Patterns › Time series

Vector autoregression (VAR) + Granger causality

A vector autoregression regresses each of several time series on p lags of all the series, with Granger-causality tests and a stability check.

What is Vector autoregression (VAR) + Granger causality?

A VAR treats a set of series symmetrically — every variable is explained by the recent history of the whole system, with no a-priori exogeneity. It's the standard reduced-form tool for multivariate macro/financial dynamics and the basis for impulse-response and forecast work.

Granger causality asks whether one variable's lags improve prediction of the others (an F-test on the excluded lags). Stability (all companion-matrix roots inside the unit circle) is required for the VAR to be stationary.

When should I use Vector autoregression (VAR) + Granger causality?

  • Modeling several interacting time series (e.g. output, prices, rates).
  • Testing lead-lag / predictive relationships.

What data does it need?

≥ 2 numeric series (row order = time) + lag order p (or automatic AIC selection) + deterministic terms.

What does it report?

Selected lag, per-equation R², AIC/BIC, stability, and Granger-causality F-tests for each variable against the rest.

What does it assume?

  • Series are stationary (difference or use a VECM if they're cointegrated).
  • Enough observations for k·p parameters per equation.

How do I interpret the result?

Granger 'causality' is predictive precedence, not structural causation. An unstable VAR signals mis-specified lags or non-stationary data.

See also

References

  • Pfaff (2008). VAR, SVAR and SVEC models. JSS 27.
  • Lütkepohl (2005). New Introduction to Multiple Time Series Analysis.