在金融领域中, 大量时间序列数据具有丰富的特征, 如股票收益具有长期记忆性, 收益分布具有尖峰、细尾等^[1], %如波动的长期记忆以及波动率结构变化等, 而这些特征有助于投资者选择最佳投资组合, 以便减少风险.为了考虑金融资产收益率序列的长记忆性, Hosking^[2]提出了ARFIMA模型, 但ARFIMA模型不能够刻画资产收益率中普遍存在的波动聚类, 波动率是资产收益不确定性的衡量, 常被用来衡量资产的风险.Bollerslev^[3]提出的GARCH模型能有效地捕捉资产收益率波动集聚现象.为了同时描述资产收益率中的长记忆性和异方差性, 因此产生了ARFIMA-GARCH模型, 该模型有利于投资者研究金融资产组合的风险水平问题.

然而, 许多工作致力于模型的提出以及参数估计, 对于模型诊断检验没有得到相应的重视.对于实践者来说, 去检验模型的准确性是一个基本的问题.在时间序列模型中, 混成检验是一种广泛有用的诊断工具, 特别对于拟合的模型.近年来, 随着金融市场日益变得复杂, 关于混成检验也相继有了一些较为成熟的结果.对于拟极大指数似然估计, Li^[4]和Zhu^[5]分别给出不同的混成检验统计量.基于高斯拟极大似然估计, Wong和Ling^[6]研究了混合混成检验统计量.

本文对于ARFIMA-GARCH模型, 在拟极大指数似然估计下, 给出了平方残差自相关函数的渐近性, 并进行了证明, 进而得到了基于平方残差自相关函数的混成检验统计量; 由实例分析可知, 该统计量有利于诊断检验由拟极大指数似然估计拟合的ARFIMA-GARCH模型, 以便确定在实际的股票市场中, 拟合的模型是否准确, 确保对未来股票市场的变化做出较准确的预测.

ARFIMA $(p, d, q)$-GARCH $(r, s)$模型形式如下

$ \begin{eqnarray} \Phi(B)(1-B)^{d}y_{t}=\Psi(B)\varepsilon_{t}, \\ \end{eqnarray} $

(2.1)

$ \begin{eqnarray} \varepsilon_{t}=\eta_{t}\sqrt{h_{t}}, h_{t}=\alpha_{0}+\sum_{i=1}^{r}\alpha_{i}\varepsilon_{t-i}^{2}+\sum_{j=1}^{s}\beta_{j}h_{t-j}, \end{eqnarray} $

(2.2)

其中 $B$为滞后算子, $d$是分数差分参数, $-1/2<d<1/2$, $y_{t}$为 $t$时刻的对数收益率, $\Phi(B)$和 $\Psi(B)$分别是 $p$阶和 $q$阶的滞后多项式, $\Phi(B)=1-\phi_{1}B-\phi_{2}B-\cdots-\phi_{p}B^{p}$, $\Psi(B)=1+\psi_{1}B+\psi_{2}B\cdots+\psi_{q}B^{q}$, $(1-B)^{d}=\sum\limits_{k=0}^{\infty}\frac{k-d-1)!}{k!(-d-1)!}B^{k}$, $\varepsilon_{t}$是信息序列, 且均值为0, $\{\eta_{t}\}$是均值为零的独立同分布随机变量序列, $\alpha_{0}>0$, $\alpha_{i}\geq0(i=1, 2, \cdots, r)$, $\beta_{j}\geq0(j=1, 2, \cdots, s)$, 对所有的 $t$都有 $\eta_{t}$和 $\varepsilon_{t}$是相互独立的, 即称随机过程 $\{\varepsilon_{t}\}$为ARFIMA $(p, d, q)$-GARCH $(r, s)$模型.

