Impulse-Response Analysis

Install and import#

npm install fintech-algorithms

import { impulseResponses } from "fintech-algorithms/statistical-time-series/multivariate-systems/impulse-response-analysis";

Signature#

impulseResponses(coefficients, horizon, impactMatrix)

Traces how a one-off shock to one variable propagates through the system over time. The headline output of any VAR — and only interpretable given the identifying assumption that produced the impact matrix.

Parameters#

Returns#

{ responses, cumulative_responses, horizon, impact_matrix, companion_spectral_radius, stability_state, … }

Responses and their cumulative sums, plus the companion spectral radius — above 1 the system is explosive and the responses diverge rather than decay, which is stated instead of silently plotted.

Errors#

• When the coefficient matrices are not square or are inconsistent in dimension — throws

Complexity: time O(horizon × k³), space O(horizon × k²).

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#

[
  [
    [0.55, 0.18],
    [-0.12, 0.42]
  ]
]

8

[
  [0.8, 0],
  [0.35, 0.55]
]

Call#

impulseResponses(coefficients, horizon, impactMatrix)

Returns#

object with 10 fields: responses, cumulative_responses, horizon, impact_matrix, companion_spectral_radius, stability_state, stability_boundary, near_boundary_threshold, …

{
  "responses": [
    [
      [0.8, 0],
      [0.35, 0.55]
    ],
    [
      [0.503, 0.099],
      [0.051, 0.231]
    ],
    [
      [0.28583, 0.09603],
      [-0.03894, 0.08514]
    ]
  ],
  "cumulative_responses": [
    [
      [0.8, 0],
      [0.35, 0.55]
    ],
    [
      [1.303, 0.099],
      [0.401, 0.781]
    ],
    [
      [1.58883, 0.19503],
      [0.36206, 0.86614]
    ]
  ],
  "horizon": 8,
  "impact_matrix": [
    [0.8, 0],
    [0.35, 0.55]
  ],
  "companion_spectral_radius": 0.502593274925,
  "stability_state": "stable",
  "stability_boundary": 1,
  "near_boundary_threshold": 0.9,
  "state": "computed",
  "reason": "ma-recursion-times-declared-impact"
}

Other exports#

This module also exports companionMatrix, companionSpectralRadius, fitVAR, choleskyLower, fitRecursiveSVAR, fitVECMFixedBeta, movingAverageMatrices, 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#

Calculation flow#

flowchart LR
    A["VAR coefficient matrices"] --> B["Phi zero equals identity"]
    B --> C["Moving-average recursion"]
    C --> D["Multiply by impact matrix"]
    D --> E["Response and cumulative paths"]

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 vector autoregression documentation — statsmodels developers
• statsmodels Impulse-Response Analysis API documentation — statsmodels developers
• Generalized Impulse Response Analysis in Linear Multivariate Models — M. Hashem Pesaran and Yongcheol Shin
• New Introduction to Multiple Time Series Analysis — Helmut Lütkepohl
• Macroeconomics and Reality — Christopher A. Sims
• Statistical Analysis of Cointegration Vectors — Søren Johansen
• Publication boundary

The rest of the Multivariate Systems family#

• VARVector autoregression: every series regressed on lags of all of them.
• Structural VARIdentifies structural shocks by Cholesky decomposition of the residual covariance.
• VECMVector error correction with a fixed cointegrating vector.
• Forecast-Error Variance DecompositionAttributes each variable's forecast error variance to the structural shocks, by horizon.