定义13.1 (ARMA模型) 设\(\{\varepsilon_t\}\) 是 \(\text{WN}(0,\sigma^2)\), 实系数多项式\(A(z)\) 和 \(B(z)\)没有公共根, 满足 \(b_0=1,\ a_p b_q\neq 0\) 和 \[\begin{align} A(z)=& 1-\sum^{p}_{j=1} a_j z^j\neq 0 , |z| \leq 1, \\ B(z)=& \sum^{q}_{j=0} b_j z^j\neq 0 ,\; |z|< 1, \tag{13.1} \end{align}\] 就称差分方程： \[\begin{align} X_t = \sum^{p}_{j=1} a_j X_{t-j} + \sum^{q}_{j=0} b_j \varepsilon_{t-j}, \quad t \in \mathbb Z, \tag{13.2} \end{align}\] 是一个自回归滑动平均模型, 简称为ARMA(\(p,q\))模型. 称满足(13.2)的平稳序列\(\{X_t\}\)为平稳解或ARMA(\(p,q\))序列.

模型写成 \[\begin{align} A(\mathscr B) X_t = B(\mathscr B) \varepsilon_t, \quad t \in \mathbb Z \tag{13.3} \end{align}\] \(A^{-1}(z) B(z)\)在\(|z| < \rho\) 解析(\(1 < \rho < \min\{|z_j|\}\), \(\{z_j\}\)为\(A(z)\)的所有根)， 可以Taylor展开 \[\begin{align} \Psi(z) \stackrel{\triangle}{=}& A^{-1}(z) B(z) = \sum_{j=0}^\infty \psi_j z^j, \quad |z| \leq \rho \tag{13.4} \end{align}\]

易见\(\psi_j = o(\rho^{-j})\), \[ A^{-1}(\mathscr B) B(\mathscr B)\varepsilon_t = \Psi(\mathscr B)\varepsilon_t = \sum_{j=0}^\infty \psi_j \varepsilon_{t-j} \] 是线性平稳列。两边用\(A(\mathscr B)\)作用，根据7中线性滤波的逆的定理7.2知 \[\begin{aligned} A(\mathscr B) \Psi(\mathscr B)\varepsilon_t = A(\mathscr B) A^{-1}(\mathscr B) B(\mathscr B) \varepsilon_t = B(\mathscr B) \varepsilon_t \end{aligned}\] 即\(\Psi(\mathscr B)\varepsilon_t\)是ARMA(\(p,q\))模型(13.2)的解。

反之，若\(\{Y_t\}\)是(13.2)的一个平稳解， 在(13.2)两边作用\(A^{-1}(\mathscr B)\)即得 \[\begin{aligned} A^{-1}(\mathscr B) A(\mathscr B) Y_t = Y_t = A^{-1}(\mathscr B) B(\mathscr B) \varepsilon_t = \Psi(\mathscr B)\varepsilon_t \end{aligned}\] 即 \[\begin{align} X_t = A^{-1}(\mathscr B) B(\mathscr B) \varepsilon_t = \Psi(\mathscr B)\varepsilon_t \tag{13.5} \end{align}\] 是ARMA(\(p,q\))模型(13.2)的唯一平稳解。

称(13.5)中的\(\{\psi_j\}\)为\(\{X_t\}\)的Wold系数。

定理13.1 由(13.5)定义的平稳序列\(\{X_t\}\) 是 ARMA(\(p,q\))模型(13.2)的惟一平稳解.

模型(13.2)的任意解可以写成 \[\begin{align} Y_t = X_t + \sum_{j=1}^k \sum_{l=0}^{r(j)-1} V_{l,j} t^l \rho_j^{-t} \cos(\lambda_j t - \theta_{l,j}), t \in \mathbb Z, \tag{13.6} \end{align}\] 其中\(\{X_t\}\)为平稳解(13.5)， \(z_1,z_2,\dots,z_k\)为\(A(z)\)的全体互不相同的零点, \(z_j = \rho_j e^{i\lambda_j}\) 有重数\(r(j)\). 随机变量\(V_{j,l}, \theta_{l,j}\) 由\(Y_0-X_0,Y_1-X_1,\dots,Y_{p-1}-X_{p-1}\)惟一决定.

