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VECM

Estimates the vector error-correction model once the Johansen test says your series are cointegrated: the long-run relationship, how fast each variable adjusts to it, and who is weakly exogenous.

What it needs

Series - Two or more I(1) series - check with the cointegration module first.

Frequently asked questions

What is a VECM model?

A vector error correction model (VECM) is a VAR for variables that are non-stationary but cointegrated. It models their short-run changes while including an error correction term that pulls them back towards their long-run equilibrium relationship.

When should I use a VECM instead of a VAR?

Use a VECM when the series are I(1) and the Johansen test finds at least one cointegrating relationship. A VAR in levels ignores the long-run constraint, and a VAR in differences throws the long-run relationship away; the VECM keeps both.

How do I interpret the alpha and beta in a VECM?

Beta is the cointegrating vector: the long-run equilibrium relationship between the variables. Alpha is the speed of adjustment: how much of last period's deviation from equilibrium each variable corrects this period. A significant negative alpha on a variable means it does the adjusting; an alpha near zero means it is weakly exogenous.

What is the error correction term?

The error correction term is last period's deviation from the long-run relationship, beta times the lagged levels. Multiplied by alpha, it is the part of each variable's change that closes the gap to equilibrium.

How many cointegrating relationships should a VECM have?

As many as the Johansen trace or maximum-eigenvalue test supports, which must be fewer than the number of variables. Zero means no cointegration (use a VAR in differences); as many as the number of variables means the series are stationary (use a VAR in levels).

The intuition

Two non-stationary series that are tied together cannot drift apart forever. A VECM formalises that: it models the CHANGES in each variable, but adds a term measuring how far the system currently sits from its long-run equilibrium. The further from equilibrium, the harder the pull back. β describes the equilibrium; α describes who does the adjusting.

When to use it

After Johansen finds one or more cointegrating relationships. It is the correct model there - a VAR in differences would discard the long-run information.

Where the estimator comes from

Start from a VAR in levels and rewrite it as \Delta y_t=\Pi y_{t-1}+\sum_i \Gamma_i \Delta y_{t-i}+\varepsilon_t. The rank of \Pi decides everything: rank 0 means no cointegration (a VAR in differences), full rank means the series were stationary all along, and intermediate rank r means \Pi=\alpha\beta' where \beta holds the r cointegrating vectors and \alpha the adjustment speeds. Johansen estimates this by reduced-rank regression, and the trace and maximum-eigenvalue statistics test the rank.

The model

\Delta \mathbf{y}_t=\boldsymbol{\alpha}\boldsymbol{\beta}^{\prime}\mathbf{y}_{t-1}+\sum_{i=1}^{p-1}\boldsymbol{\Gamma}_i\,\Delta\mathbf{y}_{t-i}+\boldsymbol{\varepsilon}_t

What it assumes

The series are I(1) - check with unit-root tests.
The cointegrating rank r is correct - from the Johansen trace test.
0 < r < number of variables. If r = 0 use a VAR in differences; if r = n they were stationary all along.
The lag length was chosen on the VAR in levels before the rank was tested. The Johansen rank test conditions on the lag order, so choosing lags after fixing the rank gets the sequence backwards.
The cointegrating relationship is stable across the sample. A structural break in the long-run relationship shows up as a rank that changes with the sample window, which is worth checking by re-running on subsamples.

Statistical properties

β is identified only up to a normalisation, which is why coefficients are reported relative to one variable.
The Granger representation theorem says cointegration and an error-correction representation are equivalent - one implies the other.
Superconsistency: β̂ converges at rate T rather than √T, so the long-run parameters are estimated very precisely.
An α that is insignificant means that variable is weakly exogenous - it pushes the system rather than being pushed by it.
Johansen critical values are sensitive to the deterministic specification; choosing it casually invalidates the test.

How to read the output

β is the long-run relationship. In logs, read it as a long-run elasticity.
α is the speed of adjustment. Negative and significant means that variable corrects towards equilibrium.
An α that is NOT significant means that variable is weakly exogenous - it drives the system rather than responding to it. That is often the most interesting finding.
The plotted error-correction term should be visibly stationary.

A worked reading

β normalised on log GDP gives (1, −0.83) on log money, so the long-run money elasticity of output is 0.83. α = −0.19 on the GDP equation and insignificant on money: output does the adjusting, correcting 19% of any gap per quarter (half-life ≈ 3.3 quarters), while money supply is weakly exogenous.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Rank is 0	No cointegration	Model in first differences - a plain VAR on Δy
Rank equals the number of variables	Everything was stationary	Use a VAR in levels
α positive and significant	Explosive - moves away from equilibrium	Check the sign convention and the normalisation
All α insignificant	Nothing adjusts	The cointegrating relation may be spurious
Results flip with the deterministic terms	Specification-driven	Justify the choice economically and report alternatives

Common mistakes

Getting the sign convention wrong - the signs in β are reversed relative to how you would write the equilibrium equation.
Picking the rank to suit your story rather than the test.
Interpreting β as causal. It is a long-run association.
Reporting the adjustment coefficients without checking which variables are weakly exogenous. A variable whose adjustment coefficient is indistinguishable from zero is not adjusting to the long-run relationship at all - it is driving it, and that is usually the interesting finding.

Where to read this properly

Engle & Granger (1987)
Johansen (1991, 1995)
Lütkepohl (2005), chapter 6
Juselius (2006), The Cointegrated VAR Model

Related Time series - several series

Yield curve (Nelson–Siegel / Svensson) · Dynamic factor model · Dynamic regression (ARIMAX) · Structural VAR · VAR · Granger causality · Cointegration · ARDL & bounds test

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VECM

What it needs

Frequently asked questions

What is a VECM model?

When should I use a VECM instead of a VAR?

How do I interpret the alpha and beta in a VECM?

What is the error correction term?

How many cointegrating relationships should a VECM have?

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Time series - several series

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