VAR models
& system modelling

From univariate to vector processes

So far every model has described a single series, for example the AR(1) \(y_t=\phi y_{t-1}+\varepsilon_t\). But economic variables rarely move in isolation: output, inflation and interest rates influence one another over time. A vector autoregression (VAR) models several series jointly. The bivariate VAR of order 1 is

$$y_t=\Phi y_{t-1}+\varepsilon_t, \qquad y_t=\begin{pmatrix}y_{1,t}\\ y_{2,t}\end{pmatrix},\ \ \Phi=\begin{pmatrix}\phi_{1,1}&\phi_{1,2}\\ \phi_{2,1}&\phi_{2,2}\end{pmatrix},\ \ \varepsilon_t=\begin{pmatrix}\varepsilon_{1,t}\\ \varepsilon_{2,t}\end{pmatrix},$$

which written out is a pair of equations in which each variable depends on the past of both,

$$y_{1,t}=\phi_{1,1}y_{1,t-1}+\phi_{1,2}y_{2,t-1}+\varepsilon_{1,t}, \qquad y_{2,t}=\phi_{2,1}y_{1,t-1}+\phi_{2,2}y_{2,t-1}+\varepsilon_{2,t}.$$

The off-diagonal coefficients \(\phi_{1,2},\phi_{2,1}\) are the whole point: they let one variable's past feed into another's present. VARs are a form of systems modelling, and their story is central to how modern macroeconometrics developed.

A little history

Systems modelling grew out of attempts to understand the business cycle after the Great Depression. Jan Tinbergen built the first large macroeconometric models — a 1937 model of the Dutch economy (16 equations, 31 variables) and a 1939 model of the US (48 equations) — work that earned the inaugural Nobel Prize in economics in 1969 (shared with Ragnar Frisch). The Cowles Commission, founded by Alfred Cowles after the 1929 crash, developed the statistical theory of systems of linear equations, and through the 1950s and 60s Lawrence Klein built ever more elaborate models of the US macroeconomy.

These models worked well until the 1970s, when two influential critiques emerged. Robert Lucas (1976) argued that the parameters of such models are functions of the economic environment and cannot be treated as constant when the model is used to evaluate policy. Christopher Sims (1980), in "Macroeconomics and Reality", argued that the identifying restrictions these models rely on are never really credible; instead he proposed treating all variables in a block as endogenous, each depending on the past of all the others — the VAR.

Simultaneous equations, structural and reduced form

To see what Sims was reacting to, consider a linear simultaneous-equations model in matrix form,

$$By_t=\Gamma x_t+u_t,$$

with \(y_t\) endogenous, \(x_t\) exogenous and \(u_t\) the errors. This is the structural form: its parameters have direct economic meaning. Assuming \(B\) is nonsingular and premultiplying by \(B^{-1}\) gives the reduced form

$$y_t=B^{-1}\Gamma x_t+B^{-1}u_t=\Pi x_t+v_t,$$

in which each endogenous variable is expressed purely in terms of the exogenous variables. The reduced-form parameters \(\Pi\) can always be estimated; but recovering the structural parameters \(B,\Gamma\) from \(\Pi=B^{-1}\Gamma\) is only possible under identification conditions, which depend on the number of exogenous variables and on exclusion restrictions. Sims' objection was precisely that these exclusion restrictions are usually imposed for convenience rather than for sound economic reasons. His alternative, the VAR, needs none of them.

The VAR(p)

A VAR of order \(p\) for a \(k\times1\) vector \(y_t\) is

$$y_t=\alpha+\Phi_1 y_{t-1}+\Phi_2 y_{t-2}+\cdots+\Phi_p y_{t-p}+\varepsilon_t,$$

where each \(\Phi_i\) is a \(k\times k\) matrix of coefficients and \(\varepsilon_t\) is vector white noise. Using the lag operator this compresses to

$$\Phi(L)\,y_t=\alpha+\varepsilon_t, \qquad \Phi(L)=I_k-\Phi_1 L-\Phi_2 L^2-\cdots-\Phi_p L^p.$$

The matrix polynomial \(\Phi(L)\) captures every dynamic interaction, both within and across variables, while leaving no dynamics in the disturbances. For a bivariate VAR(1), for example, \(\Phi(L)=\begin{pmatrix}1-\phi_{11}L & -\phi_{12}L\\ -\phi_{21}L & 1-\phi_{22}L\end{pmatrix}\).

Vector white noise

The disturbance in a VAR is a vector generalisation of white noise.

The subtlety is in the covariance structure. There is no correlation across time — \(\mathrm{Cov}[\varepsilon_{i,t},\varepsilon_{j,s}]=0\) for \(t\neq s\), for all \(i,j\) — but there can be correlation across equations at the same instant: \(\mathrm{Cov}[\varepsilon_{i,t},\varepsilon_{j,t}]=\sigma_{ij}\) need not be zero. The variance matrix \(\Sigma\) is positive definite, and its off-diagonal entries encode the contemporaneous links between the shocks. This distinction — contemporaneous correlation allowed, intertemporal correlation ruled out — is what lets the VAR lag structure carry all the dynamics.

