某私募(private fund)交易日下午四点,你的 PM 把过去 500 个交易日的策略净值推过来,问:这条曲线的均值真的稳定吗?波动率有没有结构性变化?只看一条路径,凭什么相信估出来的均值与自相关有意义?这是时间序列分析(time series analysis)的元问题。横截面统计里你有 n n n T T T
1. 随机过程与样本路径
形式化一下。一个(实值、离散时间的)随机过程(stochastic process)是定义在同一概率空间 ( Ω , F , P ) (\Omega, \mathcal{F}, P) ( Ω , F , P ) t ∈ Z t \in \mathbb{Z} t ∈ Z { X t } t ∈ Z \{X_t\}_{t \in \mathbb{Z}} { X t } t ∈ Z ω ∈ Ω \omega \in \Omega ω ∈ Ω t ↦ X t ( ω ) t \mapsto X_t(\omega) t ↦ X t ( ω ) ( X t 1 , … , X t k ) (X_{t_1}, \dots, X_{t_k}) ( X t 1 , … , X t k )
握住这个区分:样本路径是一次实现,过程是所有实现的概率法则 。估均值时你在用一条路径上的时间平均替代横截面的总体平均;只有过程在时间方向上有足够对称性,这步替换才合法。这套对称性就是平稳性——它把「时间平均 → \to →
2. 严平稳与宽平稳
按王燕《应用时间序列分析》与何书元《随机过程》的惯例,先严后宽,因为前者是直接对分布层面的对称要求,后者是它的二阶矩弱化。
严平稳(strictly stationary) :{ X t } \{X_t\} { X t } k ≥ 1 k \geq 1 k ≥ 1 ( t 1 , … , t k ) (t_1, \dots, t_k) ( t 1 , … , t k ) h ∈ Z h \in \mathbb{Z} h ∈ Z
( X t 1 + h , … , X t k + h ) = d ( X t 1 , … , X t k ) . (X_{t_1 + h}, \dots, X_{t_k + h}) \stackrel{d}{=} (X_{t_1}, \dots, X_{t_k}). ( X t 1 + h , … , X t k + h ) = d ( X t 1 , … , X t k ) . 口语化:把时间轴整体平移,统计性质纹丝不动。代价是这条要求覆盖了全分布,只用二阶矩信息无法直接验证;实操中很难直接检验。
宽平稳(weakly stationary)/ 二阶平稳(second-order stationary) :存在有限二阶矩 E [ X t 2 ] < ∞ E[X_t^2] < \infty E [ X t 2 ] < ∞
E [ X t ] = μ (与 t 无关) , V a r ( X t ) = γ ( 0 ) < ∞ , C o v ( X t , X t − k ) = γ ( k ) (仅依赖滞后 k ) . E[X_t] = \mu \text{ (与 } t \text{ 无关)}, \quad \mathrm{Var}(X_t) = \gamma(0) < \infty, \quad \mathrm{Cov}(X_t, X_{t-k}) = \gamma(k) \text{ (仅依赖滞后 } k\text{)}. E [ X t ] = μ ( 与 t 无关 ) , Var ( X t ) = γ ( 0 ) < ∞ , Cov ( X t , X t − k ) = γ ( k ) ( 仅依赖滞后 k ) . 只对一阶矩、二阶矩提要求——这正是它在实践中好用的原因:样本均值、样本自协方差直接对应它的三条条件。两者关系:严平稳 + 有限二阶矩 ⇒ \Rightarrow ⇒ 高斯过程(Gaussian process)的严平稳与宽平稳等价 ——多元正态分布完全由均值向量与协方差矩阵决定,二阶矩的不变性自动升级为整个分布的不变性。本模块从此处起,「平稳」默认指「宽平稳」,只在需要时显式区分。
3. 自协方差与自相关函数
