Part III · Chapter 7
Randomness and Dependence
White noise has no useful past information. An autocorrelated series LOOKS random but contains predictable dependence — visible only on a lag-scatter or ACF plot.
\( y_t = \phi\, y_{t-1} + \varepsilon_t \)
Learning objectives
Distinguish white noise from autocorrelated series.
Read a lag-1 scatter plot to detect dependence.
Interpret the sample ACF.
Recognize that visual randomness ≠ statistical independence.
Dependence Lab
Seed:
Interpretation
Sample ρ(1): —
Expected ρ(1): —
If model = white noise, both should be near zero. If AR(1) with φ=0.7, expect ≈ 0.7.
📐 ACF definition
\[ \rho(h) = \frac{\mathrm{Cov}(y_t, y_{t-h})}{\mathrm{Var}(y_t)} \]
ρ(0) = 1 (trivially)
ρ(h) ≈ 0 ∀ h ⇒ white noise
Geometric decay ⇒ AR(1)
±1.96/√n significance band
🔍 What to look for
- White noise: scatter is a random cloud; ACF bars within bands.
- AR(1) with positive φ: scatter forms an upward-sloping cloud; ACF decays geometrically.
- Negative φ: scatter slopes down; ACF alternates in sign.
⚠️ Pro Tip: What to Avoid
Student says
"The series looks erratic so it must be independent noise."
Why this is wrong
An AR(1) with φ = 0.8 still looks "jagged" but has strong dependence visible only in lag scatter / ACF.
Correct interpretation
Always check the ACF before declaring a series random. Independence is a statistical property, not a visual one.
📝 Mini-quiz
📋 Key Takeaways
| Series | ACF pattern | Scatter (y_t vs y_{t-1}) |
|---|---|---|
| White noise | all ≈ 0 | random cloud |
| AR(1) φ > 0 | geometric decay | upward cloud |
| AR(1) φ < 0 | alternating decay | downward cloud |
| Random walk | very slow decay | near-perfect diagonal |