(13.6)中的\(\{ Y_t \}\)与\(\{X_t \}\)当\(t \to \infty\)时无限接近： \[\begin{align} |Y_t - X_t| \leq \sum_{j=1}^k \sum_{l=0}^{r(j)-1} |V_{l,j}| t^l \rho_j^{-t} , t \to \infty \tag{13.7} \end{align}\] 可以据此模拟ARMA模型: 取初值\(Y_{-(p-1)} = \dots = Y_{-1} = Y_0 = 0\)，递推得 \[\begin{aligned} Y_t = \sum_{j=1}^p a_j Y_{t-j} + \sum_{j=0}^q b_j \varepsilon_{t-j}, t=1,2,\dots, m+n \end{aligned}\] 当\(m\)较大时取后一段\(Y_t, t=m+1,m+2,\dots,m+n\)作为ARMA(\(p,q\))模型的模拟数据。 当\(A(z)\)有靠近单位圆的根时\(m\)要取得较大。

\(\{\gamma_k\}\)可由Wold系数表示: \[\begin{align} \gamma_k = \sigma^2 \sum_{j=0}^\infty \psi_j \psi_{j+k}, \quad k=0,1,2,\dots \tag{13.8} \end{align}\] 由于\(\psi_j = o(\rho^{-j}), j\to\infty\)， 由(13.8)可得\(\gamma_k = o(\rho^{-j}), j\to\infty\)。

记\(b_j=0, jq\), \(b_0=1\)； \(\psi_j=0, j0 \end{aligned}\]

类似AR模型可推导ARMA模型的Y-W方程: 在模型方程两边同乘以\(X_{t-k}\)求期望得 \[\begin{aligned} E(X_t X_{t-k}) =& \sum_{j=1}^p a_j E(X_{t-j} X_{t-k}) + \sum_{j=0}^q b_j E(\varepsilon_{t-j} X_{t-k})\\ \text{即} & \\ \gamma_k =& \sum_{j=1}^p a_j \gamma_{k-j} + \sum_{j=0}^q b_j E(\varepsilon_{t-j} \sum_{l=0}^\infty \psi_l \varepsilon_{t-k-l}) \\ =& \sum_{j=1}^p a_j \gamma_{k-j} + \sum_{j=0}^q b_j \psi_{j-k} \sigma^2, \quad k \in \mathbb Z \end{aligned}\]

当\(k>q\)时\(\psi_{j-k}=0, j=0,1,\dots,q\)，上式为 \[\begin{aligned} \gamma_k =& \sum_{j=1}^p a_j \gamma_{k-j}, \quad k \geq q+1 \end{aligned}\]

总之 \[\begin{align} \gamma_k - \sum_{j=1}^p a_j \gamma_{k-j} = \begin{cases} \sigma^2 \sum_{j=\max(0,k)}^q b_j \psi_{j-k}, \ & kq \end{cases} \tag{13.10} \end{align}\]

对\(k>q\)的Y-W方程可以写成矩阵形式： \[\begin{align} \left[ \begin{array}{c} \gamma_{q+1} \\ \gamma_{q+2} \\ \vdots \\ \gamma_{q+p} \end{array} \right] = \left[ \begin{array}{cccc} \gamma_{q} & \gamma_{q-1} & \cdots & \gamma_{q-p+1} \\ \gamma_{q+1} & \gamma_{q} & \cdots & \gamma_{q-p+2} \\ \vdots & \vdots & & \vdots \\ \gamma_{q+p-1} & \gamma_{q+p-2} & \cdots & \gamma_{q} \\ \end{array} \right] \left[ \begin{array}{c} a_1 \\ a_2 \\ \vdots \\ a_p \end{array} \right] \tag{13.11} \end{align}\]

对比AR(\(p\))的Y-W方程，相当于\(\Gamma_p\)的\((i,j)\)元素写成\(\gamma_{i-j}\)后 给所有\(\gamma_{\cdot}\)的下标都加上\(q\)。

把系数矩阵记为\(\Gamma_{p,q}\): \[\begin{aligned} \Gamma_{p,q} =&(\gamma_{|q+i-j|})_{i,j=1,2,\dots,p} \\ =& \left[ \begin{array}{cccc} \gamma_{q} & \gamma_{q-1} & \cdots & \gamma_{q-p+1} \\ \gamma_{q+1} & \gamma_{q} & \cdots & \gamma_{q-p+2} \\ \vdots & \vdots & & \vdots \\ \gamma_{q+p-1} & \gamma_{q+p-2} & \cdots & \gamma_{q} \\ \end{array} \right] \end{aligned}\] 只要\(\Gamma_{p,q}\)可逆则可解出\(a_1,\dots, a_p\)。