设 $\theta=(\gamma^{'}, \delta^{'})^{'}$是ARFIMA $(p, q, d)$-GARCH $(r, s)$模型的未知参数, 它的真实值是 $\theta_{0}$, 其中 $\gamma=\left(d, \phi_{1}, \phi_{2}, \cdots, \phi_{p}, \psi_{1}, \psi_{2}, \cdots, \psi_{q}\right)^{'}$, $\delta=\left(\alpha_{0}, \cdots, \alpha_{r}, \beta_{1}, \cdots, \beta_{s}\right)^{'}$.令 $l=p+q+r+s+2$, 那么 $\theta$是一个 $l$维向量, 参数空间为 $\Theta=\Theta_{\gamma}\times\Theta_{\delta}$, 其中 $\Theta_{\gamma}\subset\mathbb{R}^{p+q+1}$, $\Theta_{\delta}\subset\mathbb{R}^{r+s+1}_{0}$, $\mathbb{R}=\left(-\infty, +\infty\right)$, $\mathbb{R}_{0}=\left[0, +\infty\right)$.设 $\theta_{0}$是 $\Theta$中的一个内点, 定义 $\alpha(B)=\sum\limits_{i=1}^{r}\alpha_{i}B^{i}$, $\beta(B)=1-\sum\limits_{j=1}^{s}\beta_{j}B^{j}$.

假设1   $-1/2<d<1/2$, 对于每个 $\theta\in\Theta$, 多项式 $\phi(B)$和 $\psi(B)$的所有根都在单位圆之外, 当 $\phi_{p}\neq0$或者 $\psi_{q}\neq0$时, $\phi(B)$和 $\psi(B)$没有公共根.

假设2  对于每个 $\theta\in\Theta$, $\alpha(B)$和 $\beta(B)$没有公共根, $\alpha(1)\neq0$, $\alpha_{r}+\beta_{s}\neq0$, $\sum\limits_{j=1}^{s}\beta_{j}<1$.

假设1表明了 $\{y_{t}\}$的平稳可逆性, 当 $-1/2<d<0$时, ${y_{t}}$表现出间断记忆性, 然而当 $0<d<1/2$时, ${y_{t}}$表现出长期记忆性.假设2是模型(2.2) 的可识别性条件.给出观测值 $\{y_{n}, \cdots, y_{1}\}$和初始值 $Y_{0}\equiv\{y_{0}, y_{-1}, \cdots\}$, 写参数模型如下

$ \begin{align} & \Phi (B){{(1-B)}^{d}}{{y}_{t}}=\Psi (B){{\varepsilon }_{t}}(\gamma ), \\ & {{\varepsilon }_{t}}(\gamma )={{\eta }_{t}}(\theta )\sqrt{{{h}_{t}}(\theta )},{{h}_{t}}(\theta )={{\alpha }_{0}}+\sum\limits_{i=1}^{r}{{{\alpha }_{i}}}\varepsilon _{t-i}^{2}(\theta )+\sum\limits_{j=1}^{s}{{{\beta }_{j}}}{{h}_{t-j}}(\theta ). \\ \end{align} $

设 $\hat{\theta}_{n}=(\hat{\gamma}_{n}^{'}, \hat{\delta}_{n}^{'})^{'}$, 由文献[7]可知 $\theta_{0}$的拟极大指数似然估计被定义为

$ {{{\hat{\theta }}}_{n}}=\arg \underset{\theta \in \Theta }{\mathop{\rm{min}}}\,{{L}_{n}}(\theta ),{{L}_{n}}(\theta )=\sum\limits_{t=1}^{n}{\left[ \log \sqrt{{{h}_{t}}(\theta )}+\frac{|{{\varepsilon }_{t}}(\gamma )|}{\sqrt{{{h}_{t}}(\theta )}} \right]}. $

假设3   $\eta_{t}$的中位数等于0, $E|\eta_{t}|=1$, ${\rm Var}(\eta_{t}^2)=\sum\limits_{t=1}^{n}\eta_{t}^2<\infty$, $\eta_{t}$的概率密度函数满足 $f(0)>0, \sup\limits_{x\in \mathbb{R}}f(x)<\infty$, 且在0处是连续的.

假设4   $\varepsilon_{t}$是严平稳遍历过程, 且 $E\varepsilon_{t}^2<\infty$.

假设5   $\sqrt{n}(\hat{\theta}_{n}-\theta_{0})=O_{p}(1)$.