自协方差函数(autocovariance function)记作 γ ( k ) = C o v ( X t , X t − k ) \gamma(k) = \mathrm{Cov}(X_t, X_{t-k}) γ ( k ) = Cov ( X t , X t − k ) t t t k k k
ρ ( k ) = γ ( k ) γ ( 0 ) , ρ ( 0 ) = 1 , ∣ ρ ( k ) ∣ ≤ 1 , ρ ( − k ) = ρ ( k ) . \rho(k) = \frac{\gamma(k)}{\gamma(0)}, \qquad \rho(0) = 1, \quad |\rho(k)| \leq 1, \quad \rho(-k) = \rho(k). ρ ( k ) = γ ( 0 ) γ ( k ) , ρ ( 0 ) = 1 , ∣ ρ ( k ) ∣ ≤ 1 , ρ ( − k ) = ρ ( k ) . 三条结构性质各有出处:ρ ( 0 ) = 1 \rho(0) = 1 ρ ( 0 ) = 1 ∣ ρ ( k ) ∣ ≤ 1 |\rho(k)| \leq 1 ∣ ρ ( k ) ∣ ≤ 1 X t X_t X t X t − k X_{t-k} X t − k ρ ( − k ) = ρ ( k ) \rho(-k) = \rho(k) ρ ( − k ) = ρ ( k ) k ≥ 0 k \geq 0 k ≥ 0 自协方差必须是正定核 ——对任意 ( a 1 , … , a n ) (a_1, \dots, a_n) ( a 1 , … , a n ) ( t 1 , … , t n ) (t_1, \dots, t_n) ( t 1 , … , t n ) ∑ i , j a i a j γ ( t i − t j ) ≥ 0 \sum_{i,j} a_i a_j \gamma(t_i - t_j) \geq 0 ∑ i , j a i a j γ ( t i − t j ) ≥ 0
4. 偏自相关函数 PACF
偏自相关函数(PACF)记作 ϕ k k \phi_{kk} ϕ k k X t X_t X t s p a n ( X t − 1 , … , X t − k ) \mathrm{span}(X_{t-1}, \dots, X_{t-k}) span ( X t − 1 , … , X t − k ) L 2 L^2 L 2
X ^ t ( k ) = β 1 X t − 1 + β 2 X t − 2 + ⋯ + β k X t − k , \hat{X}_t^{(k)} = \beta_1 X_{t-1} + \beta_2 X_{t-2} + \dots + \beta_k X_{t-k}, X ^ t ( k ) = β 1 X t − 1 + β 2 X t − 2 + ⋯ + β k X t − k , ϕ k k \phi_{kk} ϕ k k β k \beta_k β k ϕ k k = C o r r ( X t , X t − k ∣ X t − 1 , … , X t − k + 1 ) \phi_{kk} = \mathrm{Corr}(X_t, X_{t-k} \mid X_{t-1}, \dots, X_{t-k+1}) ϕ k k = Corr ( X t , X t − k ∣ X t − 1 , … , X t − k + 1 ) X t X_t X t X t − k X_{t-k} X t − k
一句话对照:ACF 量度总线性依赖 (含经由中间滞后的间接通道),PACF 量度直接线性依赖 (中间滞后被线性偏出后)。设想 X t → X t − 1 → X t − 2 X_t \to X_{t-1} \to X_{t-2} X t → X t − 1 → X t − 2 X t X_t X t X t − 1 X_{t-1} X t − 1 X t − 2 X_{t-2} X t − 2 X t − 1 X_{t-1} X t − 1
5. 两个原型:白噪声与随机游走
