29 条件异方差模型

29.1 资产收益率

设\(t\)为某个固定时间单位的个数（比如天数、月数、年数），以天为例，用\(P_t\)表示某金融资产在第\(t\)天的价格。令 \[ R_t = \frac{P_t - P_{t-1}}{P_{t-1}}, \] 称为第\(t\)天的简单收益率。 \(1 + R_t\)称为毛收益率。令 \[ r_t = \log(1 + R_t) = \log \frac{P_t}{P_{t-1}} = \log P_t - \log P_{t-1}, \] 称为第\(t\)天的对数收益率。

如果已知\(r_1, \dots, r_t\)，则易见 \[ \frac{P_t}{P_0} = \prod_{j=1}^t \frac{P_j}{P_{j-1}} = \prod_{j=1}^t (1 + R_j), \] 而 \[ \log \frac{P_t}{P_0} = \sum_{j=1}^t \log \frac{P_j}{P_{j-1}} = \sum_{j=1}^t r_j, \] 所以对数收益率更容易进行数学推导。

29.2 ARCH模型

对于资产收益率序列\(\{r_t \}\)，如果它是高斯过程，则最优线性预测就是最优预测，通常只要建立一个ARMA这样的线性模型就足够了。但是，在金融市场中，资产收益率常常不是正态分布的，而且其条件分布\(X_t | \mathscr F_{t-1}\)的条件方差不是恒定的，会随时间\(t\)变化，金融资产的收益率会有“波动率聚集”现象，即某一段时间的\(r_t\)波动较大，而另一端时间的\(r_t\)波动较小，所以有必要在对条件期望\(E(r_t | \mathscr F_{t-1})\)建模的同时对条件方差\(\text{Var}(r_t | \mathscr F_{t-1})\)建模。

设\(\{ \varepsilon_t \}\)是对条件均值建模后的残差，理论上，这相当于Wold分解中的新息，设其满足 \[ E(\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) = 0 . \] \(\{ \varepsilon_t \}\)是二阶平稳的宽白噪声，但这并不要求\(\{ \varepsilon_t \}\)独立，我们可以考虑能够表现波动率聚集的模型。因为\(\varepsilon_t^2\)代表了波动大小，我们尝试对其建立如下的AR(1)模型： \[ \varepsilon_t^2 = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \eta_t, \] 其中\(\alpha_0 > 0\), \(\alpha_1 \geq 0\)， \(\{\eta_t \}\)是独立同分布零均值白噪声列。这样的模型中\(\varepsilon_t^2\)和\(\varepsilon_{t-1}^2\)是正相关的，所以能够体现波动率聚集性质。设\(E\varepsilon_t^2 = \sigma^2\)，则 \[ \sigma^2 = \alpha_0 + \alpha_1 \sigma^2 + 0, \] 将\(\varepsilon_t^2\)中心化，得 \[ (\varepsilon_t^2 - \sigma^2) = \alpha_1 (\varepsilon_{t-1}^2 - \sigma^2) + \eta_t, \] 可见\(\varepsilon_t^2 - \sigma^2\)满足一个AR(1)模型，其中\(0 \leq \alpha_1 < 1\)。记\(A(z) = 1 - \alpha_1 z\)，则 \[ A(\mathscr B)(\varepsilon_t^2 - \sigma^2) = \eta_t, \] 所以\(\varepsilon_t^2\)有平稳解 \[ \varepsilon_t^2 = \sigma^2 + A^{-1}(\mathscr B) \eta_t = \sigma^2 + \sum_{j=0}^\infty \alpha_1^j \eta_{t-j} . \] 由此平稳解可见\(\eta_t\)与\(\{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots \}\)独立，从而 \[ \sigma_t^2 = \text{Var}(\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) = E(\varepsilon_t^2 | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 . \] 这就给出了条件方差的一个模型。进一步地可以设 \[ \sigma_t^2 = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dots + \alpha_p \varepsilon_{t-p}^2 . \] 注意上式中已经没有\(\eta_t\)。

显然，这个模型中的系数应使得\(\alpha_0 > 0\), \(\alpha_j \geq 0\), \(j=1,2,\dots,p\)，且多项式\(A(z) = 1 - \alpha_1 z - \dots - \alpha_p z^p\)满足最小相位性。注意对应的AR(\(p\))模型不能保证\(\varepsilon_t^2\)非负，但有下面的ARCH模型定义和下一节的平稳解结果。