假设3是LAD类估计的一般性条件, 见文献[4].假设4的充分必要条件是 $\sum\limits_{i=1}^{r}\alpha_{i}+\sum\limits_{j=1}^{s}\beta_{j}<1$, 见文献[3].假设5是为了简化的证明.在假设1-4下, 由文献[7]可知

$ \sqrt{n}\left(\hat{\theta}_n-\theta_0\right)\rightarrow_d\left(0, \frac{1}{4}\Sigma^{-1}_0\Omega_{0}\Sigma^{-1}_0\right), $

其中

$ \begin{array}{l} {\Omega _0} = E\left[ {\frac{1}{{{h_t}({\theta _0})}}\frac{{\partial {\varepsilon _t}({\gamma _0})}}{{\partial \theta }}\frac{{\partial {\varepsilon _t}({\gamma _0})}}{{\partial {\theta ^T}}}} \right] + \frac{{E\eta _t^2 - 1}}{4}E\left[ {\frac{1}{{h_t^2({\theta _0})}}\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }}{{\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }}}^\mathit{T}}} \right],\\ {\Sigma _0} = f(0)E\left[ {\frac{1}{{{h_t}({\theta _0})}}\frac{{\partial {\varepsilon _t}({\gamma _0})}}{{\partial \theta }}\frac{{\partial {\varepsilon _t}({\gamma _0})}}{{\partial {\theta ^T}}}} \right] + \frac{1}{8}E\left[ {\frac{1}{{h_t^2({\theta _0})}}\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }}{{\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }}}^T}} \right]. \end{array} $

定义残差 $\hat{\eta}_{t}=\eta_{t}(\hat{\theta}_{n})$, 那么滞后 $\iota$平方残差自相关函数被定义为

$ \hat{\rho}_{\iota}^{*} =\frac{\sum\limits_{t=\iota+1}^{n}(\hat{\eta}_{t}^2-\bar{\eta})(\hat{\eta}_{t-\iota}^2-\bar{\eta})}{\sum\limits_{t=1}^{n}(\hat{\eta}_{t}^2-\bar{\eta})^{2}}, $

其中 $\bar{\eta}=\frac{1}{n}\sum\limits_{t=1}^{n}\hat{\eta}_{t}^2$, 由假设5可知, $\hat{\theta}_{n}-\theta_{0}=o_{p}(1)$.在假设1-4下, 在文献[8]中由定理3.1和控制收敛定理, 能够表明

$ \bar{\eta}=\mu+o_{p}(1), \frac{1}{n}\sum\limits_{t=\iota+1}^{n}(\hat{\eta}_{t}^2-\bar{\eta})^{2}=\sigma_{0}^{2}+o_{p}(1), $

其中 $\mu=E\eta_{t}^2$, 因此理论上只需要考虑

$ \hat{\rho}_{\iota}=\frac{1}{n\sigma_{0}^{2}}\sum\limits_{t=\iota+1}^{n}(\hat{\eta}_{t}^2-\mu)(\hat{\eta}_{t-\iota}^2-\mu). $

(3.1)

设 $C=\left(C_{1}, C_{2}, \cdots, C_{M}\right)^{'}$, $\hat{C}=\left(\hat{C_{1}}, \hat{C_{2}}, \cdots, \hat{C_{M}}\right)^{'}$, 其中 $M$为正整数, $\iota=1, 2, \cdots, M$, $C_{\iota}=\frac{1}{n}\sum\limits_{t=\iota+1}^{n}\left(\eta_{t}^2-\mu\right)\left(\eta_{t-\iota}^2-\mu\right)$, $\hat{C}_{\iota}=\frac{1}{n}\sum\limits_{t=\iota+1}^{n}\left(\hat{\eta}_{t}^2-\mu)(\hat{\eta}_{t-\iota}^2-\mu\right)$, 由泰勒展示, 有

$ \hat{C}\simeq C+\frac{\partial C}{\partial \theta}\left(\hat{\theta}_{n}-\theta_{0}\right), $

(3.2)

其中 $\partial C/\partial \theta=\left(\partial C_{1}/\partial \theta, \partial C_{2}/\partial \theta, \cdots, \partial C_{m}/\partial \theta\right)^{'}$,