白噪声(white noise) { ϵ t } ∼ W N ( 0 , σ 2 ) \{\epsilon_t\} \sim \mathrm{WN}(0, \sigma^2) { ϵ t } ∼ WN ( 0 , σ 2 ) E [ ϵ t ] = 0 E[\epsilon_t] = 0 E [ ϵ t ] = 0 V a r ( ϵ t ) = σ 2 \mathrm{Var}(\epsilon_t) = \sigma^2 Var ( ϵ t ) = σ 2 C o v ( ϵ t , ϵ s ) = 0 \mathrm{Cov}(\epsilon_t, \epsilon_s) = 0 Cov ( ϵ t , ϵ s ) = 0 t ≠ s t \neq s t = s μ = 0 \mu = 0 μ = 0 γ ( 0 ) = σ 2 \gamma(0) = \sigma^2 γ ( 0 ) = σ 2 γ ( k ) = 0 \gamma(k) = 0 γ ( k ) = 0 k ≠ 0 k \neq 0 k = 0 ρ ( 0 ) = 1 \rho(0) = 1 ρ ( 0 ) = 1 ρ ( k ) = 0 \rho(k) = 0 ρ ( k ) = 0 k ≠ 0 k \neq 0 k = 0 不相关不等于独立 ——例如 GARCH 模型的残差是不相关白噪声,但条件方差有结构,不是独立序列(细节见模块 2.3.2 波动率与机制模型)。后续 AR / MA / ARMA 都以 { ϵ t } ∼ W N ( 0 , σ 2 ) \{\epsilon_t\} \sim \mathrm{WN}(0, \sigma^2) { ϵ t } ∼ WN ( 0 , σ 2 )
随机游走(random walk) S t = ∑ i = 1 t ϵ i S_t = \sum_{i=1}^{t} \epsilon_i S t = ∑ i = 1 t ϵ i S 0 = 0 S_0 = 0 S 0 = 0 ϵ t ∼ W N ( 0 , σ 2 ) \epsilon_t \sim \mathrm{WN}(0, \sigma^2) ϵ t ∼ WN ( 0 , σ 2 )
V a r ( S t ) = t σ 2 , C o v ( S t , S s ) = min ( t , s ) σ 2 . \mathrm{Var}(S_t) = t\sigma^2, \qquad \mathrm{Cov}(S_t, S_s) = \min(t, s)\,\sigma^2. Var ( S t ) = t σ 2 , Cov ( S t , S s ) = min ( t , s ) σ 2 . 方差随 t t t 不是 宽平稳的。它是非平稳过程的原型,模块第 4 课用单位根框架系统处理。
6. 样本估计与巴特利特置信带
实践中你拿到的是一条长度为 T T T x 1 , … , x T x_1, \dots, x_T x 1 , … , x T
x ˉ = 1 T ∑ t = 1 T x t , γ ^ ( k ) = 1 T ∑ t = k + 1 T ( x t − x ˉ ) ( x t − k − x ˉ ) , ρ ^ ( k ) = γ ^ ( k ) γ ^ ( 0 ) . \bar{x} = \frac{1}{T}\sum_{t=1}^{T} x_t, \qquad \hat{\gamma}(k) = \frac{1}{T}\sum_{t = k + 1}^{T} (x_t - \bar{x})(x_{t - k} - \bar{x}), \qquad \hat{\rho}(k) = \frac{\hat{\gamma}(k)}{\hat{\gamma}(0)}. x ˉ = T 1 t = 1 ∑ T x t , γ ^ ( k ) = T 1 t = k + 1 ∑ T ( x t − x ˉ ) ( x t − k − x ˉ ) , ρ ^ ( k ) = γ ^ ( 0 ) γ ^ ( k ) . 分母用 T T T T − k T-k T − k γ ^ \hat{\gamma} γ ^
巴特利特(Bartlett)结论 :宽平稳加温和混合条件下,对 k ≥ 1 k \geq 1 k ≥ 1 ρ ^ ( k ) \hat{\rho}(k) ρ ^ ( k ) 1 / T 1/\sqrt{T} 1/ T ± 1.96 / T \pm 1.96/\sqrt{T} ± 1.96/ T 1 / T 1/\sqrt{T} 1/ T
