VAR — Vector Autoregression

VAR models a system of time series where each variable depends on its own lags and the lags of the others — without imposing a priori causal direction. It is the standard tool for system dynamics: impulse responses (IRF), forecast error variance decomposition (FEVD), and Granger causality tests.

When to use

Use VAR when all series are stationary I(0) (or differenced). If I(1) series are cointegrated ⇒ use VECM; if I(1) but not cointegrated ⇒ VAR in differences.

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Model specification

$$Y_t = c + A_1 Y_{t-1} + A_2 Y_{t-2} + \dots + A_p Y_{t-p} + \varepsilon_t$$

where $Y_t$ is a vector of variables and $A_i$ are coefficient matrices. Choose lag $p$ via AIC/BIC/HQ.

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Post-estimation analysis

• IRF: the system's response to a shock.
• FEVD: decomposition of forecast variance by shock source.
• Granger causality: which variable helps forecast which.

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Running in EcoLab

1. Modeling module → Multivariate time series family → VAR.
2. Choose the variable set (same frequency, stationarity checked), the lag length.
3. Run; view IRF/FEVD/Granger; export the replication code.

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Replication code

Stata

* --- VAR ---
tsset time

* Estimate VAR(2)
var rate inflation output, lags(1/2)

* Granger causality test
vargranger

* Impulse Response Functions (orthogonalized)
irf create myirf, set(irf_results) step(20)
irf graph oirf, impulse(rate) response(inflation output)

* Forecast Error Variance Decomposition
irf graph fevd

R

# --- VAR ---
library(vars)

# Prepare data matrix
data_var <- cbind(rate, inflation, output)

# Select lag order
VARselect(data_var, lag.max = 8)

# Estimate VAR(2)
model <- VAR(data_var, p = 2)
summary(model)

# Granger causality
causality(model, cause = "rate")

# Impulse Response Functions
irf_res <- irf(model, impulse = "rate", n.ahead = 20)
plot(irf_res)

# Forecast Error Variance Decomposition
fevd_res <- fevd(model, n.ahead = 20)
plot(fevd_res)

Python

from statsmodels.tsa.api import VAR
import matplotlib.pyplot as plt

# Prepare data
data = df[['rate', 'inflation', 'output']]

# Estimate VAR(2)
model = VAR(data)
result = model.fit(2)
print(result.summary())

# Granger causality
for col in data.columns:
    test = result.test_causality(col, causing=data.columns.drop(col).tolist())
    print(test.summary())

# Impulse Response Functions
irf = result.irf(10)
irf.plot()
plt.tight_layout(); plt.show()

# Forecast Error Variance Decomposition
fevd = result.fevd(10)
fevd.plot()
plt.show()

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Limitations

• Many parameters (explodes with number of variables × lags).
• Structural IRFs require identification assumptions ⇒ see SVAR.

Video tutorial

Video Tutorial: Running VAR in EcoLab

See also

• VECM · SVAR · Catalog