定义29.1 (ARCH模型) 设\(\{v_t \}\)是独立同分布零均值标准白噪声WN(0,1)，非负常数\(\alpha_0, \alpha_1, \dots, \alpha_p\)满足\(\alpha_1 + \dots + \alpha_p < 1\)且\(\alpha_0>0\), \(\alpha_p>0\)，则如下模型 \[\begin{equation} \begin{cases} \varepsilon_t = [\alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dots + \alpha_p \varepsilon_{t-p}^2 ]^{1/2} v_t, \\ v_t \text{与} \{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots \} \text{相互独立}, \end{cases} \tag{29.1} \end{equation}\] 称为\(\text{ARCH}(p)\)模型。如果\(\{\varepsilon_t \}\)是严平稳白噪声且满足上述模型，则称其为\(\text{ARCH}(p)\)序列。

记 \[ \sigma_t = [\alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dots + \alpha_p \varepsilon_{t-p}^2 ]^{1/2} . \] 则模型(29.1)可以表述为 \[\begin{equation} \begin{cases} \varepsilon_t = \sigma_t v_t, \\ \sigma_t = [\alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dots + \alpha_p \varepsilon_{t-p}^2 ]^{1/2}, \\ v_t \text{与} \{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots \} \text{相互独立} . \end{cases} \tag{29.2} \end{equation}\]

对ARCH(\(p\))序列，来证明 \[ \sigma_t^2 = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dots + \alpha_p \varepsilon_{t-p}^2 = \text{Var}(\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) . \] 记\(\mathscr F_{t-1} = \sigma(\{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots\})\)，则\(v_t\)与\(\mathscr F_{t-1}\)独立而\(\sigma_t^2\)关于\(\mathscr F_{t-1}\)可测，故\(v_t\)与\(\sigma_t\)独立。于是， \[\begin{aligned} & \text{Var}(\varepsilon_t | \mathscr F_{t-1}) = E(\varepsilon_t^2 | \mathscr F_{t-1}) \\ =& E \left\{ \sigma_t^2 v_t^2 | \mathscr F_{t-1}\right\} = \sigma_t^2 E(v_t^2 | \mathscr F_{t-1}) \\ =& \sigma_t^2 E(v_t^2) = \sigma_t^2 . \end{aligned}\]

对ARCH(\(p\))序列，设\(Ev_t^4 < \infty\)， \(E\varepsilon_t^4 < \infty\)，来证明\(\{ \varepsilon_t^2 \}\)满足如下AR(\(p\))模型： \[ (\varepsilon_t^2 - \sigma^2) = \alpha_1 (\varepsilon_{t-1}^2 - \sigma^2) + \cdots + \alpha_p (\varepsilon_{t-p}^2 - \sigma^2) + \eta_t, \] 其中\(\eta_t\)为独立同分布零均值白噪声列。事实上，令 \[ \eta_t = \varepsilon_t^2 - \sigma_t^2 = \sigma_t^2(v_t^2 - 1) , \] 注意\(v_t\)与\(\{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots\}\)独立， \(v_t\)与\(\sigma_t\)独立，所以有 \[\begin{aligned} E(\eta_t) =& E(\sigma_t^2 (v_t^2 - 1)) = E(\sigma_t^2) E(v_t^2 - 1) = 0, \\ E(\eta_t \eta_{t+k}) =& E[\sigma_t^2 (v_t^2 - 1) \sigma_{t+k}^2 (v_{t+k}^2 - 1)] = E(\sigma_t (v_t^2 - 1) \sigma_{t+k}) E(v_{t+k}^2 - 1) = 0, \\ E(\eta_t^2) =& E[\sigma_t^4(v_t^2 - 1)^2] = E[\sigma_t^4] E[(v_t^2 - 1)^2] = E(\sigma_1^4) E[(v_1^2 - 1)^2] . \end{aligned}\] 这里利用了\(\{\varepsilon_t \}\)严平稳，所以\(E\sigma_t^4\)不依赖于\(t\)。上式说明了\(\{\eta_t \}\)是零均值白噪声列，从而 \[ \varepsilon_t^2 = \sigma_t^2 + \eta_t = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dots + \alpha_p \varepsilon_{t-p}^2 + \eta_t, \] 这是一个AR(\(p\))模型。

29.3 ARCH模型平稳解

引理29.1 设\(\alpha_0, \alpha_1, \dots\)是非负常数， \(\{u_t \}\)是独立同分布非负随机变量序列， \(E(u_t)=1\)。如果\(\alpha_0 > 0\)， \(c = \sum_{j=1}^\infty \alpha_j < 1\)，则有以下结论：