实战读图 :设 T = 200 T = 200 T = 200 ρ ^ ( 1 ) = 0.42 \hat{\rho}(1) = 0.42 ρ ^ ( 1 ) = 0.42 ρ ^ ( 2 ) = 0.18 \hat{\rho}(2) = 0.18 ρ ^ ( 2 ) = 0.18 ρ ^ ( 3 ) = 0.05 \hat{\rho}(3) = 0.05 ρ ^ ( 3 ) = 0.05 ± 1.96 / 200 ≈ ± 0.139 \pm 1.96/\sqrt{200} \approx \pm 0.139 ± 1.96/ 200 ≈ ± 0.139 0.42 > 0.139 0.42 > 0.139 0.42 > 0.139 0.18 > 0.139 0.18 > 0.139 0.18 > 0.139 0.05 < 0.139 0.05 < 0.139 0.05 < 0.139
7. 预告:一阶自回归的 ACF 形状
若某过程的 ACF 是几何衰减 ρ ( k ) = ϕ ∣ k ∣ \rho(k) = \phi^{|k|} ρ ( k ) = ϕ ∣ k ∣ ϕ \phi ϕ ∣ ϕ ∣ |\phi| ∣ ϕ ∣
练习
Exercise
设 X t = 0.6 X t − 1 + ϵ t X_t = 0.6\, X_{t-1} + \epsilon_t X t = 0.6 X t − 1 + ϵ t ϵ t ∼ W N ( 0 , 1 ) \epsilon_t \sim \mathrm{WN}(0, 1) ϵ t ∼ WN ( 0 , 1 ) μ = E [ X t ] \mu = E[X_t] μ = E [ X t ] γ ( 0 ) = V a r ( X t ) \gamma(0) = \mathrm{Var}(X_t) γ ( 0 ) = Var ( X t ) k = 0 , 1 , 2 , 3 k = 0, 1, 2, 3 k = 0 , 1 , 2 , 3 ρ ( k ) \rho(k) ρ ( k ) ϕ = 0.6 \phi = 0.6 ϕ = 0.6
提示 两端取期望并用平稳性
E [ X t ] = E [ X t − 1 ] E[X_t] = E[X_{t-1}] E [ X t ] = E [ X t − 1 ] 立得
μ = 0 \mu = 0 μ = 0 ;两端取方差,由
V a r ( X t ) = ϕ 2 V a r ( X t − 1 ) + σ 2 \mathrm{Var}(X_t) = \phi^2 \mathrm{Var}(X_{t-1}) + \sigma^2 Var ( X t ) = ϕ 2 Var ( X t − 1 ) + σ 2 与
V a r ( X t ) = V a r ( X t − 1 ) \mathrm{Var}(X_t) = \mathrm{Var}(X_{t-1}) Var ( X t ) = Var ( X t − 1 ) 解出
γ ( 0 ) = σ 2 / ( 1 − ϕ 2 ) \gamma(0) = \sigma^2/(1-\phi^2) γ ( 0 ) = σ 2 / ( 1 − ϕ 2 ) 。
提示 由
γ ( k ) = ϕ γ ( k − 1 ) \gamma(k) = \phi\,\gamma(k-1) γ ( k ) = ϕ γ ( k − 1 ) 递推得
ρ ( k ) = ϕ k \rho(k) = \phi^k ρ ( k ) = ϕ k ;代
ϕ = 0.6 \phi = 0.6 ϕ = 0.6 即
ρ ( 0 ) = 1 , ρ ( 1 ) = 0.6 , ρ ( 2 ) = 0.36 , ρ ( 3 ) = 0.216 \rho(0)=1, \rho(1)=0.6, \rho(2)=0.36, \rho(3)=0.216 ρ ( 0 ) = 1 , ρ ( 1 ) = 0.6 , ρ ( 2 ) = 0.36 , ρ ( 3 ) = 0.216 。(c)
∣ ϕ ∣ < 1 |\phi|<1 ∣ ϕ ∣ < 1 ,故宽平稳。
通向下一课
到这里你已经能区分严平稳与宽平稳、会算 ACF 与 PACF、会读样本 ACF 图,也见到了「白噪声 vs. 随机游走」这条平稳与非平稳的分水岭。下一课回答互补问题:哪一类参数化过程,其 ACF 与 PACF 长什么形状? ——我们将引入滞后算子 L L L