Stationarity

Strict Stationarity

\[ [x_t, x_{t+k}] \sim Dist\left(\mu, \sigma^2\right) \] \[ [x_{t+\tau}, x_{t+\tau+k}] \sim Dist\left(\mu, \sigma^2\right) \] Strict stationarity should satisfy above assumption, as it is rarely observed in natural world, in analytics, it is universally accepted to use stationarity to describe covariance stationarity.

Covariance Stationarity

Covariance (weak-form) stationarity assumption

• Constant $\mu$
• Constant $\sigma^2$
• $Cov\left(x_n, x_{n+k}\right) = Cov\left(x_m, x_{m+k}\right)$

White noise satisfy the weak-form stationarity, as:

• $\mu=0$
• $\sigma^2$ is constant
• $Cov\left(x_n, x_{n+k}\right) = Corr\left(x_m, x_{m+k}\right)\sigma_1\sigma_2=0$

Seasonality

ACF

PACF

Factor Exposure/Factor Return/Sepcific Return