Part II · Chapter 5
Smoothing and Trend Estimation
A moving average reveals the underlying trend by averaging out noise — at the cost of lag and smoothed-over turning points.
\( \tilde y_t = \frac{1}{k}\sum_{i=0}^{k-1} y_{t-i} \)
Learning objectives
Compute simple and exponential moving averages.
Choose window size k that balances smoothness vs lag.
Understand bias from large windows at turning points.
Distinguish smoothers (descriptive) from forecasters.
Smoother Lab
Seed:
Interpretation
k (SMA window): —
EWMA effective lag: —
Large k = very smooth, large lag. Large α = reactive EWMA. Use shorter window to catch turning points.
📐 Smoothing formulas
\[ \text{SMA: } \tilde y_t = \tfrac{1}{k}\sum_{i=0}^{k-1} y_{t-i} \qquad \text{EWMA: } \tilde y_t = \alpha y_t + (1-\alpha)\tilde y_{t-1} \]
k window length (SMA)
α weight on newest obs (EWMA)
Lag (SMA) ≈ (k-1)/2 periods
Lag (EWMA) ≈ (1-α)/α
🔍 What to look for
- Smoother line lags behind the actual series — visible at turning points.
- Larger k or smaller α exaggerates the lag.
- Smoothing is descriptive — use a real forecasting model (e.g., AR or ETS) to make predictions.
⚠️ Pro Tip: What to Avoid
Student says
"Bigger window is always better because the line looks cleaner."
Why this is wrong
A larger window over-smooths and hides exactly the turning points you want to detect (recessions, regime shifts).
Correct interpretation
Choose window k for the lag tolerance you have. Validate on out-of-sample turning-point detection, not visual smoothness.
📝 Mini-quiz
📋 Key Takeaways
| Method | Lag | Best for |
|---|---|---|
| SMA(k) | (k-1)/2 | Visual trend, trading signals |
| EWMA(α) | (1-α)/α | Adaptive smoothing |
| Holt | — | Trending series, basic forecast |
| Holt-Winters | — | Trend + seasonality |