$ \begin{align} & \frac{\partial {{C}_{\iota }}}{\partial \theta }=\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\frac{\partial \left( \eta _{t}^{2}-\mu \right)}{\partial \theta }}(\eta _{t-\iota }^{2}-\mu )+\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{(\eta _{t}^{2}-\mu )}\frac{\partial (\eta _{t-\iota }^{2}-\mu )}{\partial \theta } \\ & \ \ \ \ \ \ \ \ =\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\frac{\partial (\frac{\varepsilon _{t}^{2}}{{{h}_{t}}}-\mu )}{\partial \theta }}\left( \frac{\varepsilon _{t-\iota }^{2}}{{{h}_{t}}}-\mu \right)+\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\left( \frac{\varepsilon _{t}^{2}}{{{h}_{t}}}-\mu \right)}\frac{\partial (\frac{\varepsilon _{t-\iota }^{2}}{{{h}_{t-\iota }}}-\mu )}{\partial \theta } \\ & \ \ \ \ \ \ \ \ =\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\left( \frac{2{{\varepsilon }_{t}}}{{{h}_{t}}}\frac{\partial {{\varepsilon }_{t}}}{\partial \theta }-\frac{\varepsilon _{t}^{2}}{h_{t}^{2}}\frac{\partial {{h}_{t}}}{\partial \theta } \right)}\left( \frac{\varepsilon _{t-\iota }^{2}}{{{h}_{t-\iota }}}-\mu \right) \\ & \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\left( \frac{\varepsilon _{t}^{2}}{{{h}_{t}}}-\mu \right)}\left( \frac{2{{\varepsilon }_{t-\iota }}}{{{h}_{t-\iota }}}\frac{\partial {{\varepsilon }_{t-\iota }}}{\partial \theta }-\frac{\varepsilon _{t-\iota }^{2}}{h_{t-\iota }^{2}}\frac{\partial {{h}_{t-\iota }}}{\partial \theta } \right) \\ & \ \ \ \ \ \ \ \ =\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\frac{2{{\varepsilon }_{t}}}{{{h}_{t}}}}\frac{\partial {{\varepsilon }_{t}}}{\partial \theta }\left( \frac{\varepsilon _{t-\iota }^{2}}{{{h}_{t-\iota }}}-\mu \right)-\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\frac{\varepsilon _{t}^{2}}{h_{t}^{2}}}\frac{\partial {{h}_{t}}}{\partial \theta }\left( \frac{\varepsilon _{t-\iota }^{2}}{{{h}_{t-\iota }}}-\mu \right) \\ &\ \ \ \ \ \ \ \ \ \ \ +\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\left( \frac{\varepsilon _{t}^{2}}{{{h}_{t}}}-\mu \right)}\frac{2{{\varepsilon }_{t-\iota }}}{{{h}_{t-\iota }}}\frac{\partial {{\varepsilon }_{t-\iota }}}{\partial \theta }-\frac{1}{n}\sum\limits_{t=\iota +1}^{n}{\left( \frac{\varepsilon _{t}^{2}}{{{h}_{t}}}-\mu \right)}\frac{\varepsilon _{t-\iota }^{2}}{h_{t-\iota }^{2}}\frac{\partial {{h}_{t-\iota }}}{\partial \theta }, \\ \end{align} $

由遍历定理可知, 上式中最后一个等式的第一、三、四项都为 $0$, 因此有

$ \frac{\partial C_{\iota}}{\partial \theta}=-\frac{1}{n}\sum\limits_{t=\iota+1}^n\frac{\varepsilon_{t}^2}{h_{t}^2}\frac{\partial h_{t}}{\partial \theta}\left(\frac{\varepsilon_{t-\iota}^2}{h_{t-\iota}}-\mu\right)=-\frac{1}{n}\sum\limits_{t=\iota+1}^n\frac{\eta_{t}^2\left(\eta_{t-\iota}^2-\mu\right)}{h_{t}}\frac{\partial h_{t}}{\partial \theta}, $

因此当 $n\rightarrow \infty$时, $ \frac{\partial C_{\iota}}{\partial \theta}\overset{\text{a.s.}}{\longrightarrow}-\mu X_{\rho\iota}, $其中 $X_{\rho\iota}=E\left[\frac{\eta_{t-\iota}^2-\mu}{h_{t}}\frac{\partial h_{t}}{\partial \theta}\right]$.令 $X_{\rho}=\left(X_{\rho1}, X_{\rho2}, \cdots, X_{\rho M}\right)^{'}$, 则等式(3.2) 可写成

$ \hat{C}\simeq C+\left(-\mu X_{\rho}\right)\left(\hat{\theta}_{n}-\theta_{0}\right). $

(3.3)

设 $\hat{\rho}=\left(\hat{\rho}_{1}, \hat{\rho}_{2}, \cdots, \hat{\rho}_{M}\right)^T$, $\rho=\left(\rho_{1}, \rho_{2}, \cdots, \rho_{M}\right)^T$, 由等式(3.1) 和等式(3.3), 有

