What is a vector autoregression (VAR) model?

Question

AcadiFi · Accepted Answer

A vector autoregression, VAR, is a multivariate generalization of AR. It models a vector of time series, each regressed on p lags of itself and all other series. For two variables, VAR(1) is:

x_t equals c_1 plus a_{11} times x_{t-1} plus a_{12} times y_{t-1} plus epsilon_{1t}
y_t equals c_2 plus a_{21} times x_{t-1} plus a_{22} times y_{t-1} plus epsilon_{2t}

Every variable is endogenous, so VAR does not require a priori theoretical identification. It captures dynamic interactions across variables without imposing structural restrictions.

Example: macro analyst Fernanda Teixeira-Oyama at Boa Vista Research estimates a VAR(4) on quarterly GDP growth, CPI inflation, and short-term policy rate for Brazil from 2000 to 2026. The model reveals that a positive rate shock reduces inflation with a four-quarter lag and GDP with a two-quarter lag, consistent with monetary transmission theory.

```mermaid
flowchart LR
    X[Var X lags] --> Y[Var Y]
    Y[Var Y lags] --> X
    X --> X
    Y --> Y
```

Lag order selection uses AIC or BIC. All series should be stationary; if cointegrated, use a Vector Error Correction Model (VECM) instead. Once estimated, VAR models are the foundation for impulse response analysis, variance decomposition, and Granger causality tests (covered in other questions).

VAR models underpin much of modern central bank policy analysis and are a core tool at the Federal Reserve, ECB, and BoE.

What is a vector autoregression (VAR) model?

Master Level II with our CFA Course

Related Questions

Practice Questions