28 多元平稳序列介绍

28.1 多维平稳序列的概念

沿时间变化的量经常有多个，互相之间有相关性。如第1章的北京地区洪涝灾害受灾面积$X_t$和成灾面积$Y_t$有正相关，可以看成向量值时间序列 \[\begin{aligned} (X_t, Y_t), \ t \in \mathbb N \end{aligned}\]

向量值的时间序列称为多维（多元）时间序列。仅介绍多维平稳序列。

定义28.1 (多维平稳序列) 称$m$维随机序列 $\boldsymbol X_t = (X_{1t}, X_{2t}, \dots, X_{mt})^T$, $t \in \mathbb Z$是平稳序列，如果对任何$t, n \in \mathbb Z$,

(1) $E \boldsymbol X_t = \boldsymbol \mu = (\mu_1, \mu_2, \dots, \mu_m)^T$与$t$无关；
(2) $\Gamma(n) = E[(\boldsymbol X_{t+n} - \boldsymbol \mu)(\boldsymbol X_t - \boldsymbol \mu)^T]$ 与$t$无关。

这时称$\{\Gamma(n), n \in \mathbb Z \}$为平稳序列$\{ \boldsymbol X_t \}$的自协方差函数（矩阵）。

定义 \[\begin{align} \gamma_{jk}(n) =& E[(X_{j,t+n} - \mu_j) (X_{kt} - \mu_k)] \tag{28.1} \end{align}\] 则 \[\begin{align} \Gamma(n) =& (\gamma_{jk}(n))_{j,k=1,2,\dots,m}. \tag{28.2} \end{align}\]

定义自相关系数 \[\begin{align} \rho_{jk}(n) = \frac{\gamma_{jk}(n)}{\sqrt{ \gamma_{jj}(0) \gamma_{kk}(0) }} \tag{28.3} \end{align}\] 称 \[\begin{aligned} R(n) = (\rho_{jk}(n))_{j,k=1,2,\dots,m} \end{aligned}\] 为$\{ \boldsymbol X_t \}$的自相关系数（矩阵）。

把$\{ \boldsymbol X_t \}$标准化得 \[\begin{aligned} \boldsymbol Y_t =& \left( \frac{X_{1t} - \mu_1}{\sqrt{\gamma_{11}(0)}}, \frac{X_{2t} - \mu_2}{\sqrt{\gamma_{22}(0)}}, \dots, \frac{X_{mt} - \mu_m}{\sqrt{\gamma_{mm}(0)}} \right)^T \end{aligned}\] 则$\{ R(n) \}$是平稳序列$\{ \boldsymbol Y_t \}$的自协方差函数。

定理28.1 对任何$n \in \mathbb Z$,

(1) $\Gamma(-n) = [\Gamma(n)]^T$;
(2) $|\gamma_{jk}(n)| \leq [ \gamma_{jj}(0) \gamma_{kk}(0) ]^{1/2}$;
(3) 非负定性：对任何实向量 $\boldsymbol\alpha_1, \boldsymbol\alpha_2, \dots, \boldsymbol\alpha_n \in \mathbb R^m$, \[\begin{aligned} \sum_{k=1}^n \sum_{j=1}^n \boldsymbol\alpha_k^T \Gamma(k-j) \boldsymbol\alpha_j \geq 0. \end{aligned}\]

这些性质$\{R(n)\}$也满足。

证明: (1) \[\begin{aligned} \Gamma(-n) =& E\left[(\boldsymbol X_{t-n} - \boldsymbol \mu)(\boldsymbol X_t - \boldsymbol \mu)^T\right] \\ =& \left\{ E \left[ (\boldsymbol X_t - \boldsymbol \mu) (\boldsymbol X_{t-n} - \boldsymbol \mu)^T \right]\right\}^T \\ =& \left[ \Gamma(n) \right]^T \end{aligned}\]

(2) 这就是Schwarz不等式。

(3) 记$\boldsymbol Y_t = \boldsymbol X_t - \boldsymbol \mu$, 则 \[\begin{aligned} & E \left[ \sum_{j=1}^n \boldsymbol\alpha_j^T \boldsymbol Y_j \right]^2 \\ =& E \sum_{k=1}^n \sum_{j=1}^n (\alpha_k^T \boldsymbol Y_k) (\alpha_j^T \boldsymbol Y_j) \\ =& \sum_{k=1}^n \sum_{j=1}^n \boldsymbol\alpha_k^T E[\boldsymbol Y_k \boldsymbol Y_j^T] \boldsymbol\alpha_j \\ = & \sum_{k=1}^n \sum_{j=1}^n \boldsymbol\alpha_k^T \Gamma(k-j) \boldsymbol\alpha_j \geq 0 \end{aligned}\]

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例28.1 (多维白噪声) 若$m$维平稳序列$\{\boldsymbol X_t \}$满足 \[\begin{aligned} E \boldsymbol X_1 = \boldsymbol \mu, \quad \Gamma(n) = Q \delta_n \end{aligned}\] 就称$\{\boldsymbol X_t \}$是$m$维白噪声，简记为WN$(\boldsymbol\mu, Q)$。

这时$\forall k, j$, 只要$n \neq 0$，就有 \[\begin{aligned} E[(X_{kt} - \mu_k)(X_{j,t+n} - \mu_j)] = 0 \end{aligned}\] 另外，每个分量是一维白噪声。特别地，当$Q$是对角阵时，各分量是$m$个互不相关的白噪声。

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例28.2 (多维线性平稳列) 设$\{ \boldsymbol\varepsilon_t, t\in \mathbb Z \}$是$m$维WN(0,$Q$), $Q = E[\boldsymbol\varepsilon_t \boldsymbol\varepsilon_t^T]$。如果实矩阵列$\{ C_j \}$满足 \[\begin{aligned} \sum_{j=-\infty}^\infty C_j Q C_j^T < \infty \end{aligned}\] 就称 \[\begin{align} \boldsymbol X_t =& \sum_{j=-\infty}^\infty C_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \tag{28.4} \end{align}\] 为$m$维平稳线性序列。

这时可以证明(28.4)中每个分量都是均方收敛的，并且 \[\begin{align} E \boldsymbol X_t =& 0, \\ \Gamma(n) =& E[\boldsymbol X_{t+n} \boldsymbol X_t^T] = \sum_{j=-\infty}^\infty C_{j+n} Q C_j^T \tag{28.5} \end{align}\]

例28.3 (多维MA模型) 设$\{ \boldsymbol\varepsilon_t \}$是$m$维WN(0,$Q$)。如果$B_1, B_2, \dots, B_q$是$m \times m$实矩阵，满足 \[\begin{aligned} \mbox{det}(I_m + B_1 z + B_2 z^2 + \dots + B_q z^q) \neq 0, \ |z| < 1 \end{aligned}\] 就称 \[\begin{aligned} \boldsymbol X_t = \boldsymbol\varepsilon_t + B_1 \boldsymbol\varepsilon_{t-1} + \dots + B_q \boldsymbol\varepsilon_{t-q}, \ t \in \mathbb Z \end{aligned}\] 为一个$m$维MA($q$)序列。

若进一步要求 \[\begin{aligned} \mbox{det}(I_m + B_1 z + B_2 z^2 + \dots + B_q z^q) \neq 0, \ |z| \leq 1 \end{aligned}\] 则称$\{ \boldsymbol X_t \}$为一个可逆的MA($q$)序列。

