Observed Variance Ratio Calculator
Compare overlapping q-period population variance with q times the one-period population variance of entered equal-frequency returns. Version 1.0.0 exposes every aggregate window and the exact finite-sample convention, but deliberately performs no Lo–MacKinlay z test and makes no random-walk, trend, mean-reversion or trading verdict.
Enter ordered equal-frequency returns
Use signed percentage-point returns from one sampling and cost convention, oldest to newest, then select an aggregation horizon q.
Whole number of one-period returns in each overlapping aggregate.
Oldest to newest. Enter numbers without percent signs; separate with spaces, commas, semicolons or new lines. Maximum 500.
Entered variance-ratio arithmetic
Entered Return Statistical Diagnostics 1.0.0.
On smaller screens, scroll horizontally to inspect the complete audit table.
| Window | Start index | End index | Aggregate return | Deviation | Squared deviation |
|---|
How the observed variance ratio is calculated
Var₁ = Σ(r[i] − r̄)² ÷ n
Varq = Σ(R[t,q] − R̄q)² ÷ (n − q + 1)
VR(q) = Varq ÷ (q × Var₁)
Version 1.0.0 forms every overlapping q-observation return sum, calculates population variance across those windows, and divides by q times the population variance of the original one-period returns.
The number one is displayed only as the mathematical reference in the ratio. This finite-sample estimator can differ from one because of dependence, sampling variability, mean treatment, overlapping windows, heteroskedasticity, non-stationarity or the selected horizon.
Formal variance-ratio tests use additional finite-sample and variance corrections to construct test statistics. Those procedures are intentionally outside this descriptive calculator, so no p-value or hypothesis-test conclusion is shown.
Worked example from the audited fixture
The audited fixture contains 10 percentage-point returns and uses q = 3, producing 8 overlapping three-period sums.
- One-period population variance is 0.480625; overlapping three-period population variance is 0.3583984375; q times one-period variance is 1.441875.
- VR(3) = 0.3583984375 ÷ 1.441875 = 0.24856415, so the displayed difference from one is −0.75143585. These are descriptive sample quantities, not a mean-reversion or random-walk decision.
Reproduce it: select “Load audited example” above. The governed engine retains full precision and rounds only the visible interface.
How to interpret the result
- A ratio of one means the two disclosed variance quantities are equal for the entered sample and horizon.
- A ratio above or below one records only the direction of this sample comparison; it does not establish trend or mean reversion.
- The table makes each overlapping aggregate visible so data order and window construction can be audited.
- Changing q changes the windows, the multi-period variance and the number of observations contributing to that variance.
- Use formal, assumption-matched statistical testing separately when a hypothesis-test conclusion is genuinely required.
Assumptions and limits
- Enter 5 to 500 equally spaced percentage-point returns under one coherent gross-or-net convention.
- The horizon is capped at 100 and must leave at least three overlapping aggregate windows.
- Population denominators are used for both one-period and overlapping q-period variance; other estimators can produce different ratios.
- Overlapping sums share observations, and version 1.0.0 applies no heteroskedasticity or finite-sample test correction.
- The calculator does not determine stationarity, identify structural breaks or estimate a tradable forecast.
- No random-walk conclusion, persistence label, mean-reversion claim, grade, signal or recommendation is generated.
Which return diagnostic answers which question?
Distribution shape, one-lag autocorrelation, joint residual autocorrelation and multi-period variance are related but not interchangeable. The comparison below preserves each tool’s evidence requirement and prevents one statistic from being presented as a universal strategy test.
| Diagnostic | Evidence entered | Question answered | Main boundary |
|---|---|---|---|
| Return Distribution Analyzer | Entered return series | Location, spread, percentiles, skewness and raw kurtosis | Descriptive moments only. |
| Jarque–Bera | Entered n, skewness and raw kurtosis | Joint normal-reference moment statistic | Small-sample p-value withheld; no normality verdict. |
| Return Autocorrelation | Entered return series plus one lag | One sample autocorrelation coefficient | No joint multi-lag reference. |
| Ljung–Box | Ordered residuals, maximum lag and fitted orders | Joint asymptotic portmanteau reference | No residual-independence or adequacy verdict. |
| Observed Variance Ratio | Entered returns plus horizon q | Overlapping multi-period variance divided by scaled one-period variance | No corrected z test, p-value or random-walk verdict. |
Frequently asked questions
- It divides the population variance of overlapping q-period return sums by q times the population variance of the entered one-period returns.
- Version 1.0.0 sums every overlapping window of q consecutive percentage-point returns in oldest-to-newest order.
- Both calculations use population denominators: n for one-period returns and n minus q plus one for overlapping aggregate returns.
- It means the two disclosed variance quantities are equal for this entered sample and horizon; it does not prove a random walk.
- No. Sampling variation, estimator choice, dependence, heteroskedasticity, non-stationarity and structural breaks can all affect the ratio.
- No. The direction of one descriptive sample ratio is not a persistence diagnosis, price forecast or trading signal.
- No. Version 1.0.0 performs no finite-sample or heteroskedasticity-robust variance correction, z statistic, p-value or formal hypothesis test.
- No. It creates no random-walk verdict, trend or mean-reversion label, grade, forecast, signal, position instruction or recommendation.
Sources and methodology
- MIT Open Learning Library — Andrew W. Lo, Market Efficiency — Primary-author explanation of the multi-period variance-ratio intuition and its relation to autocorrelation.
- Lo and MacKinlay — Stock Market Prices Do Not Follow Random Walks — Original variance-ratio specification-test paper; cited to distinguish the full test from this descriptive estimator.
Version 1.0.0 was locked only after the governing formulas and boundaries were checked in the cited primary or standards sources. Independent fixtures recompute the displayed statistics and reference probabilities separately from the browser adapter.
Continue the return-dependence review
Verify the return and residual evidence
Reconcile the exact statement period, sampling frequency, timezone, realized results, spread, commission, financing, conversion and fitted-model preprocessing before deriving inputs. These browser calculations cannot certify that an entered sample is complete, stationary or representative.
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