（1）有唯一的严平稳遍历序列\(\{Y_t \}\)满足模型 \[\begin{equation} \begin{cases} Y_t = \sigma_t^2 u_t, \ \text{其中} \sigma_t^2 = \alpha_0 + \sum_{j=1}^\infty \alpha_j Y_{t-j} . \\ E(Y_t) < \infty, \ \{Y_{t-1}, Y_{t-2}, \dots\} \text{与} u_t \text{独立} . \end{cases} \tag{29.3} \end{equation}\]

（2）满足(29.3)的严平稳遍历序列可表示成 \[\begin{equation} Y_t = \alpha_0 \sum_{n=1}^\infty A_t(n), \ E(Y_t) = \frac{\alpha_0}{1 - c}, \ t \in \mathbb N . \tag{29.4} \end{equation}\] 其中\(A_t(0) = u_t\)，对\(n \geq 1\)，有 \[\begin{equation} A_t(n) = \sum_{i_1, i_2, \dots, i_n \geq 1} \alpha_{i_1} \alpha_{i_2} \dots \alpha_{i_n} u_t u_{t-i_1} u_{t - (i_1 + i_2)} \dots u_{t - (i_1 + \dots + i_n)} ; \tag{29.5} \end{equation}\] 若进一步假设\(c \sqrt{E(u_t^2)} < 1\)，则\(E(Y_t^2) < \alpha_0 E(u_t^2) / (1 - c \sqrt{E(u_t^2)})^2\).

参见Fan J Q, Yao Q W. Nonlinear time series: non-parametric and parametric methods. Springer-Verlag. 2005. 证明见(何书元 2023)附录1.5.

定理29.1 对于ARCH模型(29.2), 设\(c = \sum_{j=1}^p \alpha_j\), \(u_t = v_t^2\), \(A_t(n)\)按(29.5)定义，其中当\(j>p\)时\(\alpha_j = 0\)。则

（1）存在严平稳遍历白噪声\(\{\varepsilon_t\}\)满足ARCH模型(29.2), 其中 \[\begin{equation} \varepsilon_t = \left[ \alpha_0 \sum_{n=0}^\infty A_t(n) \right]^{1/2} u_t, \ E \varepsilon_t = 0, \ E \varepsilon_t^2 = \frac{\alpha_0}{1 - c} ; \tag{29.6} \end{equation}\]

（2）如果\(\{e_t \}\)也是模型(29.2)的严平稳解，则\(e_t^2 = \varepsilon_t^2\), a.s.;

（3）如果\(c \sqrt{E(v_1^4)} < 1\), 则 \[ E(\varepsilon_t^4) < \alpha_0^2 E(v_1^4) / \left[1 - c \sqrt{E(v_1^4)} \right]^2 . \]

证明：（1）取\(u_t = v_t^2\)，则\(\{u_t \}\)是独立同分布非负随机变量序列且\(E(u_t)=1\)。令\(Y_t\)为(29.4)的定义，则\(\{Y_t \}\)符合引理29.1中的结论(1)，即 \[\begin{equation} \begin{cases} Y_t = \left(\alpha_0 + \sum_{j=1}^p \alpha_j Y_{t-j} \right) u_t, \\ E(Y_t) < \infty, \quad u_t \text{与} \{Y_{t-1}, Y_{t-2}, \dots\} \text{独立} . \end{cases} \tag{29.7} \end{equation}\] 取 \[\begin{equation} \varepsilon_t = \left(\alpha_0 + \sum_{j=1}^p \alpha_j Y_{t-j} \right)^{1/2} v_t, \tag{29.8} \end{equation}\] 则\(\varepsilon_t^2 = Y_t\)，从而可知\(E(\varepsilon_t^2)<\infty\)。由定理4.1可知\(\{\varepsilon_t \}\)是严平稳遍历序列。将(29.8)中的\(Y_t\)替换成\(\varepsilon_t^2\)，可知\(\{\varepsilon_t\}\)满足如下模型 \[ \varepsilon_t = \left(\alpha_0 + \sum_{j=1}^p \alpha_j \varepsilon_{t-j}^2 \right)^{1/2} v_t . \]

根据\(Y_t\)的定义可以看出\(v_t\)与\(\{Y_{t-1}, Y_{t-2}, \dots\}\)独立，而\(\{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots\}\)仅依赖于\(\{v_{t-1}, Y_{t-1}, Y_{t-2}, \dots\}\)，所以\(v_t\)与\(\{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots\}\)独立。

