0.1 Vector Autoregression (VAR)
The AR(p) model generalizes naturally to vector-valued series, allowing each variable to depend on lagged values of itself and all other variables in the system — the workhorse model of empirical macroeconomics and multivariate financial time series.
For an \(n\)-dimensional series \(\mathbf{X}_t = (X_{1t},\dots,X_{nt})'\), \[ \mathbf{X}_t = \mathbf{c} + \Phi_1\mathbf{X}_{t-1} + \dots + \Phi_p\mathbf{X}_{t-p} + \boldsymbol{\varepsilon}_t, \qquad \boldsymbol{\varepsilon}_t \sim WN(0,\Sigma), \] with \(n\times n\) coefficient matrices \(\Phi_i\). Stationarity requires the roots of \(\det(I - \Phi_1 z - \dots -\Phi_p z^p)=0\) to lie outside the unit circle, mirroring the univariate AR condition.
Estimation. Each equation can be estimated separately by OLS (equivalent to GLS/MLE here since regressors are identical across equations), making VAR simple to fit even in moderate dimension; overfitting/dimensionality is controlled via Model Selection Criteria (AIC BIC) or shrinkage (Bayesian/“Minnesota” priors).
Granger causality. Testing whether the lags of \(X_j\) have jointly zero coefficients in the equation for \(X_i\) gives a formal test of whether \(X_j\) “Granger-causes” \(X_i\) — see Granger Causality.
Impulse response & variance decomposition. Writing the VAR in its (causal) \(VMA(\infty)\) form via the Wold representation lets one trace the dynamic effect of a one-time shock to one variable on the future path of all variables (impulse response functions), and decompose forecast-error variance by source.
Non-stationary extension. When the component series are \(I(1)\) and cointegrated, the VAR is reparametrized as a Vector Error Correction Model — see Cointegration.