The first two moments of a random vector

For a \(k\times1\) random vector \(y_t\), the mean \(\mathbb{E}[y_t]\) is the vector of individual means, and the variance–covariance matrix is

$$\mathrm{Var}[y_t]=\mathbb{E}\big[(y_t-\mathbb{E}[y_t])(y_t-\mathbb{E}[y_t])'\big],$$

whose diagonal entries are the variances of each \(y_{i,t}\) and whose off-diagonal entries are the covariances between them. It is symmetric and positive definite — positive definiteness guaranteeing that any linear combination of the elements has positive variance.

Stationarity and the VMA representation

A VAR is (second-order) stationary if its mean and its autocovariance matrices are finite and constant over time:

$$\mathbb{E}[y_t]=\mu, \qquad \mathbb{E}[(y_t-\mu)(y_t-\mu)']=\Gamma_0, \qquad \mathbb{E}[(y_t-\mu)(y_{t-\ell}-\mu)']=\Gamma_\ell\ \text{ for each }\ell.$$

Here \(\Gamma_0\) is the contemporaneous covariance matrix and \(\Gamma_\ell\) the autocovariance matrix at lag \(\ell\). A feature with no univariate analogue: for \(\ell\neq0\) the matrix \(\Gamma_\ell\) is generally not symmetric, because the cross-covariance of variable \(i\) today with variable \(j\) in the past need not equal that of \(j\) today with \(i\) in the past.

When a VAR is stationary it can be inverted into a vector moving average (VMA) of infinite order. For the VAR(1), repeatedly substituting the right-hand side into itself,

$$y_t=\alpha(I_k+\Phi_1+\Phi_1^2+\cdots)+\varepsilon_t+\Phi_1\varepsilon_{t-1}+\Phi_1^2\varepsilon_{t-2}+\cdots,$$

which converges because \(\Phi_1^{\,j}\to0\). Equivalently, via the lag operator,

$$y_t=(I_k-\Phi_1 L)^{-1}(\alpha+\varepsilon_t)=\mu+\sum_{s=0}^{\infty}\Theta_s\varepsilon_{t-s}, \qquad \Theta_0=I_k,$$

with \(\mu=(I_k-\Phi_1)^{-1}\alpha\). The general VAR(p) inverts the same way, \(y_t=[\Phi(L)]^{-1}(\alpha+\varepsilon_t)=\mu+\Theta(L)\varepsilon_t\), and the VMA coefficient matrices \(\Theta_s\) are functions of the VAR coefficients \(\Phi_1,\ldots,\Phi_p\). This representation is the foundation for the next stage of VAR analysis — the precise stationarity condition on the characteristic roots, forecasting, estimation, and impulse–response analysis — which build directly on the VMA form introduced here.

Check your understanding

Write out the two equations of a bivariate VAR(1) and say what makes it more than two separate AR(1)s.

The equations are \(y_{1,t}=\alpha_1+\phi_{11}y_{1,t-1}+\phi_{12}y_{2,t-1}+\varepsilon_{1,t}\) and \(y_{2,t}=\alpha_2+\phi_{21}y_{1,t-1}+\phi_{22}y_{2,t-1}+\varepsilon_{2,t}\). If \(\phi_{12}=\phi_{21}=0\) the system collapses into two unrelated AR(1)s. The cross-lag terms \(\phi_{12}\) and \(\phi_{21}\) — and the contemporaneous correlation in \(\Sigma\) — are what make it a genuine system.

In what sense is vector white noise both correlated and uncorrelated?

It is uncorrelated over time: \(\mathrm{Cov}[\varepsilon_{i,t},\varepsilon_{j,s}]=0\) whenever \(t\neq s\), for any equations \(i,j\). But it may be correlated across equations at the same date: \(\mathrm{Cov}[\varepsilon_{i,t},\varepsilon_{j,t}]=\sigma_{ij}\), the \((i,j)\) entry of \(\Sigma\), can be non-zero. So shocks in different equations can hit together, but no shock carries predictive information about any other period.

Why does a stationary VAR(1) have a VMA(∞) representation?

Repeated substitution gives \(y_t=\mu+\sum_{s=0}^{\infty}\Phi_1^{\,s}\varepsilon_{t-s}\). This infinite sum converges only if the powers \(\Phi_1^{\,s}\to0\) as \(s\to\infty\), which is exactly what stationarity of the VAR guarantees. The resulting VMA expresses each variable as a weighted sum of current and past shocks, with weight matrices \(\Theta_s=\Phi_1^{\,s}\).

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