只要证明\(\{\varepsilon_t\}\)是白噪声列。前面已证明\(E(\varepsilon_t^2)<\infty\)，于是 \[\begin{aligned} E(\varepsilon_t) =& E \left[ \left(\alpha_0 + \sum_{j=1}^p \alpha_j \varepsilon_{t-j}^2 \right)^{1/2} v_t \right] \\ =& E \left[ \left(\alpha_0 + \sum_{j=1}^p \alpha_j \varepsilon_{t-j}^2 \right)^{1/2} \right] E(v_t) \\ =& 0 . \\ E(\varepsilon_t^2) =& E(Y_t) = \frac{\alpha_0}{1 - c} = \frac{\alpha_0}{1 - \sum_{j=1}^p \alpha_j} . \\ E(\varepsilon_s \varepsilon_t) =& E \left[ \left(\alpha_0 + \sum_{j=1}^p \alpha_j \varepsilon_{s-j}^2 \right)^{1/2} v_s \left(\alpha_0 + \sum_{j=1}^p \alpha_j \varepsilon_{t-j}^2 \right)^{1/2} v_t \right] \\ =& E \left[ \left(\alpha_0 + \sum_{j=1}^p \alpha_j \varepsilon_{s-j}^2 \right)^{1/2} v_s \left(\alpha_0 + \sum_{j=1}^p \alpha_j \varepsilon_{t-j}^2 \right)^{1/2} \right] E(v_t) \\ =& 0 \quad(s < t) . \end{aligned}\] 即\(\{\varepsilon_t \}\)是零均值白噪声列，且是严平稳遍历序列，满足ARCH模型。

（2）如果\(\{e_t \}\)也是模型(29.4)的严平稳解，则\(\{Y_t = e_t^2 \}\)是模型(29.7)的唯一严平稳解，根据\(\varepsilon_t\)定义可知\(e_t^2 = \varepsilon_t^2\)，a.s.

（3）由引理29.1结论(3)，若\(c \sqrt{E(v_1^4)} < 1\), 即\(c \sqrt{E(u_t^2)} < 1\), 则 \[ E(\varepsilon_t^4) = E(Y_t^2) < \alpha_0^2 E(u_t^2) / \left[ 1 - c \sqrt{E(u_t^2)} \right]^2 = \alpha_0^2 E(v_1^4) / \left[ 1 - c \sqrt{E(v_1^4)} \right]^2 . \]

由定理29.1结论(3)可知，当\(c \sqrt{E(v_1^4)} < 1\), 则\(E(\varepsilon_t^4) < \infty\)，根据上一节的讨论可知这时\(\{\varepsilon_t^2 \}\)满足如下AR(\(p\))模型： \[\begin{equation} \varepsilon_t^2 = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dots + \alpha_p \varepsilon_{t-p}^2 + \eta_t, \tag{29.9} \end{equation}\] 其中\(\eta_t = \varepsilon_t^2 - \sigma_t^2 = \sigma_t^2(v_t^2 - 1)\)是零均值白噪声列, \[ \text{Var}(\eta_t) = E(\eta_t^2) = E(\sigma_t^4) E[(v_t^2 - 1)^2] . \]

对随机变量\(\xi\)，若\(E(\xi^4) < \infty\)，则定义 \[ \kappa_{\xi} = \frac{E(X- E(X))^4}{[\text{Var(X)}]^2}, \] 称\(\kappa_{\xi}\)为\(\xi\)的峰度。对正态分布的\(\xi\)，有\(\kappa_{\xi} = 3\)，称\(\kappa_{\xi} - 3\)为\(\xi\)的超额峰度。这个指标度量了随机变量分布的厚尾性，其样本的表现是异常值比正态分布更多。 ARCH模型主要用于金融资产收益率模型，金融资产收益率常常体现出厚尾分布特性，而ARCH模型一般是厚尾的。

例29.1 设\(\{\varepsilon_t \}\)是ARCH模型(29.2)的严平稳解， \(\kappa_{\varepsilon}\)是\(\varepsilon_t\)的峰度， \(\kappa_v\)是\(v_t\)的峰度，则：

（1）\(\kappa_{\varepsilon} \geq \kappa_v\)，且等号成立当且仅当\(\sigma_t^2\)为常数值，即没有条件异方差性的情形；

（2）当\(E(v_1^4) < 1\), \(E(\varepsilon_t^4)<1\)时，作为AR(\(p\))序列的\(\{\varepsilon_t^2\}\)序列的自相关系数都是非负的；特别地当\(\alpha_j > 0\)时\(\rho_{kj} > 0\), \(k=1,2,\dots\)。