$ \sqrt{n}\hat{\rho}=\sqrt{n}\rho+\sigma_{0}^{-2}\left(-\mu X_{\rho}\right)\sqrt{n}\left(\hat{\theta}_{n}-\theta_{0}\right)+o_{p}(1). $

(3.4)

定理1  如果假设1-5成立, 那么

$ \sqrt{n}\hat{\rho}=\left(\hat{\rho}_{1}, \hat{\rho}_{2}, \cdots, \hat{\rho}_{M}\right)^{'}\longrightarrow_{d}N\left(0, V\right), $

其中

$ \begin{align} & V={{1}_{M}}+\frac{{{\mu }^{2}}}{4\sigma _{0}^{4}}{{X}_{\rho }}\Sigma _{0}^{-1}{{\Omega }_{0}}\Sigma _{0}^{-1}X_{\rho }^{T}-\frac{\mu {{\kappa }_{1}}}{2\sigma _{0}^{4}}{{X}_{\rho }}\Sigma _{0}^{-1}X_{\rho }^{T}+\frac{\mu {{\kappa }_{2}}}{2\sigma _{0}^{4}}\left( X_{\rho }^{*}\Sigma _{0}^{-1}X_{\rho }^{T}+X_{\rho }^{T}\Sigma _{0}^{-1}X_{\rho }^{*} \right), \\ & X_{\rho }^{*}={{\left( X_{\rho 1}^{*}, X_{\rho 2}^{*}, \cdots, X_{\rho M}^{*} \right)}^{\mathit{'}}}, X_{\rho \iota }^{*}=E\left[\frac{\eta _{t-\iota }^{2}-\mu }{\sqrt{{{h}_{t}}}}\frac{\partial {{\varepsilon }_{t}}({{\theta }_{0}})}{\partial \theta } \right], \\ & {{\kappa }_{1}}=E\left[\eta _{t}^{2}(|{{\eta }_{t}}|-1) \right], {{\kappa }_{2}}=E\left[\rm{sgn}({{\eta }_{t}})\eta _{t}^{2} \right]. \\ \end{align} $

证  由文献[9]中的定理2.8.1, 能够得到 $ \sqrt{n}C\longrightarrow_{d}N\left(0, \sigma_{0}^{4}1_{M}\right), $其中 $l_{M}$为 $M\times M$阶单位矩阵.令

$ D_{t}=\frac{|\eta_{t}|-1}{4h_{t}}\frac{\partial h_{t}(\theta_0)}{\partial \theta}-\frac{{\rm sgn}(\eta_{t})}{2\sqrt{h_t}}\frac{\partial \varepsilon_t(\theta_0)}{\partial \theta}, $

由文献[3]可知

$ \sqrt{n}\left(\hat{\theta}_n-\theta_0\right)=\frac{\Sigma_0^{-1}}{\sqrt{n}}\sum\limits_{t=1}^nD_{t}+o_p(1), $

因此由Mann-Wald device和鞅差中心极限定理^[10]可知

$ \begin{align} & \ \ \ \rm{Cov}\left( \sqrt{\mathit{n}}\rm{(}{{{\mathit{\hat{\theta }}}}_{\mathit{n}}}\mathit{-}{{\mathit{\theta }}_{\rm{0}}}\rm{)}\mathit{,}\sqrt{\mathit{n}}\mathit{C} \right)=\mathit{E}\left[ \sqrt{\mathit{n}}\rm{(}{{{\mathit{\hat{\theta }}}}_{\mathit{n}}}\mathit{-}{{\mathit{\theta }}_{\rm{0}}})\cdot \sqrt{\mathit{n}}{{\mathit{C}}^{\mathit{T}}} \right] \\ & =\mathit{E}\left[ \frac{\Sigma _{0}^{-1}}{\sqrt{\mathit{n}}}\sum\limits_{\mathit{t}=1}^{\mathit{n}}{{{\mathit{D}}_{\mathit{t}}}}\cdot \sqrt{\mathit{n}}{{\mathit{C}}^{\mathit{T}}} \right]=\Sigma _{0}^{-1}\mathit{E}\left[ \sum\limits_{\mathit{t}=1}^{\mathit{n}}{{{\mathit{D}}_{t}}}\cdot {{\mathit{C}}^{\mathit{T}}} \right], \\ \end{align} $

