Part III · Chapter 10
Unit Roots and Differencing
If a series has a unit root, shocks accumulate forever and the series is non-stationary. Differencing — Δy_t = y_t − y_{t-1} — turns I(1) into I(0).
\( \Delta y_t = y_t - y_{t-1} \)
Learning objectives
Recognize unit-root behavior visually and via ADF.
Apply first and second differencing.
Avoid over-differencing.
Read ADF p-values correctly (H₀ = unit root).
Differencing Visualizer
Seed:
ADF intuition
—
ADF H₀: unit root (non-stationary). Reject ⇒ stationary. Apply to LEVEL first; if fails to reject, difference and retry.
📐 ADF test
\[ \Delta y_t = \alpha + \beta t + \gamma\, y_{t-1} + \sum_{i=1}^{p}\delta_i\Delta y_{t-i}+\varepsilon_t \]
H₀ γ = 0 (unit root)
H₁ γ < 0 (stationary)
5% τ_μ ≈ −2.88 (constant only)
5% τ_τ ≈ −3.44 (constant + trend)
🔍 What to look for
- RW level wanders; first difference looks like white noise.
- If you over-difference (d=2 on an AR(1)), you create artificial θ near −1 in MA.
- Always test ADF on level first; difference only if you fail to reject.
⚠️ Pro Tip: What to Avoid
Student says
"ADF p-value = 0.50 means there IS a unit root."
Why this is wrong
Failing to reject H₀ is not the same as proving H₀. It just means insufficient evidence to declare stationarity at the chosen level.
Correct interpretation
"We fail to reject the unit-root null at the 5% level; series is likely non-stationary." Pair with KPSS for confirmation.
📝 Mini-quiz
📋 Key Takeaways
| Integration | Treatment |
|---|---|
| I(0) | Use directly (ARMA) |
| I(1) | First difference → ARMA on Δy |
| I(2) | Second difference (rare in finance) |
| Over-diff signature | MA(1) θ ≈ −1 |