证明：（1） \(\varepsilon_t = \sigma_t v_t\)， \(E(\varepsilon_t) = 0\)， \(E(v_t^2) = 1\)， \[ E(\varepsilon_t^2) = E(\sigma_t^2 v_t^2) = E(\sigma_t^2) E(v_t^2) = E(\sigma_t^2). \] 从而 \[\begin{aligned} E(\varepsilon_t^4) =& E(\sigma_t^4 v_t^4) = E(\sigma_t^4) E(v_t^4) \\ \geq& [E(\sigma_t^2)]^2 E(v_t^4), \\ \kappa_{\varepsilon} =& \frac{E(\varepsilon_t^4)}{[E(\varepsilon_t^2)]^2} \\ \geq& \frac{[E(\sigma_t^2)]^2 E(v_t^4)}{[E(\sigma_t^2)]^2} \\ =& E(v_t^4) = \kappa_{v} . \end{aligned}\] 等式成立的条件是等式\(E(\sigma_t^4) = [E(\sigma_t^2)]^2\)，这当且仅当\(\sigma_t^2 = c\), a.s.

（2）这时\(E(v_t^4)<\infty\)，\(E(\varepsilon_t^4) < \infty\)，所以\(\{\varepsilon_t^2 \}\)满足AR(\(p\))模型(29.9)，令\(A(z) = 1 - \alpha_1 z - \dots - \alpha_p z^p\)，有 \[ \varepsilon_t^2 - E(\varepsilon_t^2) = A^{-1}(\mathscr B)(\eta_t) = \sum_{j=0}^\infty c_j \eta_{t-j} . \] 记\(s = \sum_{j=1}^p \alpha_j t^j\)，则 \[ A^{-1}(t) = \frac{1}{1 - s} = \sum_{k=0}^\infty s^k = \sum_{k=0}^\infty (\sum_{j=1}^p \alpha_j t^j)^k, \ t \in (0, 1] . \] 则\(\{c_j \}\)满足 \[ \sum_{j=0}^\infty c_j t^j = \sum_{k=0}^\infty (\sum_{j=1}^p \alpha_j t^j)^k, \ t \in (0, 1]. \] 可见\(c_j \geq 0\), \(j=0,1,2,\dots\)，故\(\{\varepsilon_t^2 \}\)的协方差函数 \[ \gamma_{\varepsilon_t^2}(k) = \sigma_{\eta}^2 \sum_{n=0}^\infty c_n c_{n+k} \geq 0 . \] 即\(\{\varepsilon_t^2 \}\)的相关系数都是非负的。如果对某个\(j_0\)满足\(\alpha_{j_0} > 0\)，则\(\sum_{k=0}^\infty (\sum_{j=1}^p \alpha_j t^j)^k\)中\(t^{k j_0}\)系数为正值，从而\(c_{k j_0} > 0\)，这时 \[\begin{aligned} \gamma_{\varepsilon_t^2}(k j_0) \geq& \sigma_{\eta}^2 \sum_{m=0}^\infty c_{m j_0} c_{m j_0 + k j_0} = \sigma_{\eta}^2 \sum_{m=0}^\infty c_{m j_0} c_{(m+k) j_0} > 0 . \end{aligned}\]

上面的例子说明ARCH模型对应的\(\{\varepsilon_t \}\)序列通常是厚尾分布的，而且\(\{\varepsilon_t^2 \}\)都是正相关的。

如果金融收益率序列\(\{r_t \}\)是平稳列，有Wold分解，其新息序列为\(\{\varepsilon_t \}\)，设其满足鞅差条件： \[ E(\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) = 0, \] 则可设\(\{\varepsilon_t \}\)满足ARCH(\(p\))模型，这个模型保证了波动率聚集效应（即\(\{\varepsilon_t^2 \}\)序列相关性为正相关），厚尾分布，条件方差随时间变化，称为ARCH效应。对实际数据建模时，可以对平稳数据先建立ARMA模型，将这样的模型看做是关于条件期望\(E(r_t | r_{t-1}, r_{t-2}, \dots)\)的模型，然后将模型的残差看做是新息序列\(\{\varepsilon_t \}\)，可以通过检验\(\{\varepsilon_t^2 \}\)是否白噪声列来检验是否具有ARCH效应，如果有ARCH效应则对\(\{\varepsilon_t \}\)建立ARCH(\(p\))模型，作为条件方差\(\text{Var}(r_t | r_{t-1}, r_{t-2}, \dots)\)的模型。

29.4 ARCH模型参数估计

办法是设定\(v_t\)的分布，称为“条件分布”，然后定义似然函数，进行最大似然估计。对\(v_t\)可以考虑使用正态分布、t分布等类型，为了进行模型选择，可以计算AIC准则值，以及对拟合残差进行残差诊断。

