Which test should I use to detect mean reversion or trending?

Quick read of regime → Hurst exponent (one number, easy to interpret). Designing trades at a specific frequency → autocorrelation at the relevant lag. Formal statistical test / publication-grade evidence → variance ratio test (gives a z-statistic and p-value). For real strategy work, run all three and check they agree.

How is the Hurst exponent related to autocorrelation?

Loosely: autocorrelation measures one lag at a time, Hurst integrates across all lags. For an exactly self-similar process, the Hurst exponent H relates to autocorrelation as H = 1 - (slope of ln(autocorr) vs ln(lag))/2. In practice they often agree — strongly mean-reverting series have H < 0.5 AND negative lag-1 autocorrelation. They can disagree when the autocorrelation structure is non-monotonic across lags.

What is the variance ratio test?

VR(k) = Var(k-period returns) / (k × Var(1-period returns)). Under random walk this equals 1. VR > 1 = trending (variance grows faster than k); VR < 1 = mean-reverting. The test (Lo & MacKinlay 1988) compares the observed VR to a z-statistic for statistical significance.

Which one is the most statistically rigorous?

Variance ratio test. It explicitly gives you a p-value for "is this series random walk vs mean-reverting/trending?" Hurst is more of a descriptive statistic — useful but harder to interpret statistically. Lag-by-lag autocorrelation can be tested with Ljung-Box or Box-Pierce statistics. For academic / hedge-fund publication purposes, variance ratio is the cleanest.

When does the Hurst exponent mislead?

Three failure modes. (1) Small samples (< 100 observations) — Hurst values can be off by ±0.1 or more. (2) Non-stationary series — strong trends or regime breaks inflate Hurst above 1.0. (3) Aggregated data — daily returns aggregated from intraday can show a Hurst that doesn't match either frequency. Always check R-squared of the log-log fit; if it's below 0.85, treat the Hurst value with caution.

When does autocorrelation mislead?

Autocorrelation at a single lag can hide long-memory structure. A series can have near-zero lag-1 autocorrelation but still be strongly trending or mean-reverting at longer horizons — the signal is in the slow decay across many lags, not in any one. Autocorrelation also assumes stationarity; if the underlying mean shifts (regime change), the computed autocorr is a mix of within-regime and across-regime effects.

When does the variance ratio test mislead?

VR is sensitive to the choice of k. VR(2) might say "random walk" while VR(20) says "trending" — this is real information about the time scale of the autocorrelation but easy to misread. Also: VR assumes a constant-variance underlying. Heteroskedastic series (which is most financial data) need the robust variant of the test. Pure VR also doesn't distinguish between mean-reverting and trending — only between random walk and not-random-walk. Sign of (VR-1) tells you which.

Can I use these on crypto?

Yes, and crypto is often the most interesting case. Bitcoin shows H ≈ 0.55-0.70 on daily data (strongly trending). Altcoins vary widely. The risk is non-stationarity — crypto has had multiple regime breaks (2017 mania, 2018 crash, 2020-2021 surge, 2022 winter). Always compute these on rolling windows, not lifetime aggregates.

How do these connect to specific trading strategies?

H > 0.55 or VR > 1.0 (significant): favor momentum / breakout / moving-average crossover strategies. H < 0.45 or VR < 1.0 (significant): favor mean-reversion / Bollinger / RSI strategies. H near 0.5 or VR ≈ 1: neither — look for carry, fundamentals, or other alpha sources. The lag-by-lag autocorrelation pattern tells you the time scale to trade at.

Is there a calculator for these?

The Hurst Exponent Calculator computes H via R/S analysis with R-squared fit-quality readout. Autocorrelation and the variance ratio test aren't standalone calculators on QuantOracle yet (likely additions to the roadmap). For now, you can build either from raw returns: autocorr is just Pearson correlation between r_t and r_{t-k}; VR is the ratio of period-k variance to k × period-1 variance.

Hurst Exponent vs Autocorrelation vs Variance Ratio Test

Q: What is the variance ratio test?

VR(k) = Var(k-period returns) / (k × Var(1-period returns)). Under random walk this equals 1. VR > 1 = trending (variance grows faster than k); VR < 1 = mean-reverting. The test (Lo & MacKinlay 1988) compares the observed VR to a z-statistic for statistical significance.

Q: Which one is the most statistically rigorous?

Variance ratio test. It explicitly gives you a p-value for "is this series random walk vs mean-reverting/trending?" Hurst is more of a descriptive statistic — useful but harder to interpret statistically. Lag-by-lag autocorrelation can be tested with Ljung-Box or Box-Pierce statistics. For academic / hedge-fund publication purposes, variance ratio is the cleanest.

Q: When does the Hurst exponent mislead?