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矩阵系数多项式：

设$A_1, A_2, \dots, A_p, B_1, B_2, \dots, B_q$是$p+q$个$m\times m$实矩阵，记 \[\begin{aligned} A(z) =& I_m - A_1 z - A_2 z^2 - \dots - A_p z^p \\ B(z) =& I_m + B_1 z + B_2 z^2 + \dots + B_q z^q \end{aligned}\] 形如$A(z), B(z)$的多项式被称为矩阵系数多项式，实际是$m^2$维向量值函数，每个分量是多项式。

如果对任意满足 \[\begin{aligned} A(z)= C(z) A_1(z), \ B(z) = C(z) B_1(z) \end{aligned}\] 的$m \times m$矩阵系数多项式$C(z)$必有$\mbox{det}(C(Z))=$常数，就称$A(z)$和$B(z)$是左互素的。

例28.4 (多维ARMA的不可识别性) 设$\{ \boldsymbol\varepsilon_t \}$是WN(0,$Q$)。称$m$维平稳序列$\{ \boldsymbol X_t \}$满足$m$维平稳可逆的ARMA($p,q$)模型，如果对任何$t \in \mathbb Z$， \[\begin{align} \boldsymbol X_t = \sum_{j=1}^p A_j \boldsymbol X_{t-j} + \varepsilon_t + \sum_{j=1}^q B_j \varepsilon_{t-j} \tag{28.6} \end{align}\] 其中多项式$A(z), B(z)$满足 \[\begin{aligned} (1) & A(z), B(z) \text{左互素}; \\ (2) & \mbox{det}(A(z)B(z)) \neq 0, \ |z| \leq 1. \end{aligned}\]

用推移算子把模型方程写成 \[\begin{align} A(\mathscr B) \boldsymbol X_t = B(\mathscr B) \boldsymbol\varepsilon_t, \ t \in \mathbb Z \tag{28.7} \end{align}\]

对于一维ARMA，自协方差函数可以唯一决定模型参数；对于多维ARMA不能。

在模型(28.7)中，如果$\mbox{det}(A(z)) = c$是常数，则$A^{-1}(z)$仍然是一个矩阵多项式，并且 $\mbox{det}(A^{-1}(z))=c^{-1}$。于是 \[\begin{aligned} \boldsymbol X_t = A^{-1}(\mathscr B) B(\mathscr B) \varepsilon_t = B_1(\mathscr B) \varepsilon_t, \ t \in \mathbb Z \end{aligned}\] 其中$B_1(z)=A^{-1}(z)B(z)$是矩阵系数多项式，满足 \[\begin{aligned} \mbox{det}(B_1(z)) = c^{-1} \mbox{det}(B(z)) \neq 0, \ |z| \leq 1 \end{aligned}\] 就有了两个模型。所以ARMA模型参数是不唯一的。

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28.2 多维平稳序列的均值和自协方差函数的估计

28.2.1 均值的估计

设$\{ \boldsymbol X_t \}$是$m$维平稳序列, $\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_N$是观测值。均值$\boldsymbol\mu$的估计为 \[\begin{aligned} \hat{\boldsymbol\mu}_N = (\hat\mu_1, \hat\mu_2, \dots, \hat\mu_m)^T = \frac{1}{N} \sum_{t=1}^N \boldsymbol X_t. \end{aligned}\]

由分量的相应结论可得：

定理28.2 如果$\{ \boldsymbol X_t \}$的每个分量序列 $\{ X_{jt}: t\in\mathbb Z \}$都是严平稳遍历序列，则当$N\to\infty$时， \[\begin{aligned} \hat{\boldsymbol\mu}_N \to \boldsymbol\mu, \ \text{a.s.} \end{aligned}\]

定理28.3 如果自协方差函数$\Gamma(n) \to 0$, 当$n \to\infty$，则 \[\begin{aligned} E|\hat{\boldsymbol\mu}_N - \hat{\boldsymbol\mu}|^2 \to 0, \ \text{当$N\to\infty$时} \end{aligned}\] 其中$|\hat{\boldsymbol\mu}_N - {\boldsymbol\mu}|^2 = \sum_{j=1}^m (\hat\mu_j - \mu_j)^2$。

这是均方收敛，推出依概率收敛，即$\hat{\boldsymbol\mu}_N$是$\boldsymbol\mu$的相合估计。

证明： \[\begin{aligned} E |\hat{\boldsymbol\mu}_N - \hat{\boldsymbol\mu}|^2 =& \sum_{j=1}^m E (\hat\mu_j - \mu_j)^2 \\ =& \sum_{j=1}^m E \left( \frac{1}{N} \sum_{t=1}^N X_{jt} - \mu_j \right)^2 \\ =& \sum_{j=1}^m \frac{1}{N^2} E \left( \sum_{t=1}^N (X_{jt} - \mu_j) \right)^2 \\ =& \sum_{j=1}^m \frac{1}{N^2} \sum_{l=1}^N \sum_{k=1}^N \gamma_{jj}(l-k) \\ =& \sum_{j=1}^m \frac{1}{N^2} \sum_{k=1-N}^{N-1}(N - |k|) \gamma_{jj}(k) \\ \leq& \sum_{j=1}^m \frac{1}{N} \sum_{k=1-N}^{N-1} |\gamma_{jj}(k)| \to 0, \ (N\to\infty) \end{aligned}\]

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定理28.4 如果 \[\begin{aligned} \sum_{k=0}^\infty | \gamma_{jj}(k) | < \infty, \ j=1,2,\dots,m \end{aligned}\] 则当$N\to\infty$时 \[\begin{aligned} N E | \hat{\boldsymbol\mu}_N - \boldsymbol\mu |^2 \to \sum_{j=1}^m \sum_{k=-\infty}^\infty \gamma_{jj}(k). \end{aligned}\]

证明按定理28.3的证明，有 \[\begin{aligned} & N E | \hat{\boldsymbol\mu}_N - \boldsymbol\mu |^2 \\ =& \sum_{j=1}^m \frac{1}{N} \sum_{k=1-N}^{N-1}(N - |k|) \gamma_{jj}(k) \\ =& \sum_{j=1}^m \sum_{k=1-N}^{N-1} \gamma_{jj}(k) - \sum_{j=1}^m \frac{1}{N} \sum_{k=1-N}^{N-1} |k| \gamma_{jj}(k) \\ \to& \sum_{j=1}^m \sum_{k=-\infty}^{\infty} \gamma_{jj}(k). \end{aligned}\]

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定理28.5 设$\{\boldsymbol\varepsilon_t \}$是$m$维独立同分布的 WN($\boldsymbol 0, Q$)， $m \times m$矩阵$C_n = (c_{j,k}(n))$的每个元素$\{c_{j,k}(n)\}$对$n\in\mathbb Z$绝对可和， $m$维平稳序列$\{ \boldsymbol X_t \}$为线性平稳列 \[\begin{aligned} \boldsymbol X_t =& \boldsymbol\mu + \sum_{j=-\infty}^\infty C_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \end{aligned}\] 如果 \[\begin{aligned} \Sigma =& \left( \sum_{j=-\infty}^\infty C_j \right) Q \left( \sum_{j=-\infty}^\infty C_j^T \right) \neq 0 \end{aligned}\] 则 \[\begin{aligned} \sqrt{N}(\hat{\boldsymbol\mu}_N - \boldsymbol\mu) \stackrel{\mbox{d}}{\longrightarrow} \text{N}(0,\Sigma) \end{aligned}\]