29.5 GARCH模型

ARCH模型容易理解，有严平稳遍历解，但是在实际数据建模时，往往需要比较高阶才能良好拟合。为此，类似于从AR模型推广到ARMA模型，将ARCH模型推广为如下的GARCH模型（广义自回归条件异方差模型）。仍设\(\{r_t \}\)为平稳的资产收益率， \(\{\varepsilon_t \}\)为其新息，满足\(E(\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) = 0\)。

定义29.2 (GARCH模型) 设\(\{v_t \}\)为独立同分布WN(0,1)列，非负常数\(\alpha_i\), \(\beta_j\)满足条件 \[\begin{equation} \sum_{i=1}^p \alpha_i + \sum_{j=1}^q \beta_j < 1, \quad \alpha_0 \alpha_p \beta_q > 0 . \tag{29.10} \end{equation}\] 称 \[\begin{equation} \begin{cases} \varepsilon_t = \sigma_t v_t, \\ \sigma_t^2 = \alpha_0 + \sum_{i=1}^p \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^q \beta_j \sigma_{t-j}^2, \\ v_t \text{与} \{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots \} \text{独立} . \end{cases} \tag{29.11} \end{equation}\] 为GARCH(\(p,q\))模型，其中\(\sigma_t^2 = \text{Var}(\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots)\)。如果\(\{\varepsilon_t \}\)是严平稳白噪声，满足模型(29.11)，则称\(\{\varepsilon_t \}\)是GARCH(\(p,q\))序列。

下面讨论模型(29.11)的严平稳解存在性。记\(h = \max(p,q)\)，当\(i>p\)时定义\(\alpha_i=0\), 当\(j>q\)时定义\(\beta_j = 0\)。定义两个多项式 \[ \alpha(t) = \sum_{i=1}^p \alpha_i t^i, \quad \beta(t) = \sum_{j=1}^q \beta_j t_j , \ t \in [-1, 1] . \] 由(29.10)，可以展开\(\frac{1}{1 - \beta(t)}\)为 \[ \frac{1}{1 - \beta(t)} = \sum_{k=0}^\infty [\beta(t)]^k, \] 于是 \[\begin{equation} [1 - \beta(t)]^{-1} \alpha(t) = \alpha(t) \sum_{k=0}^\infty [\beta(t)]^k = \sum_{n=1}^\infty c_n t^n . \tag{29.12} \end{equation}\] 因为\(\alpha(t)\)和\(\beta(t)\)都是非负系数的多项式，所以\(\sum_{k=0}^\infty [\beta(t)]^k\)也是非负系数的，从而\(c_n\)都非负，且存在正值。

由(29.10)可知\(\alpha(1) + \beta(1) < 1\)，所以 \[\begin{equation} 0 < c = \sum_{n=1}^\infty c_n = \frac{\alpha(1)}{1 - \beta(1)} < 1 . \tag{29.13} \end{equation}\]

定理29.2 在GARCH(\(p,q\))模型中，记\(u_t = v_t^2\)， \(\{c_n \}\), \(c\)分别由(29.12)和(29.13)定义。引入 \[\begin{equation} \begin{cases} A_t(0) = u_t, \\ A_t(n) = \sum_{i_1, i_2, \dots, i_n \geq 1} c_{i_1} c_{i_2} \dots c_{i_n} u_{t} u_{t - i_1} u_{t - i_2} \dots u_{t - i_n} . \end{cases} \tag{29.14} \end{equation}\] 则有以下结果：

（1）GARCH模型(29.11)有如下的严平稳遍历解 \[\begin{equation} \begin{cases} \varepsilon_t = \left[ \alpha_0 \sum_{n=0}^\infty A_t(n) \right]^{1/2} v_t , \\ E(\varepsilon_t) = 0, \quad E(\varepsilon_t^2) = \frac{\alpha_0}{ 1 - \sum_{i=1}^p \alpha_i - \sum_{j=1}^q \beta_j} . \end{cases} \tag{29.15} \end{equation}\]

（2）如果\(\{e_t \}\)也是(29.11)的严平稳解，则\(e_t^2 = \varepsilon_t^2\), a.s.

（3）如果\(c \sqrt{E(v_1^4)} < 1\)，则\(E(\varepsilon_t^4) < \alpha_0^2 E(v_1^4) / (1 - c \sqrt{E(v_1^4)})^2\) .

