VarianceRatio

The Lo–MacKinlay variance-ratio test on the spread a − b — does the spread mean-revert, random-walk, or trend?

Quick reference

Field	Value
Family	Price Statistics
Input type	(f64, f64) (a pair of prices a, b)
Output type	f64
Output range	>= 0; 1 is the random-walk null
Default parameters	period, q both required (q >= 2, period >= q + 2)
Warmup period	period
Interpretation	VR < 1 mean reversion; VR ≈ 1 random walk; VR > 1 momentum / trend.

Formula

Each update forms the spread sₜ = aₜ − bₜ and compares the variance of q-step changes against q times the variance of one-step changes:

rₜ    = sₜ − sₜ₋₁                                  (one-step changes)
VR(q) = Var(Σ of q consecutive r) / (q · Var(r))

Under a random walk the variance of returns grows linearly with the horizon, so VR(q) = 1. Departures reveal autocorrelation structure:

• VR(q) < 1 — mean reversion (negatively autocorrelated changes): the spread's moves partly cancel, the regime pairs traders exploit.
• VR(q) ≈ 1 — a random walk: no exploitable structure.
• VR(q) > 1 — momentum / trending (positively autocorrelated changes).

The estimator uses overlapping q-step windows. When the one-step changes have zero variance (a flat spread) the ratio is undefined and the indicator returns the null value 1. The output is always >= 0.

Source: crates/wickra-core/src/indicators/variance_ratio.rs.

Parameters

Name	Type	Default	Valid range	Source	Description
period	usize	none	>= q + 2	variance_ratio.rs:66	Look-back window of spreads.
q	usize	none	>= 2	variance_ratio.rs:66	Aggregation horizon (number of steps). Bad parameters error with Error::InvalidPeriod.

Inputs / Outputs

From crates/wickra-core/src/indicators/variance_ratio.rs:

use wickra::{Indicator, VarianceRatio};
// VarianceRatio: Input = (f64, f64), Output = f64
const _: fn(&mut VarianceRatio, (f64, f64)) -> Option<f64> =
    <VarianceRatio as Indicator>::update;

Python streams as update(a, b) -> float | None and batches over two equal-length arrays. Node streams as update(a, b) and batches over a[], b[].

Warmup

warmup_period() == period. The unit test accessors_and_metadata pins warmup_period() == 60 for period = 60; warmup_returns_none pins the first Some at the period-th pair.

Edge cases

• Mean reversion. An oscillating spread has negatively autocorrelated changes and VR < 1; pinned by oscillating_spread_is_below_one. Perfectly alternating changes drive VR to 0 (alternating_changes_give_zero_ratio).
• Random walk / flat. A flat spread has zero one-step variance, so the ratio is undefined and returns the null 1; pinned by flat_spread_returns_one.
• Non-negative. The output is clamped to >= 0; pinned by output_non_negative.
• Reset. reset() clears the spread window.

Examples

Rust

use wickra::{BatchExt, Indicator, VarianceRatio};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut vr = VarianceRatio::new(60, 2)?;
    // High-frequency oscillating spread -> mean reversion -> VR < 1.
    let pairs: Vec<(f64, f64)> = (0..200)
        .map(|t| {
            let b = 100.0 + f64::from(t);
            (b + 2.0 * (f64::from(t) * 2.5).sin(), b)
        })
        .collect();
    let last = vr.batch(&pairs).into_iter().flatten().last().unwrap();
    println!("{last:.6}");
    Ok(())
}

Output:

0.201245

Python

import numpy as np
import wickra as ta

t = np.arange(200)
b = 100.0 + t
a = b + 2.0 * np.sin(t * 2.5)
print(round(ta.VarianceRatio(60, 2).batch(a, b)[-1], 6))

Output:

0.201245

Node

const ta = require('wickra');
const b = Array.from({ length: 200 }, (_, t) => 100 + t);
const a = b.map((bv, t) => bv + 2 * Math.sin(t * 2.5));
console.log(new ta.VarianceRatio(60, 2).batch(a, b).at(-1).toFixed(6));

Output:

0.201245

Interpretation

The variance ratio is a formal test of the random-walk hypothesis on the spread. A VR materially below 1 (here ~0.20) is strong evidence of mean reversion — the spread's moves cancel over the horizon q, exactly the structure a relative-value trade harvests. A VR above 1 warns of momentum: a fade would be fighting the trend. Sweep q to see at what horizon the structure lives — reversion at q = 2 but not q = 10 means the edge is short-lived. Pair with SpreadHurst for a second opinion on the regime.

Common pitfalls

• 1 is the null, not a measurement. A flat spread also returns exactly 1. Confirm with the spread's own variance before reading 1 as "random walk".
• Horizon sensitivity. VR(q) depends on q; reporting a single q hides the horizon structure. Scan several q values when screening.

References

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.

See also

• Indicator-SpreadHurst — a complementary regime classifier.
• Indicator-OuHalfLife — reversion time scale.
• Indicators-Overview — the full taxonomy.