见(Brockwell and Davis 1987)。

28.2.2 自协方差函数的估计

设$\{ \boldsymbol X_t \}$是$m$维平稳序列, $\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_N$是观测值。自协方差函数$\Gamma(n)$的估计为 \[\begin{aligned} \begin{cases} \hat\Gamma(n) = \frac{1}{N} \sum_{t=1}^{N-n} ( \boldsymbol X_{t+n} - \hat{\boldsymbol\mu}_N) ( \boldsymbol X_{t} - \hat{\boldsymbol\mu}_N)^T, & 0 \leq n \leq N-1 \\ \hat\Gamma(-n) = \hat\Gamma^T(n), & 1 \leq n \leq N-1 \end{cases} \end{aligned}\] 是一维情况的推广，具有良好统计性质。

若$\hat\gamma_{jk}(n)$为$\hat\Gamma(n)$的第$(j,k)$元素，则相关系数估计为 \[\begin{align} \hat\rho_{jk}(n) = \frac{\hat\gamma_{jk}(n)} {\sqrt{\hat\gamma_{jj}(0) \hat\gamma_{kk}(0)}} \tag{28.8} \end{align}\] 自相关系数矩阵的估计为 \[\begin{align} \hat R(n) = (\hat\rho_{jk}(n)). \tag{28.9} \end{align}\]

利用严平稳序列的遍历定理可得：

定理28.6 在定理28.5的条件下，对固定的$n$，当$N\to\infty$时 \[\begin{aligned} \hat\Gamma(n) \to \Gamma(n),\ \text{a.s.}, \quad \hat R(n) \to R(n), \ \text{a.s.} \end{aligned}\]

关于$\hat R(n)$的渐近分布有(见(Brockwell and Davis 1987)定理11.2.2)：

定理28.7 设$\{ Z_{1t} \}$和$\{ Z_{2t} \}$都是一维独立同分布的零均值白噪声，彼此相互独立， $\{a_j \}$和$\{ b_j \}$是绝对可和的实数列， \[\begin{aligned} X_{1t} =& \sum_{j=-\infty}^\infty a_j Z_{1,t-j}, \ t \in \mathbb N_+, \\ X_{2t} =& \sum_{j=-\infty}^\infty b_j Z_{2,t-j}, \ t \in \mathbb N_+, \end{aligned}\] 则

(1) 对$k \geq 0$, 当$N \to \infty$时， \[\begin{aligned} \sqrt{N}\hat\rho_{12}(k) \stackrel{\text{d}}{\longrightarrow} \text{N}(0,\sigma_{11}) \end{aligned}\]
(2) 对$h, k \geq 0$且$h \neq k$， \[\begin{aligned} \sqrt{N}( \hat\rho_{12}(h), \hat\rho_{12}(k)) \stackrel{\text{d}}{\longrightarrow} \text{N}(0, \Sigma) \end{aligned}\] 其中 \[\begin{aligned} \Sigma =& \left(\begin{array}{cc} \sigma_{11} & \sigma_{12} \\ \sigma_{12} & \sigma_{11} \end{array}\right) \\ \sigma_{11} =& \sum_{j=-\infty}^\infty \rho_{11}(j) \rho_{22}(j) \\ \sigma_{12} =& \sum_{j=-\infty}^\infty \rho_{11}(j) \rho_{22}(j+k-h). \end{aligned}\]

28.3 VAR模型

定义28.2 设$\{ \boldsymbol\varepsilon_t \}$是$m$维WN(0,$Q$), $A_1, A_2, \dots, A_p$是$m \times m$实矩阵，使得 \[\begin{align} \det\left( I_m - \sum_{j=1}^p A_j z^j \right) \neq 0, \ |z| \leq 1 \tag{28.10} \end{align}\] 称如下的模型 \[\begin{align} \boldsymbol X_t = \sum_{j=1}^p A_j \boldsymbol X_{t-j} + \boldsymbol\varepsilon_t, \ t \in \mathbb Z \tag{28.11} \end{align}\] 是一个$m$维AR($p$)模型；如果平稳序列$\{ \boldsymbol X_t \}$满足$m$维AR($p$)模型，就称$\{ \boldsymbol X_t \}$是一个$m$维的AR($p$)序列。

记 \[\begin{aligned} A(z) = I_m - \sum_{j=1}^p A_j z^j \end{aligned}\] 模型为 \[\begin{align} A(\mathscr B) \boldsymbol X_t = \boldsymbol\varepsilon_t, \ t \in \mathbb Z \tag{28.12} \end{align}\] 用$A(z)$的伴随矩阵表示$A^{-1}(z)$后，可知$A^{-1}(z)$的每个元素在$|z| \leq 1$有泰勒展式，于是 \[\begin{align} A^{-1}(z) = \sum_{j=0}^\infty C_j z^j, \ |z| \leq 1. \tag{28.13} \end{align}\] 其中$C_n=(c_{jk}(n))$, $\{c_{jk}(n), n \in \mathbb N_+ \}$以负指数速度趋于零。类似一维情况可得 \[\begin{align} \boldsymbol X_t =& A^{-1}(\mathscr B) \boldsymbol\varepsilon_t = \sum_{j=0}^\infty C_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \tag{28.14} \end{align}\] 是模型(28.11)的唯一平稳解。

例28.5 (一维AR的马氏化) 将一维AR($p$)序列表示成VAR。

一维AR($p$) \[\begin{align} X_t =& \sum_{j=1}^p a_j X_{t-j} + \varepsilon_t, \ t \in \mathbb Z, \tag{28.15} \\ \{ \varepsilon_t \} & \text{为WN}(0,\sigma^2) \end{align}\] 令 \[\begin{aligned} \boldsymbol X_t =& (X_t, X_{t-1}, \dots, X_{t-p+1})^T \\ \boldsymbol \varepsilon_t =& (\varepsilon_t, 0, \dots, 0)^T \\ A =& \left(\begin{array}{ccccc} a_1 & a_2 & \cdots & a_{p-1} & a_p \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 \end{array}\right) \end{aligned}\] 则得到$p$维AR(1)模型： \[\begin{align} \boldsymbol X_t = A \boldsymbol X_{t-1} + \boldsymbol\varepsilon_t, \ t \in \mathbb Z. \tag{28.16} \\ \end{align}\] 其中 \[\begin{aligned} & \det(I_p - Az) \\ =& \det\left(\begin{array}{ccccc} 1-a_1z & -a_2z & \cdots & -a_{p-1}z & -a_pz \\ -z & 1 & \cdots & 0 & 0 \\ 0 & -z & \cdots & 0 & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & -z & 1 \end{array}\right) \\ =& 1 - a_1 z - a_2 z^2 - \dots - a_p z^p \neq 0, \ |z| \leq 1. \end{aligned}\] 计算上面行列式时，按$j=2, 3, \dots, p$的顺序把第$j$列乘以$z^{j-1}$后加到第一列，则第一列只有$(1,1)$元素非零。

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28.3.1 多维AR序列的自协方差函数

由平稳解(28.14)可知 \[\begin{aligned} E[ \boldsymbol X_t \boldsymbol\varepsilon_{t+n}^T] = 0, n \geq 1. \end{aligned}\] 这体现出模型有因果性。于是对$n\geq 0$ \[\begin{align} \Gamma(n) =& E(\boldsymbol X_{t+n} \boldsymbol X_t^T) \\ =& E \left[ \left( \sum_{j=1}^p A_j \boldsymbol X_{t+n-j} + \boldsymbol\varepsilon_{t+n} \right) \boldsymbol X_t^T \right] \\ =& \sum_{j=1}^p A_j \Gamma(n-j) + E(\boldsymbol\varepsilon_{t+n} \boldsymbol X_t^T) \tag{28.17} \end{align}\] 从而 \[\begin{align} A(\mathscr B) \Gamma(n) = 0, \ n \geq 1. \tag{28.18} \end{align}\]

