VECM (Vector Error Correction Model)

VECM is a multivariate model used when several time series are non-stationary I(1) yet cointegrated — that is, a long-run equilibrium relationship exists among them. VECM extends VAR by adding an error correction term that captures the speed at which the variables return to long-run equilibrium after a shock.

In EcoLab, VECM belongs to the Time Series group. Unlike ARDL (a single equation, suitable when there is one clear dependent variable), VECM is appropriate when you suspect a multi-directional simultaneous relationship among the variables.

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When should you use VECM?

• You have two or more variables that are all I(1) and suspect a long-run (cointegrating) relationship.
• You care about system dynamics (impulse responses, variance decomposition), not just a single equation.
• Cointegration has been confirmed by the Johansen test (trace / max-eigenvalue).

If all variables are I(0) → use a plain VAR. If they are I(1) but NOT cointegrated → use a VAR in differences.

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Model specification

Reduced VECM form:

$$\Delta Y_t = \Pi \, Y_{t-1} + \sum_i \Gamma_i \, \Delta Y_{t-i} + \varepsilon_t$$

• $\Pi = \alpha \beta'$: the matrix $\beta$ contains the cointegrating vectors (long-run relationships); $\alpha$ is the speed of adjustment.
• The rank of $\Pi = r$ = the number of cointegrating relationships, determined by the Johansen test.
• $\Gamma_i$: short-run dynamics.

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Testing workflow

1. Unit-root tests (ADF/PP/KPSS): confirm the variables are all I(1).
2. Lag selection for the base VAR (AIC/BIC/HQ).
3. Johansen cointegration test (trace & max-eigenvalue) → determine the rank $r$.
4. Estimate the VECM with rank $r$; check the sign and significance of $\alpha$ (error correction).
5. Residual diagnostics: autocorrelation (LM), normality, stability; analyze IRF and variance decomposition (FEVD).

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Running in EcoLab

1. Data Collection module: obtain the relevant time series (same frequency).
2. Modeling module → Time Series group → VAR/VECM; choose the system of variables.
3. Declare the lag length and the cointegration rank (or let the system suggest them from Johansen).
4. Read the results: the long-run vector $\beta$, the adjustment coefficient $\alpha$, IRF/FEVD; get the code from the Replication Code tab.

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Input / output example

Input (illustrative): quarterly series of lgdp (log GDP), lm2 (log money supply), lcpi (log price).

Output (format, illustrative figures — not real results):

Component	Value	Note
Johansen trace (r=0)	rejected	at least one relationship
Johansen trace (r≤1)	not rejected	r = 1
$\alpha$ (lgdp)	−0.18***	adjusts toward equilibrium
Long-run vector $\beta$	lgdp − 0.7·lm2 + 0.5·lcpi	equilibrium relationship

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Replication code

Stata

* --- VECM ---
tsset time

* Step 1: Johansen cointegration rank test
vecrank y1 y2 y3, lags(2)

* Step 2: Estimate VECM with rank r = 1
vec y1 y2 y3, rank(1) lags(2)

* Long-run cointegrating vector (beta)
* Speed-of-adjustment coefficients (alpha)

* IRF from VECM
irf create vecirf, set(vec_irf) step(20)
irf graph oirf

R

# --- VECM ---
library(urca)
library(vars)

# Step 1: Johansen cointegration test
data_mat <- cbind(y1, y2, y3)
jo_test  <- ca.jo(data_mat, type = "trace", K = 2,
                  ecdet = "const")
summary(jo_test)

# Step 2: Convert to VECM (rank = 1)
vecm <- cajorls(jo_test, r = 1)
summary(vecm$rlm)

# Convert VECM to VAR-level for IRF
var_from_vec <- vec2var(jo_test, r = 1)
irf_res <- irf(var_from_vec, n.ahead = 20)
plot(irf_res)

Python

from statsmodels.tsa.vector_ar.vecm import VECM, coint_johansen
import matplotlib.pyplot as plt

data = df[['y1', 'y2', 'y3']]

# Step 1: Johansen cointegration test
johansen = coint_johansen(data, det_order=0, k_ar_diff=1)
print("Trace statistic:", johansen.lr1)
print("Critical values:", johansen.cvt)

# Step 2: Estimate VECM with rank 1
model = VECM(data, k_ar_diff=1, coint_rank=1)
result = model.fit()
print(result.summary())

# IRF
irf = result.irf(20)
irf.plot()
plt.tight_layout(); plt.show()

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Limitations and notes

• Sensitive to lag selection and the deterministic terms (constant/trend) in Johansen.
• Requires a sufficiently long sample; Johansen results are unreliable with short samples.
• Interpreting cointegrating vectors requires normalization and grounding in economic theory.
• Does not handle I(2) variables; check the order of integration first.

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Video tutorial

Video Tutorial: Running VECM in EcoLab

See also

• ARDL Model — single-equation alternative
• Worked example: FDI and growth (ARDL)
• Estimation & Econometric Modeling