解出\(a_1,\dots,a_p\)后令 \[\begin{aligned} Y_t = A(\mathscr B) X_t = B(\mathscr B)\varepsilon_t, \quad t \in \mathbb Z \end{aligned}\] 则\(\{Y_t\}\)是一个MA(\(q\))序列，其自协方差函数为\(q\)步截尾，且 \[\begin{aligned} \gamma_y(k) =& E(Y_t Y_{t-k}) \\ =& \sum_{j=0}^p \sum_{l=0}^p \phi_j \phi_l E(X_{t-j} X_{t-l}) \\ =& \sum_{j=0}^p \sum_{l=0}^p \phi_j \phi_l \gamma_{k+l-j}, \quad 0\leq k \leq q \end{aligned}\] 可以用§12的方法唯一解出\(b_1,\dots, b_q, \sigma^2\)。

于是，只要\(\Gamma_{p,q}\)可逆, 则ARMA(\(p,q\))序列的自协方差函数和 ARMA(\(p,q\))模型的参数\((\boldsymbol{a}_p^T, \boldsymbol{b}_q^T, \sigma^2)\) 相互惟一决定。

如果\(\Gamma_{p,q}\)可逆则由(13.11)可以解出\(a_1, \dots, a_p\)。

定理13.2 设\(\{\gamma_k\}\)为ARMA(\(p,q\))序列\(\{X_t\}\) 的自协方差函数列,则\(m\geq p\)时\(\Gamma_{m,q}\)可逆。

证明:

用反证法然后由引理13.1导出矛盾。

设\(\Gamma_{m,q}\)(\(m\times m\)矩阵)不满秩， 则存在\(\boldsymbol{\beta}=(\beta_0,\beta_1, \dots, \beta_{m-1})^T \neq 0\)使得 \(\Gamma_{m,q} \boldsymbol{\beta}=0\)，即 \[\begin{align} \sum_{l=0}^{m-1} \beta_l \gamma_{q+k-l} = 0, \quad k=0,1,\dots, m-1 \tag{13.12} \end{align}\] 注意当\(k\geq m\)时\(q+k-l > q\)，所以这时 \(\gamma_{q+k-l} = \sum_{j=1}^p a_j \gamma_{q+k-l-j}\)， 所以取\(k=m\)有 \[\begin{aligned} \sum_{l=0}^{m-1} \beta_l \; \gamma_{q+k-l} =& \sum_{l=0}^{m-1} \beta_l \sum_{j=1}^p a_j \gamma_{q+k-l-j} \\ =& \sum_{j=1}^p a_j \sum_{l=0}^{m-1} \beta_l \; \gamma_{q+(k-j)-l} \\ =& 0 \quad (\text{由(13.12)及}0 \leq k-j = m-j \leq m-1) \end{aligned}\]

递推得上式当\(k>m\)时也成立。因此 \[\begin{aligned} \sum_{l=0}^{m-1} \beta_l \; \gamma_{q+k-l} = 0, \quad k \geq 0. \end{aligned}\]

令\(Y_t = \sum_{l=0}^{m-1} \beta_l X_{t-l}\)则\(\{Y_t\}\)是零均值平稳列， 利用 \[\begin{aligned} E(Y_t X_{t-q-k}) = \sum_{l=0}^{m-1} \beta_l \gamma_{q+k-l} = 0, \quad k \geq 0 \end{aligned}\] 可知\(\{Y_t\}\)的自协方差\(q-1\)步截尾: \[\begin{aligned} E(Y_t Y_{t-q-k}) = 0, \quad k \geq 0 \end{aligned}\] 所以\(\{Y_t\}\)是MA(\(q'\))(\(q' \leq q-1\))序列， 存在\(\{\eta_t\} \sim \text{WN}(0,s^2)\)使得 \[\begin{aligned} \sum_{l=0}^{m-1} \beta_l X_{t-l} = \sum_{j=0}^{q'} d_j \eta_{t-j} \end{aligned}\] 与引理13.1矛盾。

○○○○○○

定理13.3 设零均值平稳序列\(\{X_t\}\)有自协方差函数\(\{\gamma_k\}\). 又设实数\(a_1,a_2,\cdots,a_p\) \((a_p\neq 0)\) 使得\(A(z)=1-\sum_{j=1}^p a_j z^j\) 满足最小相位条件, 另外 \[\begin{align} \gamma_k - \sum^{p}_{j=1} a_j\gamma_{k-j}=\begin{cases} c \neq 0 , \ & k=q, \\ 0, & k>q, \end{cases} \tag{13.13} \end{align}\] 则\(\{X_t\}\)是一个ARMA\((p',q')\)序列, 其中\(p'\leq p , \ q' \leq q\). \