另外 \[\begin{align} E(\boldsymbol\varepsilon_t \boldsymbol X_t^T) = E \left( \sum_{j=0}^\infty \boldsymbol\varepsilon_t \boldsymbol\varepsilon_{t-j}^T C_j^T \right) = Q C_0^T = Q \tag{28.19} \end{align}\]

总之有$m$维情况下自协方差函数矩阵的Yule-Walker方程 \[\begin{align} \begin{cases} \Gamma(0) = \sum_{j=1}^p A_j \Gamma(-j) + Q, &\\ \Gamma(n) = \sum_{j=1}^p A_j \Gamma(n-j), & n \geq 1 \\ \end{cases} \tag{28.20} \end{align}\]

把(28.20)的第二式写成分块矩阵形式： \[\begin{align} \left(\begin{array}{c} \Gamma^T(1) \\ \Gamma^T(2) \\ \vdots \\ \Gamma^T(n) \end{array}\right) = \left(\begin{array}{cccc} \Gamma(0) & \Gamma(1) & \cdots & \Gamma(p-1) \\ \Gamma(-1) & \Gamma(0) & \cdots & \Gamma(p-2) \\ \vdots & \vdots & & \vdots \\ \Gamma(-p+1) & \Gamma(-p+2) & \cdots & \Gamma(0) \end{array}\right) \left(\begin{array}{c} A_1^T \\ A_2^T \\ \vdots \\ A_p^T \end{array}\right) \tag{28.21} \end{align}\] 系数矩阵是向量 \[\begin{aligned} (\boldsymbol X_p^T, \boldsymbol X_{p-1}^T, \dots, \boldsymbol X_1^T)^T \end{aligned}\] 的协方差矩阵，如果它正定则$A_1, A_2, \dots, A_p$和$Q$可以由 \[\begin{aligned} \Gamma(0), \Gamma(1), \dots, \Gamma(p) \end{aligned}\] 唯一决定，且满足最小相位条件(28.10)。

28.3.2 多维AR序列的参数估计

28.3.2.1 Y-W方法

在Y-W方程(28.20)中，把$\Gamma(n)$用样本自协方差函数$\hat\Gamma(n)$代替，得到样本Y-W方程，可解得模型的Y-W估计 \[\begin{aligned} (\hat A_1, \hat A_2, \dots, \hat A_p, \hat Q). \end{aligned}\] 解法有类似一维时Levinson递推那样的递推公式，见(谢衷洁 1990)。只要Y-W方程中的系数矩阵正定，则Y-W估计满足最小相位条件(28.10)。

28.3.2.2 最小二乘

把观测值$\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_N$满足的模型写成 \[\begin{align} \boldsymbol X_t^T = \sum_{j=1}^p \boldsymbol X_{t-j}^T A_j^T + \boldsymbol\varepsilon_t^T, \ t=p+1, p+2, \dots, N \tag{28.22} \end{align}\] 引入 \[\begin{aligned} \boldsymbol X_n =& \left(\begin{array}{cccc} \boldsymbol X_p^T & \boldsymbol X_{p-1}^T & \cdots & \boldsymbol X_1^T \\ \boldsymbol X_{p+1}^T & \boldsymbol X_p^T & \cdots & \boldsymbol X_2^T \\ \vdots & \vdots & & \vdots \\ \boldsymbol X_{n-1}^T & \boldsymbol X_{n-2}^T & \cdots & \boldsymbol X_{n-p} \end{array}\right)_{(n-p)\times mp} \\ \boldsymbol Y_n =& \left(\begin{array}{c} \boldsymbol X_{p+1}^T \\ \boldsymbol X_{p+2}^T \\ \vdots \\ \boldsymbol X_n^T \end{array}\right)_{(n-p)\times m} \ \boldsymbol E_n = \left(\begin{array}{c} \boldsymbol \varepsilon_{p+1}^T \\ \boldsymbol \varepsilon_{p+1}^T \\ \vdots \\ \boldsymbol \varepsilon_n^T \end{array}\right)_{(n-p)\times m} \\ \boldsymbol A =& \left(\begin{array}{c} \boldsymbol A_1^T \\ \boldsymbol A_2^T \\ \vdots \\ \boldsymbol X_p^T \end{array}\right)_{mp\times m} \end{aligned}\] 可以把(28.22)写成 \[\begin{aligned} \boldsymbol Y_n = \boldsymbol X_n \boldsymbol A + \boldsymbol E_n \end{aligned}\] 求$A_1, A_2, \dots, A_p$的最小二乘估计$\hat{\boldsymbol A}$，就是求$\hat{\boldsymbol A}$使得 \[\begin{aligned} S(\boldsymbol A) = (\boldsymbol Y_n - \boldsymbol X_n \boldsymbol A)^T (\boldsymbol Y_n - \boldsymbol X_n \boldsymbol A) \end{aligned}\] 最小。这里$S(\boldsymbol A)$是$m \times m$矩阵。对两个对称矩阵$A$和$B$，当且仅当$A-B$非负定且不等于零矩阵时称$A>B$。 $\hat{\boldsymbol A}$满足方程 \[\begin{aligned} (\boldsymbol X_n^T \boldsymbol X_n) \hat{\boldsymbol A} = \boldsymbol X_n^T \boldsymbol Y_n \end{aligned}\] 当$(\boldsymbol X_n^T \boldsymbol X_n)$可逆时 \[\begin{aligned} \hat{\boldsymbol A} = (\boldsymbol X_n^T \boldsymbol X_n)^{-1} \boldsymbol X_n^T \boldsymbol Y_n \end{aligned}\]

事实上，若$\hat{\boldsymbol A}$满足下式 \[\begin{aligned} (\boldsymbol X_n^T \boldsymbol X_n) \hat{\boldsymbol A} = \boldsymbol X_n^T \boldsymbol Y_n \end{aligned}\] 则对任何与$\hat{\boldsymbol A}$同阶的$\boldsymbol B$，有 \[\begin{aligned} & (\boldsymbol Y_n - \boldsymbol X_n \boldsymbol B)^T(\boldsymbol Y_n - \boldsymbol X_n \boldsymbol B) \\ =& (\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A} + \boldsymbol X_n \hat{\boldsymbol A} - \boldsymbol X_n \boldsymbol B )^T \\ & \cdot (\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A} + \boldsymbol X_n \hat{\boldsymbol A} - \boldsymbol X_n \boldsymbol B ) \\ =& (\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A})^T(\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A}) + (\boldsymbol X_n (\hat{\boldsymbol A} - \boldsymbol B))^T (\boldsymbol X_n (\hat{\boldsymbol A} - \boldsymbol B)) \\ \geq& (\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A})^T(\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A}) \end{aligned}\]