证明：（1）令\(c_0 = \alpha_0 / (1 - \beta(1))\)。 \(\{c_n \}\), \(\{u_t \}\)满足引理29.1的条件，所以模型 \[ \begin{cases} Y_t = \left(c_0 + \sum_{j=1}^\infty c_j Y_{t-j} \right) u_t, \\ u_t \text{与} \{ Y_{t-1}, Y_{t-2}, \dots \} \text{独立} \end{cases} \] 的唯一的严平稳遍历解是 \[ Y_t = c_0 \sum_{n=0}^\infty A_t(n) . \] 并且\(E(Y_t) = c_0 / (1 - c) < \infty\). 定义 \[\begin{equation} \varepsilon_t = \left( c_0 + \sum_{j=1}^\infty c_j Y_{t-j} \right)^{1/2} v_t, \tag{29.16} \end{equation}\] 则\(\varepsilon_t^2 = Y_t\)，由定理4.1可知\(\{\varepsilon_t \}\)是严平稳遍历序列，由\(A_t(n)\)定义可以看出\(v_t\)与\(\{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots\}\)独立，于是 \[ E(\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) = E\left( c_0 + \sum_{j=1}^\infty c_j \varepsilon_{t-j}^2 \right)^{1/2} E(v_t) = 0, \] 又 \[\begin{aligned} \sigma_t^2 =& \text{Var}(\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) \\ =& E(\varepsilon_t^2 | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) \\ =& E(Y_t^2 | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) \\ =& \left( c_0 + \sum_{j=1}^\infty c_j \varepsilon_{t-j}^2 \right) E(v_t^2 | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) \\ =& \left( c_0 + \sum_{j=1}^\infty c_j \varepsilon_{t-j}^2 \right) E(v_t^2) \\ =& c_0 + \sum_{j=1}^\infty c_j \varepsilon_{t-j}^2 . \end{aligned}\] 由定理4.1可知\(\{\sigma_t \}\)也是严平稳遍历序列。上式表明ARCH(\(p,q\))模型是一种无穷阶的ARCH模型。用推移算子可以将上式写成 \[ \sigma_t^2 = c_0 + [1 - \beta(\mathscr B)]^{-1} \alpha(\mathscr B) \varepsilon_t^2 . \] 由\(c_0 = \frac{\alpha_0}{1 - \beta(1)} = [1 - \beta(\mathscr B)]^{-1} \alpha_0\)可见 \[ \sigma_t^2 = [1 - \beta(\mathscr B)]^{-1} \alpha_0 + [1 - \beta(\mathscr B)]^{-1} \alpha(\mathscr B) \varepsilon_t^2 = [1 - \beta(\mathscr B)]^{-1} (\alpha_0 + \alpha(\mathscr B) ) \varepsilon_t^2 . \] 在上式两边作用\(1 - \beta(\mathscr B)\)得 \[ [1 - \beta(\mathscr B)] \sigma_t^2 = \alpha_0 + \alpha(\mathscr B) \varepsilon_t^2 , \] 可写成 \[ \sigma_t^2 = \alpha_0 + \alpha(\mathscr B) \varepsilon_t^2 + \beta(\mathscr B) \sigma_t^2, \] 即 \[ \sigma_t^2 = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dots + \alpha_p \varepsilon_{t-p}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_q \sigma_{t-q}^2 . \] 即严平稳遍历序列\(\{\varepsilon_t \}\)满足GARCH模型(29.11)， \[ E(\varepsilon_t^2) = E(Y_t) = \frac{c_0}{1 - c} = \frac{\alpha_0 / (1 - \beta(1))}{1 - \frac{\alpha(1)}{1 - \beta(1)}} = \frac{\alpha_0}{1 - \beta(1) - \alpha(1)} . \] 与定理29.1证明类似可见\(\{\varepsilon_t \}\)是白噪声列。

（2）和（3）的证明与定理29.1证明类似。

令\(h = \max(p,q)\)， \[ A(z) = 1 - \alpha(z) - \beta(z), \quad B(z) = 1 - \beta(z), \] 则 \[ A(z) = 1 - \sum_{j=1}^h (\alpha_j + \beta_j) z^j, \quad B(z) = 1 - \sum_{j=1}^q \beta_j z^j . \] 条件(29.10)保证\(A(t)\)满足最小相位性， \(B(t)\)满足可逆性。事实上，记\(a_j = \alpha_j + \beta_j\)，则\(a_j \geq 0\)， \(\sum_{j=1}^h a_j = \sum_{j=1}^p \alpha_j + \sum_{j=1}^q \beta_j < 1\)。于是对任意复数\(z\)满足\(|z| \leq 1\)，有 \[ \left| \sum_{j=1}^h a_j z^j \right| \leq \sum_{j=1}^h a_j |z|^j \leq \sum_{j=1}^h a_j < 1, \] 所以\(1 - \sum_{j=1}^h a_j z^j\)在\(|z| \leq 1\)时没有零点，即\(A(z)\)满足最小相位条件，类似可知\(B(z)\)满足可逆条件。

