Patterns › Time series

Vector error-correction model (VECM)

Vector error-correction model estimates the cointegrating vector(s) β and the adjustment speeds α for cointegrated I(1) series.

What is Vector error-correction model (VECM)?

When series are cointegrated, a pure VAR-in-differences throws away the long-run level relationship. A VECM adds an error-correction term — the lagged deviation from equilibrium (β'y) — to each differenced equation, so the system is pulled back toward its long-run path.

The Johansen procedure estimates the cointegrating rank r and the normalized cointegrating vector β (the equilibrium relation). The loadings α measure how fast each equation corrects: a negative own-loading means the variable moves to close the gap after a shock.

When should I use Vector error-correction model (VECM)?

  • Modeling ≥ 2 cointegrated I(1) series (confirm with the Johansen test first).
  • Estimating both the long-run equilibrium and the speed of adjustment to it.

What data does it need?

≥ 2 numeric I(1) series (time-ordered) + cointegration rank r + lag length K + deterministic (ecdet) term.

What does it report?

The normalized cointegrating vector(s) β and the adjustment-speed matrix α, one loading per equation.

What does it assume?

  • Series are individually I(1) and cointegrated at the chosen rank.
  • Results are sensitive to K and the deterministic specification — mirror the cointegration test's settings.

How do I interpret the result?

β gives the long-run relation (β'y ≈ 0 in equilibrium). A negative, significant α on a variable's own equation confirms it error-corrects; a near-zero α means that variable is weakly exogenous.

See also

References

  • Johansen (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models.
  • Pfaff (2008). Analysis of Integrated and Cointegrated Time Series.