Three failure modes. (1) Small samples (< 100 observations) — Hurst values can be off by ±0.1 or more. (2) Non-stationary series — strong trends or regime breaks inflate Hurst above 1.0. (3) Aggregated data — daily returns aggregated from intraday can show a Hurst that doesn't match either frequency. Always check R-squared of the log-log fit; if it's below 0.85, treat the Hurst value with caution.

Q: When does autocorrelation mislead?

Autocorrelation at a single lag can hide long-memory structure. A series can have near-zero lag-1 autocorrelation but still be strongly trending or mean-reverting at longer horizons — the signal is in the slow decay across many lags, not in any one. Autocorrelation also assumes stationarity; if the underlying mean shifts (regime change), the computed autocorr is a mix of within-regime and across-regime effects.

Q: When does the variance ratio test mislead?

VR is sensitive to the choice of k. VR(2) might say "random walk" while VR(20) says "trending" — this is real information about the time scale of the autocorrelation but easy to misread. Also: VR assumes a constant-variance underlying. Heteroskedastic series (which is most financial data) need the robust variant of the test. Pure VR also doesn't distinguish between mean-reverting and trending — only between random walk and not-random-walk. Sign of (VR-1) tells you which.

Q: Can I use these on crypto?

Yes, and crypto is often the most interesting case. Bitcoin shows H ≈ 0.55-0.70 on daily data (strongly trending). Altcoins vary widely. The risk is non-stationarity — crypto has had multiple regime breaks (2017 mania, 2018 crash, 2020-2021 surge, 2022 winter). Always compute these on rolling windows, not lifetime aggregates.

Q: How do these connect to specific trading strategies?

H > 0.55 or VR > 1.0 (significant): favor momentum / breakout / moving-average crossover strategies. H < 0.45 or VR < 1.0 (significant): favor mean-reversion / Bollinger / RSI strategies. H near 0.5 or VR ≈ 1: neither — look for carry, fundamentals, or other alpha sources. The lag-by-lag autocorrelation pattern tells you the time scale to trade at.

Q: Is there a calculator for these?

The Hurst Exponent Calculator computes H via R/S analysis with R-squared fit-quality readout. Autocorrelation and the variance ratio test aren't standalone calculators on QuantOracle yet (likely additions to the roadmap). For now, you can build either from raw returns: autocorr is just Pearson correlation between r_t and r_{t-k}; VR is the ratio of period-k variance to k × period-1 variance.

Test	Best for…	Output
Hurst exponent	Quick regime classification (trend vs mean-revert vs random walk)	One number 0-1; ≈0.5 means random walk
Autocorrelation	Finding the time-scale of structure (which lag to trade on)	Correlations at each lag, plus the decay pattern
Variance ratio test	Formal hypothesis test (publication, allocator pitch)	VR(k) plus z-statistic and p-value

What each test actually measures

Hurst exponent: one number across all lags

The Hurst exponent (Hurst, 1951) is a single number between 0 and 1 that summarizes the long-memory structure of a time series. Computed via R/S analysis: for each window size n, take the range of cumulative deviations from the mean divided by the standard deviation. The rescaled range R/S(n) scales as a power of n, and the exponent of that power-law is the Hurst exponent.

H = 0.5 → random walk (Brownian motion, no memory). H > 0.5 → persistent / trending. H < 0.5 → anti-persistent / mean-reverting. The beauty of Hurst is its compactness — one number. The cost is that you lose all information about which time scales the persistence operates at.

Autocorrelation: lag-by-lag picture

ρ(k) = Corr(r_t, r_{t-k})

Autocorrelation measures linear correlation between observations separated by lag k. Positive lag-1 autocorrelation means an up-day tends to be followed by another up-day (short-term trending). Negative lag-1 autocorrelation means up-days tend to be followed by down-days (mean reversion). Plotting autocorrelation across many lags shows the structure: how fast it decays, whether it's monotonic, whether there are seasonal/cyclic patterns.

Variance ratio test: formal hypothesis test

VR(k) = Var(k-period returns) / (k · Var(1-period returns))

Under random walk, returns are independent and variance scales linearly with horizon: Var(k-period) = k · Var(1-period), so VR = 1. The Lo & MacKinlay (1988) test formalizes this with a z-statistic for the null hypothesis VR = 1. Significant VR > 1 means trending (positive autocorrelation accumulates); significant VR < 1 means mean-reverting (negative autocorrelation accumulates). It's the cleanest test statistically — you get a p-value, you can defend the finding in a paper or pitch deck.

A concrete example: three series, three results

Three 1000-observation series, same volatility, very different memory structures:

Series	True regime	Hurst	Lag-1 autocorr	VR(10) [p-value]
A. Random walk	GBM	~0.51	~0.00	1.02 [p=0.48]
B. AR(1) trending	r_t = 0.2·r_{t-1} + ε	~0.68	~0.20	1.45 [p<0.001]
C. OU mean-reverting	Ornstein-Uhlenbeck	~0.30	~-0.25	0.55 [p<0.001]

All three tests agree directionally in each case. The differences become visible at the margins — for a series that's "trending at short lags, mean-reverting at long lags," Hurst might say ≈0.5 (averaging cancels), autocorrelation shows the structure lag by lag, and VR depends sharply on which k you choose.

