Augmented Dickey-Fuller

Install and import#

npm install fintech-algorithms

import { adf } from "fintech-algorithms/statistical-time-series/diagnostics/augmented-dickey-fuller";

Signature#

adf(values, lags, criticalValue)

Tests for a unit root. The null is that the series *has* one — so failing to reject is not evidence of stationarity, merely absence of evidence against a unit root. That asymmetry is the most misread thing in applied time series.

Parameters#

Returns#

{ method, variant, nobs, lags, gamma, gamma_standard_error, statistic, decision, … }

The statistic, the coefficient and its standard error, and an explicit decision — so the conclusion is separable from the arithmetic.

Errors#

• When the sample is too short for the requested lag order — throws

Complexity: time O(n × lags²), space O(n).

Worked example#

executed Captured by running this function on the input its own test provides. Real output of real code — but not asserted against a published figure.

Input#

[0.49366418, 1.54242291, -0.82556495, 0.47513288, 0.50704417, -0.3816439]

Showing 6 of 96 elements.

1

-2.86

Call#

adf(values, lags, criticalValue)

Returns#

object with 15 fields: method, variant, nobs, regression_nobs, lags, gamma, gamma_standard_error, statistic, …

{
  "method": "adf",
  "variant": "constant-only-fixed-lag",
  "nobs": 96,
  "regression_nobs": 94,
  "lags": 1,
  "gamma": -0.3307327402574888,
  "gamma_standard_error": 0.08555486242368153,
  "statistic": -3.8657386721007945,
  "critical_value": -2.86,
  "reject_null": true,
  "state": "reject-unit-root",
  "reason": "statistic-below-boundary",
  "coefficients": [0.09192051133561, -0.3307327402574888, -0.0052913498392460345],
  "residual_sse": 84.25503279621253
}

Showing 14 of 15 fields.

Other exports#

This module also exports acf, pacf, kpss, ljungBox, zivotAndrews, runDiagnostic. Every module additionally exports run as an alias of its primary function, and a meta object carrying its catalog id, domain, family, shape and article URL.

Diagrams#

Calculation flow#

flowchart TD
  A["Ordered finite series"] --> B{"Contract valid?"}
  B -- "No" --> X["Stop: explicit invalid state"]
  B -- "Yes" --> C["Difference after warm-up"]
  C --> D["Build fixed-lag constant-only OLS rows"]
  D --> E["Compute gamma t-ratio"]
  E --> F["Compare with nonstandard boundary"]
  F --> G{"Declared strict decision rule"}
  G --> H["Report machine state, null, and limitation"]
  H --> I["Compare with KPSS"]

How it works#

This page states the contract — how to call it correctly. The article explains the concept: why it works, and where it breaks.

Read the article →

References#

• statsmodels.tsa.stattools.adfuller
• Distribution of the Estimators for Autoregressive Time Series with a Unit Root
• Evidence decisions

The rest of the Diagnostics family#

• ACFAutocorrelation at each lag.
• PACFPartial autocorrelation: correlation at each lag with the intervening lags removed.
• KPSSTests stationarity with the null *reversed* relative to ADF: here the null is that the series is stationary.
• Ljung-BoxTests whether a group of autocorrelations is jointly zero.
• Zivot-Andrews Break TestA unit-root test that allows one structural break at an unknown date, found by searching.