28.3.3 VAR的预测

例28.5证明了一维AR($p$)模型可以化为$p$维AR(1)模型。进一步地，任何一个$m$维AR($p$)模型可以写成一个$mp$维的AR(1)模型。考虑$m$维AR($p$)模型 \[\begin{align} \boldsymbol X_t = \sum_{j=1}^p A_j \boldsymbol X_{t-j} + \boldsymbol\varepsilon_t, \ t \in \mathbb Z \tag{28.23} \end{align}\] 记 \[\begin{aligned} \boldsymbol Y_t =& \left(\begin{array}{c} \boldsymbol X_t \\ \boldsymbol X_{t-1} \\ \vdots \\ \boldsymbol X_{t-p+1} \end{array}\right)_{mp \times 1}, \quad \boldsymbol \eta_t = \left(\begin{array}{c} \boldsymbol \varepsilon_t \\ \boldsymbol 0 \\ \vdots \\ \boldsymbol 0 \end{array}\right)_{mp \times 1} \\ \boldsymbol A =& \left(\begin{array}{ccccc} A_1 & A_2 & \cdots & A_{p-1} & A_p \\ I_m & \boldsymbol 0 & \cdots & \boldsymbol 0 & \boldsymbol 0 \\ \boldsymbol 0 & I_m & \cdots & \boldsymbol 0 & \boldsymbol 0 \\ \vdots & \vdots & & \vdots & \vdots \\ \boldsymbol 0 & \boldsymbol 0 & \cdots & I_m & \boldsymbol 0 \end{array}\right)_{mp \times mp} \end{aligned}\] 则$m$维AR($p$)模型(28.23)可以写成$mp$维AR(1): \[\begin{aligned} \boldsymbol Y_t = \boldsymbol A \boldsymbol Y_{t-1} + \boldsymbol \eta_t, \ t \in \mathbb Z \end{aligned}\] 其中，系数矩阵$\boldsymbol A$满足 \[\begin{align} & \det(I_p - \boldsymbol A z) \\ =& \det\left(\begin{array}{ccccc} I_p - A_1z & -A_2z & \cdots & -A_{p-1}z & -A_pz \\ -I_p z & I_p & \cdots & 0 & 0 \\ 0 & -I_p z & \cdots & 0 & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & -I_p z & I_p \end{array}\right) \\ =& \det(I_p - A_1 z - A_2 z^2 - \dots - A_p z^p) \neq 0, \ |z| \leq 1. \tag{28.24} \end{align}\] 且$\{ \boldsymbol \eta_t \}$为$mp$维零均值白噪声。

于是，只需要研究多维AR(1)的预报问题。对$m$维随机向量$\boldsymbol Y=(Y_1, Y_2, \dots, Y_m)^T$定义 \[\begin{aligned} L(\boldsymbol Y | \boldsymbol Z) = [ L(Y_1 | \boldsymbol Z), L(Y_2 | \boldsymbol Z), \dots, L(Y_m | \boldsymbol Z) ] \end{aligned}\] 其中$L(Y_k | \boldsymbol Z)$是$\boldsymbol Z$对$Y_k$的最佳线性预测。设$\{ \boldsymbol X_t \}$是$m$维AR(1)序列，满足 \[\begin{aligned} \boldsymbol X_t = A \boldsymbol X_{t-1} + \boldsymbol \varepsilon_t, \ t \in \mathbb Z. \end{aligned}\] 考虑用$\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n$对$\boldsymbol X_{n+k}$进行最佳线性预测。

利用平稳解的因果性，即$E(\boldsymbol X_t \boldsymbol \varepsilon_{n+k})=0$($k \geq 1$)可得 \[\begin{aligned} & L(\boldsymbol X_{n+1} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(A \boldsymbol X_n + \boldsymbol\varepsilon_{n+1} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(A \boldsymbol X_n | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& A \, L(\boldsymbol X_n | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& A \boldsymbol X_n \end{aligned}\] 对$k \geq 1$有 \[\begin{aligned} & L(\boldsymbol X_{n+k} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(A \boldsymbol X_{n+k-1} + \boldsymbol\varepsilon_{n+k} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(A \boldsymbol X_{n+k-1} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& A \, L(\boldsymbol X_{n+k-1} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& \cdots = A^k \boldsymbol X_n \end{aligned}\]

总之有 \[\begin{aligned} & L(\boldsymbol X_{n+k} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(\boldsymbol X_{n+k} | \boldsymbol X_n) = A^k \boldsymbol X_n, \ n,k \in \mathbb N_+ \end{aligned}\]

28.4 多维平稳序列的谱分析

28.4.1 多维平稳序列的谱函数

设$\{ \boldsymbol X_t = (X_{1t}, X_{2t})^T: \ t \in \mathbb Z \}$是一个2维零均值平稳序列。对给定复数$z$定义 \[\begin{aligned} Y_t = X_{1t} + z X_{2t}, \ t \in \mathbb Z \end{aligned}\] 易见$E Y_t = 0$, \[\begin{aligned} \gamma_z(k) \stackrel{\triangle}{=}& E(Y_{t+k} \bar Y_t) = E[ (X_{1,t+k} + z X_{2,t+k}) (X_{1t} + \bar z X_{2t}) ] \\ =& \gamma_{11}(k) + z \gamma_{21}(k) + \bar z \gamma_{12}(k) + |z|^2 \gamma_{22}(k) \end{aligned}\] 都不依赖于$t$，所以$\{ Y_t \}$是一个复值平稳序列。设$\{ Y_t \}$有谱函数$F_z$, 则对$n \in \mathbb Z$, \[\begin{aligned} \gamma_z(n) = \int_{-\pi}^\pi e^{in\lambda} d F_z(\lambda), \ F_z(-\pi) = 0. \end{aligned}\]

当$z=\pm 1, \pm i$时得 \[\begin{aligned} \gamma_1(n) =& \gamma_{11}(n) + \gamma_{21}(n) + \gamma_{12}(n) + \gamma_{22}(n) \\ \gamma_{-1}(n) =& \gamma_{11}(n) - \gamma_{21}(n) - \gamma_{12}(n) + \gamma_{22}(n) \\ \gamma_{i}(n) =& \gamma_{11}(n) + i \gamma_{21}(n) - i \gamma_{12}(n) + \gamma_{22}(n) \\ \gamma_{-i}(n) =& \gamma_{11}(n) - i \gamma_{21}(n) + i \gamma_{12}(n) + \gamma_{22}(n) \end{aligned}\] 可以解出 \[\begin{aligned} \gamma_{12}(n) =& \frac14[ \gamma_1(n) - \gamma_{-1}(n) + i \gamma_i(n) - i \gamma_{-i}(n)] \\ \gamma_{21}(n) =& \frac14[ \gamma_1(n) - \gamma_{-1}(n) - i \gamma_i(n) + i \gamma_{-i}(n)] \end{aligned}\]

用$F_{11}(\lambda)$和$F_{22}(\lambda)$分别表示$\{ X_{1t} \}$和 $\{ X_{2t} \}$的谱函数，并引入 \[\begin{aligned} F_{12}(\lambda) =& \frac14[ F_1(\lambda) - F_{-1}(\lambda) + i F_i(\lambda) - i F_{-i}(\lambda)] \\ F_{21}(\lambda) =& \frac14[ F_1(\lambda) - F_{-1}(\lambda) - i F_i(\lambda) + i F_{-i}(\lambda)] \end{aligned}\] 引入矩阵函数 \[\begin{align} \boldsymbol F(\lambda) =& \left(\begin{array}{cc} F_{11}(\lambda) & F_{12}(\lambda) \\ F_{21}(\lambda) & F_{22}(\lambda) \end{array}\right), \ \lambda \in [-\pi, \pi] \tag{28.25} \end{align}\] 则有 \[\begin{align} \Gamma(n) =& E(\boldsymbol X_{t+n} \boldsymbol X_t^T) = \int_{-\pi}^\pi e^{in\lambda} d \boldsymbol F(\lambda) \\ =& \left(\begin{array}{cc} \int_{-\pi}^\pi e^{in\lambda} d F_{11}(\lambda) & \int_{-\pi}^\pi e^{in\lambda} d F_{12}(\lambda) \\ \int_{-\pi}^\pi e^{in\lambda} d F_{21}(\lambda) & \int_{-\pi}^\pi e^{in\lambda} d F_{22}(\lambda) \end{array}\right), \ n \in \mathbb Z \tag{28.26} \end{align}\]

