Seasonal differencing does not remove periodic variance
= Seasonal differencing does not remove periodic variance
If $X_t=c_t\varepsilon_t$ with a deterministic periodic scale $c_t=c_{t-S}$ and <strong white noise> of variance $\sigma^2$, then $\Delta_SX_t=c_t(\varepsilon_t-\varepsilon_{t-S})$. Its variance is $2\sigma^2c_t^2$, still seasonal when $c_t^2$ varies. A periodic scale model or variance standardization is more appropriate than blindly applying <differencing>.