SpreadAr1Coefficient

The first-order autoregression coefficient ρ of the spread a − b — a direct measure of pairs cointegration / mean-reversion strength.

Quick reference

Field	Value
Family	Price Statistics
Input type	(f64, f64) (a pair of prices a, b)
Output type	f64
Output range	unbounded; 0–1 for stationary spreads, ~1 at a unit root
Default parameters	period is required (>= 3)
Warmup period	period
Interpretation	ρ → 0 strong mean reversion; ρ → 1 random walk (not cointegrated); ρ > 1 explosive.

Formula

Each update forms the spread sₜ = aₜ − bₜ and fits the AR(1) model over the trailing window by ordinary least squares of the level on its own lag:

sₜ = ρ · sₜ₋₁ + c + εₜ
ρ  = cov(sₜ₋₁, sₜ) / var(sₜ₋₁)

ρ is the raw stationarity statistic many pairs-trading screens threshold on directly. It is the complement of OuHalfLife: the OU half-life is −ln(2) / ln(ρ) for 0 < ρ < 1. When the spread is flat over the window (var(sₜ₋₁) = 0) the slope is undefined and the indicator returns 0.

Source: crates/wickra-core/src/indicators/spread_ar1_coefficient.rs.

Parameters

Name	Type	Default	Valid range	Source	Description
period	usize	none	>= 3	spread_ar1_coefficient.rs:60	Look-back window of spreads. < 3 errors with Error::InvalidPeriod.

Inputs / Outputs

From crates/wickra-core/src/indicators/spread_ar1_coefficient.rs:

use wickra::{Indicator, SpreadAr1Coefficient};
// SpreadAr1Coefficient: Input = (f64, f64), Output = f64
const _: fn(&mut SpreadAr1Coefficient, (f64, f64)) -> Option<f64> =
    <SpreadAr1Coefficient 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() == 30 for period = 30; warmup_returns_none pins the first Some at the period-th pair.

Edge cases

• Random walk. A spread that grows by exactly 1 each bar has a unit root, so ρ = 1; pinned by random_walk_spread_has_rho_near_one.
• Mean reversion. A fast sinusoidal spread is stationary, so 0 < ρ < 1; pinned by mean_reverting_spread_has_rho_below_one.
• Flat spread. A constant spread has zero level variance, so ρ = 0 (undefined); pinned by flat_spread_returns_zero.
• Reset. reset() clears the spread window.

Examples

Rust

use wickra::{BatchExt, Indicator, SpreadAr1Coefficient};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut ar1 = SpreadAr1Coefficient::new(20)?;
    // Spread a - b grows by exactly 1 each bar -> unit root -> rho = 1.
    let pairs: Vec<(f64, f64)> =
        (0..40).map(|t| (2.0 * f64::from(t), f64::from(t))).collect();
    let last = ar1.batch(&pairs).into_iter().flatten().last().unwrap();
    println!("{:.6}", last);
    Ok(())
}

Output:

1.000000

Python

import numpy as np
import wickra as ta

a = np.array([2.0 * i for i in range(40)])
b = np.array([float(i) for i in range(40)])
print(round(ta.SpreadAr1Coefficient(20).batch(a, b)[-1], 6))

Output:

1.0

Node

const ta = require('wickra');
const a = Array.from({ length: 40 }, (_, i) => 2 * i);
const b = Array.from({ length: 40 }, (_, i) => i);
console.log(new ta.SpreadAr1Coefficient(20).batch(a, b).at(-1).toFixed(6));

Output:

1.000000

Interpretation

Use SpreadAr1Coefficient to screen pairs for tradeability: ρ well below 1 means the spread mean-reverts and a stat-arb entry has an edge; ρ near 1 means the spread wanders like a random walk and there is no reliable reversion. Pair it with OuHalfLife (holding-period sizing) and PairSpreadZScore (entry/exit timing).

Common pitfalls

• Sign of the spread. ρ is computed on a − b; swapping the inputs leaves ρ unchanged (the AR(1) coefficient is symmetric in the spread's sign), but downstream z-scores are not — keep the leg order consistent.
• ρ = 0 as a sentinel. A flat spread also returns 0; distinguish "strong reversion" from "degenerate flat" via the spread's own variance if needed.

References

Roll's and Engle-Granger's cointegration framework; the AR(1) unit-root intuition underlies the Augmented Dickey-Fuller test (Dickey & Fuller, 1979).

See also

• Indicator-Cointegration — ADF-style cointegration test.
• Indicator-PairSpreadZScore — spread entry timing.
• Indicators-Overview — the full taxonomy.