称$\boldsymbol F(\lambda)$为2维平稳序列$\{ \boldsymbol X_t \}$的谱函数矩阵。由于矩阵$\boldsymbol F(\lambda)$的每个元素都是$[-\pi, \pi]$上分布函数的线性组合，所以都是有界变差右连续函数。另外$\boldsymbol F(\lambda)$还是Hermite矩阵： \[\begin{aligned} \boldsymbol F^*(\lambda) = \boldsymbol F(\lambda) \end{aligned}\] 其中星号表示共轭转置。

还可证明$\boldsymbol F(\lambda)$关于$\lambda$单调不减，即$\forall \lambda_1 < \lambda_2, \lambda_1, \lambda_2 \in [-\pi, \pi]$, $\boldsymbol F(\lambda_2) - \boldsymbol F(\lambda_1)$非负定。

当$\boldsymbol F(\lambda)$的每个元素的实部和虚部都是连续函数，并且除去有限个点外导函数连续时，称 \[\begin{align} \boldsymbol f(\lambda) =& \left(\begin{array}{cc} f_{11}(\lambda) & f_{12}(\lambda) \\ f_{21}(\lambda) & f_{22}(\lambda) \end{array}\right) \\ =& \left(\begin{array}{cc} F_{11}'(\lambda) & F_{12}'(\lambda) \\ F_{21}'(\lambda) & F_{22}'(\lambda) \end{array}\right), \ \lambda \in [-\pi, \pi] \tag{28.27} \end{align}\] 为$\{ \boldsymbol X_t \}$的谱密度矩阵, 称$f_{12}(\lambda)$是$\{ X_{1t} \}$和$\{ X_{2t} \}$的互谱密度。

因为$\boldsymbol F(\lambda)$是Hermite矩阵，并且单调不减，所以$\boldsymbol f(\lambda)$是Hermite非负定的。这时 \[\begin{align} \Gamma(n) =& \int_{-\pi}^\pi e^{in\lambda} \boldsymbol f(\lambda) d\lambda \\ \stackrel{\triangle}{=}& \left(\begin{array}{cc} \int_{-\pi}^\pi e^{in\lambda} f_{11}(\lambda) d\lambda & \int_{-\pi}^\pi e^{in\lambda} f_{12}(\lambda) d\lambda \\ \int_{-\pi}^\pi e^{in\lambda} f_{21}(\lambda) d\lambda & \int_{-\pi}^\pi e^{in\lambda} f_{22}(\lambda) d\lambda \end{array}\right), \ n \in \mathbb Z \tag{28.28} \end{align}\]

定理28.8 设$m$维平稳序列$\{ \boldsymbol X_t \}$有自协方差函数$\{\Gamma(n) \}$，则存在唯一的$m \times m$函数矩阵 \[\begin{aligned} \boldsymbol F(\lambda) = (F_{jk}(\lambda)), \ \lambda \in [-\pi, \pi], \end{aligned}\] 使得

(1) $\Gamma(n) = \int_{-\pi}^\pi e^{in\lambda} d \boldsymbol F(\lambda)$, $n \in \mathbb Z$;
(2) $\boldsymbol F^*(\lambda) = \boldsymbol F(\lambda)$, $\boldsymbol F(-\pi) = \boldsymbol 0$, 当$\lambda_1 < \lambda_2$时$\boldsymbol F(\lambda_2) - \boldsymbol F(\lambda_1)$非负定；
(3) $\boldsymbol F(\lambda)$中的每个元素$F_{jk}(\lambda)$是有界变差和右连续的。

$\boldsymbol F(\lambda)$叫做$\{ \boldsymbol X_t \}$的谱函数矩阵。

如果$\boldsymbol F(\lambda)$的每个元素$F_{jk}(\lambda)$的实部和虚部都是连续函数，且除去有限个点外导函数连续，就称 \[\begin{aligned} \boldsymbol f(\lambda) = (f_{jk}(\lambda)) \stackrel{\triangle}{=} (F_{jk}'(\lambda)) \end{aligned}\] 为$\{ \boldsymbol X_t \}$的谱密度矩阵。这时 \[\begin{align} \Gamma(n) =& \int_{-\pi}^\pi e^{in\lambda} \boldsymbol f(\lambda) d \lambda \\ =& \left( \int_{-\pi}^\pi e^{in\lambda} \boldsymbol f_{jk}(\lambda) d \lambda \right)_{m \times m}, \ n \in \mathbb Z \tag{28.29} \end{align}\]

称实变复值函数是绝对连续函数，如果其实部和虚部都是绝对连续函数。当$\boldsymbol F(\lambda)$的每个元素都是绝对连续函数时， $\boldsymbol f(\lambda) = (F_{jk}'(\lambda))$就是$\{ \boldsymbol X_t \}$的谱密度矩阵。

定理28.9 设$m$维平稳序列$\{ \boldsymbol X_t \}$有自协方差函数 $\Gamma(n) = (\gamma_{jk}(n))$, 如果 \[\begin{aligned} \sum_{n=-\infty}^\infty | \gamma_{jk}(n) | < \infty, \ j,k=1,2,\dots, m \end{aligned}\] 则 \[\begin{aligned} \boldsymbol f(\lambda) =& \frac{1}{2\pi} \sum_{n=-\infty}^\infty \Gamma(n) e^{-in\lambda}, \ \lambda \in [-\pi, \pi] \end{aligned}\] 是$\{ \boldsymbol X_t \}$的谱密度矩阵。

28.4.2 多维平稳序列的谱表示

设$\{ \boldsymbol X_t = (X_{1t}, X_{2t}, \dots, X_{mt})^T$ 是$m$维零均值平稳序列，则其每个分量$\{ X_j(t) \}$是一位零均值平稳序列，有谱表示 \[\begin{align} X_{jt} =& \int_{-\pi}^\pi e^{it\lambda} d Z_j(\lambda), \ t \in \mathbb Z \tag{28.30} \end{align}\] 中$\{ Z_j(\lambda) \}$是$[-\pi, \pi]$上右连续的正交增量过程。

记 \[\begin{align} \boldsymbol Z(\lambda) =& (Z_1(\lambda), Z_2(\lambda), \dots, Z_m(\lambda))^T, \ \lambda \in [-\pi, \pi], \tag{28.31} \end{align}\] 有 \[\begin{align} \boldsymbol X_t =& \int_{-\pi}^\pi e^{in\lambda} d \boldsymbol Z(\lambda) \ \lambda \in [-\pi, \pi]. \tag{28.32} \end{align}\] 这称为$m$维平稳序列$\{ \boldsymbol X_t \}$的谱表示。

(28.32)中的$\boldsymbol Z(\lambda)$有下列性质：

(1) 正交增量性：对$-\pi \leq \lambda_1 < \lambda_2 \leq \lambda_3 < \lambda_4 \leq \pi$, \[\begin{aligned} E\left[ (\boldsymbol Z(\lambda_2) - \boldsymbol Z(\lambda_1)) (\boldsymbol Z(\lambda_4) - \boldsymbol Z(\lambda_3))^* \right] = 0. \end{aligned}\]
(2) 右连续性：当$\delta \downarrow 0$时 \[\begin{aligned} E\left[ (\boldsymbol Z(\lambda + \delta) - \boldsymbol Z(\lambda)) (\boldsymbol Z(\lambda + \delta) - \boldsymbol Z(\lambda))^* \right] \to 0. \end{aligned}\]
(3) $\boldsymbol F(\lambda) = E[ \boldsymbol Z(\lambda) \boldsymbol Z^*(\lambda)]$ 是$\{ \boldsymbol X_t \}$的谱函数矩阵，满足 \[\begin{aligned} \boldsymbol F(\lambda_2) - \boldsymbol F(\lambda_1) =& E\left[ (\boldsymbol Z(\lambda_2) - \boldsymbol Z(\lambda_1)) (\boldsymbol Z(\lambda_2) - \boldsymbol Z(\lambda_1))^* \right], \ \lambda_1 < \lambda_2 \end{aligned}\]