又

$ \begin{array}{l} E[\sum\limits_{t = 1}^n {{D_t}} \cdot {C_\iota }] = E\left[ {\sum\limits_{t = 1}^n {\left( {\frac{{|{\eta _t}| - 1}}{{4{h_t}}}\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }} - \frac{{{\rm{sgn}}({\eta _t})}}{{2\sqrt {{h_t}} }}\frac{{\partial {\varepsilon _t}({\theta _0})}}{{\partial \theta }}} \right)} \cdot } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\;\frac{1}{n}\sum\limits_{s = \iota + 1}^n {\left( {\eta _s^2 - \mu } \right)} \left( {\eta _{s - \iota }^2 - \mu } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{1}{n}E\left[ {\sum\limits_{t = 1}^n {\left( {\frac{{|{\eta _t}| - 1}}{{4{h_t}}}\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }}} \right)} \cdot \sum\limits_{s = \iota + 1}^n {\left( {\eta _s^2 - \mu } \right)} \left( {\eta _{s - \iota }^2 - \mu } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{1}{n}E\left[ {\frac{{{\rm{sgn}}({\eta _t})}}{{2\sqrt {{h_t}} }}\frac{{\partial {\varepsilon _t}({\theta _0})}}{{\partial \theta }}) \cdot \sum\limits_{s = \iota + 1}^n {\left( {\eta _s^2 - \mu } \right)} \left( {\eta _{s - \iota }^2 - \mu } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{1}{n}\sum\limits_{t = 1}^n {\sum\limits_{s = \iota + 1}^n E } \left[ {\frac{{|{\eta _t}| - 1}}{{4{h_t}}}\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }}\left( {\eta _s^2 - \mu } \right)\left( {\eta _{s - \iota }^2 - \mu } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{1}{n}\sum\limits_{t = 1}^n {\sum\limits_{s = \iota + 1}^n E } \left[ {\frac{{{\rm{sgn}}({\eta _t})}}{{2\sqrt {{h_t}} }}\frac{{\partial {\varepsilon _t}({\theta _0})}}{{\partial \theta }}\left( {\eta _s^2 - \mu } \right)\left( {\eta _{s - \iota }^2 - \mu } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{1}{n}\sum\limits_{t = 1}^n E \left[ {\frac{{|{\eta _t}| - 1}}{{4{h_t}}}\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }}\left( {\eta _t^2 - \mu } \right)\left( {\eta _{t - \iota }^2 - \mu } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{1}{n}\sum\limits_{t = 1}^n E \left[ {\frac{{{\rm{sgn}}({\eta _t})}}{{2\sqrt {{h_t}} }}\frac{{\partial {\varepsilon _t}({\theta _0})}}{{\partial \theta }}\left( {\eta _t^2 - \mu } \right)\left( {\eta _{t - \iota }^2 - \mu } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = E\left[ {\frac{{|{\eta _t}| - 1}}{{4{h_t}}}\frac{{\partial {h_t}({\theta _0})}}{{\partial \theta }}\left( {\eta _t^2 - \mu } \right)\left( {\eta _{t - \iota }^2 - \mu } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - E\left[ {\frac{{{\rm{sgn}}({\eta _t})}}{{2\sqrt {{h_t}} }}\frac{{\partial {\varepsilon _t}({\theta _0})}}{{\partial \theta }}\left( {\eta _t^2 - \mu } \right)\left( {\eta _{t - \iota }^2 - \mu } \right)} \right], \end{array} $

则 $ E[\sum\limits_{t=1}^{n}D_t\cdot C_\iota] =\frac{\kappa_1}{4}X_{\rho\iota}-\frac{\kappa_2}{2}X_{\rho\iota}^*, $那么 $ {\rm Cov}\left({\sqrt{n}(\hat{\theta}_n-\theta_0), \sqrt{n}C^T}\right)=\Sigma_0^{-1}E\left[\frac{\kappa_1}{4}X_{\rho}-\frac{\kappa_2}{2}X_{\rho}^*\right], $因此 $\sqrt{n}\rho$的协方差为

