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VAR

Install and import#

bash
npm install fintech-algorithms
ts
import { fitVAR } from "fintech-algorithms/statistical-time-series/multivariate-systems/var";

Signature#

fitVAR(values, lags, includeIntercept)

Vector autoregression: every series regressed on lags of all of them. The natural model when variables move together, and the base every impulse-response and variance-decomposition result is computed from.

Parameters#

NameTypeNotes
valuesnumber[][]Multivariate series, one row per observation and one column per variable. Column order is meaningful — everything downstream refers to variables by position.
lagsnumberLag order. Parameters grow with the square of the number of variables, so this is where a VAR runs out of data.
min: 1 · integer: true
includeInterceptbooleanWhether to fit a constant term.
optional

Returns#

{ intercept, coefficients, sigma_u_mle, residuals, fitted, one_step_forecast, effective_observations, … }

Coefficients and the residual covariance, which is the input to structural identification.

Errors#

  • When observations are fewer than the parameters to estimate — throws

Complexity: time O(n × (k × lags)²), space O((k × lags)²).

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
[
  [-1.4163056264, -0.7075620912],
  [-1.4144491277, -0.24959813],
  [-1.915874322, -0.31233481]
]

Showing 3 of 120 elements.

lags
1

Call#

fitVAR(values, lags, includeIntercept)

Returns#

object with 16 fields: intercept, coefficients, sigma_u_mle, residuals, fitted, one_step_forecast, effective_observations, variables, …

{
  "intercept": [-0.070907764383, -0.013615076949],
  "coefficients": [
    [
      [0.555831819774, 0.20542641337],
      [-0.118628539474, 0.408179850969]
    ]
  ],
  "sigma_u_mle": [
    [0.726815513707, 0.33092608263],
    [0.33092608263, 0.420249111781]
  ],
  "residuals": [
    [-0.410961687007, -0.115184732022],
    [-1.00749667636, -0.364632839724],
    [-0.403977225473, -0.149480679886]
  ],
  "fitted": [
    [-1.003487440693, -0.134413397978],
    [-0.90837764564, 0.052298029724],
    [-1.199973495027, 0.086173519486]
  ],
  "one_step_forecast": [0.452328292866, -0.113742494174],
  "effective_observations": 119,
  "variables": 2,
  "lags": 1,
  "row_sum_stability_bound": 0.761258233144,
  "companion_spectral_radius": 0.501247229166,
  "stability_state": "stable",
  "stability_boundary": 1,
  "near_boundary_threshold": 0.9
}

Showing 14 of 16 fields.

Other exports#

This module also exports companionMatrix, companionSpectralRadius, choleskyLower, fitRecursiveSVAR, fitVECMFixedBeta, movingAverageMatrices, impulseResponses, forecastErrorVarianceDecomposition. 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#

VAR — system map

Calculation flow#

VAR system flow 
flowchart LR
    A["Aligned stationary rows"] --> B["Lagged design matrix"]
    B --> C["Equation-wise OLS"]
    C --> D["A matrices and residual covariance"]
    D --> E["Forecast plus diagnostics"]

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#

The rest of the Multivariate Systems family#