满足上述(1), (2), (3)的$m$维随机过程$\{ \boldsymbol Z(\lambda) \}$叫做右连续的正交增量过程。

定理28.10 对$m$维零均值平稳序列$\{ \boldsymbol X_t \}$，有右连续的正交增量过程 $\{ \boldsymbol Z(\lambda) \}$使得 $\boldsymbol Z(-\pi) = \boldsymbol 0$和(28.32)成立。如果正交增量过程$\{ \boldsymbol\xi(\lambda) \}$也满足上述的条件，则 \[\begin{aligned} P(\boldsymbol\xi(\lambda) = \boldsymbol Z(\lambda)) = 1, \ \lambda \in [-\pi, \pi]. \end{aligned}\]

例28.6 (多维线性平稳序列的谱密度) 设$\{ \boldsymbol \varepsilon_t \}$是$m$维WN(0, $Q$), $Q = E[\boldsymbol\varepsilon_t \boldsymbol\varepsilon_t^T]$, $\{ A_j \}$是一列$m \times m$实值矩阵，满足 \[\begin{aligned} \sum_{j=-\infty}^\infty A_j Q A_j^T < \infty, \end{aligned}\] 则 \[\begin{aligned} \boldsymbol X_t = \sum_{j=-\infty}^\infty A_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \end{aligned}\] 是$m$维零均值平稳序列。

自协方差函数为 \[\begin{aligned} \Gamma(n) = E[ \boldsymbol X_{t+n} \boldsymbol X_n ] = \sum_{j=-\infty}^\infty A_{j+n} Q A_j^T, \ n \in \mathbb Z \end{aligned}\]

本例中，对两个实矩阵$A, B$，用$A \leq B$表示$A$和$B$的对应元素比较全部成立小于等于关系，用$[A]$表示把$A$的所有元素都取绝对值组成的矩阵。这时必有 \[ [AB] \leq [A] [B] \] 如果要求$\sum_{j=-\infty}^\infty [A_j] < \infty$, 则 \[\begin{aligned} \sum_{n=-\infty}^\infty [\Gamma(n)] =& \sum_{n=-\infty}^\infty \left[\sum_{j=-\infty}^\infty A_{j+n} Q A_j^T \right] \\ \leq & \sum_{n=-\infty}^\infty \sum_{j=-\infty}^\infty [A_{j+n}] [Q] [A_j^T] \\ \leq& \left(\sum_{n=-\infty}^\infty [A_n] \right) [Q] \left(\sum_{j=-\infty}^\infty [A_j^T] \right) < \infty \end{aligned}\] 从而$\{ \boldsymbol X_t \}$有谱密度 \[\begin{aligned} \boldsymbol f(\lambda) =& \frac{1}{2\pi} \sum_{n=-\infty}^\infty \Gamma(n) e^{-in\lambda} \\ =& \frac{1}{2\pi} \sum_{n=-\infty}^\infty \sum_{j=-\infty}^\infty A_{j+n} Q A_j^T e^{-in\lambda} \\ =& \frac{1}{2\pi} \left( \sum_{k=-\infty}^\infty A_k e^{-ik\lambda} \right) Q \left( \sum_{j=-\infty}^\infty A_j e^{-ij\lambda} \right)^* \end{aligned}\]

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例28.7 (VARMA序列谱密度) 考虑平稳可逆的$m$维ARMA($p,q$)模型的谱密度。模型为 \[\begin{aligned} A(\mathscr B) \boldsymbol X_t = B(\mathscr B) \boldsymbol\varepsilon_t, \ t \in \mathbb Z, \end{aligned}\]

$\det(A(z)) A^{-1}(z)$是$A(z)$的伴随矩阵，仍是矩阵系数多项式。于是$A^{-1}(z) B(z)$的每个元素是有理多项式。将$A^{-1}(z) B(z)$的每个元素进行泰勒展开后得$A^{-1}(z) B(z)$泰勒展开式 \[\begin{aligned} A^{-1}(z) B(z) = \sum_{j=0}^\infty C_j z^j, \ |z| \leq 1. \end{aligned}\] 其中的系数矩阵满足 \[\begin{aligned} \sum_{j=0}^\infty [C_j] < \infty. \end{aligned}\]

由例28.6，平稳解 \[\begin{aligned} \boldsymbol X_t = A^{-1}(\mathscr B) B(\mathscr B) \boldsymbol\varepsilon_t = \sum_{j=0}^\infty C_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \end{aligned}\] 有谱密度矩阵 \[\begin{aligned} \boldsymbol f(\lambda) =& \frac{1}{2\pi} \left( \sum_{k=-\infty}^\infty C_k e^{-ik\lambda} \right) Q \left( \sum_{j=-\infty}^\infty C_j e^{-ij\lambda} \right)^* \end{aligned}\]

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例28.8 (二次相干函数) 设二维零均值平稳序列$\{ \boldsymbol X_t \}$有谱表示 \[\begin{aligned} \boldsymbol X_t = \int_{-\pi}^\pi e^{it\lambda} d \left(\begin{array}{c} Z_1(\lambda) \\ Z_2(\lambda) \end{array}\right) \end{aligned}\]

取$\lambda \in [-\pi, \pi]$, 对充分小的正数$\Delta\lambda$，定义 \[\begin{aligned} \Delta Z_k(\lambda) = Z_k(\lambda + \Delta\lambda) - Z_k(\lambda), \ k=1,2 \end{aligned}\] 则有 \[\begin{aligned} & \boldsymbol F(\lambda + \Delta\lambda) - F(\lambda) \\ =& E\left[ \left(\begin{array}{c} \Delta Z_1(\lambda) \\ \Delta Z_2(\lambda) \end{array}\right) (\Delta \bar Z_1(\lambda) , \Delta \bar Z_2(\lambda)) \right] \\ =& \left(\begin{array}{cc} E|\Delta Z_1(\lambda)|^2 & E[\Delta Z_1(\lambda) \Delta \bar Z_2(\lambda)] \\ E[\Delta \bar Z_1(\lambda) \Delta Z_2(\lambda)] & E|\Delta Z_2(\lambda)|^2 \end{array}\right) \end{aligned}\]

如果$\boldsymbol F'(\lambda)$连续，则有 \[\begin{aligned} d \boldsymbol F(\lambda) =& \boldsymbol f(\lambda) d\lambda = \left(\begin{array}{cc} E|d Z_1(\lambda)|^2 & E[d Z_1(\lambda) d \bar Z_2(\lambda)] \\ E[d \bar Z_1(\lambda) d Z_2(\lambda)] & E|d Z_2(\lambda)|^2 \end{array}\right) \end{aligned}\] $d Z_1(\lambda)$和$d Z_2(\lambda)$的相关系数为 \[\begin{aligned} \rho_{12}(\lambda) =& \frac{E[d Z_1(\lambda) d \bar Z_2(\lambda)]} {\sqrt{E|d Z_1(\lambda)|^2 E|d Z_2(\lambda)|^2}} \\ =& \frac{f_{12}(\lambda) d\lambda} {\sqrt{f_{11}(\lambda) d\lambda \cdot f_{22}(\lambda) d\lambda}} \\ =& \frac{f_{12}(\lambda)}{\sqrt{f_{11}(\lambda) f_{22}(\lambda)}} \end{aligned}\] 其中规定$0/0=0$。

