How does a VAR model capture multivariate dynamics?

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AcadiFi · Accepted Answer

A Vector Autoregression (VAR) model treats each of k variables as a linear function of its own lags and lags of all other variables in the system. A VAR(p) is yt = c + Φ1 yt-1 + ... + Φp yt-p + et where yt is a k-vector, Φi are k×k coefficient matrices, and et is a k-vector of errors with covariance Σ.

Example VAR(1) on (inflation, unemployment, federal funds rate): each variable depends on one lag of all three. Estimation is OLS equation-by-equation (efficient under symmetric lag structure). Identification requires no parameter restrictions within reduced form but does require restrictions for structural interpretation.

Applications include:
• Granger causality tests—whether X helps predict Y beyond Y's own past.
• Impulse response functions (IRF)—effect of a one-unit shock in one variable on all others over time.
• Variance decomposition—share of forecast error variance attributable to each shock.
• Forecasting multivariate systems with feedback.

VARs suffer the curse of dimensionality: k variables with p lags estimate kp + k(k-1)p/2 parameters. Bayesian VARs with Minnesota priors shrink coefficients toward random walk, improving out-of-sample forecasts. Large VARs require factor models or LASSO-type regularization. For non-stationary cointegrated systems, use VECM (Vector Error Correction Model). Financial applications include term-structure modeling, macro-finance links, and risk factor dynamics across asset classes.

How does a VAR model capture multivariate dynamics?

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