Part III ยท Chapter 8
Stationarity
A stationary process has stable mean, stable variance, and dependence that only depends on the GAP between observations โ not on absolute time.
\( E[y_t]=\mu,\; \mathrm{Var}(y_t)=\sigma^2,\; \mathrm{Cov}(y_t,y_{t-h})=\gamma(h) \)
Learning objectives
State the three conditions for weak stationarity.
Recognize stationary vs nonstationary patterns visually.
Use rolling mean / std to diagnose.
Connect non-stationarity to the need for differencing.
Process selector
Seed:
Stationarity verdict
โ
Rolling mean drifting? variance growing? ACF slow-decaying? โ signs of non-stationarity.
๐ Weak stationarity
\[ \forall t,\; E[y_t]=\mu,\; \mathrm{Var}(y_t)=\sigma^2,\; \mathrm{Cov}(y_t,y_{t-h})=\gamma(h) \]
ฮผ constant mean over time
ฯยฒ constant variance over time
ฮณ(h) covariance depends only on lag h
RW: Var(y_t)=tฯยฒ variance grows linearly โ non-stationary
๐ Diagnostic checklist
- Plot the series โ does the mean look stable?
- Rolling mean & std โ do they drift or grow?
- ACF โ does it die quickly (stationary) or slowly (unit root)?
- ADF / KPSS tests (formal) โ covered in Ch 10.
โ ๏ธ Pro Tip: What to Avoid
Student says
"Random walk has zero drift, so its mean is zero, so it's stationary."
Why this is wrong
Although a driftless RW has E[y_t]=0, its variance Var(y_t)=tฯยฒ GROWS with time, so it violates weak stationarity.
Correct interpretation
All three conditions (mean, variance, covariance structure) must hold for stationarity. RW fails on variance.
๐ Mini-quiz
๐ Key Takeaways
| Process | Stationary? | Why |
|---|---|---|
| White noise | YES | iid with finite variance |
| AR(1), |ฯ|<1 | YES | shocks decay |
| Random walk | NO | variance grows |
| Trend-stationary | NO until detrended | mean increases with t |
| AR(1), ฯ=1 | NO (unit root) | shocks accumulate |