命题29.1 如果\(E(v_1^4) < \infty\), \(E(\varepsilon_1^4) < \infty\)，则GARCH(\(p,q\))序列\(\{ \varepsilon_t^2 \}\)满足如下ARMA(\(p,q\))模型： \[\begin{equation} A(\mathscr B) \varepsilon_t^2 = \alpha_0 + B(\mathscr B) \eta_t, \tag{29.17} \end{equation}\]

\[ \] 其中\(\{\eta_t \}\)是严平稳的零均值白噪声列。

证明：令\(\eta_t = \varepsilon_t^2 - \sigma_t^2\)，则\(\{\eta_t \}\)为严平稳列， \(\varepsilon_t^2 = \sigma_t^2 + \eta_t\)，代入到(29.11)中可得 \[\begin{aligned} \varepsilon_t^2 =& \sigma_t^2 + \eta_t \\ =& \alpha_0 + \alpha(\mathscr B) \varepsilon_t^2 + \beta(\mathscr B) \sigma_t^2 + \eta_t \\ =& \alpha_0 + \alpha(\mathscr B) \varepsilon_t^2 + \beta(\mathscr B)(\varepsilon_t^2 - \eta_t) + \eta_t \\ =& \alpha_0 + [\alpha(\mathscr B) + \beta(\mathscr B)] \varepsilon_t^2 + \eta_t - \beta(\mathscr B) \eta_t \\ =& \alpha_0 + [1 - A(\mathscr B)] \varepsilon_t^2 + B(\mathscr B) \eta_t, \end{aligned}\] 这就是模型(29.17)，且满足最小相位条件和可逆性条件。只要再证明\(\{\eta_t \}\)是零均值白噪声列。 \[ \eta_t = \varepsilon_t^2 - \sigma_t^2 = \sigma_t^2 (v_t^2 - 1), \] 由\(v_1\)四阶矩有限和\(\varepsilon_1\)四阶矩有限可知\(\eta_t\)二阶矩有限，由\(\{\eta_t \}\)严平稳可知\(\{\eta_t \}\)宽平稳。由\(v_t\)与\(\{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots \}\)独立， \(\sigma_t\)由\(\{\varepsilon_{t-1}, \varepsilon_{t-2}, \dots \}\)决定，可知\(v_t\)与\(\sigma_t\)独立，从而 \[\begin{aligned} E(\eta_t) =& E(\sigma_t^2) E(v_t^2 - 1) = 0, \\ E(\eta_t \eta_{t+k}) =& E(\sigma_t^2 (v_t^2 - 1) \sigma_{t+k}^2) E(v_{t+k}^2 - 1) = 0 . \end{aligned}\] 由于\(E(\varepsilon_1^4) < \infty\), \(E(v_1^4) < \infty\)，必有\(E(\sigma_1^4) < \infty\)，这是因为 \[ \sigma_t^2 = E(\varepsilon_t^2 | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots), \] 由条件Jensen不等式有 \[ \sigma_t^4 = [E(\varepsilon_t^2 | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots)]^2 \leq E(\varepsilon_t^4 | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots), \text{ a.s.}, \] 从而 \(E(\sigma_t^4) \leq E(\varepsilon_t^4) < \infty\)。由\(\{\sigma_t \}\)的严平稳性可知 \[ E(\eta_t^2) = E(\sigma_t^4) E[(v_t^2 - 1)^2] < \infty, \] 且不依赖于\(t\)，从而\(\{\eta_t \}\)是严平稳的零均值白噪声列。

GARCH序列也具有厚尾性。

命题29.2 设\(\{\varepsilon_t \}\)是GARCH(\(p,q\))序列， \(\kappa_{\varepsilon}\)是\(\varepsilon_t\)的峰度， \(\kappa_v\)是\(v_t\)的峰度，则

（1）\(\kappa_{\varepsilon} \geq \kappa_v\)，等号成立当且仅当\(\sigma_t\)为常数(a.s.);

（2）如果\(E(v_1^4) < \infty\), \(E(\varepsilon_1^4) < \infty\)，则\(\{ \varepsilon_t^2 \}\)的自相关系数\(\rho_k = \text{Corr}(\varepsilon_t^2, \varepsilon_{t+k}^2)\)都非负，且当\(\alpha_j > 0\)时所有的\(\rho_{kj} > 0\)， \(k=1,2,\dots\)。

证明略。

References

———. 2023. 应用时间序列分析. 2nd ed. 北京大学出版社.