Hill Plot Analysis for SPY
The Hill plot shows how the tail exponent α varies with the number of order statistics k used in the estimation. Stable regions (highlighted) indicate reliable α estimates for the power law tail behavior.
Daily Analysis: Day-to-day percentage returns
Standard deviation: 1.22% | k (2σ exceedances): 336 | Sample size: 6940 observations
Both Tails (Absolute Returns)
Analyzes the magnitude of all returns regardless of direction. Lower α indicates fatter tails (more extreme events).
Loading Hill plot...
Right Tail (Positive Returns)
Analyzes upside movements only. A fatter right tail means more frequent large positive moves.
Loading Hill plot...
Left Tail (Negative Returns)
Analyzes downside movements only. A fatter left tail means more frequent large negative moves (crash risk).
Loading Hill plot...
How to Interpret Hill Plots
The α (alpha) Value
- α < 2: Very fat tails (infinite variance)
- 2 ≤ α < 3: Fat tails (finite variance, infinite skewness)
- α ≥ 3: Moderate tails (finite skewness)
- α → ∞: Approaches normal distribution
Reading the Plot
- Stable region: Where α stabilizes (highlighted area)
- Small k: High variance, unreliable estimates
- Large k: Bias from non-tail observations
- Optimal k: Balance in the stable region
All Timeframes Summary
| Timeframe | Observations | Std Dev | k (2σ exceedances) | Positive Returns | Negative Returns |
|---|---|---|---|---|---|
| Daily | 6,940 | 1.22% | 336 | 3,743 | 3,172 |
| Weekly | 1,204 | 2.30% | 57 | 666 | 538 |
| Monthly | 165 | 4.42% | 8 | 99 | 66 |