Common mistake 1 with VAR

Too many lags. A 4-variable VAR with 8 lags needs 132 parameters and will overfit any normal macro sample.

Common mistake 2 with VAR

Treating the Cholesky ordering as innocuous. Reordering the variables changes the impulse responses - say what your ordering assumes.

Common mistake 3 with VAR

Calling Granger causality "causality" in the write-up.

Common mistake 4 with VAR

Reading an impulse response as the effect of a policy. It is the response to a statistical innovation - the part of a variable the system could not predict - which is not the same as a deliberate, announced change.

EconStatsCraft › VAR

VAR

Vector autoregression: lag-length criteria, every equation, Granger causality in both directions, stability roots, impulse responses and variance decomposition.

What it needs

Endogenous variables - Two or more, all stationary.

The intuition

Sims's objection in 1980 was that the identifying restrictions macroeconomists used to pin down "which variable causes which" were incredible. A VAR refuses to impose them: every variable depends on the lags of every variable, and you let the data describe the joint dynamics. The cost is that you cannot interpret anything structurally until you put SOME assumption back in.

When to use it

Several macro series that plainly influence each other - output, inflation and interest rates - where imposing a causal ordering in advance would be arbitrary.

Where the estimator comes from

y_t=A_1y_{t-1}+\dots+A_p y_{t-p}+\varepsilon_t with E[\varepsilon_t\varepsilon_t']=\Sigma. Because every equation has identical regressors, equation-by-equation OLS is efficient - no system estimator needed. Inverting to the \mathrm{MA}(\infty) form y_t=\sum_j \Psi_j\varepsilon_{t-j} gives the impulse responses. But \varepsilon_t are correlated reduced-form residuals, so a Cholesky factorisation \Sigma=PP' is imposed to make the shocks orthogonal - and that ordering IS an economic assumption about which variable cannot respond within the period.

The model

\mathbf{y}_t=\mathbf{A}_1\mathbf{y}_{t-1}+\dots+\mathbf{A}_p\mathbf{y}_{t-p}+\boldsymbol{\varepsilon}_t,\qquad E[\boldsymbol{\varepsilon}_t\boldsymbol{\varepsilon}_t^{\prime}]=\boldsymbol{\Sigma}

What it assumes

All series are stationary. If they are I(1) and cointegrated, use a VECM instead.
Enough lags to leave the residuals white.
For impulse responses: an identification assumption, usually a Cholesky ordering.
The variables belong in the same system at the same frequency. Mixing a monthly series with a quarterly one by interpolating the slower one invents variation the VAR will then model as if it were real.

Statistical properties

Stability requires all eigenvalues of the companion matrix to lie inside the unit circle, equivalently all inverse roots outside it.
Parameters grow as n²p: a 4-variable VAR with 4 lags needs 68 parameters, which is a lot for 200 quarters.
Cholesky impulse responses depend on the variable ordering; report the ordering and ideally show robustness to it.
Forecast-error variance decomposition allocates each variable's forecast uncertainty across shocks at each horizon.
Granger causality within a VAR is a joint test on lag coefficients, and remains predictive, not structural.

How to read the output

Do not try to read individual VAR coefficients - there are too many and they are not interpretable one at a time.
Read the impulse responses: how does variable A react over time to a one-off shock to variable B?
The variance decomposition says which shocks drive most of the forecast error at each horizon.
Stability: all inverse roots must have modulus above 1, otherwise the responses explode.

A worked reading

A one-standard-deviation shock to the interest rate lowers GDP growth by 0.31pp after 3 quarters, with the effect dying out by quarter 8, and explains 22% of GDP growth's forecast-error variance at that horizon. That is conditional on the Cholesky ordering placing output before the interest rate.

Symptom → diagnosis → what to do

If you see this	It means	Do this
A series is I(1)	Spurious VAR	Difference, or use a VECM if they are cointegrated
An inverse root has modulus below 1	Unstable	Reduce lags; check for unit roots
Impulse responses explode	Same instability	Same fix
Wide impulse-response bands	Too many parameters for the sample	Fewer lags or fewer variables; consider a Bayesian VAR
Results change a lot on reordering	The identification is doing the work	Say so; consider sign restrictions or an external instrument

Common mistakes

Too many lags. A 4-variable VAR with 8 lags needs 132 parameters and will overfit any normal macro sample.
Treating the Cholesky ordering as innocuous. Reordering the variables changes the impulse responses - say what your ordering assumes.
Calling Granger causality "causality" in the write-up.
Reading an impulse response as the effect of a policy. It is the response to a statistical innovation - the part of a variable the system could not predict - which is not the same as a deliberate, announced change.

Where to read this properly

Sims (1980), "Macroeconomics and Reality"
Lütkepohl (2005), New Introduction to Multiple Time Series Analysis
Stock & Watson (2001), "Vector Autoregressions"
Kilian & Lütkepohl (2017), Structural Vector Autoregressive Analysis

Related Time series - several series

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

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VAR

What it needs

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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