证明: 设\(Y_t=A(\mathscr B) X_t = X_t-\sum^{p}_{j=1}a_j X_{t-j}\). 则\(\{Y_t\}\)是零均值平稳序列, 满足 \[ E(Y_t X_{t-k})=\gamma_k -\sum^{p}_{j=1} a_j \gamma_{k-j} =\begin{cases} c \neq 0, & k=q ,\\ 0, & k>q. \end{cases} \] 所以有 \[\begin{aligned} \gamma_y(k) =& E(Y_t Y_{t-k}) =E\big [ Y_t (X_{t-k} - \sum_{j=1}^p a_j X_{t-k-j}) \big]\\ =& \begin{cases} c \neq 0, \ & k=q ,\\ 0, & k>q. \end{cases} \end{aligned}\] 说明\(\{Y_t\}\)的自协方差函数是\(q\)后截尾的.

由定理12.2知道, \(\{Y_t\}\)为一个MA\((q)\)序列, 即存在单位圆内没有根的\(q\)阶实系数多项式\(B(z)\)使得\(B(0)=b_0=1\) 和 \[\begin{align} A(\mathscr B) X_{t}=Y_t = B(\mathscr B)\varepsilon_t, \quad t\in \mathbb Z, \tag{13.14} \end{align}\] 其中\(\{\varepsilon_t\}\) 是 \(\text{WN}(0,\sigma^2)\).

如果\(A(z)\)和\(B(z)\)没有公因子, 上述模型就是所需要的ARMA(\(p,q\))模型. 否则设公因子是\(C(z)\), 则有 \(A(z)=C(z)A'(z)\), \(B(z)=C(z)B'(z).\) 这时(13.14)变成 \[ C(\mathscr B)A'(\mathscr B)X_t =C(\mathscr B)B'(\mathscr B)\varepsilon_t. \] 两边乘\(C^{-1}(\mathscr B)\)(显然\(C(z)\)也满足最小相位条件) 后得到所需ARMA(\(p',q'\))模型: \[ A'(\mathscr B)X_t =B'(\mathscr B)\varepsilon_t. \]

○○○○○○

由于ARMA序列的\(\{\gamma_k\}\)绝对可和，以及平稳解的线性序列表达式， 可得ARMA(\(p,q\))序列(2.6)有谱密度 \[\begin{align} f(\lambda) =& \frac{1}{2\pi} \sum_{k=-\infty}^\infty \gamma_k e^{-ik\lambda} \\ =& \frac{\sigma^2}{2\pi} \left| \sum_{j=0}^\infty \psi_j e^{ij\lambda} \right|^2 = \frac{\sigma^2}{2\pi} \left| \frac{B(e^{i\lambda})}{A(e^{i\lambda})} \right|^2 \tag{13.15} \end{align}\] 形如(13.15)的谱密度被称为有理谱密度.

定义13.2 在ARMA\((p,q)\)模型的定义 13.1 中, 如果进一步要求\(B(z)\)在单位圆上无根: \[\begin{align} B(z)=1 + \sum^{q}_{j=1} b_j z^j \neq 0, |z|\leq 1 \tag{13.16} \end{align}\] 则称ARMA(\(p,q\))模型(13.16)为可逆的ARMA模型, 称相应的平稳解为可逆的ARMA(\(p,q\))序列.

从定理12.3(最小序列的谱密度条件）知道可逆的ARMA(\(p,q\))序列是最小序列.

对于可逆的ARMA(\(p,q\))模型(13.16)， 由于\(B^{-1}(z)A(z)\)在\(\{z: |z|\leq\rho\}\) \((\rho>1)\) 内解析, 所以有Taylor展式： \[\begin{align} B^{-1}(z)A(z)=\sum^{\infty}_{j=0} \varphi_j z^j,\ \ |z|\leq\rho, \tag{13.17} \end{align}\] 其中\(|\varphi_j|=o(\rho^{-j})\),当\(j\to\infty\),从而可以定义 \(B^{-1}(\mathscr B)A(\mathscr B)=\sum^{\infty}_{j=0}\varphi_j \mathscr B^j\). 在(13.17)两边乘以\(B^{-1}(\mathscr B)\), 得到： \[\begin{align} \varepsilon_t =B^{-1}(\mathscr B)A(\mathscr B)X_t =\sum^{\infty}_{j=0} \varphi_j X_{t-j}, \ \ t\in \mathbb Z. \tag{13.18} \end{align}\] (13.18)是(13.5)的逆转形式, 表明可逆ARMA(\(p,q\))序列和它的噪声序列可以相互线性表示.

○○○○○○

先给出一些有关的自定义R函数。