$ \begin{eqnarray*} &&{\rm var}\left(\sqrt{n}\rho\right)=\sigma_0^{-4}{\rm var}(\sqrt{n}\hat{C})\\ &=&\sigma_0^{-4}[{\rm var}\left(\sqrt{n}C\right)+{\rm var}\left(-\mu X_\rho\sqrt{n} \left(\hat{\theta}_n-\theta_0\right)\right)\\ &&+{\rm cov}\left(\sqrt{n}C, -\mu X_\rho\sqrt{n}\left(\hat{\theta}_n-\theta_0\right)\right) +{\rm cov}\left(-\mu X_\rho\sqrt{n}\left(\hat{\theta}_n-\theta_0\right), \sqrt{n}C\right)]\\ &=&\sigma_0^{-4}[\sigma_0^{4}1_M+\mu^2X_\rho\frac{1}{4}\Sigma_0^{-1}\Omega_0\Sigma_0^{-1}X_\rho^{T} -\mu\left(\frac{\kappa_1}{4}X_\rho-\frac{\kappa_2}{2}X_\rho^{*}\right)\Sigma_0^{-1}X_\rho^{T}\\ &&-\mu X_\rho^{T}\Sigma_0^{-1}\left(\frac{\kappa_1}{4}X_\rho-\frac{\kappa_2}{2}X_\rho^{*}\right)]\\ &=&1_{M}+\frac{\mu^2}{4\sigma_{0}^{4}}X_{\rho}\Sigma_{0}^{-1}\Omega_{0}\Sigma_{0}^{-1}X_{\rho}^T -\frac{\mu \kappa_{1}}{2\sigma_{0}^{4}}X_{\rho}\Sigma_{0}^{-1}X_{\rho}^T +\frac{\mu \kappa_{2}}{2\sigma_{0}^{4}} \left(X_{\rho}^{*}\Sigma_{0}^{-1}X_{\rho}^T+X_{\rho}^T\Sigma_{0}^{-1}X_{\rho}^{*}\right), \end{eqnarray*} $

即完成了定理的证明.

在定理1中, 通过样本均值来估计 $V$, 记为 $\hat{V}$, 在假设1-4下, 表明 $ \hat{V}=V+o_p(1), $因此由定理1, 下面的结论是直接成立的.

结论1  如果假设1-5成立, 那么当 $n\rightarrow\infty$时, $ Q\left(M\right)=n\hat{\rho}^T\hat{V}^{-1}\hat{\rho}\longrightarrow_d\chi^2(M). $对于拟极大指数似然估计, 在结论1中称 $Q\left(M\right)$为基于平方残差自相关函数的混成检验统计量, 并且可用 $Q\left(M\right)$来诊断检验由拟极大指数似然估计拟合的ARFIMA-GARCH模型.

在拟极大指数似然估计下, 为验证基于平方残差自相关函数的混成检验统计量的有效性, 以中证800指数的股票价格为例, 选取从2007年1月4日到2008年12月31日共488个数据(数据来自http://quotes.money.163.com/1399906.html).根据ADF检验法, 可知该序列是平稳的.由图 1和图 2可以看出, 序列显然存在异方差性.通过MATLAB软件, 利用R/S检验法可以计算出中证800指数收益率序列的Hurst指数, 即 $H=0.5405$, 由于 $H>0.5$, 因此该序列存在长记忆性.由以上可知, 对该序列建立ARFIMA(1, $d$, 0)-GARCH(1, 1) 模型, 使用拟极大指数似然估计方法, 得到的拟合模型为

$ \begin{align} & (1-0.0427B){{(1-B)}^{0.0003}}{{y}_{t}}={{\varepsilon }_{t}}, \\ & {{\varepsilon }_{t}}={{\eta }_{t}}\sqrt{{{h}_{t}}}, {{h}_{t}}=0.0836+0.2349\varepsilon _{t-1}^{2}+0.5000{{h}_{t-1}}. \\ \end{align} $

然而, 在实际中拟合的模型是否有误差, 是否真正符合实际数据, 就需要对拟合后的模型进行诊断检验.因此, 可利用基于平方残差自相关函数的混成检验统计量( $Q\left(M\right)$)对上述拟合模型进行诊断检验.由卡方分布表可知, 在0.05显著性水平下, $\chi^2(6)=12.592$, $\chi^2(12)=21.026$.通过MTALAB软件计算可得, $Q\left(6\right)=4.3431<\chi^2(6)$, $Q\left(12\right)=8.2062<\chi^2(12)$.从而可以看出, 上述拟合的ARFIMA(1, $d$, 0)-GARCH(1, 1) 模型是准确的, 符合实际数据, ARFIMA(1, $d$, 0)-GARCH(1, 1) 模型适合于拟合该时间段内的中证800指数收益率序列.