fintech-algorithms
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PACF

Install and import#

bash
npm install fintech-algorithms
ts
import { pacf } from "fintech-algorithms/statistical-time-series/diagnostics/pacf";

Signature#

pacf(values, maxLag)

Partial autocorrelation: correlation at each lag with the intervening lags removed. Read together with the ACF it identifies model order — a PACF cutting off after lag p suggests AR(p).

Parameters#

NameTypeNotes
valuesnumber[]Observation series in chronological order, oldest first.
maxLagnumberHighest lag to compute.
min: 1 · integer: true

Returns#

{ method, variant, coefficients, acf_coefficients, recursion, confidence_95 }

Partial coefficients with the Durbin–Levinson recursion steps that produced them.

Errors#

  • When maxLag is not less than the sample size — throws

Complexity: time O(maxLag²), space O(maxLag).

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#

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

Showing 6 of 96 elements.

maxLag
12

Call#

pacf(values, maxLag)

Returns#

object with 10 fields: method, variant, nobs, max_lag, coefficients, acf_coefficients, recursion, confidence_95, …

{
  "method": "pacf",
  "variant": "biased-yule-walker-levinson-durbin",
  "nobs": 96,
  "max_lag": 12,
  "coefficients": [
    1,
    0.6643847368068891,
    -0.0011391366900339354,
    -0.013487015741398519,
    -0.1763888054329689,
    -0.1217089519082119
  ],
  "acf_coefficients": [
    1,
    0.6643847368068891,
    0.4407707648102877,
    0.28488438813871975,
    0.08546397187939134,
    -0.07801591085987973
  ],
  "recursion": [
    {
      "lag": 1,
      "reflection": 0.6643847368068891,
      "prediction_variance": 0.5585929214980406
    },
    {
      "lag": 2,
      "reflection": -0.0011391366900339354,
      "prediction_variance": 0.5585921966497681
    },
    {
      "lag": 3,
      "reflection": -0.013487015741398519,
      "prediction_variance": 0.5584905889562045
    }
  ],
  "confidence_95": 0.2000416623272929,
  "state": "estimated",
  "reason": "positive prediction variance through requested lag"
}

Other exports#

This module also exports acf, adf, 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#

PACF — diagnostic anatomy

Calculation flow#

PACF decision flow 
flowchart TD
  A["Ordered finite series"] --> B{"Contract valid?"}
  B -- "No" --> X["Stop: explicit invalid state"]
  B -- "Yes" --> C["Compute biased ACF"]
  C --> D["Seed reflection and variance"]
  D --> E["Run Levinson-Durbin recursion"]
  E --> F["Inspect incremental lag coefficient"]
  F --> G{"Declared strict decision rule"}
  G --> H["Report machine state, null, and limitation"]
  H --> I["Compare with Augmented Dickey-Fuller"]

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.acf
  • statsmodels.tsa.stattools.pacf
  • Evidence decisions

The rest of the Diagnostics family#

  • ACFAutocorrelation at each lag.
  • Augmented Dickey-FullerTests for a unit root.
  • 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.