace{(1-\phi_{1}L-\cdots-\phi_{p}L^{p})}_{\phi(L)}y_{t}=\underbrace{(1+\theta_{1}L\cdots+\theta_{q}L_{q})}_{\theta(L)}\varepsilon_{t}$.

其中AR模型就是$y_{t}$和其滞后项之间的关系, 而MA模型是$y_{t}$和white noise的滞后项之间的关系. 宏观中经常用MA模型做impulse response, 即探寻t时刻误差项的增加量来判断t+k时刻y的增量. 这个方法的坏处就是不知道实际的误差项的值, 只能用估计值来代替.

Stationary condition for AR

可以将AR模型转化为MA模型, eg: for AR(1): $\begin{aligned}y_{t}=\phi y_{t-1}+\epsilon_{t}&=\phi(\phi y_{t-2}+\epsilon_{t-1})+\epsilon_{t}\\&=\phi^{j+1}y_{t-j-1}+\sum_{i=0}^{j}\phi^{i}\epsilon_{t-i}\end{aligned}$. 只要令$|\phi|<1\wedge$ t began from $-\infty$,则$j\to\infty \leftrightarrow \phi^{j+1}y_{t-j-1}\to 0$. $\to y_{t}=\sum_{i=0}^{\infty}\phi^{i}\epsilon_{t-i}\to$ MA($\infty$). 此时有$E(y_{t})=0, V(y_{t})=E[\sum\phi\epsilon_{t-i}]^{2}=E[\epsilon_{t}+\phi\epsilon_{t-1}+\phi^{2}\epsilon_{t-2}+\ldots\phi^{\infty}\epsilon_{t-\infty}]^{2}$

$=E[\epsilon_{t}^{2}]+\phi E[\epsilon_{t-1}^{2}]+\phi^{2}E[\epsilon_{t-2}^{2}]+\underbrace{\sum_{i}\sum_{j}\phi^{i}\phi^{j}E[\epsilon_{t-i}\epsilon_{t-j}]}_{=0}=\sigma_{\epsilon}^{2}[1+\phi^{2}+\cdots+\phi^{\infty}]=\frac{1}{1-\phi^{2}}\sigma_{\epsilon}^{2}$. 可得V是有限常数.

同理, $Cov(y_{t},y_{t-k})=...=\frac{\phi^{k}\sigma_{\varepsilon}^{2}}{1-\phi^{2}}=r(k)\not\sim t, if\ |\phi|<1$.

得结论: $|\phi|<1$是AR(1)平稳的充要条件.

stationary condition for MA

eg: MA(1). $y_{t}=\theta_{1}\epsilon_{t-1}+\epsilon_{t}\rightarrow\begin{cases}E[y_{t}]=0\\V[y_{t}]=E[y_{t}^{2}]=E[\theta_{1}\varepsilon_{t-1}+\varepsilon_{t}]^{2}=\theta_{1}^{2}E[\varepsilon_{t-1}^{2}]+E[\varepsilon_{t}^{2}]+2\theta_{1}E[\epsilon_{t-1}\epsilon_{t}]=(1+\theta_{1}^{2})\sigma_{2}^{2},\forall\theta_{1}\\Cov[y_{t},y_{t-1}]=\theta_{1}\sigma_{\epsilon}^{2}, Cov[y_{t},y_{t-k}]=0,for\ k>1\end{cases}$.

可知V为有限常数, Cov也是有限常数. 所以MA(1)一定是stationary.

stationary condition for ARMA

ARMA的平稳性由其AR部分决定,因为MA部分永远平稳. 只要AR部分满足其平稳性条件, 则ARMA平稳. 具体为: $\phi(z)=0$中的解, 每个z的绝对值都要大于1.

Invertible

AR是一定可以转化为MA的, 而MA是否可以转化为AR由系数决定.

eg: MA(1): $y_{t}=\theta_{1}\epsilon_{t-1}+\epsilon_{t}=(1+\theta_{1}L)\epsilon_{t}\to 1+\theta_{1}z=0$, solution: $|z|>1$ for invertible.

ARMA的invertible由其MA的部分决定. 只要$\theta(z)=0$的每个解都满足$|z|>1$, 则ARMA invertible.

注意一下, 这个特征根的意思是: 将$y_{t}$化为关于$\epsilon_{t}$的函数, 然后其系数就是特征方程. 特征根就是代替L的字母. 如果出现复数, 那么其绝对值是实数部分和复数的norm的欧几里得距离.