Hurst Exponent vs Variance Ratio Tests

Both the Hurst exponent and variance ratio tests assess whether financial returns exhibit persistence (predictable continuation) or mean reversion (predictable reversal). They approach this question differently: Hurst examines how fluctuation magnitude scales with time, while variance ratios compare variances at different horizons. Understanding their relationship and respective strengths helps analysts choose the appropriate tool.

Variance Ratio Test Fundamentals

The variance ratio test, popularized by Lo and MacKinlay (1988), compares the variance of k-period returns to k times the variance of one-period returns:

VR(k) = Var(r_k) / (k × Var(r_1))

For a random walk (no serial correlation), VR(k) equals 1 at all horizons. Deviations indicate:

• VR > 1: Positive serial correlation (persistence)
• VR < 1: Negative serial correlation (mean reversion)

Statistical tests determine whether observed deviations from 1 are significant.

The Mathematical Connection

Hurst and variance ratios are mathematically related. For a process with Hurst exponent H:

VR(k) ≈ k^(2H-1)

This means:

• H = 0.5 implies VR = 1 (random walk)
• H > 0.5 implies VR > 1 (persistence)
• H < 0.5 implies VR < 1 (mean reversion)

In theory, you could derive H from variance ratios or vice versa. In practice, different estimation methods and sample properties mean they can give somewhat different results.

Key Differences

Horizon-Specific Information

A key advantage of variance ratios is horizon-specific information. VR(5) tells you about 5-period persistence; VR(20) tells you about 20-period persistence. These can differ:

• Short-term mean reversion + long-term persistence is possible
• Different market microstructure at different horizons
• Persistence may exist only at specific timescales

The Hurst exponent summarizes persistence across all scales into a single number, potentially masking horizon-specific behavior.

When Hurst Is Preferred

• You need a single summary measure of persistence
• Comparing persistence across multiple instruments or time periods
• Regime identification (trending vs mean-reverting)
• Long-range dependence is the primary interest
• Connection to fractal/self-similarity concepts is relevant

When Variance Ratios Are Preferred

• Specific investment horizons matter (e.g., 5-day vs 20-day)
• Testing random walk hypothesis formally
• Different behavior at different horizons is suspected
• Working within finance-native methodology
• Heteroscedasticity-robust testing is important

Practical Synthesis

Using both methods together provides richer insight:

1. Calculate Hurst exponent for overall persistence measure
2. Calculate variance ratios at multiple horizons (e.g., 2, 5, 10, 20, 40)
3. Check consistency: Does the implied H from VR match the direct estimate?
4. Examine horizon patterns: Does VR vary systematically with k?
5. If Hurst and VR disagree significantly, investigate why

Agreement between methods strengthens confidence; disagreement prompts deeper analysis.

Variance Ratio Test Statistics

Multiple variance ratio test statistics exist:

• Lo-MacKinlay: Original test, assumes homoscedasticity or heteroscedasticity
• Wright: Rank-based test, robust to non-normality
• Joint tests: Test VR = 1 at multiple horizons simultaneously
• Wild bootstrap: Robust to various data issues

For financial data, heteroscedasticity-robust versions are generally preferred.

Interpretation Example

Consider the following results for daily equity returns:

• Hurst = 0.56 (95% CI: 0.52-0.60)
• VR(2) = 1.08 (significant at 5%)
• VR(5) = 1.15 (significant at 5%)
• VR(20) = 1.02 (not significant)

Interpretation: Mild persistence overall (H slightly above 0.5), concentrated at short horizons (VR elevated at 2-5 days but not at 20 days). This suggests short-term momentum that dissipates over longer periods.

Conclusion

The Hurst exponent and variance ratio tests measure related but not identical properties. Hurst provides a single summary of persistence across scales; variance ratios provide horizon-specific information. Using both methods together yields a more complete picture of return dynamics than either alone. For comprehensive analysis of market persistence, treat them as complements rather than substitutes.