When each one wins

Hurst: when you need a single number for screening

Hurst's strength is compression: 5,000 returns into one number. If you're screening hundreds of assets for "which ones are trending," Hurst gives you a ranked list immediately. It's also the most intuitive to communicate to non-quantitative stakeholders — "H of 0.65 means this market is trending" is easier than "the lag-1 autocorrelation is 0.20 and the lag-5 is 0.08."

Use Hurst when you need: regime classification across many assets, rolling-window regime detection through time, a single number for dashboards, intuitive communication.

Autocorrelation: when you need to design the trade

If you're going to run a 5-day mean-reversion strategy, lag-1 autocorrelation doesn't tell you what you need. You need the lag-5 autocorrelation. The autocorrelation plot across lags 1-50 is your design surface — strongly negative lag-3 means trade 3-day reversion; near-zero lag-3 but strongly negative lag-20 means trade 20-day reversion.

Use autocorrelation when you need: lag-specific time scale of the signal, decay pattern shape, identification of the right trade horizon.

Variance ratio: when you need a p-value

Hurst is descriptive. Autocorrelation is descriptive. Variance ratio is a hypothesis test. It gives you the z-statistic and p-value for "is this series random walk?" — which is exactly what you need for academic publication, regulator submission, or allocator pitch deck. It's also the most robust to heteroskedasticity (use the Lo-MacKinlay heteroskedasticity-consistent variant).

Use VR when you need: formal hypothesis test, publication-grade evidence, allocator pitch material, robustness to heteroskedasticity.

When they disagree (what it means)

The most common disagreement: Hurst says one thing and autocorrelation/VR say another. Three reasons:

Multi-scale structure. The series is mean-reverting at short lags and trending at long lags. Hurst averages these and lands near 0.5; autocorrelation and VR at the relevant lag/horizon show the real structure. Trust the lag-specific metrics.
Non-stationarity. The series has regime breaks or trends in its mean. Hurst can inflate above 1.0; autocorrelation gets contaminated. The fix: compute on rolling windows or split at suspected regime breaks. Don't trust any of them on non-stationary lifetime data.
Insufficient sample size. Under 100-200 observations all three are noisy. Hurst is the most sample-sensitive. If they disagree on a short sample, don't conclude — get more data.

The decision rule

Quick regime read across many assets → compute Hurst on each. Use the Hurst exponent calculator and rank by H.
Designing a specific trade frequency → compute autocorrelation at multiple lags. Trade the lag where autocorrelation is strongest (most negative for mean-reversion, most positive for momentum).
Validating a finding before deployment → run variance ratio test at the relevant horizon. If p > 0.05 don't trust the apparent signal — it could be noise.
For a serious research workflow: compute all three on rolling windows. Chart H and VR through time alongside cumulative return. Look for regime changes — points where these flip across 0.5 / 1.0 thresholds.

Asset class reference values

Asset	Typical H (daily)	Lag-1 autocorr	Best strategy style
S&P 500 index	0.55 - 0.60	slightly positive	Mild trend / buy-and-hold
Major FX (EUR/USD)	0.48 - 0.52	≈ 0	Neither — look for carry
Short-term interest rates	0.30 - 0.40	strongly negative	Mean reversion / carry
Single equities (avg)	0.40 - 0.55	slightly negative (intraday)	Short-term reversion / mid-term momentum
Bitcoin (daily)	0.55 - 0.70	positive	Trend-following
Commodity futures	0.50 - 0.65	positive at multi-day	Trend-following / CTA

Related calculators

Hurst Exponent Calculator — R/S analysis with R-squared fit-quality readout
Sharpe Ratio Calculator — confirm that whatever strategy you pick has real edge
Probabilistic Sharpe Ratio Calculator — significance test for that edge, parallels VR test for return-distribution properties
Monte Carlo Simulation Calculator — generate synthetic series with known H to validate your test pipeline

References

Hurst, H. E. (1951). "Long-term storage capacity of reservoirs." Transactions of the American Society of Civil Engineers 116, 770-808.
Mandelbrot, B. (1972). "Statistical methodology for non-periodic cycles: from the covariance to R/S analysis." Annals of Economic and Social Measurement 1, 259-290.
Lo, A. W. & MacKinlay, A. C. (1988). "Stock market prices do not follow random walks: evidence from a simple specification test." Review of Financial Studies 1(1), 41-66.
Box, G. E. P. & Pierce, D. A. (1970). "Distribution of residual autocorrelations in autoregressive-integrated moving average time series models." Journal of the American Statistical Association 65(332), 1509-1526.
Ljung, G. M. & Box, G. E. P. (1978). "On a measure of lack of fit in time series models." Biometrika 65(2), 297-303.