通常称 \[\begin{aligned} K_{12}^2(\lambda) = |\rho_{12}(\lambda)|^2 = \frac{|f_{12}(\lambda)|^2}{f_{11}(\lambda) f_{22}(\lambda)} \end{aligned}\] 为$\{ \boldsymbol X_t, t \in \mathbb Z \}$的二次相干函数。 $K_{12}^2(\lambda)$描述了$\{ X_{1t} \}$和$\{ X_{2t} \}$ 在角频率$\lambda$处的线性相关性强弱。如果$K_{12}^2(\lambda)=1$, 说明$\{ X_{1t} \}$和$\{ X_{2t} \}$在角频率$\lambda$处线性相关。

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例28.9 (线性滤波的二次相干函数) 设平稳序列$\{ X_{1t} \}$有谱密度$f(\lambda)$, $\{ h_j \}$是绝对可和线性滤波器，输出过程为 \[\begin{aligned} X_{2t} = \sum_{j=-\infty}^\infty h_j X_{1, t-j}, \ t \in \mathbb Z \end{aligned}\]

则$\{X_{2t}\}$有谱密度（见定理6.5） \[\begin{aligned} f_{22}(\lambda) =& |h(\lambda)|^2 f(\lambda) \end{aligned}\] 其中 \[\begin{aligned} h(\lambda) = \sum_{j=-\infty}^\infty h_j e^{-ij\lambda}. \end{aligned}\]

计算得 \[\begin{aligned} \gamma_{12}(n) =& E(X_{1, t+n} X_{2t}) \\ =& \sum_{j=-\infty}^\infty h_j E(X_{1,t+n} X_{2,t-j}) \\ =& \sum_{j=-\infty}^\infty h_j \gamma_{11}(n+j) \\ =& \sum_{j=-\infty}^\infty h_j \int_{-\pi}^\pi e^{i(n+j)\lambda} f(\lambda) d\lambda \\ =& \int_{-\pi}^\pi e^{in\lambda} h(-\lambda) f(\lambda) d\lambda \end{aligned}\] 所以$\{ X_{1t} \}$和$\{ X_{2t} \}$的互谱密度为 \[\begin{aligned} f_{12}(\lambda) = h(-\lambda) f(\lambda), \ \lambda \in [-\pi, \pi]. \end{aligned}\] 于是2维平稳序列$\boldsymbol X_t = (X_{1t}, X_{2t})^T$有退化的谱密度矩阵 \[\begin{aligned} \boldsymbol f(\lambda) =& \left(\begin{array}{cc} f(\lambda) & h(-\lambda)f(\lambda) \\ h(\lambda)f(\lambda) & |h(\lambda)|^2 f(\lambda) \end{array}\right) \end{aligned}\] 这时二次相干函数 \[\begin{aligned} K^2(\lambda) \equiv 1, \ \forall \lambda \in [-\pi, \pi]. \end{aligned}\]

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28.4.3 谱密度矩阵的估计

二维零均值平稳序列$\{ \boldsymbol X_t \}$的观测值$\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_N$的周期图定义为 \[\begin{aligned} I_N(\lambda) =& \frac{1}{2\pi N} \left( \sum_{j=1}^N \boldsymbol X_j e^{-ij\lambda} \right) \left( \sum_{j=1}^N \boldsymbol X_j e^{-ij\lambda} \right)^*. \end{aligned}\] 周期图$I_n(\lambda)$的值是$2 \times 2$方阵，可以对周期图平滑得到谱密度矩阵估计。

设$\lambda_j = 2j\pi / N$, 对$\lambda \in [0, \pi]$, 用$g(N, \lambda)$表示$\{ \lambda_j: 0 \leq j \leq N/2 \}$中距离$\lambda$最近的$\lambda_j$（若左右两个距离相等取左边一个）。取$M_N = O(\sqrt{N})$且$\lim_{N \to\infty} M_N = \infty$。设$W_N(k)$为满足下列条件的实值权函数： \[\begin{aligned} (1) & W_N(k) = W_N(-k), \ W_N(k) \geq 0, \ |k| \leq M_N; \\ (2) & \sum_{|k| \leq M_N} W_N(k) = 1; \\ (3) & \lim_{N \to\infty} \sum_{|k| \leq M_N} W_N^2(k) = 0. \end{aligned}\] 谱密度矩阵的平滑周期图估计为 \[\begin{aligned} \hat{\boldsymbol f}(\lambda) =& \sum_{|k| \leq M_N} W_N(k) I_N(g(N,\lambda) + \lambda_k), \ \lambda \in [0,\pi], \\ \hat{\boldsymbol f}(\lambda) =& [\hat{\boldsymbol f}(-\lambda) ]^T, \ \lambda \in [-\pi, 0]. \end{aligned}\]

设 \[\begin{aligned} \hat{\boldsymbol f}(\lambda) =& \left(\begin{array}{cc} \hat f_{11} & \hat f_{12} \\ \hat f_{21} & \hat f_{22} \end{array}\right) \end{aligned}\] 二次相干函数$K_{12}^2(\lambda)$的估计定义为 \[\begin{aligned} \hat K_{12}^2(\lambda) =& \frac{| \hat f_{12}(\lambda) |^2} {\hat f_{11}(\lambda) \hat f_{22}(\lambda)}. \end{aligned}\]

28.5 附录：补充

28.5.1 多维ARMA不可辨识的例子

只要$\mbox{det}(A(z))$为不依赖于$z$的非零常数，则多维ARMA模型参数不能从$\{ \Gamma(n) \}$唯一确定。

例如，考虑2维$VAR(1)$。模型为 \[ \boldsymbol X_t = A \boldsymbol X_{t-1} + \boldsymbol\varepsilon_t \] 其中 \[ A = \left(\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right) \] 特征多项式的行列式为 \[\begin{aligned} \mbox{det}(I - A z) =& \left| \begin{array}{cc} 1 - a_{11} z & -a_{12}z \\ -a_{21}z & 1 - a_{22}z \end{array}\right | \\ =& 1 - (a_{11} + a_{22}) z + (a_{11} a_{22} - a_{12} a_{21}) z^2 \end{aligned}\] 令$z$和$z^2$系数等于零，则 \[ a_{22} = -a_{11}, \ a_{12} a_{21} = - a_{11}^2 \] 取$a_{11} \neq 0$, $a_{22} = -a_{11}$, 取$a_{12} \neq 0$，取$a_{21} = - \frac{a_{11}^2}{a_{12}}$则有$\mbox{det}(I - A z)=1$不依赖于$z$。这时 \[\begin{aligned} (I - A z)^{-1} =& \left( \begin{array}{cc} 1 - a_{22} z & a_{21}z \\ a_{12}z & 1 - a_{11}z \end{array}\right) \\ =& I + \left( \begin{array}{cc} - a_{22} & a_{21} \\ a_{12} & - a_{11} \end{array}\right) z \end{aligned}\] 令 \[ B = \left( \begin{array}{cc} - a_{22} & a_{21} \\ a_{12} & - a_{11} \end{array}\right) \] 则模型可以写成 \[ \boldsymbol X_t = (I - A \mathscr B)^{-1} \boldsymbol\varepsilon_t = (I + B \mathscr B) \boldsymbol\varepsilon_t \] 又可以写成一个2维MA(1)模型。

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References

Brockwell, P. J., and R. A. Davis. 1987. Time Series: Theory and Methods. Springer-Verlag.

谢衷洁. 1990. 时间序列分析. 北京大学出版社.