# PACF

`D09-F01-A02` · Statistical Time Series → Diagnostics · archetype `record-transform` · difficulty 3/5 · verification **contract**

Full page: https://docs.thefintechbuilder.com/statistical-time-series/diagnostics/pacf/
Agent skill: `npx skills add IslamBaraka90/Fintech-Algorithms-Library` — https://docs.thefintechbuilder.com/guides/agent-skill/

## Install and import

```bash
npm install fintech-algorithms
```

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

## Signature

```ts
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

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `values` | `number[]` | yes | Observation series in chronological order, oldest first. |
| `maxLag` | `number` | yes | Highest 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

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`:

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

Showing 6 of 96 elements.

`maxLag`:

```json
12
```

### Call

```ts
pacf(values, maxLag)
```

### Returns

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

```json
{
  "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

`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.

## Verification and provenance

Tier: **contract**.

The module loads, the entry point is callable and its declared signature matches the compiled code. The example below is real captured output, but no independently published figure asserts the numbers.

Both tiers guarantee the signature. Full explanation: https://docs.thefintechbuilder.com/guides/verification/

Generated from the docs.json payload shipped inside fintech-algorithms@0.13.1.
The signature and parameter list are checked against the compiled implementation at build time,
so a description that contradicts the code fails the build rather than reaching this file.

## Links

- Article (how it works, step by step): https://thefintechbuilder.com/statistical-time-series/diagnostics/pacf/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/statistical-time-series/diagnostics/pacf/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/statistical-time-series/llms.txt